Using a serial dilution experiment to estimate the density of organisms
by Milton Wayne Loyer
A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF
PHILOSOPHY in Statistics
Montana State University
© Copyright by Milton Wayne Loyer (1981)
Abstract:
The serial dilution assay is a standard microbiological method for determining the density of organisms
in a solution. This paper presents alternatives to current standard serial dilution confidence interval,
point estimate and design recommendations.
Original exact confidence intervals are given which are narrower than those available in standard
tables. Point estimates are given which have smaller mean squared error than the standard most
probable number (MPN) maximum likelihood estimator. An algorithm is given which, for the
techniques discussed and within certain researcher-chosen constraints, identifies the optimal design and
the most efficient estimator.
This paper also gives the solution to the general finite population serial dilution problem, discusses
finite population analogs of the confidence interval and point estimate techniques discussed, and
compares the finite and the infinite population models.
The computer programs which were used to obtain the confidence intervals, point estimates and tables
presented in the text are given in the Appendix. These programs, including the one for identifying the
optimal design, generalize to any number of dilutions, any number of samples per dilution and any
dilution factor. USING A SERIAL DILUTION EXPERIMENT
TO ESTIMATE THE DENSITY OF ORGANISMS
by
MILTON WAYNE LOYER
A thesis submitted in partial fulfillment
of the requirements for the degree
of
DOCTOR OF PHILOSOPHY
in
Statistics
Approved:
/<0sZZMJ-'
HeadyMajor Department
Chairman, Examining Committee
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
March, 1981
iii
ACKNOWLEDGMENT
I wish to thank my thesis advisor Dr. Martin A. Hamilton for his
advice and assistance throughout my graduate work.
Thanks are also due
to the Montana State University Research-Creativity Development
Committee for funding computer work connected to the thesis and to
Messiah College for assisting in the preparation of the final
manuscript.
I also wish to acknowledge my wife and my parents for their
encouragement and support throughout my education.
iv
TABLE OF CONTENTS
CHAPTER
PAGE
1. INTRODUCTION..........................
I
2. CONFIDENCE INTERVALS ........................................
5
2.1
2.2
2.3
2.4
2.5
Woodward's Method ......................................
DeMan's M e t h o d ..........................................
Methods of Combining Independent Results ................
The Method of Minimum Expected
W i d t h ....................
Approximate Methods
....................................
5
9
11
15
19
3. POINT E S T I M A T E S ............................................... 22
3.1 The M P N ........................
22
3.2 Alternative Procedures ..................................
26
■ 3.3 Bias and MSE Comparisons .. . . ............................ 35
4. DESIGN CONSIDERATIONS
......................................... 49
4.1 The Single Dilution Experiment ..........................
4.2 A Design Algorithm for theSerial Dilution Experiment . .
5. THE' FINITE POPULATION M O D E L .......................... ..
50
53
. .
60
5.1 The General F o r m u l a .................................... 62
5.2 Point and Interval Estimation . . . . . ................
64
6 . S U M M A R Y ....................................................... 70
FOOTNOTES......................................................... 72
A P P E N D I X ................................. .... .................74
BIBLIOGRAPHY
.112
V
LIST OF TABLES
TABLE
PAGE
1. 95% Confidence Intervals ....................................
2.
3
MPN R e s u l t s ............................................... . 2 4
3. Point E s t i m a t e s ..............................................27
4.
Expected Values and MSE V a l u e s ................................ 28
5.
Selected Expected Values and MSE Values
...................
36
6 . Expected Values and MSE V a l u e s ............................. 38-47
7. MPN Expected Values and MSE V a l u e s ............................ 52
8 . MSE C o m p a r i s o n s ..............................................57
9.
10.
Infinite and Finite Population R e s u l t s ......................... 65Infinite and Finite Population Results ....................... 68
vi
LIST OF FIGURES
FIGURE
PAGE
1. Distribution of Possible Sample Results ....................
2. Output for the Program of Appendix I V ...................... 55
7
vii
ABSTRACT
The serial dilution assay is a standard microbiological method
for determining the density of organisms in a solution. This paper
presents alternatives to current standard serial dilution confidence
interval, point estimate and design recommendations.
Original exact confidence intervals are given which are narrower
than those available in standard tables. Point estimates are given
which have smaller mean squared error than the standard most probable
number (MPN) maximum likelihood estimator. An algorithm is given which
for the techniques discussed and within certain researcher-chosen
constraints, identifies the optimal design and the most efficient
estimator.
This paper also gives the solution to the general finite
population serial dilution problem, discusses finite population analogs
of the confidence interval and point estimate techniques discussed, and
compares the finite and the infinite population models.
The compqter programs which were used to obtain the confidence
intervals, point estimates and tables presented in the text are given
in the Appendix. These programs, including the one for identifying
the optimal design, generalize to any number of dilutions, any number
of samples per dilution and any dilution factor.
I. INTRODUCTION
Halvorson and Ziegler (1933a) state "the use of dilution methods
...dates back to the early days of science" and note that Pasteur, for
example, was using serial dilution techniques about 1875.
Typically,
one seeks to estimate the number of organisms per unit volume of
solution under the assumptions that (I) the organisms are randomly
distributed throughout the solution and (2) each sample from the
solution, when incubated in the culture medium, is certain to exhibit
fertility whenever the sample contains one or more organisms.
If the
solution averages X organisms per unit volume and z is the dilution
(multiple of the unit volume selected for analysis), then, under the
Poisson probability model, P(sterile sample) = e
In practice, one
guards against obtaining samples which are likely to be either all
sterile or all fertile by using more than one dilution.
Letting
equal the number of fertile samples in n_ trials at the i
dilution,
P(X.Fr); = (ni) (l-e"'Xzi)r (e"Xzi)ni"r .
1
.
r
.
In'the first definitive study of the problem of estimating X
using serial dilutions, McCrady (1915) described the estimate X, the
value of X that maximizes the probability of obtaining the specific
arrangement of fertile and sterile samples observed.
z,
■
McCrady called
,
•
X .,the "most probable number" (MPN) and presented the procedure, which
today is known as maximum likelihood (ML) estimation, as Bayes
a
estimation with an improper uniform prior on X.
To justify the
procedure, he cites, among others, distinguished late nineteenth
2
century mathematician Richard L. Edgeworth who stated, "The
assumption that any probability constant about which we know nothing
in particular
is as likely to have one value as another, is grounded
upon the rough but solid experience that such constants do, as a
matter of fact, as often have one value as another."
For k dilutions, the likelihood function is given by
(1.1)
LCx1,x2 ,. . . =
H(^i)(1-e ^Zi)Xi(e ^Zi)ni Xi
i
and the maximum likelihood estimate for X , still most commonly
referred to as the MPN, is the X which solves Z(x_^z^e ^Zi)/(l-e ^Zi) =
Z(n_-x^)z^, which simplifies (deMan 1977) to
(1 .2)
^nizi = ^xIzV
•
For k>l, the solution to (1.2) must be obtained by iterative methods.
Several programs (e.g., Parnow 1972) to obtain the MPN for any k, any
and any n^ are readily available.
While the methods of all sections of this paper generalize to
any k, any z^ and any n^ (except in Chapter 5 where it is required
that Zn^z/hL), the numerical examples given are for the commonly
encountered case of k=3 decimal dilutions z^=(.l)
i=l,2,3.
with n^=3 for
The 64 possible (Xi,Xa,Xg) sample results will be referred
to by the codes 000,001,...,332,333.
For these k, z^ and n^, the
first three columns of Table I summarize the results presented and
recommended by standard reference works.
3
TABLE I
95% Confidence Intervals,: .n=3, Z^=.I :
re s u lt
MPNa
000
001
002
003
010
011
012
013
020
021
022'
023
030
031
032
033
. 100
101
. 102
102
HO
111
112
113
120
121
122
123
130
131
132
133
200
201
202
203
210
211
212
213
220
221
•222 ‘
223
230
231
232
233
300
301
302
303
310
311
312
313
320
321
322
323
330
331
332
333
0 .0
3 .0
6 .0
9 .0
3 .0
6 .1
9 .2
12
6 .2
9 .3
12
16
9 .4
13
16
. 19
3 .6
7 .2
'u
15
7 .3
11
15
19
11
15
20
24
16
20
24
29
9 .1
14
20
26
15
20
27
34
21
28
35
42
29
. 36
44
53
23
39
64
95
■ 43
75
120
160
93
150
210
290
240
460
1100
Woodward^
Com bining^
In d e p e n d e n t
R e s u lt s
deManC
0 -9 £
0-9
< 1 -1 7
.0 8 5 -1 3
Minimum6
E x p ected
W idth
0-12
2 -1 5
0-13
< 1 -1 7
< 1 -1 6
7-19
2-10
2 -2 2
4 -1 7
1 6 -2 0
1 7 -1 7 '''
.0 8 5 -2 0
.8 7 -2 1
<1-21
2 -2 7
< 1-24
3 -2 8
26-28
< 1-25
.8 8 -2 3
3 -3 6
2-28
4 -3 4
1-3 0 .
7-35
35-35
3 -2 0
4 -3 5
6-41
5-32
1 7 -3 7
6-42
1 8 -3 2
2 -3 8
5-48
1 -4 2
5 -5 0
2 7 -5 0
<1-37
11-14
3 -5 5
9-64
36-65
5 -4 2
8-62
11-74
7-61
1 8 -7 1
5 1 -7 2
10-32
1 1 -7 7
1 9 -6 3
40-74
3 .5 -1 2 0
6 .9 -1 3 0
15 -3 8 0
< 10-130
1 0 -1 8 0
2 0-230
3 -1 3 7
1 0-175
4 2 -1 8 3
4 -1 2 0
14-69
7 .1 -2 1 0
14 -2 3 0
30 -3 8 0
1 0 -2 1 0
2 0-280
4 0 -3 5 0
7-200
21-180
15 -3 8 0
30-440
35-470
3 0 -3 8 0
5 0-500
8 0-640
1 10-790
<100-1400
100-2400
300-4800
5 -2 5 7
1 5 -3 2 0
5 1-340
1 9 5-345
1 1-456
26-594
69-659
208-687
27-1612
54->1800
115-*»1800
298-»
2 .7 - 3 6
‘
1 .0 - 3 6
2 .7 - 3 7
2 .8 -4 4
7-89
3 .5 -4 7
10 -1 5 0
36-1300
71-2400
150-48X10
460—
'
5 -5 0 ,
7-60
'
1 2-360
38-400
120-260
26-990
70-2000
140-4070
370-”
^A m erican P u b lic H e a lth A s a o c ia c lo n (1 9 7 0 , p ag e 101)
A m erican P u b lic H e a lth A s s o c ia tio n (1 9 7 1 , p ag e 6 7 6 ); s e e s e c t i o n 2 .1
^jdeMan (1 9 7 7 ); s e e s e c t i o n 2 .2
s e e s e c t i o n 2 .3
e s e e s e c t i o n 2 .4
^one-sided 95% confidence interval
01 z
=
.001
4
This paper examines presently recommended serial dilution
interval estimation (Chapter 2), point estimation (Chapter. 3) and
design (Chapter 4) techniques.
In each chapter, alternatives are
developed and compared to the currently standard methods.
In
Chapter 5, the exact solution is given to the finite population
serial dilution problem.
2. CONFIDENCE INTERVALS
Sections 2.1-2.4 present two commonly used and two new methods
for constructing exact 100(l-a)% confidence intervals.
Several
approximate confidence interval techniques are discussed briefly in
section '2.5.
The 95% confidence intervals obtained by the methods in
sections 2.1-2.4 are given in the final four columns of Table I.
As
apparent from the discussion below, the methods of sections 2.1-2 .3
can be used to construct one-sided confidence intervals, and for these
methods the endpoints given in Table I may be used as the endpoints
for appropriate 97.5% one-sided confidence intervals.
2.1 Woodward's Method
Perhaps the most commonly used 95% confidence intervals are those
given by the American Public Health Association (1971, page 676).
Prepared by Woodward (1957) and appearing in the "Woodward" column of
Table I, these intervals are the accepted norm by which other
procedures are often judged (e.g., Martins and Selby 1980).
Woodward
ranked each of the 64 possible X 1X2X 3 outcomes according to the
magnitude of the MPN and then constructed 95% confidence intervals
(i.e., approximate intervals, since the outcome space is discrete) by
testing H q IX=Xo for selected Xq values in [0,=°).
For a given X 1X2X 3
outcome. Woodward rejected H q IX=Xq if and only if that X 1X2X 3 outcome
produced an
MPN in the lower 2.5% or the upper 2.5% of the
probability distribution of MPN's generated under Hq .
The set of all
Aq iS not rejected for any X 1X2X 3 outcome comprise the two-sided 95%
confidence interval associated with that outcome.
Figure I illustrates the Woodward method.
When testing Hq :A=21
vs. H^:Af21, one obtains the distribution of MPN1s shown.
Rejecting
the .025 most extreme results in each tail. Woodward rejects A=21. for
X 1X2X 3=OOO,001,010,100,002,Oil,020,101 in the lower tail and for
X 1X2X 3=SSS,332,331,'323,330,322,313,321,312,303 in the upper tail.
The
Woodward■95% confidence intervals of Table I for these results (when
given) should not include the value 21.
This is true for all cases
except XiX2X 3=IOl, which represents an error in Woodward's
calculations.
Nor is this the only error in Woodward's work, as deMan
(1975) notes.
"In the table presented by Woodward (1957)," he states,
"a few mistakes were also found, but they were minor.
Undoubtedly,
this, table should have been given more attention than it apparently
b
r e c e i v e d . T h e program used to generate the probabilities for
Figure I is. given in Appendix I.
..
The MPN's in Figure I were obtained '
.,by Newton's method within the program of Appendix III.
Two additional comments regarding Woodward's confidence intervals
need to be made.
First, a caveat given by Woodward but often omitted
by those reproducing his tables should be repeated.
For the 000 (333)
result, Woodward rejects Hq :A=Aq if and only if the MPN is in the
upper (lower) 5% of the. sampling distribution and presents only upper
(lower) 95% confidence intervals.
,,
■
7
FIGURE I
Distribution of Possible Sample Results (arranged by
magnitude of MPN )
n=3, :
I Zn= .01 z = .001
A=21
'2.
MPN
result
i-
0 0 0 '.0 0 0 9
0.00
3 .0 1
3 .0 5
3 .5 7
6.02
010 .0 0 0 6
100
I .0 1 9 7
002 .0000
6.11
011 .0000
001 .0001
6 .1 9
020 .0002
7 .2 3 101 .0 0 1 3
7 .3 6 H O 3 3 .0 1 3 8
9 .0 5
003 .0 0 0 0
9 .1 8
012 .0000
9 .1 8 200
9 .3 1
021 .0000
9 .4 4
0300000
1 0 .9 9
102 .0000
1 1 .1 8 1 1 1 ’ .0 0 0 9
1 1 .3 8 1 2 0 ] . 0 0 3 2
1 2 .2 6 013 1.0 0 0 0
1 2 .4 3 0 2 2 1.0 0 0 0
1 2 .6 1
0 3 1 1.0 0 0 0
1 4 .3 3
201 n
.0 0 9 0
1 4 .6 8
1 4 .8 4
1 0 3 .0 0 0 0
1 5 .1 1
ST
112 1.0 0 0 0
1 5 .3 9 1 2 1 1.0 0 0 2
p
to
1 5 .5 6 023 .0 0 0 0
1 5 .6 9 130 .0 003
O
M
1 5 .7 9
032 .0 0 0 0
2.
1 8 .9 9
033 .0 0 0 0
1 9 .1 4
113 .0 0 0 0
S
1 9 .5 0 122 .0 0 0 0
1 9 .8 9
131 .0 0 0 0
1 9 .9 2
202 .0 002
I
2 0 .4 7
i
211 I .0 0 6 3
I
2 1 .0 7
220
I .0 2 3 2
2 3 .1 2 300 _____________
2 3 .7 2 123 .0 0 0 0
2 4 .2 0 132 .0 0 0 0
2 5 .9 9
203 .0 0 0 0
2 6 .7 8 212 .0001
2 7 .6 3 221 .0 0 1 5
2 8 .5 5
230 .0 0 1 8
2 8 .6 1 133 .0 0 0 0
3 3 .6 1
213 1 .0000
3 4 .7 7 222 1.0000
3 6 .0 4 2 3 1 1.0 0 0 1
3 8 .5 0 301 i____ ) .0 2 1 5
4 2 .4 2
223 !.0 0 0 0
4 2 .7 3
310 I_____________
4 4 .0 8 232 1.0 0 0 0
5 2 .5 7 2 3 3 LOOOO
6 3 .5 6 302 !.0 0 0 5
7 4 .8 9 3 1 1 '
| .0 151
9 3 .2 8 320 i
I .0554
9 5 .3 8
303 0000
1 1 5 .2 2
312 !.0 0 0 3
1 4 9 .3 6
321 1 .0 0 3 5
1 5 8 .8 0 313 I. 0000
2 1 4 .6 6 3 2 2 i . 0001
2 3 9 . 79 330 I ] . 0043
2 9 1 .7 2 323 TOOOO
4 6 2 .1 8 331 1.0003
1 0 9 8 .9 5 332 .0 0 0 0
“
333 i . 0000
_____ [
T
.1 4 1 5
.0 9 9 2
1 .3 3 7 9
V
1 .2 3 6 9
8
Secondly, while Woodward's method provides a 95% confidence
interval for each of the X 1X2X 3 possible outcomes, his 1957 table
includes confidence intervals only for what he determines to-be the 22
most likely X 1X2X 3 outcomes'.
The remaining 42 X 1X2X 3 outcomes he
calls "improbable" and recommends that they not be used for making
inferences.
result 003 —
In other words, there are some X 1X2X 3 outcomes (e.g ., the
no organisms present in the more concentrated .1 or .01
dilutions, but organisms present in all three samples at the weakest
.001 dilution) for which Woodward's method gives a 95% confidence
interval in which he apparently does not have 95% confidence.
The
last two methods of Table I eliminate this subjectivity by inherently
failing to give confidence intervals (i.e., by giving empty confidence
intervals) for improbable results.
A further inspection of Woodward's method reveals some serious
practical flaws.
Note from Figure I that ordering the X 1X2X 3 results
by the magnitude of the MPN'does not yield a unimodal sampling
distribution.
According to most statistical inference texts (e.g.,
Cox' and Hinkley 1974, page 66), this means that the MPN is not an
acceptable test statistic since more extreme values of the MPN do not
necessarily give stronger evidence of departure from Hq .
Fisher
(1956, page 98) objected to a procedure of Bartlett for similar
reasons since his statistic "does not increase or decrease
monotonically for changes in the weight of the evidence."
9
The difficulty caused by a multi-modal sampling distribution
can be seen from Figure I.
Woodward's rejection region includes the
result 100 for which P(X1X2X 3=IOO) = .0197 but fails to include the
less likely result H O for which P(X1X2X 3=IlO) = .0138.
In fact,
Woodward cannot place in his rejection region any result, no matter
how unlikely, which gives an MPN larger than 3.57 unless the result
100 were already in the rejection region.
Woodward's intervals, then,
form a "staircase" based on the magnitude of the MPN so that the lower
(upper) confidence limit associated with one sample result cannot be
lower (higher) than the limit associated with another sample result
yielding a lower (higher) MPN.C
Consequently, the width of an
interval is not determined by the precision associated with the sample
result, and preliminary calculations verify that the
actual level of
Woodward's intervals is greater than 95%.
2.2 DeMan's Method
Another set of commonly used confidence intervals is given by
deMan (1975) and appears in the "deMan" column of Table I.
Even
though deMan uses the term "confidence interval," his procedure does
not meet the necessary and sufficient conditions given by Neyman
(1941), the originator of the concept of confidence intervals as
presently employed.
In the opinion of many authors (e.g., von Mises
1942), however, this is not necessarily to deMan's detriment.
DeMan
does, in fact, provide the limits of the middle 95% of the likelihood
10
distribution for each X^XgXs result or, equivalently, the Bayesian
interval for X under an improper uniform prior.
While the posterior
function f (X;xi.,x2 ,xg) is defined continuously for
deMan used
discrete approximations in both directions from the MEN and truncated
the posterior distribution whenever an additional tail histogram
area contributed less than .000005 of the cumulative total.
DeMan's method, like Woodward's, provides a 95% confidence
interval for each of the 64 possible XjX2X 3 results. Also like
Woodward, deMan sates that "MEN tables should be restricted to results
having a defined minimal probability" and gives no confidence
intervals for "improbable" XjX2X 3 results.
In their original
articles, deMan and Woodward agree in all but two cases on what is
improbable (deMan considers the result 312 improbable, but not the
result 211).
It is clear that each finds himself deciding which of
his 95% intervals he chooses not to accept with nominal level 95
per cent.
If one desires to use a Bayesian procedure, of course, he is not
limited to an improper uniform prior.
In general, the more specific
prior information the researcher has (or is willing to assume) about
X , the narrower he can make his "confidence interval."
Even if the
researcher begins with complete ignorance about X, however, the
uniform prior may not be the appropriate prior.
Box and Tiao (1973)
discuss Bayesian interval estimation in general and define a
11
1'noninformalive prior," based on Fisher's information, that they
recommend for the researcher with little or no prior information.
2.3 Methods of Combining Independent Results
Since each of the three dilutions gives results independent of
those of the other dilutions, the total serial dilution experiment
yields three independent point estimates and three independent
confidence intervals for X.
First impressions might suggest
constructing V . 95 confidence intervals Ci, Cg and Cg for each of the
three dilutions and using the intersection C i H C z ^ C z as an
experiment-wide 95% confidence interval.
This is certainly
statistically acceptable and has the advantage of permitting certain
unlikely results to produce empty confidence intervals whenever
Ci H C 2 ^ C 3= 0.
There are, however, at least two disadvantages
major enough to discourage the use of this procedure.
First, the three independent confidence intervals have at most
three distinct lower endpoints and three distinct upper endpoints.
This means that the 64 possible XiX2X 3 results generate a maximum of
32=9 distinct non-empty confidence intervals.
Certainly, there exists
the possibility of different XiX2X 3 results yielding identical
confidence intervals.
This would not be undesirable if the minimal
sufficient statistic were some function of the X^ (e.g., Y=%X_) that
could assume only some number of values considerably less than 64.
12
Here, unfortunately, the minimal sufficient statistic is the ordered
triple (X1,X2 ,X3) and none of the 64 possible X1X2X 3
outcomes gives
the same information as any of the other outcomes.
Secondly, one intuitively has more confidence in the interval
associated with the strongest concentration Z 1 since it represents, in
some sense, the largest sample size.
Or should one have more
confidence in the interval associated with the weakest concentration
z 3 since it represents, in some sense, the finest scrutiny of the
solution?
What if two of the three intervals agree and. the third
appears to be an outlier?
While the weighting of estimates is
usually associated with point estimation, one can't help but feel that
merely intersecting three intervals obtained at three different levels
of examination would be naive and inefficient.
Fisher (1932) introduced a method of combining the results of
independent tests using p-values,.and Lancaster (1976) gives an
updated review of the procedure. Here, one combines the p-values
associated with H :A=A
o o
experiment-wide p-value.
for each of the three dilutions to obtain ah
The method uses the well-known (e.g., Hogg
and Craig 1970, pages 349,104 and 159) facts
(i) For any continuous random variable Y with distribution
function F(Y), the random variable W=F(Y) has a uniform
[0 ,1] distribution.
(ii) If Y is a random variable having a uniform [0,1]
13
distribution, then W=-2[In(Y)] has a chi-square distribution
with 2 degrees of freedom.
(iii) If Yi sY2,...,Y^. are independent random variables each
having a chi-square distribution with r^ degrees of freedom,
then W=YifY2+..,+Y^ has a chi-square distribution with
r=r1+ r2+ . ..+r^ degrees of freedom.
When the test H :X=X vs. H :X>X
o
o
o
o
at the single dilution z yields
m fertile samples in n trials, the associated p-value is given by
(2 .1)
p = Z (n)(l-e-XoZ)x (e~XoZ)n~x
^
x=m x
= P(X>m|X=Xo)
= I - P(X<m|X=Xq)
= I - F(X)
(ignoring the lack of continuity, a correction for which will be given
later).
Since p = I-F(X) has a uniform [0,1] distribution whenever
W = F(X) has a uniform [0,1] distribution, (i), (ii) and (iii) above
imply that an experiment-wide p-value for three independent dilutions
can be obtained by calculating the probability that -2%[ln(p^)] is
greater than a random variable having a chi-square distribution with
6 degrees of freedom.
Collecting for each X iX2X 3 result the Xq 1s for
which one fails to reject H :X=X
o o
vs. H :X>X at the a=.025 level
a
o
(i.e., for which the p-value is greater than .025) and for which one.
fails to reject H :X=X
o o
vs. H :X<X at the a=.025 level, one
a
o
constructs two-sided 95% confidence intervals.
14
Rosenthal (1978) reviews several methods of combining the results
of independent studies and for k=3 independent results such as those
in the serial dilution problem, he recommends Fisher's method.
Rosenthal does, however, remind his readers of two disadvantages
inherent in the method.
First, for several independent p-values just
slightly lower than .50, Fisher's method might not yield an overall
significant p-value when such simple tests as the sign test would.
Secondly, independent trials with strongly significant results in
opposite directions could cause Fisher's method to support the
significance of either outcome.
While it is precisely this second
phenomenon that proves to be the desired advantage in the serial
dilution problem by causing empty confidence intervals for improbable
results, not all authors would see this as an asset.
Cox and Hinkley
(1974, page. 225)., for example, discuss procedures that give empty
confidence intervals and say, "the assertion that [the parameter]
lies in a null region is certainly false...
What is the point of
making assertions known to be false?"
I.
Lancaster (1949) gibe's a continuity correction for Fisher s
;■ i
method which, according to'recent surveys (e.g., Rosenthal 1978),
I
others have not been able to improve upon since.
Moses (1956)
discusses the Lancaster-corrected Fisher technique in detail.
.I
Confidence intervals obtained using Fisher's technique and Lancaster's
!
correction for ,continuity are given in the "Combining Independent
15
Results" column of Table I.
While Woodward's method and deMan's
method yield confidence intervals for all possible X iX2X 3 results
(recall that those authors give intervals only for what they consider
"probable" and acceptable results), the method of combining
independent results described above yields only the confidence
intervals given in Table I.
X iX2X 3 results for which no confidence
interval is given are results inconsistent with any value of Xq .
The
program used to obtain the "Combining Independent Results” entries
in Table I is given, along with an example, in Appendix II.
2.4 The Method of Minimum Expected Width
It is axiomatic that methods giving narrower confidence intervals
ar;e to be preferred over competing methods (all other considerations
being equal). While the combination method of section 2.3 certainly
gives narrower confidence intervals than Woodward's or deMan's method
whenever it yields an empty interval, Table I indicates other X iX2X 3
results (e.g., 120) for which the combination method gives the
narrowest non-empty confidence interval of the three methods.
This,
prompts a search for yet another method which deliberately seeks to
minimize the widths of the 95% confidence intervals.
It will be ,
apparent, however, that such a method produces only two-sided
intervals and does not have the ability of the Woodward, deMan or
combination method to use upper or lower 95% confidence interval
16
endpoints as endpoints of corresponding 97.5% one-sided
confidence intervals.
The method of minimum expected width, like Woodward’s method,
starts by testing H q :A=Aq
vs
.
at the a=.05 level, considering
the probability distribution of the 64 possible X 1X2X 3 results under
H
and rejecting H
if the observed result falls within the extreme
5% of the sampling distribution.
The two methods differ according to
how each determines whether a particular X 1X 2X 3 result lies within the
extreme 5% of all results possible under
.
Most textbook presentations of the minimum expected width
technique are confined to continuous unimodal problems, for which the
technique gives the shortest confidence interval (e.g., Guenther
1969).
Mood, Graybill and Boes (1974, page 383) describe the
technique for this case and contrast it with the usual equal tails
method of constructing confidence intervals.
Larson and Marx (1981,
page 290) note that the technique, unlike the equal tails method,
preserves the likelihood ratio criterion.
Sterne (1954) first applied
the technique in a discrete problem when he constructed 1-a^
confidence intervals for the binomial parameter p by forming rejection
regions of size a<(X
Hq .'P=Pq .
that contained as many points as possible under
Crow (1956) indicates that this technique does achieve
the minimum possible expected width among all non-randomized
techniques meeting the usual Neyman confidence interval definition.
17
In general, a 95% minimum expected width confidence interval may
be formed as follows.
For the parameter X and the statistic T having
the sampling distribution f (t;X), increase r until /f(t;X )dt = .05,
R
o
where R={T|f(t;X )<r}.
H :X=X
o o
R then forms the rejection region for testing
vs. H :XfX and the collection of X Ts for which a particular
a
o
o
value of T is not in R forms the confidence interval associated with
that particular T.
In Figure I, /f(t;X )dt = ZP(XiX2Xs;X =21) = .0496
R
0
R
°
when r=.0138 and the rejection region R is the set of all 64 XiX2X 3
outcomes except. 100,200,210,220,300,301,310,311 and 320.
Accordingly,
the "Minimum Expected Width" column of Table I includes X=21 in
precisely the confidence intervals associated with 100,200,210,220, .
300,301,310,311 and 320.
The confidence intervals appearing in the "Minimum Expected
Width" column of Table I were obtained following the procedure
described above and using the program employed to construct Figure I
(and given in Appendix I) to produce the sampling distributions for
various X 's .
XiXzX3 results for which no confidence interval is
given are those results which, as with the combination method in
section 2.3, were inconsistent with all values of X.
In this author’s
opinion, the combination method and minimum expected width method are
significant improvements over currently used techniques and should be
the recommended serial dilution analyses for one-sided and two-sided
confidence intervals respectively.
There are, however, some general
18
cautions and areas for further work associated with the method of
minimum expected width of which the reader should be made aware.
First, while the confidence intervals given in the "Minimum
Expected Width" column of Table I are generally narrower than those in
the other columns (recall that the deMan intervals, while generally
wider than those of this section anyway, are not true Neyman
confidence intervals at all) , there are a few specific exceptions
(e.g., for XiXaX3=IOO).
This reminds one that the method guarantees
minimum expected width over all 64 possible X iX2X 3 results and not
necessarily the minimum width confidence interval for each X iX2X 3
result individually.
Secondly, the failure of the sampling distribution to be unimodal
can cause the minimum expected width confidence interval to be the
union of disjoint intervals.
For the k, z^, n^ and a of this chapter,
this occurred only once, and to two significant digits the confidence
interval properly associated with the result X iX2X 3=SSO is
26 U [32,990].
Since this would not be an acceptable result to most
researchers, this author recommends the procedure used in Table I of
extending the confidence intervals across any such gaps from the
lowest included X value to the highest included X value.
Santner and
Snell (1980), in the first paper applying the minimum expected width
technique to a multi-modal distribution, and, hence, the first paper
addressing this problem, describe an algorithm that makes adjustments
19
that guarantee continuous minimum expected width intervals.
Finally, as discussed in Chapter 3, the MPN is a positively
biased point estimate for A.
Blyth and Hutchinson (1960) note that
the minimum expected width confidence interval technique presented
here is also biased and that minimum expected width unbiased
•
confidence intervals for discrete distributions require randomization,
a technique unacceptable to most researchers.
After examining the
endpoints of the confidence intervals in Table I and considering the
positive bias of the MPN discussed in Chapter 3, this author
conjectures that the minimum expected width confidence intervals
are not only generally narrower but also less biased than the other
competing intervals considered.
Since, however, there may exist a
non-randomized confidence interval procedure with greater.expected .
width but less bias than the minimum expected width technique, bias
investigations of these and other methods might be appropriate.
2.5 Approximate Methods
The methods of sections 2.1-2.4 use the true probabilities
generated under the Poisson assumptions to provide exact confidence
intervals which, by definition, are to be preferred over approximate
methods advocated before the availability of advanced computing
techniques.
Various approximate methods proposed by such authorities
as Neyman (Matuszewski, Neyman and Supinska 1935), Haldane (1939),
20
Fisher and Yates (1943), Cochran (1950), Ferguson (1958) and Finney
(1978) , however, continue to enjoy widespread use and do offer useful
insights into the problem.
Eisenhart and Wilson (1943) adequately
review the pre-computer history of the MPN X, the search for estimates
of the variance of X and In(X), and approximate confidence intervals.
They conclude, "With regard to further research, it seems highly
desirable to construct mathematically exact charts giving .95 and .99
confidence intervals for the bacterial density for the case of several
tubes at a single dilution, and for the case of one or more tubes at
each of the successive dilutions."
Woodward's (1957) exact tables,
a direct response to the above challenge, are preceded by further
warnings about the inadequacies of continuous approximations when n=3
and n=5 samples are used at each dilution.
Inspection of Figure I
indicates the degree of departure of the true sampling distribution of
X and In(X) from any normal approximation.
As a final note, one needs to be on guard against computer
generated confidence intervals which, although sometimes even
narrower than those of sections 2 .1-2 .4, are based on only
approximate methods.,
Parnow (1972), for example, gives a program to
determine confidence intervals based on the normal approximation to
In(X) .
Not only is his estimate
^
=
based on asymptotic
theory, but his starting estimate of oC, taken from Haldane (1939) ,
21
uses the very crude approximations
= I//l(A) and I(A) =
-S2In(L)/BA2, where L is the likelihood function defined by equation
(1.1) evaluated at the observed (X1,X2 ,X3) with A estimating A.
Even
r
for large values of k and n^, Parnow1s method involves at least five
distinct approximations.
For the k<5
.<10 typically encountered
- and n I-
in practice, one should certainly prefer exact confidence intervals.
3. POINT ESTIMATES
The MPN is precisely defined as the solution to equation (1.2).
Being the ML estimate for X , the MPN is asymptotically unbiased and
asymptotically fully efficient.
In fact, Cochran (1950) states, "The
limiting distribution of the MPN has the smallest standard deviation
that can be achieved by any method of estimation...
in seeking further for a more precise estimate."
There is no point
Because, however,
the MPN is tedious to compute, there has been a steady stream of
alternatives and adjustments starting with Wolman and Weaver (1917)
ever since McCrady (1915) introduced the concept.
In addition, modern
computing techniques are allowing authors to discover just how biased
the MPN really is for the small n encountered in practice.
Section 3.1 examines the small sample behavior of the MPN,
section 3.2 discusses alternatives and adjustments proposed in the
literature as well as some original estimates, and section 3.3
compares the bias and the mean squared error (MSE) of the estimates
presented.
The numerical examples are, as in Chapter 2, for the
commonly encountered and tabled k=3 decimal dilutions zi=.l, Z2= .01
and z 3= .001 with n I=^n2=113=3 .
3.1 The MPN
Most review articles and textbooks on statistical methodology in
the biological sciences (e.g., Eisenhart and Wilson 1943 and Finney
1978) begin the serial dilution problem by examining the single
23
dilution experiment for which f(X;X) = (^)(1-e ^Z)x (e ^Z)n X .
As
first noted by Finney (1952), Fisher's information I(A) =
- E [92ln(f)/3A2] = nz^/(e^Z-l) is maximized for Az=I.59.
This suggests
selecting z=l.59/A, where A represents the researcher's best a pVtori
guess for A, would provide the most efficient single solution design
and selecting z=l.59/A as the middle dilution would provide the most
efficient k=3‘ serial dilution design.
Indeed, this is the current
recommendation of most authorities (e.g., Finney 1978).
The design of Chapter I, then, with Z 1=.!, Z2= . 01 and Z g=-OOl
should be appropriate (if not optimal in some sense) for A 's near 160.
Table 2 gives the expected value, variance and MSE of the MPN for
A=10(10)300.
As the MPN is infinite for XiXzX3=333, only the reduced
sample space consisting of the remaining 63 possible X 1X 2X 3 results .
-was used for all calculations.
The "T" (for "Total") column gives,
for each ^ , the sum of the probabilities over this restricted space.
Even though some of the estimators to be considered give finite
point estimates when XiXaXs=SSS, this will be the procedure throughout
all of Chapter 3.
Table 2 was constructed from the output of the
program given in Appendix III.
Inspection of Table 2 reveals several disturbing facts.
First,
for the k, z^ and n^ of the example, the MPN has a positive bias of
about 45%.
While the positive bias of the MPN is well known (e.g.,
Eisenhart and Wilson 1943 and Thomas and Woodward 1955), there has ■
24
TABLE 2
MPN Results: n=3, z^=.l z^=.01 z^=.OOl
X
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
T
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.999
.999
.999
.998
.998
.997
.997
.996
.995
.995
.995
.993
.992
.990
.989
E(MPN)
13.73
29.17
43.68
57.26
70.56
84.06
■97.95
112.28
127.01
142.06
157.34
172.77
188.26
203.74
219.14
234.40
249.49
264.35
278.96
293.28
307.32
321.04
334.43
347.50
360.25
372.66
384.74
.988
396.50
.987
.985
407.95
419.08
VAR(MPN)
165
656
1504
2720
4304
6248
8538
11208
14038
17179
20526
24034
27665
31376
35136
38912
42673
46401
50073
53677
57194
60618
63943
67161
70267
73263
76621
78919
81578
84129
MSE(MPN)
179
740
1691
3018
4727
6827
9318
12186
15408
18948
22767
26819
31059
35439
39916
44447
48992
53516
57987
62378
66665
70827
74849
78717
82422
85955
89313
92491
95490
98309
25
been very little investigation concerning the exact magnitude of the
bias for small n .
For k=3 decimal dilutions, however, McCarthy,
Thomas and Delaney (1958) empirically estimate the bias to be 30% for
n=5 samples per dilution and Salama, Koch and Tolley (1978) calculate
exact biases of about 10% for n=10.
In addition, Thomas (1942) and
Johnson and Brown (1961) give mathematical approximations expressing
the bias in the serial dilution problem as being inversely related to
n.
Halvorson and Ziegler (1933c) state, "Evidence has been presented
in support of the thesis that when three effective dilutions are used
to determine the bacterial population, the accuracy is... dependent
only on the number of tubes used in each dilution."
Unfortunately,
neither the "Standard Methods" volumes (American Public Health
Association 1966, 1967, 1970, 1971) nor current literature providing
MPN tables (e.g., deMan 1975,1977) warn the researcher of the
magnitude of the bias in their tabled MPN values for small n.
Secondly, note that even the large bias discussed above does not
significantly increase MSE(MPN) over VAR(MPN).
This suggests that the
MPN suffers extremely large variability, a fact long known to
researchers in the field.
Olson, Turbak and McFeters (1979), for
example, bemoan "the large confidence intervals inherent in the MPN
procedure" and Georgia (1942) even proposed that this variability be
acknowledged by abandoning the MPN in favor of the "most probable
range" (MPR).
in addition, one standard reference (American Public
26
Health Association 1966, page 139) cautions, "It is desirable to
remember that, unless a large number of portions of samples are
examined, the precision of the [MPN] test is rather low."
Indeed,
the Rao-Cramer lower variance bound for unbiased estimators,
[I(X)] 1= [Enz^/(e^Zi-l)] ^, is 10731 for A=160 and suggests that
considerable improvement over the MPN is possible.
Finally, notice that without exception MSE(MPN)>A2.
This means
that an estimator which completely ignores the experimental data and
constantly guesses X=O (even when some of the samples are fertile)
would, for this example, dominate the MPNl
Clearly, while the MPN
might possess desirable asymptotic properties, its behavior for small
n calls for consideration of alternative techniques.
3.2 Alternative Procedures
Table 3 gives the point estimates associated with each of the 64
possible X 1X 2X 3 results and Table 4 gives the expected value and MSE
for X=IO(IO)300 for each procedure discussed in this section.
The
program used to generate the point estimates, expected values and MSE
values for the procedures in sections 3.2.1 to 3.2.4 is given in
Appendix III.
While the program is given for the k, n_^ and
section 3.1', it readily generalizes.
of
The estimates for.the procedures
of section 3.2.5 were obtained as indicated in the text and their
expected values and MSE values were calculated in the usual way.
27
TABLE 3
P o in t
re s u lt
E s ti m
MPNa
OOO
0 .0 0 0
OOi
3 .008
002
6.024
003
9 .0 5 0
010
3.049
O il
6.108
012
9 .1 7 7
013
12.2 5 5
020
6.194
021
9 .3 0 7
022
12.4 3 0
023
15.5 6 5
030
9 .4 4 1
031
12.611
032
15.7 9 3
033
18.9 8 6
100
3.571
101
7 .2 3 3
102
10.988
103
14:839
HO
7.357
111
11.183
112
15.109
19.1 3 6
113
120
11.384
121
15.391
122
19.504
123
23.718
130 • 15.684
131
1 9 .886
132
24.198
133
28.611
200
9 .178
201
1 4 .327
202
1 9 .9 2 0
203
2 5 .990
210
1 4 .689
211
20.474
212
26. 781
213
33.608
220
2 1 .065
221
27.632
222
3 4 .771
. 42.421
223
230
2 8 .551
231
3 6 .036
232
4 4 .081
233
52.571
23.116
300
301
3 8 .500
302
63.558
303
9 5 .376
310
4 2 .729
311
74.085
312
115.215
313
158.79 7
320
9 3 .2 8 0
321
1 4 9.357
322
214.657
291.724
323
330
239.790
331
462.183
332 1098.950
333
v>
a t e s :
Acb
0 .0 0
2.3 0
4 .6 1
6 .9 2
2 .3 3
4 .6 7
7 .0 2
9 .3 7
4.7 4
7.12
9 .5 0
11 .9 0
7.22
9.6 4
1 2 .0 8
1 4 .5 2
2.7 3
5.5 3
8 .4 0
1 1 .3 5
5.63
8 .5 5
1 1 .55
1 4 .63
8 .7 0
1 1 .77
1 4 .91
18.14
1 1 .99
15.21
1 8 .5 0
2 1 .88
7.0 2
1 0 .9 6
1 5 .2 3
1 9 .87
1 1 .2 3
1 5 .66
20.48
2 5 .70
1 6 .1 1
21.13
26.59
32.44
2 1 .83
2 7 .56
33 .7 1
4 0 .2 0
1 7 .6 8
29.44
4 8 .6 0
72.93
32.67
5 7 .26
8 8 .1 0
121.42
71.33
114.21
164.14
223.07
183.36
353.41
640.32
to
n = 3 ,
V
Z 1= .
V
0 .0 0
0 .0 0 0
3 .5 0
3.008
8 .5 8
6 .024
1 7 .3 5
9 .0 5 0
3 .5 0
3 .049
8 .5 8
6 .1 0 8
1 7 .3 5
9 .1 7 6
3 6 .2 3
1 2 .254
8 .5 8
6 .195
1 7 .3 5
9 .3 0 7
3 8 .2 3
1 2 .4 3 0
8 7 .4 8
1 5 .5 6 2
1 7 .3 5
9.4 4 4
3 8 .2 3
1 2 .6 1 3
8 7 .4 8
1 5 .7 9 3
1 8 0 .2 3
1 8 .9 8 3
3 .5 0
3 .5 9 0
8 .5 0
7.1 9 6
1 7 .3 5
1 0 .8 1 7
38.2 3
1 4 .454
8 .5 8
7.339
1 7 .3 5
1 1 .034
38.2 3
1 4 .7 4 5
87.4 8
1 8 .4 7 3
1 7 .3 5
1 1 .2 6 4
38.2 3
1 5 .0 5 5
8 7 .4 8
1 8 .8 6 3
1 8 0 .2 3
22.689
38.2 3
1 5 .3 8 5
8 7 .4 8
1 9 .2 7 8
1 8 0 .2 3
2 3 .192
426 .7 2
27.124
8 .5 8
9 .5 0 3
1 7 .3 5
14.3 0 9
3 8 .2 3
1 9 .1 5 1
8 7 .4 8
24.031
1 7 .3 5
1 4 .8 2 3
38.23
1 9 .8 4 5
8 7 .4 8
24.909
180 .2 3
3 0 .0 1 5
38.2 3
2 0 .620
8 7.48
2 5 .8 9 0
180 .2 3
31.2 0 8
4 2 6.72
36.575
8 7 .4 8
2 6 .9 9 8
180 .2 3
32.5 5 6
4 2 6.72
38.169
1098.66
4 3 .8 4 0
1 7 .3 5
28.6 1 8
3 8.23 ' 38.749
8 7.48
4 9 .2 1 2
1 8 0.23
6 0 .0 3 0
3 8.23
4 5 .7 0 6
8 7 .4 8
58.4 1 7
180.23
71.750
4 2 6 .7 2
8 5 .7 7 5
8 7.48
75.993
1 8 0 .2 3
9 4 .9 1 6
4 2 6.72
115.659
1098.66
138.633
1 8 0.23
1 8 9 .8 3 2
4 2 6.72
271.244
1098.66
438.397
«»
Cu
.0 1
1
z_ = .0 0 1
Z2 =
V
V
V
1 .4 0
0 .0 0
0
3 .02
2.09
9
6 .51
5 .97
18
1 4 .0 2
6 .8 5
30
3 .0 2
2 .2 0
5
5 .9 8
6 .51
14
1 4 .0 2
6 .8 5
23
3 0 .2 1
1 2 .3
35
6.51
5.99
10
1 4 .0 2
1 2 .1
19
3 0 .2 1
28
1 2 .3
6 5 .1 0
1 5 .1
39
1 4 .0 2
1 2 .1
17
3 0.21
26
1 2 .3
6 5 .1 0
1 5 .1
34
1 5 .1
1 4 0.24
45
2
3 .0 2
3 .94
6 .5 1
6 .0 3
15
1 2 .1
1 4 .0 2
28
3 0 .2 1
1 5 .1
44
6.44
6 .5 1
10
1 2 .1
1 4 .0 2
23
30.2 1
. 1 5 .1
35
6 5 .1 0
1 5 .1
51
1 4 .0 2
1 2 .2
17
30.2 1
1 5 .1
29
6 5 .1 0
1 5 .1
41
56
140 .2 4
2 7 .8
1 5 .1
26
30.2 1
1 5 .1
38
6 5 .1 0
2 7.8
49
1 4 0.24
65
302.14
2 8 .0
9 .2 2
6
6 .5 1
1 4 .0 2
24
1 2 .4
15.1
43
30.21
67
6 5 .1 0
2 7 .8
16
1 4 .0 2
13.7
1 5 .1
36
3 0.21
6 5 .1 0
55
27.8
2 8 .0
80
1 4 0.24
26
1 5 .5
30.21
46
6 5 .1 0
27.9
65
140.24
2 8 .0
302.14.
29.5
90
6 5 .3 0
41
2 8 .0
140.24
2 8 .0
61
3 0 2.14
60.9
79
6 5 0.95
60.9
105
15
1 4 .0 2
2 0 .6
59
30.21
28.7
6 1 .0
113
6 5 .1 0
187
140.24
124
4 2 .1
40
3 0 .2 1
118
6 3 .3 0
64.8
190
140.24
124
152
291
302.14
6 5 .1 0
84
9 3.4
196
138
140.24
3 0 2 .lt
155
305
466
294
650.95
207
140 .2 4
211
632
447
302 .1 4
M 3
650 .9 5
1058
1402.43 >1800
>1800
^ th e MPN; soe s u c ti o n 3 .1
th e MPN w ith Thoiiuis1 c o r r e c t i o n to r b i a s ; s e e s e c t i o n 3 .2 .1
^ F i s h e r ’ s e s ti m a t e ; s e e s e c t io n J . 2 . 2
Thom as1 e s ti m a t e ; s e e s e c t io n 3 .2 .3
Lth e Johnson'-Brown Spearm an e s ti m a t e ; s e e s e c t i o n 3 .2 .4
^ th e e s ti m a t e from W oodward's C.T. m ethod; s e e s e c t i o n 3 .2 .5
!=Che e s ti m a t e from th e co m b in atio n C . I . m ethod; s e e s e c t io n 3 .2 .5
.
TABLE 4
Expected Values and MSE Values:
X
T
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
1 80
190
2 00
210
2 20
230
2 40
2 50
2 60
270
280
2 90
300
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
.9 9 9
.999
.999
.998
.9 9 8
.997
.997
.996
.995
.995
.994
.993
.992
.9 9 0
.9 8 9
.988
.9 8 7
.985
E(A )*
1 3 .7
29.2
4 3 .7
57.3
7 0 .6
8 4 .1
9 8 .0
112 .3
1 2 7 .0
142 .1
157 .3
1 7 2 .8
188 .3
203. 7
219.1
234.4
24.9.5
264.4
2 7 9.0
2 9 3.3
307.3
321 .0
334.4
347.5
360.3
372. 7
384.7
396 .5
4 0 8 .0
419 .1
ECy=
1 0 .5
1 2 .9
2 7 .0
2 2 .3
3 3 .4
4 0 .5
4 3 .8
53.3
5 4 .0
65.9
7 8 .6
6 4 .3
7 4.9
9 1 .7
1 0 5 .2
85.9
9 7 .1
11 8 .9
1 0 8 .6
13 2 .9
1 2 0 .3
14 7 .1
1 3 2 .1
1 6 1 .4
1 4 4 .0
175 .7
1 5 5 .8
1 9 0 .0
1 6 7 .6 ' 20 4 .3
17 9 .2
218.4
19 0 .8
2 3 2.4
2 0 2.1
2 4 6.2
213 .3
2 5 9.8
224.3
2 7 3.2
235 .0
286.4
2 4 5.5
299 .3
311.9
2 5 5.7
3 2 4 .2
265.7
2 7 5.5
3 3 6.3
348 .1
2 8 5 .0
2 9 4.2
3 5 9 .7
303 .2
370.9
3 1 1.9
3 8 1.9
3 9 2 .6
3 2 0 .5
n=3, z^=.l z^^.Ol Z^=-OOl
A f
E(AT ) d E ( I g ) = E(A1 ) E ( A 2 ) 8 MSE(A) MSE(Ac ) MSE(Af ) MSE(At ) MSE(Ag ) MSE(A1 ) MSE(A2 )
1 4 .9
3 0 .2
4 2 .3
5 2 .1
6 1 .0
6 9 .6
7 8 .2
86.9
9 5 .6
104 .4
1 1 3 .2
1 2 1 .8
1 3 0 .4
1 3 8 .7
1 4 6 .8
1 5 4 .7 -"
162 .4
169 .8
176 .9
183 .7
1 9 0 .2
1 9 6 .5
202 .5
208.3
213 .8
2 1 9.1
2 2 4.1
228.9
23 3 .5
237.9
1 0 .3
2 1 .2
3 1 .4
4 0 .9
5 0 .3
5 9 .7
69.3
79.1
8 9 .0
9 8 .9
108 .9
11 8 .9
1 2 8 .8
1 3 8 .5
1 4 8 .1
1 5 7 t6
1 6 6 .8
1 7 5 .8
1 8 4 .6
1 9 3 .2
201 .6
20 9 .7
217 .6
225 .2
23 2 .6
2 3 9.9
24 6 .8
253 .6
26 0 .2
266 .6
1 3 .1
2 7 .6
4 1 .4
54.3
6 6 .8
79.4
9 2 .2
105 .4
118.9
1 3 2 .8
1 4 6 .8
161 .1
175 .5
1 8 9.9
204.2
218 .6
232 .8
246 .8
260.7
2 7 4.4
28 7 .8
3 0 0.9
3 1 3 .8
326.4
33 8 .8
3 5 0 .8
36 2 .5
37 4 .0
38 5 .1
39% .0
1 1 .6
2 6 .6
4 1 .6
5 6 .3
70.9
8 6 .0
1 0 1 .8
1 1 8 .3
1 3 5 .5
1 5 3 .4
171 .9
1 9 0 .8
210 .1
229 .6
2 4 9.2
2 6 8.9
288 .5
30 8 .0
3 2 7 .4
34 6 .5
365 .3
3 8 3 .8
4 0 2 .0
4 1 9.9
4 3 7 .3
4 5 4 .4
4 7 1 .1
4 8 7 .3
503 .2
5 1 8 .7
1 79
7 40
1691
3018
4727
6827
9318
12186
15408
18948
22767
26819
3 1 059
35439
399 1 6
44447
48992
53516
57987
62378
66665
70827
74849
787 1 7
82422
85955
■89313
92491
95490
98309
^ t h e MPN; s e e s e c t i o n 3 . 1
t h e MPN w i t h T h o m a s ' c o r r e c t i o n f o r b i a s ; s e e s e c t i o n 3 . 2 . 1
^ F is h e r 's e s tim a te ; see s e c tio n 3 .2 .2
T hom as' e s t i m a t e ; s e e s e c t i o n 3 . 2 . 3
- t h e Johnson-B row n Spearm an e s t i m a t e ; s e e s e c t i o n 3 . 2 . 4
t h e e s t i m a t e f r o m W o o d w a r d 's C . I . m e t h o d ; s e e s e c t i o n 3 . 2 . 5
^ th e e s t i m a t e from t h e c o m b in a tio n C . I . m ethod; s e e s e c t i o n 3 . 2 . 5
97
389
8 91
1605
2532
3671
5015
6550
8259
10119
12107
14199
16369
18595
20852
23121
25382
27620
29821
.3 1 9 7 2
34065
36093
38049
39930
43 733
43459
45108
46681
48179
49607
155
6 58
1490
2633
4109
5947
8164
10763
13732
17047
20676
24579
28713
33032
37492
42049
46662
51294
55910
60481
649 7 9
69382
7 3 671
77831
81849
85716
89423
92967
96344
99553
196
521
875
1302
1819
2413
3058
3728
4399
5051
5671
6247
6775
7252
7681
8065
8412
8727
9020
9301
9579
9864
10167
10496
10862
11273
11739
12267
12864
13539
86
' 3 44
759
1317
2010
2828
3758
4787
5897
7073
8299
9560
10842
12134
13424
14704
15968
17211
18430
19621
20785
21922
23033
24120
25186
26234
27269
28293
29311
30328
157
656
1465
2565
3983
5746
7870
10354
13185
16334
19769
23447
27325
31359
35504
39716
43957
48190
52383
56507
60538
64454
68240
718 8 2
7 5 369
78693
81851
8 4 839
87656
90302
173
832
2057
3917
6514
9939
14248
19459
25552
32477
40159
48508
57425
66804
76 542
86538
96695
106927
117152
127299
137306
147118
156691
165984
174968
183618
191913
199842
207392
214560
29
3.2.1 Bias Corrections
In general, the bias of an estimator X is a function of the true
parameter value and given by b (X) = E(X)-X.
The positive bias of the
MPN is well known and was noted for the numerical example of this
paper in section 3.1.
Thomas and Woodward (1955) compared MPN
estimates to "exact" plate counts obtained by collecting organisms as
the samples passed through a membrane filter (ME).
"A considerable
part of the disparity between MF and MPN values," they conclude, "may
be attributed to the fact that mathematically considered, the MPN
tends to overestimate the true density; on the average, MPN values are
greater than the true density."
McCarthy et al. (1958) made ten
replicate n=5 MPN determinations and also used "exact" plate counts
for their true X's.^
Their empirical estimate was that E(MPN) =
1.29X.
Thomas (1955) uses a log-normal approximation to estimate E(MPN)
6*805/r^ and recommends the multiplicative bias correction e *®05/n^
Not being able to discover a better correction factor, this author
recommends the Thomas adjustment for any k and z^.
Such bias
corrected values for the example of this paper are given in the X^.
columns of Tables 3 and 4.
In a deliberate attempt to develop an estimator with less bias
■
than the MPN, Salama et al. (1978) use expansions to obtain a X for
A
which E(X) = X+0(l/n2).
'
'
Unfortunately, their k=3 dilution numerical
30
example uses middle dilution Z2=-OOl, which suggests their design
would be appropriate for X 1s near 1600, while their bias and MSE
calculations cover A=25(25)100(50)200(100)1200.
Chapter 4 discusses
the problems associated with X's outside the optimal range of the
experimental design.
Limited comparisons this author made for the
Salama et al. design, however, indicate that their rather complicated
estimator does not perform significantly better than the much simpler
MPN with Thomas' bias correction.
Further, subsequent sections will
indicate that other non-MPN alternatives achieve even better results
than does the Thomas bias-corrected MPN.
3.2.2 The Fisher Estimate
Fisher (1922) proposes an estimator based on W=E(n.-X.), the
total number of sterile responses.
Since each dilution represents ah
independent experiment, E(W) = En_P(X/=0) = En_e ^Z± and the X that
solves W = En^e ^Zi would be an estimator both reasonable and
relatively simple to compute.
Fisher showed his estimator to be
87.71% efficient in the sense that for large n, the variance of the
MPN (which is asymptotically fully efficient) is 87.71% that of hisestimator.
Values of his estimator, designated X^ in Tables 3 and 4*
for the example of this paper were obtained by Newton's method using,
the program of Appendix III.
Fisher and Yates (1943) provide
A
additional discussion and tables for obtaining A
.
■r
for n trials at each
31
of k>4 two-fold, four-fold and decimal dilutions.
3.2.3 The Thomas Estimate
Thomas (1942) gives the estimate
(n^-X^)z_^] [Zn_z^],
which is the geometric mean of the estimates obtained using e""^Z =
I-Az and e Az = l/eAz = 1/(1+Az) in equation (1.2).
For X 1X2X 3=IlO
and the n and z^ of Table 3, for example, Thomas' estimate is A
2/iZ( .233) (.333) = 7.339.
=
Thomas recommends his pre-computer
estimator for its remarkable ability to approximate the MFN (which
must be found iteratively), as seen by comparing the MPN and A^
columns of Table 3.
Apparently unknown to Thomas, however, his A
also enjoys considerably less variance than the MPN and deviates
from the MPN so as to effect a significant reduction in bias.
A
The net result, as shown in Table 4, is that A1
^ is the estimator,
for the example of this paper, with by far the smallest MSE for
values between 40 and 300.
In fact, for A=160, A^ approaches the Rao-
Cramer lower variance bound [1+b'(A)]2/I(A) for estimators of its
bias.
Empirically estimating b '(A=IbO) from Table 4 to be
[-7.6-(-3.2)]/[170-150] = -.22, one calculates the Rao-Cramer lower
bound to be 6479, to which VAR(A^) = 8037 favorably compares.
Finally, while Thomas gives A^Z/EX^
as an approximate standard
error for his estimator, he recognizes that the distribution of the
estimator follows no tabled distribution and suggests no confidence
32
interval procedure..
3.2.4 The Johnson-Brown Estimate
Johnson and Brown (1961) develop an estimator based on the
Spearman (1908) technique which, like Fisher's estimator, uses only ■
the total number of fertile responses and is also 87.71% efficient.
Their analysis, which requires n_=n and’Z^=ZjI
for i=l,2,...,k,
produces the estimate
(3.1)
Ag = {2n/[2n+ln(d)ln(2)]}e~Y“ln(zi)+Cln(d)][(SXi/n)_*5],
where 2n/[2n+ln(d)ln(2)] is a multiplicative bias correction, y=.57722
is Euler's constant and d is commonly called the dilution factor.
The Johnson-Brown Spearman estimator enjoys both practical use
(e.g., Masover, Benson and Hayflick 1974) and further discussion
(e.g., Church and Cobb 1975) in the literature.
In fact, Cornell
(1965) states, "The work of Johnson and Brown prompted several
investigations which are summarized here."
Cornell and Speckman
(1967) discuss and compare by simulation several procedures, including
Johnson and Brown's, for estimating (for different values of z) the
parameter A in the model with expectation 1-e
“Az
.
Finally, Mantel
(1967) discusses arithmetic (as opposed to the usual logarithmic)
and arbitrary spacing of the z^'s.
Identified as A , the Johnson-Brown Spearman estimator performs
b
quite well for the example of Tables 3 and 4.
Despite the fact that
33
it ignores the pattern of the X^'s and, consequently, is not based on
the minimal sufficient statistic, it is second in MSE only to Thomas'
of section 3.2.3.
3.2.5 Estimates derived from Confidence Intervals
Typical mathematical statistics texts (e.g., Bickel and Doksum
1977, page 155) note that 1-a confidence intervals must include the
entire parameter space when a=0 and, in general, decrease their
coverage of the parameter space as a, the probability of error,
’
increases.
P.
'
As a increases toward 1.00, the confidence interval
decreases toward the empty set and, as illustrated by the combination
method and tfre minimum expected width method of Chapter 2, may achieve
the empty, set even before a increases to .05.
'
This suggests that
increasing a until the 1-a confidence interval decreases to a single ,
I
point in the parameter space (i.e., finding the smallest non-empty
confidence interval associated with a particular sample result) would.
be a reasonable point estimation procedure.
This is equivalent to
: finding the point contained in every non-empty confidence interval.,
Applying this procedure to each of the true confidence interval
techniques of Chapter 2 (i.e., the Woodward method, the.combination
method and the minimum expected width method) yields three additional •
.■''
competitive point estimates for X.
e
'
.
■
Figure I gave the distribution of possible sample results
■ ; ' :V
34
arranged by magnitude of the MPN and illustrated Woodward's technique
for obtaining confidence intervals; As a increas.es toward 1.00 in the
test of hypothesis, the only sample result for which one fails to
reject
:X=Xq is the median sample result.
Although the discrete
nature of the distribution leads to a range of X values, a unique X
can be obtained for each X iXaX3 sample result by finding the Xq
associated with the distribution for which that X iXaX3 is the exact
median in the sense that the sum of all P(X*X*X*|X=Xq) for
MPN(X*X*X*) < MPN(X1X2X 3) plus one half P(X1X 2X 3IX=Xq) is exactly
.5000.
These values were calculated, for each X 1X2X 3 sample result and
are designated X 1 in Tables 3 and 4.
The point estimate associated with confidence intervals obtained
by the combination method is the Xq for which the p-value for the
alternative H :X<X equals the p-value for the alternative H :X>X .
a
o
a
o
The smallest non-empty confidence interval occurs for an a equal to
twice that common p-value.
The sample output accompanying Appendix II
indicates that for X 1X2X 3=SOl this occurs at X=59.
These values were
caluclated for each X 1X2X 3 sample result and are designated X
in
Tables 3 and 4.
The procedure for determining the point estimate associated with
the minimum expected width method for obtaining confidence intervals
is well-defined but extremely tedious.
For a given X 3X2X 3 result, one
must find the largest p-value for testing H :X=X , and the X
at which
35
that p-value occurs is the desired point estimate.
It is conjectured
that this procedure leads to the MPN, as this author has found such to
be the case for every X jX2X 3 examined.
For' the result XiXaXg=MO, for
example, Figure I indicates that the p-value for testing H q :A=21 is
.1290.
A check of similar figures gives a p-value of .1091 for
testing H :A=20 and a p-value of .1257 for testing H :A=22. The MPN
o
o
for X 1X2X 3=220 is 21.
3.3 Bias and MSE Comparisons
Recall that both Woodward and deMan cautioned against certain
X X X
12
3
results that "are not mentioned in the table and are always
unacceptable" (deMan 1975).
While Table 4 was constructed under the
obvious restriction of excluding X jX2X 3=SSI, one might be more
interested in
results.
a table constructed excluding all "unacceptable"
Table 5 gives selected expected values and MSE values over
the severely truncated sample space consisting of only those 18 X jX2X 3
results having non-empty minimum expected width confidence intervals.
Note that the "T" values are naturally somewhat (but, perhaps
surprisingly, not significantly) smaller than those of Table 4 and
that the expected values and MSE values are essentially unchanged.
Since additional truncation of the sample space appears not to affect
the "rankings" of the estimators, subsequent comparisons will continue
to use the sample space obtained by eliminating only the X JX2X 3=nJn2n 3
TABLE 5
Selected Expected Values and MSE Values: n=3, Z 1=.!
Z^=-Ol
Z3= .001
Truncated Sample Space: only those 18 X 1X2X 3 results with non-empty minimum expected ,
width confidence intervals
X
10
50
100
150
T
.979
.992
.992
200
.991
.990
250
300
.987
.982
E(X)* E (ic )b E(Ig)C E(X1)d ECX^)^ MSE(X) M S E ( X )
13.8
70.6
142.2
219.5
293.6
360.4
419.1
10.5
54.0
108.7
167.8
224.5
275.6
320.5
10.1
49.9
98.5
147.6
192.6
231.9
265.8
13.1
66.8
132.8
204.7
275,0
339.4
396.6
11.4
70.6
153.1
249.4
346.9
437.8
519.2
182
4733
19008
40048
62549
82635
98583
MSE(Xg) M S E ( X ) MSE(X2)
159
172
98
84
2535
10150
20910
32050
41845
49768
1984
3993
6493
6995
13241
19351
24910
30114
16449
35752
56814
75680
90591
32635
77019
127955
175675
215255
*the MPN; see section 3.1
bthe MPN with Thomas' correction for bias; see section 3.2.1
Johnson-Brown Spearman estimate; see section 3.2.4
dthe
dthe estimate from Woodward's C. I. method; see section 3.2.5
ethe estimate from the combination C.I. method ; see section 3 .2.5
37
result.
Furthermore, since the estimators A 1 and A2 are tedious to
compute and did not perform well in the example of this paper, they
will not be included in further comparisons.
As noted in section 3.2, Thomas' A1
^ is without question the
preferred estimator for the example of Table 4.
There is no guarantee
Zn
that some other estimator, however, would not perform better than A^
for other
or n^ values.
Table 6 compares the expected values and
the MSE values for the estimators of the program in Appendix III
(i.e. , the MPN,
^
^
and
for n=3,5,10 and for k=3 two-fold,
four-fold and decimal dilutions (i.e. , for dilution factors d=2,4,10)
centered at Z2=-Ol.
Each of these nine experimental designs, which
cover virtually all k=3 serial dilution settings found in the
literature, should be appropriate for A=160 and the comparisons will
be made, as in Table 4, for A=IO(IO)300.
The MSE values of Table 6 follow a definite pattern illustrated
by Tables 6.1-6.3, which vary the dilution factor while maintaining
n=3 samples per dilution.
For dilution factor d=10 (Table 6.1), A
performs best as measured by MSE.
enjoys the smallest MSB.
the superior estimator.
For d=4 (Table 6.2), however, Ag
And for d=2 (Table 6.3), A^ appears to be ,
As n increases, each estimator, as expected,
performs better than it did for the previous n.
For any single
dilution factor, however, the "rankings" of the estimators do not
change as n increases.
38
TABLE 6
Expected Values and MSE Values
Ic=3 dilutions with middle dilution Z^=-Ol.
n = number of samples per dilution
d = the dilution factor
X = the MPN; see section 3.1
A = the MPN with Thomas' correction for bias; see section 3.2.1
Ap= Fisher's estimate; see section 3.2.2
Ap= Thomas' estimate; see section 3.2.3
Ag= the Johnson-Brown Spearman estimate; see section 3.2.4
Tables 6.1 - 6.9 vary n and d as indicated below.
d
10
4
A,
T
6.1
6.4
6.7
6.2
6.5
6.8
6.3
6.6
6.9
4
.— — i---- n
3
5
10
TABLE 6.1
n = 3
d =10
A
10
20
30
40
50
60
70
80
90
100
as
CO
T
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
n o 1.000
120 1.000
130 . .999
140
.999
150
.999
160
.998
170
.998
180
.997
190
.997
200
.996
210
.995
220
.995
230
.994
240
.993
250
.992
260
.990
270
i 989
280
.988
290
.987
300
.985
E(Ap)
E(At )
E(Ag)
MSE(A)
12.92
13. 73 • 1 0 .5 0
22.31 ■ 27.03
29.17
40.48
43.68
33.40
57.26
4 3 .7 8
53.26
70.56
53.96
65.87
84.06
64.27
78.64
74.90
9 7 .9 5
91.74
112.28
8 5 .8 5 105.17
9 7 .1 2 118.91
127.01
142.06 108.63 132.90
157.34 120.31 147.07
172.77 132.11 161.35
188.26 143.96 175.68
203.74 155.79 190.00
219.14 167.57 204.26
234.40 179.24 218.40
249.49 190.77 232.40
264.35 202.13 246.23
278.96 213.30 259;84
293.28 224.26 273.23
307.32 234.99 286.38
321.04 245.48 299.27
334.43 255. 73. 311.89
3 47.50 265.72 324.24
360.25 275:46 '3 3 6 .3 2
372.66 284.95 348.12
384.74 294.19 359.65
396.50 303.19 370.91
407.95 311.94 381.91
4 1 9 .0 8 320.45 392.64
14.85
1 0 .2 7
21.17
31.36
4 0 .9 1
5 0 .2 8
59.71
6 9 .3 0
79.07
88.97
9 8 .9 4
1 08.92
1 18.88
1 28.76
179
740
1691
3018
4727
6827
9318
12186
15408
18948
22767
26819
31059
35439
39916
44447
48992
53516
57987
E(A)
E ftcJ
3 0 .2 4
4 2 .3 0
52.12
60.98
69.57
78.18
86.87
9 5 .6 3
104.42
113.17
121.83
130.36
138.70
146.83
1 5 4 .7 3
162.37
169.75
176.85
1 83.68
190.24
196.52
202.53
208.: 29
213 . 79
219.06
224.09
228.90
233.51
237.91
138.51
148.12
1 57.55
166.79
175.82
184.63
1 9 3 .2 0
201.55
209.67
2 17.55
225.21
232.63
239.85
246.84
253.63
260.22
266.62
62378
66665
70827
74849
78717
82422
85955
89313
92491
95490
98309
MSE(Xc )
97
389
891
1605
2532
3671
5015
6550
8259
10119
12107
14199
16369
18595
20852
23121
25382
27620
29821
31972
34065
36093
38049
39930
41733
43459'
45108
46681
48179
49607
MSE(Xf )
155
658
1490
2633
4109
5947
8164
10763
13732
17047
20676
24579
28713
33032
37492
42049
46662
51294
55910
60481
64979
69382
73671
77831
81849
85716
89423
92967
96344
99553
MSE(A^)
196
521
875.
1302
1819
2413
3058
3728
4399
5051
5671
6247
6775
7252
7681
8065
8412
8727
9020
9301
9579
9864
10167
10496
10862
11273
11739
12267
12864
13539
MSE(Xg )
86
344
759
1317
2010
2828
3758
4787
5897
7073
8299
9560
10842
12134
13424
14704
15968
17211.
18430
19621
20785
21922
23033
24120
25186
26234
27269
28293
29311
30328
TABLE 6.2
n = 3
d = 4
o
X
T
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.999
.998
.997
.996
.994
.992
.989
.986
.982
.977
.972
.967
.961
.954
.947
.939
.931
.922
.913
.904
.894
.884
.874
E(X)
E(Xc)
E(Xf)
E(Xt)
11.54
23.70
36.40
49.46
62.74
76.07
89.37
102.55
115.56
128.36
140.92
153.23
165.25
176.97
188.37
199.43
210.15
220.50
230.49
240.11
249.36
258.24
266.75
274.90
282.71
290.17
297.29
304.10
310.60
316.80
8.82
18.12
27.83
37.82
47.97
58.17
68.34
78.41
88.36
98.15
107.76
117.17
126.36
135.32
144.04
152.50
160.69
168.61
176.25
183.60
190.67
197.46
203.97
210.21
216.17
221.88
227.33
232.53
237.50
242.24
11.37
23.13
35.26
47.64
60.18
72.77
85.34
97.84
110.20
122.41
134.42
146.19
157.71
168.94
179.85
190.44
200.67
210.55
220.06
229.20
237.97
246.37
254.42
262.11
269.46
276.47
283.17
289.56
295.65
301.46
11.75
24.42
37.67
51.10
64.42
77.46
90.13
102.41
114.30
125.80
136.93
147.70
158.-11
168.17
177.89
187.26
196.28
204.95
213.28
221.27
228.91
236.23
243.22
249.90
256.27
262.34
268.13
273.64
278.89
283.89
E(Xs)
MSE(X)
116
12.65
20.68
337
29.42
708
1264
38.37
47.26
2019
55.92
2963
64.28
4068
72.32
5297
6608
80.04
7959
87.43
94.50
9311
101.27
10631
107.74
11890
113.92
13065
14139
119.81
125.42
15097
130.76
15933
135.84
16641
17222
140.66
17678
145.24
18015
149.57
153.68
182.39
18359
157.57
161.26
18387
18333
164.75
168.05
18209
18027
171.17
17800
174.14
176.94 ' 17540
179.60
17259
MSE(Xc) MSE(Xf) MSE(Xt) MSE(Xg)
68
192
395
691
1089
1584
2162
'2802
3484
4186
4890
5578
6239
6862
7441
7974
8460
8900
9300
9664
10000
10317
10622
10926
11238
11568
' 11926
12321
12764
13262
109
311
654
1172
1878
2764
3805
4965
6205
7486
8769
10024
11221
12340
13364
14282
15087
15775
16348
16810
17166
1742517597
17691
17720
17695
17629
17534
17422
■ 17304
127
376
763
1287
1933
■2680
3500
4362
5239
6103
6934
7712
8424
9059
9610
10075
10454
10750
10967
11112
11194
11222
11207
11159
11090
11012
10935
10872
10832
10826
54
156
322
539
794
1077
1378
1691
2012
2336
2665
2999
3344
3702
4082
4490
4935
5424
5967
6572
-7249
8006
8852
9793
10839
11995
13270
14668
16197
17860
TABLE 6.3
n = 3
d = 2
A
T
10 1.000
20 1.000
30 1.000
40 1.000
50 1.000
60
.999
70
.999
80
.997
90
.994
100
.990
HO
.984
120
.976
130
.967
140
.955
150
.941
160
.925
170
.907
180
.888
190
.867
200
.846
210
.823
220
.799
230
.775
240
.750
250
.725
260
.700
270
.675
280
.650
290
.625
300
.601
E(A)
10.87
21.90
33.12
44.53
56.15
67.94
79.84
91.72
103.48
114.98
126.12
136.80
146.96
156.55
165.56
173.98
181.82
189.11
195.88
202.14
207.94
213.31
218.27
222.87
227.13
231.08
234.74
238.13
241.29
244.22
E(Xc)
E(Af)
E(At)
8.32
10.84
10.93
21.80
16.75
22.14
25.32 . 32.90
33.69
44.18
45.66
34.05
55.63
58.07
42.94
67.20
70.92
51.95
61.05
78.84
84.12
90.43
97.57
70.14
79.13 101.85 111.09
87.92 112.97 124.52
96.44 123.69 137.70
104,61 133.92 150.49
112.37 143.61 162.79
119.71 152.71 174".52"
126.59 161.21 185.62
133.03 169.12 196.09
139.03 176.46 205.91
144.61 183.23 215.10
149.78 189.49 223.68
154.57 195.25 231.66
159.00 200.56 239.10
163.11 205.44 246.01
166.90 209.94 252.43
170.42 214.08 258.41
173.68 217.89 26.3.96
176.69 221.40 269.13
179.49 2.24.64 273.94
182.09 227.63 278.42
184.50 230.39 282.60
186.74 232.94 286.49
E(Xs)
MSE(A)
MSE(Ac) MSE(Af). MSE(At) MSE(Ag)
76
23.58
126
177
29.06
289
3.09
501
34.68
478
778
40.33
688
1129
45.93
937
1556
51.37
1219
2044
56.61
■ 1519
61.59
2569
1824
66.26
3099
2120
3601
70.61
2399
74.62
4049
2657
4422
78.30
2895
4709
81.66
3120
84.70
4906
3340
5017
87.47
3567
89.96
5053
• 3816
5026
92.22
4100
94.26
4953
4433
4850
96.10
4830
4735
97.77
4626
5303
99.28
5865
4540
100.64.
4493'
6527
101.88
' 7301
4500.
103.01
4575 . 8194
104.03
9216
4731
104.97
10375
105.82
4979
5330 ’ 11678
106.60
13130
107:31 .. '5794
14738
6380
107.97
129
124
306
284
555
490
908
759
1392
1100
2022
1512
2790
1980
3666
2478
4603
2973
5548
3437
6451
3841
7268
4169
7967
4413
8528
4572
8940
4654
9206
4670
9331
4636
9330
4569
9217
4488
9013
4413
8737
4361
8409
4349
8047
4394
4510 .
7673
7303
4713
' 5013
. 6954
6642
5423
5952
6381
6185
6610
6066
7405
214
153
143
176
249
360
512
712
968
1289
1689
2181
2777
3489
4331
5311
6439
7723
9170
10786
12576
14545
16696
-19032
21556
24270
27176
30275
33569
37059
TABLE 6.4
n = 5
'd =10
A
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
T
E(A)
E(Ap)
E(At)
E(Ag)
1.000
11.83
11.54
10.07
1.000
25.25
21.50
24.03
1.000
38.18
32.50 . 36.13
1.000
50.02
42.58
47.55
61.29
52.18
58.68
1.000
72.52
1.000
61.73
69.84
1.000
83.99
71.50
81.23
1.000
95.85
81.59
92.88
1.000 108.10
92.03 104.80
1.000 120.74 102.78 116.97
1.000 133.72 113.83 129.36
1.000 146.99 125.13 141.93
1.000 160.50 136.63 154.66
1.000 174.20 148.30 167.50
1.000 188.05 160.08 180.43
1.000 201.99 171.96 193.43
1.000 216.00 183.88 206.46
1.000 230.03 195.82 219.50
1.000 244.04 207.75 232.54
1.000 258.02 219.65 245.54
1.000 271.93 231.49 258.50
1.000 285.75 243.26 271.40
1.000 299'.46 254.93 284.22
1.000 313.06 266.50 296.96
1.000 326.51 277.96 309.61
1.000 339.83 289.29 322.16
.999 352.99 300.49 334.61
.999 365.99 311.56 346.95
.999 378.83 322.49 359.18
.999 391.51 333.29 371.30
12.59
26.93
38.62
47.55
55.04
62.02
68.95
76.01
83.26
90.71
98.31
106.04
113.84
l'21.6g"
129.47
137.21
144.86
152.39
159.76
166.97
174.00
180.83
187.46
193.89
200.12
206.14
211.97
217.59
223.03
228.29
9 .9 6
20. 78
30.79
40.03
4 9 .0 3
58.12
67.44
77.01
86.81
9 6 .7 7
106.85
1 17.00
127.16
1 3 7 .3 0 '
147.37
1 57.36
167.23
176.97
186.57
196.02
205.30
214.42
223.37
232.15
240.77
249.23
257.52
265.67
273.66
281.51
E<ic)
MSE(A)
MSE(Ac )
MSE(Af )
64
270
598
1031
1587
2288
3151
4191
5418
6840
8462
10283
12303
14518
16921
19503
22255
25165
28222
31412
34724
38145
41661
45261
48933
52664
56446
60266
64116
67986
44
178
391
681
1063
1547
2144
. 2858
3693
4653
5739
6951
8286
9742
11315
12998
■ 14787
16673
18651
20711
22847
25051
27316
29635
32000
34406
36845
39312
41902
44309
58
258
590
1019
1546
2188
2964
3894
4996
6287
7778
9478
11392
13520
15860
18406
21150
24082
27189
30460
33880
37435
41111
44895
48772
52729
56754
60834
64957
69113
MSE(At )
MSE(Ag )
42
82
165
254
361
366
632
459
980
598
1404
812.
1901
1106
2465,
1471
3092
1893
3776
2359
4512
2853
5296
3362
6123
3877
6989
4387
7891
4887
8825
5375
9789
5850
10780
6312
11796
6766
7216 ■ 12834
13895
7668
14975
8128
16075
8603
17192
9102
18327
9631
• 19478
10197
20645
10810
21828
11474
23026
12197
24240
12985
TABLE 6.5
n = 5
d = 4
m
A
T
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
190
200
210
220
230
240
250
260
270
280
290
300
.999
.999
.999
.998
.997
.997
.995
.994
.992
.991
.988
.986
.983
.980
.976
.973
.968
E(A)
10.82
21.94
33.39
45.09
56.95
68.88
80.82
92.75
104.65
116.54
128.44
140.37
152.34
164.35
176.38
188.44
200.50
212.53
224.51
236.40
248.19
259.83
271.31
282.61
293.69
304.55
315.17
325.53
335.63
345.46
E(Af)
E(Xt)
9.21
10.76
21.73
18.68
28.42 ■ 32.90
44.23
38.38
48.48
55.67
67.14
58.64
78.63
68.80
90.13
78.95
89.08 101.65
99.21 113.21
109.34 124.81
119*. 50 136.46
129.68 148.17
139.91 159.92
150.16 171.69
160.42 183.48
170.68 195.26
180.93 206.99
191.12 218.66
201.25 230.22
211.28 241.67
221.19 252.97
230.97 264.10
240.58 275.03
250.02 285.76
259.26 296.26
268.30 306.53
277.12 316.55
285.72 326.32
294.09 335.83
10.92
E»c>
22.40
34.35
46.53
58.65
70.53
82.09
93.32
104.27
114.97
125.50
135.90
1.46. 2L.
156.44
166.62
176.75
186.81
196.81
206.71
216.50
226.17
235.70
245.06
254.24
263.24
272.02
280.60
288.95
297.07
304.96
E«s>
12.76
20.61
29.26
38.20
47.13
55.87
64.37
72.62
80.63
88.40
95.97
103.34
110.51
117.51
124.32
130.95
137.39
143.64
149.70
155.56
161.23
166.70
171.97
177.04
181.93
186.62
191.13
195.45
199.60 ■
203.58
MSE(A)
MSE(Ac) MSE(Af) MSE(At) MSE(Ag)
41
108
211
357
554
807
1121
' 1500
1946
2562
3047
3699
4412
5180
5994
8968
6844
10252
7717
11579
8604
12929
9491
14286
10369
15631
11227
16948
12055
182.20
19434 • 12847
13597
20577
14298
21639
14950
22613
15549
23492
16096
24273
16592
24953
17040
25532
56
150
299
515
810
1191
1662
2230
2899
3670
4544
5519
6587
7741
55
146
291 •
506
800
1181
1653
2220
2886
3652
4520
5487
6548
7693
8913
10192
11515
12865
14225
15578
16907
18197 .
19434
20607
21706
22723
23652
24490
25235
25886
59
167
335
559
831
1147
1505
1907
2358
2860
3413
4017
4666
5355
6075
6817
7570
8325
9071
9798
10498
11164
11789
12369
12902
13384
13818
14203
14542
14838
33
84
176
299
448
620
813
1027
1260
1512
1783
2072
2379
2705
3051
3418
3809
4227
4675
5157
5679
6245
6862
7533
8267
■ 9068
9943
10898
11939
13072
TABLE 6.6
n = 5
d =.2
X
T
E(X)
10
20.
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
1.000
1.000
1.000'
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.999
.998
.997
.994
.991
.987
.981
.974
.966
.956
.944
.931
.916
.901
'.884
.865
.846
10.49
21.07
31.72
42.48
53.35
64.35
75.48
86.76
2 80
290
300
9 8 .1 7
109.70
121.32
132.96
144.55
156.04
167.32
178.35
189.04
199.35
209.24
218.69
227.67
236.17
244.21
251.79
258.92
265.62
271.90
.8 2 6 277.80
.805 283.33
.784 288.50
E(Xc )
E(Xp)
E(Xt)
8.93
17.93
27.01
36.17
45.42
54.78
64.26
73.86
83.57
93.39
103.28
113.18
123*06
132.83
142.44
151.82
160.93
169.71
178.13
186.17
193.81
201.05
207.90
214.35
220.42
226.12
231.47
236.49
241.19
245.60
10.48
21.02
31.64
42.34
53.13
64.03
75.05
86.20
97.46
108.82
120.23
131.65
143.01
154.23
165.23
175.96
186.33
196.32
205.87
214.96
223.58
231.73
239.41
246.62
253.39
259.73
265.66
271.21
276.38
281.22
10.52
21.20
32.08.
43.20
54.60
66.33
78.41
90.87
103.73
116.96
130.54
144.40
158.45
1-721 59186.70
200.67
214.41
227.82
240.82
253.36
265i39
276.89
287.85
298.25
308.11
317.43
326.23
334.53
342.35
349.71
E(Xs )
MSE(X)
69
154
- 258
386
546
743
988
1288
1648
2068
2541
3055
3589
4123
8 8 .4 8
4636
91.89
5106
95.05
5518
97.96
5860
100.64
6126
103.09
105.34
6314
6427
107.40
6471
109.29
6456
111.Ol
6392
112.5&
6292
114.02 '
6171
115.34
6042
116.55
5920
117.65
5818
118.66
5750
119.59
24.06
29.48
35.08
40.74
46.37
51.90
57.28
62.47
67.44
72.17
76.65
80.86
84.80
MSE(Xc) MSE(Xf) MSE(Xt) MSE(Xg)
51
115
193
290
408
552
727
938
1187
1474
1794
2138
2496
2853
3199
3523
- 3818
4081
4312
4514
4694
. 4859
5021
. 5189
5378
5597
5861
6180.
6566
7030
68
153
256
385
545
■ 744
992
1294
1657
2079
2551
3060
3585
4105
4599
5047
5434
5749
5989
6153
6246
6275
6251
6187
6098
5997
5899
5820
5774
5774
• 215
70
131
159
97
275
■ 104
428
152
630
239
901
369
1260
5.46
1734
776
2342
1066
3099
1425
..4007
1860
5053
2381
6211
2999
7444
3723
8709
9962
4564
5530
11160
6631
12267
' 7875
13251
9271
14092
10824
14775
15293 . 12542
14430
15645
1649315837
18734
15876
21158
15775
23767
15549
26564
15212
29552
14783
32731
14277
TABLE 6.7
n =10
d =10
A
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
. T
1.000
1.000
1.000
1.000
1 .0 0 0
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
E(A)
E<V
E(Ap)
10.75
9 .9 1
20.66
22.40
34.24
31.59
4 5 .3 6 • 41.85
55.82
5 1 .5 0
60.93
66.03
70.40
76.30
80.04
86.75
97.42
89.89
108.32
99.95
119.44 110.20
130.74 120.62
142.21 131.21
153.84 141.94
165.61 152.80
177.50 163.77
1 89.50 174.85
201.60 186.01
213.77 197.23
225.99 208.51
238.26 219.83
250.54 231.16
262.83 242.50
275.10 253.82
287.34 265.12
299.54 276.38
311.69 287.58
323.77 298.73
335.78 309.81
347.71 320.82
10.68
21.93
33.10
43.77
54.16
64.51
74.99
85.61
96.39
107.31
118.35
129.50
140.74
152.07
163.48
174.95
186.47
198.03
209.61
221.21
232.82
224.41
255.98
267.53
279.04
290.51
301.94
313.31
324.63
335.90
E(Ap)
E (J s )
9 .8 0
20.64
30.65
39.74
48.51
5 7 .3 6 .
66.47
75.88
85.57
95.49
88.33 105.57
94.75 115.76
-101.32 126.01
108.01 136.27
114.82 146.50
121.70 156.65
128.62 166.72
135.57 176.67
142.50 186.48
149.38 196.14
156.21 205.65
162.94 215.00
169.55 224.18
176.04' 233;19
182.39 242.05
188.58 250.74
194.61 259.28
200.48 267.67
206.18 275 .9 2 '
211.70 284.03 ■
11.17
24.09
35.85
44.88
52.01
5 8 .2 3
64.14
70.01
75.97
82.07
MSE(A)
MSE(Ac )
MSE(Ap )
20
89
213
16
71
168
292
444
626
841
1090
1374
1698
2066
2482
2952
3480
4067
4716
5427
6199
7031
7922
8869
9869
10919
12017
13161
14347
15573
16839
18142
19482
20
89
221
392
591
817
1073
1363
1695
2075
2512
3013
3584
4232
.4961
5773
6672
7657
368
552
771
1028
1326
1670
2064
2516
3031
3615
4275
5012
5829
6727
7706
8763
9897
11103
12379
13721
15124
16586
18102
19669
21285
22947
24652
8729
9886
11125
12444
13841
15312
16853
18461
20133
21866
23656
25503
MSE(At ) ' MSE(Ag )
26
111
174
187
199
251
356
516
731
997
1310
1666
2058
2483
2933
3404
3892
4394
4907
5432
5968
6517
7084
7671
8284
8928
9610
10336
11112
11945
18
. 72
154
269
421
612
838
1097
1386
1702
2042
2405
2787
3188
3606
4038
4486
4947
5422
5911
6414
6931
7464
8012
8578
9163
9767
10393
11040
11712
TABLE 6.8
n =10
d = 4
IO
X
T
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.999
.999
.999
E(X)
10.38
20.88
31.52
42.30
53.19
64.14
75.11
86.04
96.94
107.79
118.62
129.45
140.28
151.16
162.08
173.06
184.11
195.23
206.43
217.70
229.04
240.45
251.91
263.42
274.97
286.54
298.14
309.75
321.35
332.94
E(Xci)
MSE(X)
12.90
10.43
10.36
9.58
20.63
21.17
20.81
19.27
32.22
29.23
31.36
29.09
38.17
43.49
41.99
39.03
54.78
47.13
52.68
49.08
65.90
55.93
63.39
59.18
76.72
64.50
69.30
74.09
72.82
87.20
84.79
79.39
97.33
80.91
89.44
95.47
88.78
99.46 106.15 107.15
96.45
109.45 116.84 116.74
119.43 127.56 126.14 103.96
129/43 138.32 135.41 111.29
139.46 149.13 144 .-61• 118.48
149.54 159.99 153.76 125.51
159.67 170.90 162.91 132.39
169.87 181.87 172.06 139.13
180.13 192.89 181.23 145.72
190.47 203.97 190.44 152.17
200.87 215.10 199.67 158.47
211.33 226.28 208.94 164.62
221.85 237.50 218.24 170.63
232.42 248.76 227.57 176.49
243.04 260.05 236.93 182.21
253.70 271.37 246.30 187.78
264.38 282.71 255.69 193.21
275.08 294.06 265-08 198.50
285.79 305.42 274.48 203.65
296.50 316.77 283.86 208.66
307.19 328.11 293.24 213.54
24
62
117
194
299
434
598
793
1018
1275
1566
1893
2259
2665
3116
3615
4163
4764
5421
6135
6907
7739
8630
9579
10584
11643
12750
13903
15094
' 16317
E(Ic)
E(Xf)
E(Xt)
MSE(Xn) MSE(Xt) MSE(Xrp) MSE(Xc)
21
52
98
162
247
355
487
644
826
1034
1270
1536
1833
2163
2529
2932
3374
3858
4385
4956
5573
6235
6944
7697
8493
9331
10206
11115
12055
13019
24
. 62
118
200
312
456
631
839
1078
1350
1656
1997
2377
2798
3264
3778
4343
4963
5640
6376
7173
8031
8951
9932
10971
12065
13210
14402
15635
16902
25
67
' 131
219
327
448
577
712
856
1012
1185
1379
1596
1841
2114
2418
2753
3121
5321
3956
4424
4926
5464
6036
6642
7282
7955
8658
9391
10151
20
39
82
142
215
302
403
518
648
795
959
1143
1346
1570
1817
2089
2388
2717
3077
3472
3905
4378
4896
5460
6075
6744
7471
8258
9110
10029
TABLE 6.9
n =10
d = 2
<1-
X
T
10
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
.999
.999
.998
.997
.995
.993
.990
.986
.982
.976
.970
.962
.953
E(X)
E(Xc)
9.45
10.24
18.92
20.51
30.82
28.43
37.98
41.17
51.56
47.58
62.01
57.22
72.51
66.91
83.08
76.65
93.70
86.46
104.40
96.33
115.17 106.26
126.02 116-.27
136.95 126.36
147.96 136.52
159.06 146.76
170.24 157.07
181.49 167.45
192.79 177.88
204.11 188.33
215.44 198.77
226.72 209.19
237.93 219.52
249.00 229.75
259.92 239.81
270.62 249.69
281.07 259.33
291.24 268.72
301.10 277.81
310.63 286.60
319.80 295.06
E(Xf)
E(Xt)
10.25
10.23
20.49
20.59
31.05
' 30.78
41.11
41.67
51.48
52.47
63.46
61.89
74.68
72.35
86.13
82.86
97.83
93.43
104.06 109.79
114.76 122.03
125.52 134.55
136.36 14.7.39,
147.28 160.54
158.27 174.04
169.34 187.88
180.46 202.07
191.64 216.59
202.84 231.44
214.02 246.56
225.16 261.91
236.22 277.43
247.14 293.05
257.89 308.69
268.42 324.28
278.70 339.74
288.69 354.99
298.36 369.98
307.69 384.64
316.66 398.93
E(Xs)
24.45
29.84
35.41
41.07
46.72
52.29
57.73
62.99
68.05
72.88
77.48
81.84
85.97
89.87 '
93.55
97.01
100.27
103.33
106.20
108.89
111.41
113.76
115.95
117.99
119.89
121.66
123.29
124.81
126.22 ■
127.52
MSE(X)
32
71
117
172
236
314
405
513
641
791
969
1178
1423
1708
2036
2410
2830
3291
3787
4309
4844
5380
5902
6396
6850
7252
7595
7874
8085
8230
MSE(Xc) MSE(Xf) MSE(Xt) MSE(Xg)
28
61
101
149
205
271
349
‘ 440
546
671
816
986
1183
1412
1674
1971
2303
2666
3057
3466
3886
4307
4717
5107
5469
5796
6083
6329
6532
6697
32
71
117
173
239
319
414
527
661
819
1006
1226
1482
1780
2121
2508
2940
3412
3917
4445
4984
5520
6039
6526
6969
7358
7687
7949
8144
■ 8272
32
72
122
185
263
363
489
647
847
1099
1415
1811
2308
2925
3688
4616
5729
7037
8544
10241
12109
14120
16239
18423
20628
22809
24923
26930
28795
30490
217
117
64
52
79
146
255
410
618
885
1220
1629
2120'
2702
3381
4166
5063
6081
7224
8502
9919
11482
. 13197
. 15070
17106
19311
21688
24241
26976
29894
48
While it may be disappointing that no single estimator performs
best across all dilutions, it should not be surprising.
As the
dilution factor decreases to I (d=l is equivalent to using the single
dilution z=.01 for all 3n=9 samples), the Fisher estimator
approaches the MPN and the Johnson-Brown Spearman estimator approaches
the constant value A
= e ^ In Cz1) = e ^ ln(.01) _ 5 5 3 5 . As the
b
dilution factor increases, it can be shown algebraically that the
Thomas estimator
shrinks and the Johnson-Brown estimator Xg grows.
In short, the dilution factor affects each estimator in a
particular manner and it seems, in general, that A
"high" dilution
factors and A
C
performs best at
factors, X^ performs best at "intermediate" dilution
performs best at "low" dilution factors.
The choice of
the dilution factor, however, moves one into the area of design
•considerations.
Chapter 4 will examine current design
recommendations, design suggestions in light of Table 6, and the final
selection of an estimator.
4. DESIGN CONSIDERATIONS
Early serial dilution investigations (e.g., Fisher 1922) suggest
that most researchers of the day simply used large numbers of
dilutions over a range wide enough to be certain of obtaining at least
one dilution for which some but not all of the samples were fertile.
Matuszewski et al. (1935) noted that "the most accurate predictions
are obtained" when between 59% and 66% of the total number of samples
are fertile and recommended trying to select dilutions accordingly.
Fisher and Yates (1943) advocated that the "two-fold dilution series
should be used, with correspondingly fewer [samples] at each level, in
preference to a four-fold or ten-fold series covering the same range."
Stevens (1958) proposed, a test to determine whether suspicious results
(e.g., X 1X2X 3X 4X 5=SOSOS for n/=3 and z^=(.5)i) are suitable for
further analysis.
Most current methods books recommend using k=3 decimal dilutions
with n=3 or n=5 samples per dilution and that even when more than
three dilutions are used to be certain of avoiding either all fertile
or all sterile results, "the results from only three of these are used
in computing the MPN" (American Public Health Association 1971).
The
more statistical works, however, echo Finney's design statements that
(I) "If N, the total number of samples is fixed, the ideal allocation
would be to use all [samples] at the dilution giving 1.59 organisms
per sample" (Finney 1978, page 436) and (2) for serial dilutions, "The
dilution factor should be as small as practicable; 2 and 4 are
50
definitely preferable to 10" (Finney 1978, page 437) .
In this chapter, it is established that the above present
recommendations are not necessarily correct.
As in Chapter 3, the
discussion commences with a consideration of the single dilution
experiment (section 4.1).
Finally, a workable algorithm is given for
determining, under certain researcher-chosen constraints, an efficient
serial dilution design (section 4.2).
4.1 The Single Dilution Experiment
Ever since Finney (1952) first noted that Fisher's information
I(X) for the single dilution problem was maximized for Xz=I.59,
statements like (I) in section 4.1 have abounded in the literature
(Mantel 1975).
Unfortunately, Xz=I.59 is not optimal for the small
values of n encountered in practice.
Before Finney solved the single dilution problem asymptotically,
Halvorson and Ziegler (1933b) set out "to show how [X], as well as the
number of tubes, influences the accuracy of the results."
Using the
coefficient of variation as their measure of accuracy, they note that,
"An examination of this table and graph shows that the point of
maximum accuracy varies with the number of tubes.
With 10 tubes, the
maximum accuracy is obtained when 70% of the tubes show growth [i.e.,
for Xz=I.2], but with 100 tubes, the maximum accuracy is obtained when
78% of the tubes show growth [i.e., for Xz=I.5]."
While their
51
conclusion, "Theoretically, the maximum accuracy is obtained with a
bacterial population of approximately 1.2 to 1.5 organisms per
[sample], this range shifting toward the higher values as the number
of tubes is increased," fails to identify the exact asymptotic bound
as 1.59, it is unfortunate that their small sample work has been
largely ignored.
A further consideration discussed by neither Finney nor Halvorson
and Ziegler is the effect of the MPN's sizable positive bias when MSE
and not variance is used to judge precision.
f
Table 7 gives expected
•values and MSE values for the single dilution design using all n=9
samples at either z=.008, z=.010 (Finney's optimal design for A=160)
or z=.012.
While the varying degree of truncation caused by
eliminating the result for which all samples were fertile makes
comparisons across dilutions difficult, note that MSE(A) near A=160
is actually smallest for z=.012 (i.e ., for Az>l.59).
It should be noted that the figures in Table 7 indicate that
VAR(A|A=160, z=.012) < VAR(A|A=160,
z =.010)
< VAR(A|A=160,
z =.008),
which seems to contradict the previously mentioned Halvorson and
Ziegler result.
tions
The latter's work, however, uses binomial approxima­
without actually considering each of the n+1 possible sample
results.
Because Table 7 eliminates the result for which all samples
were fertile, Halvorson and Ziegler's conclusions cannot be directly
compared with those of this paper.
At any rate, simply choosing
52
TABLE 7
MPN Expected Values and MSE Values for the Single Dilution: n=9
z=.008
X
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
T
E(X)
z=.010
MSE(X)
165
10.63
1.000
21.33. • 353
1.000
1.000
32.10
' 567
1.000
42.97
813
53.92
1094
1.000
1.000
64.96
1412
1765
1.000
76.07
87.20 . 2141
.999
2527
.998
98.28
.995 109.26 • 2904
.992 120.03
3254
3559
.987 130.51
3806
.980 140.63
3989
.971 150.33
4105
.960 159.55
4157
.947 168.27
4153
.931 176.46
4106
.912 184.12
.892 191.26
4029
3937
.869 197.88
3849
.844 204.02
3781
.817 209.68
3749
.789 214.91
3770
.760 219.72
'3859
.730 224.15
4031
.699 228.22
.688 231.96
4299
4674
.637 235.40
5169
.606 238.56
5792
.575 241.45
T
1.000
1.000
1.000
1.000
1.000
.999
.998
.995
.991
.984
.974
.960
.943
.922
.897
.869
.837
.803
.767
.730
.691
.652
.614
.575
.537
.501
.465
.431
.399
.368
z=.012
E(X)
MSE(X)
10.64
21.37
32.19
43.14
54.19
65.31
76.42
87.41
98.14
108.50
118.36
127.64
136.29
144.28
151.62
158.30
164.38
169.88
174.85
179.32
183.35
186.98
190.23
193.16
195.79
198.16
200.29
202.20
203.92
205.47
134
292
479
700
958
1248
1556
■ 1859
2134
2362
■ 2527
2627
2663
2646
2592
2520
2451
2406
2405
2470
2616
2862
3221
3707
4331
5103
6032
• 7125
8389
9830
T
E(X)
1.000
10.65
21.40
1.000
1.000
32.29
43.31
1.000
54.42
.999
65.52
.998
76.45
.994
87.01
.987
97.02
.976
.960 106.37
.939 114.95
.912 122.75
.880 129.75
.844 136.01
.803 141.57
.760 146.48
.715 150.82
.668 154.64
.621 158.01
.575 160.97
.530 163.57
.486 165.86
.445 167.88
.405 169.66
.368 171.22
.334 172.60
.302 173.82
.273 174.89
.245 175.84
.221 176.67
MSE(X)
114
' 252
421
627
867
1123
1370
1581
1736
1824
1849
1825
1770
1710
1670
1675
1748
1910
2180
2574
3106
3788
4630
5640
6826
8194
9747
11491
13428
15562
53
Az=I-. 59 clearly does not necessarily yield the optimal single dilution
design.
Observing the "T" column in Table 7 for z=.010 reveals another
difficulty with the Az=I.59 "ideal allocation."
Even if the
researcher's a pvtovi guess of A=160 should happen to be exactly•
correct, he can expect to obtain usable experimental results only
86.9% of the time; 13.1% of the time he will observe all the samples
fertile and be unable to calculate a meaningful point estimate for A.
If, moreover, the researcher guessed too low so that A=300 were the
true density, he would obtain unusable results a full 63.2% of the
time!
This illustrates, of course, the wisdom of the serial dilution
design, which protects against obtaining samples either all fertile or
all sterile.
■Finally, note from Table 7 just how large a change in the T and
the MSE values occurs for such a small change of .002 in the dilution.
It appears that the choice of the dilution(s) for the problem of
estimating the density of organisms needs to be a carefully considered
one.
4.2 A Design Algorithm for the Serial Dilution Experiment
The discussion of Table 7 in section 4.1 suggests that one could
characterize the serial dilution problems as one of minimizing MSE
while controlling P (all samples fertile)=I-T.
It seems, then, that
54
the first design consideration of the researcher should be
determining p, the risk he is willing to assume of obtaining all
fertile samples.
This being accomplished, one needs an estimate of
X
, the largest possible value the researcher believes that X could
max
reasonably assume.
The program in Appendix IV allows the user to input n (the number
of samples per dilution), ^max and p (the maximum acceptable
probability of obtaining all samples fertile, which will occur, of
course, when X=X
).
max
For various dilution factors, the program
outputs z2, the middle dilution of a k=3 serial dilution experiment
satisfying the input constraints.
For any larger choice of z^,
P(X1X2X 3=SSS) will be greater than p.
Any smaller choice of Z2 will,
in general, increase the MSB values of the estimators.
While the
program is given for k=3, it readily generalizes to any k.
Figure 2
gives the output of the program for n=3, ^max=300, p=.01 and
d=l(.5)10.
Now the researcher must decide which of the (d,z) pairs meeting
his n, X
and p constraints gives the smallest MSB.
max
Recall that the
program in Appendix III generates expected values and MSB values for
the estimators of sections 3.2.1 to 3.2.4 for designs with any
dilution factor d and any middle dilution z . Using the program in
Appendix III, then, for each (d,z) pair and over the X values of
interest, the researcher may note which (d,z) pair achieves the
55
FIGURE 2
Output for the Program of Appendix IV
HOW MANY TUBES PER DILUTION?
?3
WHAT IS THE MAX EXPECTED L?
?300
WHAT IS THE RISK FOR HAVING ALL TUBES FERTILE?
?.01
INPUT THE STARTING D, ENDING D AND JUMPSIZE
?1,10, .5
D
Z
.00305
1.0
.00314
1.5
.00334
2.0
.00359
2.5
.00389
3.0
.00422
3.5
.00457
4.0
.00492
4.5
5.0
.00527
5.5
.00563
.00598
6.0
6.5 '.00633
.00688
7.0
.00703
7.5
.00738
8.0
.00774
8.5
.00809
9.0
,00844
9.5
.00880
10.0
56
smallest MSE and for which estimator that smallest MSE occurs.
Appendix V gives the output obtained using the program in
Appendix III over the (d,z) pairs of Figure 2 for X=IO(10)300.
that in each case T = .990 = l-,p for X = X
= 300.
max
Note
For the
researcher's best a pvtovi guess of X=30, for example, he would
construct from Appendix V a summary chart similar to Table 8 giving
the MSE for each of the three competing estimators (the discussion
z\
zs
of section 3.3 eliminated X and X^ from further consideration) at each
(d,z) design.
Note that the bias-corrected MPN X^ achieves its
minimum MSE of 506 for the design (d,z)=(4.5,.00492); the Thomas
estimator X^ achieves its minimum MSE of 977 for the design (d,z)=
(4.0,.00457); the Johnson-Brown Spearman estimator Xg achieves its
minimum MSE of 400 also for the design (d,z)=(4.0,.00457).
The
researcher should use the estimator Xg and the design with dilution
factor 4.0 centered at z2=.00457.
Note that in both p=P(all samples fertile) and MSE this design is
superior to the original example of Chapters 1-3 of
Z 1=.!,
Z 2=. 01
and
Zg=.001 which, according to Table 4 has p = P(X1X2X 3=SSS) = .015 and
MSE(Xg IX=30) = 759.
Note also that the selected dilution factor of
4.0 contradicts the previously mentioned advice of Fisher and Yates
(1943) and Finney (1978).
When constructing a summary chart similar to Table 8, however,
the researcher may not wish to be so specific so as to design the
xI
57
TABLE 8
MSE Comparisons:
d
Z2
maximum A = 300
best guess A =
30
MSE(Ac)
MSE(Ax)
MSE(Ag)
1.0
.00305
799
1332
23741
1.5
.00314
757
1268
5798
2.0
.00334
684
1161
1923
2.5
.00359
618
1071
848
3.0
.00389
566
1009
512
3.5
.00422
530
979
414
4.0
.00457
511
977
400
4.5
.00492
506
997
416
5.0
.00527
514
1027
444
5.5
.00563
532
1059
477
6.0
-.0059 8
559
1087
511
6.5
.00633
591
1106
545
7.0
.00688
629
1113
580
7.5
.00703
670
1109
613
8.0
.00738
713
1093
646
8.5
.00774
756
1067
678
9.0
.00809
800
1034
708
9.5
.00844
844
997
737
10.0
.00880
886
957
765
58
experiment for precisely A=30.
Recall, for example, from the single
dilution analysis the problems that can arise when the true A value is
far from the researcher's best a pviovi. guess.
In this case, the
researcher could construct his summary chart by entering, for example,
the maximum MSE(A|10<A<100) instead of MSE(A)A=30) for each estimator.
While this author recommends the preceding algorithm and finds it
to be both simple and reliable, a few additional comments and cautions
need to be given.
First, it should be noted that under every
reasonable set of constraints imposed in connection with this paper,
(I) the three estimators (i.e., A^,, A^ and A ) all achieved their
minimum MSE values at approximately the same (d,z) pair and (2) the
Johnson-Brown Spearman estimator A^ proved consistently to be the
preferred estimator.
As there are no obvious analytical reasons why
either of the above should be true without exception, however, they
are presented as "observations" and not "conjectures."
Secondly, Table 8 indicates that the Thomas estimator A^ MSE
value actually peaks for d=7.0 and begins to decrease as d increases.
One wonders whether it will ever drop below the 400 value attained
/N
by Ag for d=4.0.
^
I
The answer, unfortunately, is yes, as MSE(A^|A=30).
drops to 376 for (d,z)=(60,.04853) and keeps on dropping.
In this
author's opinion, however, the previously recommended (4.0,.00457)
design should still be preferred because (I) dilution factors much
greater than 10 reflect too much uncertainty to be of practical
.
59
concern and indicate the need for a "preliminary investigation" and
(2) the behavior of all the estimators, as mentioned in section 3.3,
degenerates for both very large and very small (i.e., near d=1.00)
dilution factors.
The first statement is a matter of opinion, and
Seligman and Mickey (1964), for example., state, "Not infrequently,
however, the confidence in the pre-existing estimate is such that a
1,000-fold dilution interval is employed."
The second statement
recalls the fact that Iim(X)=O. As d increases, there will be some
^ d-*»
point at which MSE(X^) is artificially minimized before climbing back
to MSE(Xt) = (best guess)2 . A similar phenomenon occurs for the
Johnson-Brown Spearman estimator which, as previously noted,
-Y—3_ (z )
approaches the constant e
1 as d approaches I.
researcher's best guess happens to be close to e
If the
1
1 , there will
— ITi(Z )
be some point at which MSE(Xg) is artificially minimized before
climbing back to MSE(Xg) = ((best guess)-e ^ -*-n (zi^)2 i
5. THE FINITE POPULATION MODEL
Each technique described in Chapters 1-4 assumes that the
’s
are independent responses or, equivalently, that the samples come
from an infinite population.
This is certainly acceptable when one
is monitoring organism levels in, for example, water systems or dairy
products.
Suppose, however, the parameter X is the number of
organisms in a specific finite volume and not the average number per
volume in some conceptually infinite population.
In this case, the
number of fertile samples cannot exceed X and a point estimate or
confidence interval value less than the total number of observed
fertile responses is not appropriate.
Furthermore, since the infinite
model considers the given volume a random sample, it incorporates
additional variation due to sampling into point estimates and
confidence intervals.
Olson, Turbak and McFeters (1979),, for example, employed membrane
diffusion chambers to study the survival of organisms in mine waters.
After using serial dilution techniques to estimate the number of
organisms in a small volume, they placed the volume in mine water in
a chamber that allowed the mine water but not the organisms to pass
freely in and out of the chamber.
Periodically, they used serial
dilution techniques to estimate the number of organisms surviving in
the chamber.
While the authors were investigating a clearly finite
population (i.e., they were trying to estimate the true count in a
61
particular chamber at a particular time), they were forced to turn
to standard (i .e ., infinite population) MPN tables for point estimates
and confidence intervals and concluded "the large confidence intervals
inherent in the MPN procedure make a more definite statement difficult
to justify."
It is interesting to note that McCrady1s (1915) original serial
dilution paper dealt exclusively with finite populations.
Acknowledging that his mathematical analysis represents only an
approximation, he states
When more than one volume is to be drawn from the
[population], these formulae demand that for each draw
the initial conditions must be the same. That is, after
the first volume has been drawn, this volume, together
with its contained B . coli, must be replaced in the
[population] before drawing the next volume. Such a
procedure is obviously impossible in practice.
But perhaps, when the first volume has been drawn,
it may be assumed that a proportionate number of the B .
coli have also been drawn in this volume. If so... the
value of the general factor... has remained practically
unchanged.
But even if this assumption is not justified,
calculation will show that the error due to non-replacement
is, in general, negligible.
Although later in the paper he partially works out one specific
example correctly accounting for this non-replacement, there is no
evidence that he was aware of the exact solution to the general
finite population problem.
Starting with Greenwood and Yule (1917), papers consider only
the simpler and, in truth, more useful infinite model.
Several
62
authors (e.g., Cochran 1950), however, use a finite population
situation to motivate the serial dilution problem, state that "this
is closely approximated by" the infinite model, and then proceed to
discuss mathematically only the latter.
Section 5.1 develops the exact mathematical formulation of the
general finite population problem.
In section 5.2, point estimation
and interval estimation results are given and compared with those
obtained under the usual infinite model.
Except where otherwise
noted, the examples follow Chapters 1-4 in using the commonly employed
and tabled k=3 serial dilution experiment with n/=3 and z^=(.l)^
• for i=l,2,3.
5.1 The General Formula
According to Johnson and Kotz (1977), Polya once maintained that
"any problem of probability appears comparable to a suitable problem
about bags containing balls."
The finite population serial dilution
problem is, in fact, a variation of the classical occupancy problem
of urn modeling.
Imagine that each of YXL balls is placed at random
into one of n equally likely urns and that X denotes the number of
urns thus occupied.
The probability that exactly r urns are occupied
(see, for example, Johnson and Kotz 1977) is given by
(5.1)
P(X-r) = ( ^ 1I0 C - D r 1 (^)(i/n)Y
r=l,2, .. .,min(n,Y) .
Suppose there are n^ equally likely urns of type 1=1,2,3 and
63
that the probability associated with each type i urn is
so that
P (being placed in a type i urn) = n_z^ and P (not being placed in any
urn) = 1-Zn^z^.
If Y_^ represents the total number of balls placed
in type i urns, then after X balls have been placed (Y1,Y2,Y3) has
a multinomial distribution with parameters pu=n^z^ and X.
In
addition, letting !(X1,X2,X3=r1,r2,r3) = f (X1X2X 3) ,
(5.2)
I(XiZ2X 3)
E„
-y y y I(X1X2X 3Jy1y2y3)
I 2 3
ECf(X1Iy1) K X 2 Iy2/
)£(X
|y3)]
- <"3 1v
x'I X.
^2 X
"3 __,ZX-i-j-k.X1.,X,..X
(”0 (JU(^)1I0 S-J0C-1)
1
2
3
J
E[(i/ni)Y i(j/n2)Y2(k/n3)Y 3] .
The joint moment generating function for the multinomial
distibution of (Y1,Y2,Y3) with parameters P 1, P2 , P 3 and X indicates
that E[AY iBY 2CY 3] = [p1A+p2B+p3C+(l-p^-p2-p3)]^ for any constants A,
B and C.
Here, E[ ( i / n ^ ^ (j7n2)Y2 (k/n3)Y 3] = [l-Znz+iz^ j z2+kz3
.
Substituting into equation (5.2) gives
(5.3)
! ( X 1XzX,) = (%^)(;=) ( : ^)i;wjZok2w(-l)^^~' "'~^(^')(j = )(k=)
(1-Znz+iz^ j z2+kz,) \
the exact probability function for the finite population serial
dilution problem with X total organisms and n_^ samples at the z^
dilution for i=l,2,3.
64
5.2 Point and Interval Estimation
The method of combining independent results (section 2.3) and
Fisher's
estimate (section 3.2.2) require independent X^'s and,
consequently, cannot be used for finite populations.
Each of the
other techniques of Chapters 1-4, however, can be applied in the
finite population situation by using equation (5.3) instead of
equation (1.1) to obtain the sampling distribution of the X jX2X 3
values.
The program used to generate these probabilities is given in
Appendix VI and is the finite analog of the program in Appendix I .
Table 9 compares the MPN point estimates and the deMan and
minimum expected width interval estimates of the infinite and finite
models for the 18 X jX2X 3 results with non-empty minimum expected width
confidence intervals.
In both the infinite and the finite model,
these 18 X 1X2X 3 results include about 98% of the probability for
10<X<300. Note that for the finite model the MPN's are generally
slightly smaller and the confidence intervals are generally slightly
narrower.
The fact that the differences are so minor is noteworthy
in that this design involves sampling Zn_z^=.333 of the populationEven when the portion of the finite population sampled is quite
significant, apparently, the difference between the infinite and the
finite analyses is negligible.
Since the Olson et al. (1979) mine
water problem of this chapter, for example, involved sampling only
very small portions of the population at each step, then, applying the
65
TABLE 9
Infinite and Finite Population Results: n=3, Z1=.!
.95 deMan
interval estimates
MPN values
result
infinite3
000
010
100
HO
200
201
210
220
300
301
310
311
320
321
322
330
331
332
0.0
3.1
3.6
7.4
9.2
14.3
14.7
21.1
23.1
38.5
42.7
74.9
93.3
149.4
214.7
239.8
462.2
1099.0
finite^
0
3
3
7
9
14
14
20
22.
37
42
74
93
149
214
239
461
1098
infinite3
<1-17
<1-21
2-28
2-38
5-48
5-50
8-62
<10-130
10-180
10-210
20-280
30-380
50-500
80-640
<100-1400
100-2400
300-4800
finite^
z2=.01
Z3=-OOl
.95 minimum expected
width intervals
infinite^
0-9
0-13
1-14
2-10
1-18
<1-25
3-20
2-24
2-35
<1-37
5-45
11-14
5-46
5-42
8-59
10-32
8-126
4-120
15-174
14-69
16-210
7-200
27-278
21-180
12-360
33-383
38-400
55-503
. 86-637 130-260
91-1393 26-990
178-2405 70-2000
381-4785 140-4070
finite^
0-11
2-9
1-21
3-19
2-36
10-14
5-40
10-32
4-120
15-69
8-200
20-170
12-360
37-400
130-250
33-990
70-1990
150-4070
.fone decimal accuracy
k integer accuracy
Cvalues
values
values
for results 000-222 are given to
for results 300-322 are given to
for results 330-333 are given to
the nearest integer
the nearest 10
the nearest 100
^values less than 100 are given to the nearest integer
Values greater than 100 are given to the nearest 10 •.
66
finite population analysis will not affect their conclusions.
The finite population deMan Bayesian intervals in Table 9 were
obtained using the program in Appendix VII.
Recall that deMan (1975)
employed a truncated likelihood function to obtain his intervals for
the infinite model.
The sum of the finite model likelihood values
across all posssible X's (i.e., from A=EX to A=00), however, converges
analytically to
X X X
Ef(XiX 1X2X a) = ( ^ ) ( ^ ) ( ^ ) . E^. E ^ ( Z ) ) (h)(%:)
(5.4)
[(l-Enz+iz1+jz2+kz3)^X]/(Znz-Iz1-jz2-kz3)
so that the program in Appendix VII provides exact intervals.
It can
also be shown that the exact mean of the deMan Bayesian posterior
distribution in the finite population case is given by
<5'5>
1M o f w ? ) - -
XX + tl/Ma !X1X2X,)] (^l><”o (^) J o jIokI?
I(Xi)(X2) (X3) [i_£nz+izi+jz2+kz3), Z^X- xJ/
(Znz-Iz1-Jz2-Mz3)2 .
Even though the positive skew of the likelihood distribution and the
positive bias of the MPN (the mode of the likelihood distribution)
prevent the mean and the median of the posterior distribution from
being useful point estimates for A, the program in Appendix VII gives
these values for completeness.
The difference between the infinite and the finite population
67
models is greatest as
z_^ appraoches I, at which point all of the
•finite population is included in the sample.
Table 10 considers
'such an extreme case of the finite population problem and makes the
same comparisons as Table 9 but in the case for which Z 1=.!, z2=.03,
z 3=.003.
While the finite model intervals are, as in Table 9,
narrower than their infinite counterparts, even when Tkuz^=.999 of the
population is selected for observation, the finite analysis appears to
serve mainly to prevent lower interval estimate endpoints from falling
below the observed number of fertile tubes when ZX (and, by inference,
X) is small.
Taken together, Tables 9 and 10 suggest that the finite
population model results are essentially identical to those of the
infinite model when Zn^z^ is small and differ from those of the
infinite model as Zn_z^ approaches I only when X is small.
Chapters
A
1-4 suggested that the Johnson-Brown Spearman estimator Xg was
generally the "best" infinite population estimator, however, so that
•
'
one should examine the finte analog of Xg before reaching any
tentative conclusions.
Using equation (5.3) instead of (1.1), this author mimicked the.
work of Johnson and Brown (1961) and obtained the following results.
(i) For large X, the finite and infinite models yield the
same non-bias-corrected.point estimate X =
e-y-ln(zx)+[ln(d)][(ZX/n)-.5]
1
68
TABLE 10
Infinite and Finite Population Results: n=3,
.95 deMan
interval estimates
MPN values
result
000
010
100
HO
200
201
210
220
300
301
310
311
320
321
322
330
331
332
infInitea
Z 1=.!
finite^
0.0
1.0
1.2
2.5
3.1
4.8
4.9
7.0
7.7
12.8
14.2
25.0
31.1
49.8
71.6
79.9
154.1
366.3
infinite0
0
<.3-5.7
I
<.3-7.0
I
.7-9.3
2
2
.7-13
4 ' 1.7-16
1.7-20
4
6
2.7-31
<3.3-43
7
3.3-60
12
3.3-70
13
24
6.7-93
10-127
30
17-167
49
27-213
71
<33-467
79
33-800
153
100-1600
366
Z2=. 01
.95 minimum expected
width intervals
finite'3 ■ infInited
0-0
1-2
1-4
2-6
2-9
3-13
3-13
4-17
3-40
5-56
5-68
9-91
10-126
18-165
28-210
30-463
59-800
127-1593
Z 3=.001
finited
0-4
<1-3
<1-8
<1-6
<1-12
4-4
. 2-14
4-10
2-41
0-0
1-1
1-4
2-5
2-10
4-5
3-12
5-11
3-40
5-23
5-23
3-66
4-66
■ 9-58
6-110 .
13-130
41-84 ■
10-330
24-660
49->1000
7-60
4-120
13-130
42-87
11-330
24->600
47->600
.^one decimal accuracy
^integer accuracy
Cvalues from Table 9.divided by 3
dValues less than 100 are given to the nearest integer
values greater than 100 are given to the nearest 10
'
69
(i.i.) When Zn^z^ is small, COV(X^, )
is negative but
negligible and both the finite and the infinite models yield
the multiplicative bias correction given in section 3.2.4 and
produce the final estimator of equation (3.1).
These results, the development of which is sketched in Appendix VIII,
support the suggestions of Tables 9 and 10 given above.
6. SUMMARY
Even though the serial dilution assay is a standard micro­
biological method for determining the density of organisms in a
solution, there has been very little mathematical investigation into
the statistical appropriateness of the decimal dilution design, the
MPN point estimate and the Woodward (1957) confidence intervals that
have been long accepted as standard procedure.
This paper, the only
recent review of the serial dilution problem, has been the first to
compare exact MSE values of competing estimators, the first to provide
an algorithm to determine efficient serial dilution designs, and the
.first to give the exact formulation of the finite population serial
dilution problem.
In addition, original serial dilution point
estimates and exact confidence intervals were given and compared with
the MPN and previously proposed alternatives.
It is the author's conclusion that the serial dilution design and
estimation recommendations of standard textbooks (e.g., Finney 1978)
and reference works (e.g., American Public Health Association 1971)
are not adequate and, specifically,
(i) The combining independent results (section 2.3) and
minimum expected width (section 2.4) confidence interval methods
should be the recommended techniques for obtaining one-sided and
two-sided confidence intervals respectively.
(ii) The algorithm of section 4.2 should be used to Identify
both the optimal serial dilution design and the most efficient
71
point estimate.
(iii)
The finite population probabilities given by equation
(5.3) should be used when applicable, especially when X is small
and many non-fertile samples are likely.
General computer programs that allow the researcher to put the
recommendations of this paper into practice (or to compare completely
new procedures to those in this paper) appear in the appendices.
addition, caveats and areas for further work have been identified
throughout the paper.
In
FOOTNOTES
aMcCradytS paper spells out and applies ML estimation several
years before Fisher popularized the concept during his "spectacular
dispute" with Karl Pearson, who preferred the method of moments
estimation (Owen 1976) . Credit for the first clear and explicit
formulation of ML estimation by differentiating the likelihood
function, however, belongs to Daniel Bernoulli in 1777. Pearson and
Kendall (1970) give a translation and discussion of Bernoulli's paper
and of Euler's rebuttal (which was published in the same 1777 volume)
of Bernoulli's ML ideas.
^The MP N 's given in Figure I, calculated by the program given in
Appendix III, also reveal two minor errors in the American Public
Health Association (1970) MPN's given in Table I. The MPN's
corresponding to H O and 200 should be 7.4 and 9.2 respectively.
cNote that this is not the case for the other three 95%
confidence intervals given in Table I. The issue here is whether
Woodward is really following the accepted testing procedure of
rejecting the null hypothesis for the "most extreme" a= .05 of the
results when he chooses to reject for "likely" results and not for
"unlikely" ones.
^While it is not within the scope of this paper to discuss the
biological appropriateness of either the serial diultion or the MF
technique for various organisms, their frequent comparison in the
literature deserves some comment. Most authors which consider this
aspect of the problem (e.g., Middlebrooks, Middlebrooks, Johnson,
Wight, Reynolds and Venosa 1978) stress that neither procedure should
be regarded as absolute and note that "one technique does not appear
to be more reliable than the other." DeMan (1977), however, opinions
that "because microbiological standards for foods are becoming
increasingly severe, in the future MPN [serial dilution] methods will
probably often have to replace plate counts." Futhermore, one
standard reference (American Public Health Association 1966, page 139)
notes that "the limitations of the plate count method are well known"
and "a more accurate method for the enumeration of smallnumbers of
organisms is the MPN technique."
73
eThis technique can certainly be applied to deMan’s Bayesian
intervals also, and the resulting point estimate would be the median
of the posterior distribution. Since, however, deMan's posterior
distributions are unimodal and positively skewed, the median of each
posterior distribution would be larger than its mode (which is the
MPN), and the resulting point estimate would be even more positively
biased than the MPN. The same would be true for the estimate obtained
by using the mean of the posterior distribution.
fWhile Halvorson and Ziegler (1933c) were actually the first to
note the positive bias of the MPN, they do not mention the bias as a
design factor when selecting dilutions.
APPENDICES
APPENDIX I
n n o n n n o o o
PROGRAM TO GENERATE EXACT SAMPLING DISTRIBUTIONS FOR LAMBDA
FROM I TO 100 FOR N TUBES AT EACH OF 3 DILUTIONS
A IS THE BINOMIAL PROBABILITY W/O THE COEFFICIENT
D IS THE BINOMIAL COEFFICIENT
X IS THE NUMBER OF FERTILE TUBES
Z IS THE DILUTION
AMAT SAVES THE VALUES FOR OUTPUT A PAGE AT A TIME (10 LAMBDA
VALUES BY 64 TUBE COMBINATIONS)'
5
9
30
40
60
DIMENSION A(3),0(0:10),X(3),Z(3),AMAT(66,40)
OUTPUT 'INPUT THE 3 Z VALUES'
INPUT Z(I),Z(2),Z(3)
OUTPUT 'INPUT N (# TUBES PER Z)'
INPUT N
D(O) = 1.0
DO 5 I=O5N-I
D (1+1)=D(I)*(N-I)/(1+1)
L=I
DO 60 IM=I,40,4
IF (L>100) GO TO 99
AMAT(I1IM)=L
C=O
KOUNT=Z
DO 40 Xl=O5N
DO 40 XZ=O5N
DO 40 X3=0,N
X(I)=Xl
X(Z)=XZ
X(3)=X3
CN = D(Xl)*D(XZ)*D(X3)
DO 30 1=1,3
A(I) = (I-EXP(-Z(I)*L))**X(I)*(EXP(-Z(I)*L))**(N-X(T) )
F=CN*A (I)*A(2)*A(3)
AMAT(KOUNT,IM)=Xl
AMAT(KOUNT,IM+1)=X2
AMAT(KOUNT,IM+2)=X3
AMAT(KOUNT,IM+3)=F
KOUNT=KOUNT+!
C=C+F
CONTINUE
AMAT(66,IM)=C
L=L+1
CONTINUE
/
75
APPENDIX I (CONTINUED)
200
70
300
400
99
WRITE (108,200) AMAT(1,1),AMAT(1,5),AMAT(1,9),
C AMAT(1,13),AMAT(1,17),AMAT(I,21),AMAT(I,25),
C AMAT(1,29),AMAT(1,33),AMAT(1,37)
FORMAT (T7,13,9(9X,I3))
DO 70 1=2,65
WRITE (108,300) (AMAT(I1J),1=1,40)
FORMAT (X,10(2X,3I1,F7.4))
WRITE (108,400) AMAT(66,I),AMAT(66,5),AMAT(66,9),
C AMAT(66,13),AMAT(66,17),AMAT(66,21),AMAT(66,25),
C AMAT(66,29),AMAT(66,33),AMAT(66,37)
FORMAT (T7,F6.4,9(6X,F6.4))
GO TO 9
END
76
APPENDIX II
C
C
C
C
C
C
C
C
C
PROGRAM TO FIND .95 C.I. ENDPOINTS BY THE FISHER-LANCASTER
METHOD FOR 3 TUBES AT EACH OF THE DILUTIONS .1,.01,.001
THIS PROGRAM ALSO IDENTIFIES THE ASSOCIATED POINT ESTIMATE
D IS THE BINOMIAL COEFFICIENT
F IS THE BINOMIAL PROBABILITY
PU AND PL ARE THE UPPER AND LOWER P-VALUES •
CU-AND CL ARE THE UPPER AND LOWER CHI-SQUARE VALUES
14.45 IS THE CHI-SQUARE VALUE WITH 6 DF AND .025 BEYOND
4
n O- o o
5
DIMENSION D(0:4) ,F(0:4,3) ,PU(3) ,PL(3),CU(3) ,CL(3) ,X(3),Z:(3)
REAL L
D(O)=I
D(D=3
D(2)=3
D(3)=l
Z(I)=-I
Z (2)=.01
Z(3)=.001
OUTPUT 'HOW MANY FERTILE TUBES PER DILUTION?'
INPUT X ( 1 ) , X ( 2 ) , X ( 3 )
OUTPUT
L UPPER TAIL LOWER TAIL'
OUTPUT '
CHI**2
CHI**2'
L=1.00
CONTINUE
DO 10 1=0,3
DO 10 J=I,3
I INDEXES THE # OF FERTILE TUBES PER DILUTION
J INDEXES THE DILUTIONS
F(I1J)=D(I)*(1-EXP(-1*Z(J)))**I
C
*(EXP(-L*Z(J)))**(3-I)
10 CONTINUE
DO 40 J=l,3
S=O
N=X(J)
DO 11 I=O1N
11 S=S+F(I1J)
IF (N.EQ.O) GO TO 19
Sl=O
DO 15 I=O1N-I
15 Sl=SlEF(I1J)
PU(J)=(S+Sl)/2
77
APPENDIX II (CONTINUED)
19
20
21
25
29
30
40
C.
C
C
C
CU(J)=-2*LOG(PU(J))
GO TO 20
CU(J)=2-2*LOG(S)
■CONTINUE
T=O
DO 21 I=N,3
T=T+F(1,J)
Tl=O
IF (N.EQ.3) GO TO 29
DO 25 I=N+1,3
T1=T1+F(I,J)
PL(J) = (TH-Tl)/2
CL(J)=-2*LOG(PL(J))
GO TO 30
CL(J)=2-2*LOG(T)
CONTINUE
CONTINUE
CHU=CU(1)+CU(2)+CU(3)
CHL=CL(I)+CL(2)+CL(3)
THE FOLLOWING LINES ELIMINATE OUTPUT NOT NEAR THE POINT
ESTIMATE OR THE ENDPOINTS OF THE .95 C.I. '
IF ABS(CHU-14.45).LE.1.00) GO TO 50
IF ABS(CHL-14.45).LE.1.11) GO TO 50
IF ABS(CHU-CHL).LE.1.00) GO TO 50
GO TO 60
50 WRITE (108,200) L,CHU,CHL
200 FORMAT (3X,14,X,2F11.4)
60 . IF ((CHU.GE.16).AND.(CHL.LE.13)) GO TO 4
. L=L+I
GO TO 5
END
78
APPENDIX II (CONTINUED)
SAMPLE OUTPUT AND EXPLANATION
HOW MANY FERTILE TUBES PER DILUTION?
?3,0,1
L UPPER TAIL LOWER TAIL
CHI**2
CHI**2
8
9
10
11
2.6784
2.9001
3.0145
15.4329
14.7117
14.0869
13.5394
58
59
60
7.0247
7.0895
7.1541
7.1454
7.1049
7.0653
17!
175
176
177
14.3686
14.4319
14.4951
14.5583
4.8804
4.8700
4.8596
4.8492
2.7878
From the output above, one rejects H :A=X in favor of H :A<A at
o o
a
o
the .025 level for A >176 and one rejects H :A=A in favor of H :A>A
o—
o o
a
o
at the .025 level for A <9. Hence the two-tailed 95% confidence
o—
interval associated with XiXzXa=SOl extends from 10 to 175 as
indicated in the "Combining Independent Results" column of Table I.
(As noted in the program, the
.025 beyond it is 14.45.
value with 6 degrees of freedom and
One rejects the null hypothesis when the
calculated X2 value falls above 14.45.)
79
APPENDIX III
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
PROGRAM TO CALCULATE POINT ESTIMATES, EXPECTED VALUES AND
MSE VALUES FOR FIVE SERIAL DILUTION ESTIMATORS
MPN: CALCULATED IN LOOP 25, CALLED M IN PROGRAM AND OUTPUT
BIAS-CORRECTED MPN: CALCULATED BY ADJUSTING THE MPN, CALLED
C IN PROGRAM AND OUTPUT
FISHER ESTIMATE: CALCULATED IN LOOP 20, CALLED Y IN THE
PROGRAM AND F IN THE OUTPUT
JOHNSON-BROWN SPEARMAN ESTIMATE: CALCULATED IN LOOP 20, CALLED
B IN THE PROGRAM AND S IN THE OUTPUT
THOMAS ESTIMATE: CALCULATED IN LOOP 25, CALLED T IN THE
PROGRAM AND OUTPUT
C IS THE BINOMIAL COEFFICIENT
K IS A DUMMY VARIABLE FOR NEWTON ITERATIONS
THIS PROGRAM IS FOR 3 DILUTIONS, N TUBES PER DILUTION AND
DILUTION FACTOR D
THE PROGRAM DIMENSION STATEMENT MUST BE ADJUSTED FOR EACH N
DIMENSION B(0:3N-1),C(0:N),K(30),Y(0:3N-1)
REAL K,L,M
INTEGER Xl,X2,X3
El(W)=EXP(-Z1*W)
E2(W)=EXP(-Z2*W)
E3(W)=EXP(-Z3*W)
Fl(W)=XlAZl/(1-El(W))+X2*Z2/(l-E2(W))+X3*Z3/(1-E3(W))
C -N*(Z1+Z2+Z3)
Cl(W)=-Xl*Zl**2*El(W)/(I-El (W))**2
C - X 2 * Z 2 * * 2 * E 2 (W)/(1-E2(W))**2
C -X3*Z3**2*E3(W)/(1-E3(W))**2
F2(W)=N*El(W)+N*E2(W)+N*E3(W)-TN
G2(W)=-N*Zl*El(W)-N*Z2*E2(W)-N*Z3*E3(W)
REWIND I
OUTPUT 'WHAT IS N?'
INPUT N
OUTPUT 'WHAT IS THE MIDDLE DILUTION?'
INPUT Z2
OUTPUT rWHAT IS THE DILUTION FACTOR?'
INPUT D
OUTPUT 'INPUT STARTING L , ENDING L , JUMPSIZE'
INPUT Al,A2,A3
80
APPENDIX III (CONTINUED)
5
10
20
C
C
C
C
C
C
Zl=Z2*D
Z3=Z2/D
C(O)=I.0
DO 5 1=0,N-I
C(T-HL)=C(I) *(N-I) / (1+1)
DO 20 B=O,3*N-I
TN=3*N-I
K(I)=20
DO 10 J=I ,19
K(J+1)=K(J)-F2(K(J))/G2(K(J))
IF (K(J+l).LE.0) K(J+1)=.01
IF (ABS(K(20)-K(19)).GE..01) GO TO 990
Y(I)=K(20)
B(I)=(1/Z1)*EXP(-.57722-.5*LOG(D)+((LOG(D))/N)*1)
B(I)=B(I)*(2*N)/(2*N+LOG(D)*LOG(2))
CONTINUE
THE FISHER (Y) AND JOHNSON-BROWN (B) ESTIMATES ARE NOW STORED
IN MATRICES
THE MPN AND THOMAS ESTIMATES WILL NOW BE COMPUTED AND SENT
TO A FILE
15
16
17
100
25
DO 25 Xl=O,N
DO 25 X2=0,N
DO 25 X3=0,N
IF (Xl.EQ.N) IF (X2.EQ.N) IF (X3.EQ.N)
S=X1+X2+X3
Sl=(N-Xl)*Z1+(N-X2)*Z2+(N-X3)*Z3
S2=N*(Z1+Z2+Z3)
T=S/((S1*S2)**.5)
IF (X1.EQ.0) IF (X2.EQ.0) IF (X3.EQ.O)
K(l)=20
'
DO 15 J=l,29
K(J+1)=K(J)-F1(K(J))/GKK(J))
IF (K(J+1).LE.O) K(J+1)=.01
IF (ABS.(K(30)-K(29)) .GE. .01) GO TO 991
M=K(30)
GO TO 17
M=O.00
WRITE (1,100) M,T
FORMAT (2F8.3)
CONTINUE
GO TO 25
GO TO 16
,
81
APPENDIX III (CONTINUED)
C
C
C
C
C
C
THE MPN (M) AND THOMAS (T) ESTIMATES HAVE NOW BEEN STORED
IN A FILE
THE PROGRAM WILL NOW CALCULATE EXACT PROBABILITIES, EXPECTED
VALUES, VARIANCES AND MSE VALUES FOR ALL THE ESTIMATORS
OUTPUT '
L
T
E(M)
E(C)
E(F)
E(S)
E(T)
CMSE(C)
MSE(F)
MSE(S)
MSE(T)'
DO 80 L=Al,A2,A3
REWIND I
TO=O ■
SM=O
SF=O
SB=O
ST=O
SSM=O
SSF=O
SSB=O
SST=O
DO 70 Xl=O,N
DO 70 X2=0,N
DO 70 X3=0,N
S=X1+X2+X3
IF (Xl.EQ-.N) IF (X2.EQ.N) IF (X3.EQ.N) GO TO 70
P=C(Xl)*C(X2)*C(X3)* (1-El(L))**X1*(El(L))**(N-Xl)
C
*(1-E2(L))**X2*(E2(L))**(N-X2)
C
* (1-E3(L))**X3*(E3(L))**(N-X3)
READ (1,200) M,T .
200 FORMAT (2F8.3)
TO=TOfP
PM=P*M
PF=P*Y(S)
PB=P*B(S)
PT=PaT
PPM=PMaM
PPF=PFa Y(S)
PPB=PBa B(S)
PPT=PTaT
SM=SMfPM
SF=SFfPF
SB=SBfPB
ST=STfPT
MSE(M)
82
APPENDIX III (CONTINUED)
SSM=SSMfPPM
SSF=SSFfPPF
SSB=SSBfPPB
SST=SSTfPPT
70 CONTINUE'
EM=SM/TO
EF=SF/TO
EB=SB/TO
ET=ST/TO'
.•VM=SSM/T0-EM**2
‘VF=SSF./T0-EF**2
VB=SSB/TO-EB**2
VT=SST/TO-ET**2
•'SEM=VMf(EM-L) **2
SEF=VFf(EF-L)**2
SEB=VBf(EB-L)**2
SET=VTf(ET-L)**2
.TH=EXP(-.805/N)
■e c =e m *t h
, VC=VM*TH**2
:SEC=VCf(EC-L)**2
WRITE (108,300) L,TO,EM1EC,EFsEB,SEM,SEC,SEF,SEB,SET
■ 300 FORMAT (I5,F6.3,5F7.2,5110)
80 CONTINUE •
GO TO 91
990 OUTPUT 'FISHER ESTIMATE DOES NOT CONVERGE'
GO TO 91
• 991 OUTPUT 'MPN ESTIMATE DOES NOT CONVERGE'
91 END
83
APPENDIX IV
C
C
C
C
C
C
PROGRAM TO FIND Z (THE MIDDLE DILUTION OF A K=3 DILUTION
SERIAL DILUTION EXPERIMENT) FOR A GIVEN DILUTION FACTOR
THAT WILL
-KEEP P (ALL FERTILE TUBES) TO SOME MAXIMUM INPUT VALUE
-ACCEPT L MAX (THE MAXIMUM REASONABLE VALUE LAMBDA ASSUMES)
DIMENSION Z (30)
REAL J 5L 5M 5N
El(W)=I-EXP(-L*D*W)
E2(W)=1-EXP(-L*W)
E3(W)=I-EXP(-1*W/D)
Fl(W)=El(W)*E2(W)*E3(W)-R**(1/N)
F2(W)=El(W)*E2(W) * (L/D)*EXP(-L*W/D)
C +El(W)*E3(W)*L*EXP(-1*W)
C +E2(W)*E3(W)*L*D*EXP(-L*D*W)
OUTPUT: 'HOW MANY TUBES PER DILUTION? '
INPUT N '
OUTPUT 'WHAT IS THE MAX EXPECTED L ? '
INPUT L
OUTPUT 'WHAT IS THE RISK FOR HAVING ALL TUBES FERTILE?'
INPUT R
OUTPUT 'INPUT THE STARTING D 5 ENDING D AND JUMPSIZE'
. INPUT A 5B 5C
OUTPUT '
D
Z'
. ZD=-(l/L)*LOG(-1-R** (1/3*N)))
DO 20 D=A5B 5C
Z(I)=ZD
DO 10 1=1,29
' Z (I+1)=Z(I)-F1(Z(I))/F2(Z(I))
IF (Z(I+1).LE.O) Z(I+1)=.000001
10 CONTINUE
IF (ABS(Z(30)rZ(29)).GE..0001) GO TO 999
ZD=Z(30)
WRITE (108,100) D 5ZD
100 FORMAT (F6.3,F8.5)
•' 20 ' CONTINUE
GO TO 30
'999 OUTPUT 'Z DOES NOT CONVERGE'
30 END
WHAT IS N?
T3
APPENDIX .V
TABLE A
WHAT IS THE MIDDLE DILUTION?
7.00305
WHAT IS THE DILUTION FACTOR?
? 1.0
INPUT STARTING L f ENDING L f JUMP SIZE
?10 F300 F10
L
T
E(M)
E(C)
E(F)
E(S)
10 1.000
10.61
8.11
10.62 184.08
20 1.000
21.25
16.25
21.25 184,08
30 1.000
31.91
24.40
31.91 184.08
40 1.000
42.59
32.57
42.60 184,08
50 1.000
53.31
40.76
53.31 184.08
60 1.000
64.05
48.98
64.05 184.08
70 1.000
74.82
57.21
74.83 184.08
80 1.000
85.63
65.48
85.63 184.08
90 1.000
96.47
73,76
96.47 184.08
100 1.000 107,34
82.08 107.34 184.08
H O 1.000 118.25
90.42 118.25 184.08
120 1.000 129.19
98.79 129.19 184.08
130 1.000 140.17 107.18 140.17 184.08
140 I ,000 151.18 115.60 151.18 184.08
150 1.000 162.22 124.05 162.22 184.08
160 i .oOo 173.30 132.51 173.30 184,08
170 I .000 184.39 141.00 184.39 184.08
180 1.000 195.51 149.50 195.51 184.08
190
.999 206.64 158.01 206.64 184.08
200
.999 217.77 166.52 217.77 184.08
210
.999 228.89 175.02 228,89 184.08
220
.998 240.00 183.52 240.00 184.08
230
.998 251.09 192.00 251.09 184.08
240
.997 262.14 200.44 262.14 184.08
250
.996 273.14 208.85 273.14 184.08
260
.996 284.07 217.22 284.07 184.08
270
.994 294.94 225.52 294.94 184.08
280
.993 305.71 233.76 305.71 184.08
290 ..992 316.39 241.93 316.39 184.08
300
.990 326.96 250.01 326.96 184.08
*STOP* O
E(T)
10.62
21.29
31.99
42.75
53.56
64.43
75.37
86.38
97.47
108.65
119.92
131.28
142.75
154.33
166.02
177.82
189.73
201.74
213.87
226.09
238.39
250.78
263.24
275.75
288.31
300.89
313.48
326.06
338,61
351.13
MSE(M)
418
857
1316
1798
2304
2835
3395
3984
4604
5258
5948
6674 .
7438
8239
9078
9952
10860
11798
12763
13749
14751
15762
16776
17786
18784
19763
20716
21636
22516
23350
MSE(C)
248
514
799
1102
1426
1770
2135
2521
2931
3364
3821
4303
4809
5339
5894
6471
7069
7688
8324
8975
9639
10313
10993
11677
12363
13047
13727
14401
15068
15727
MSE(F)
418
857
1316
1798
2304
■ 2835
3395
3983
4604
5258
5948
6674
7438
8239
9077
9952
10860
11798
12763
13749
14751
15762
16776
17786
18784
19763
20716
21636
22516
23350
MSE(S)
30305
26923
23741
20760
17978
15396
13015
10833
8851
7070
5488
4106
2925
1943
1161
' 580
198
16
34
253
671
1289
2108
3126
4344
5763
7381
9199
11218
13436
MSE(T)
420
863
1332
1830
2360
2926
3535
4190
4898
5666
6499
7404
8387
9454
10608
11852
13188
14615
16131
17732
19412
21164
22979
24845
26751
28685
30634
32582
34517
36425
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00314
WHAT IS THE DILUTION FACTOR?
? 1 .5
INPUT STARTING Lr ENDING Lr JUMP SIZE
?10 r300 rIO
L
T
E(M)
E(C)
E(S)
E(F)
10 1.000
10.69
10.69
96.99
8.17
20 1.000
21.41
16.37 21.39 101.01
30 1.000
32.16
24.59 32.12 105.01
40 1.000
42.94
32.84
42.89 109.01
50 I .000
53.76
41.11
53.68 112.99
60 1.000
64.61
49.41
64.51 116.96
70 1.000
75.51
57.74
75.38 120.89
80 1.000
86.44
66.10
86.28 124.81
90 1.000
97.41
74.49
97.22 128.68
100 1.000 108.43
82.91 108.20 132.53
H O 1.000 119.50
91.37 119.22 136.33
120 1.000 130.61
99.87 130.29 140.10
130 1.000 141.77 108.40 141.40 143.82
140 1.000 152.97 116.97 152.55 147.49
150 1.000 164.21 125.56 163.74 151.11
160 1.000 175.49 134.19 174.96 154.68
170 1.000 186.81 142.84. 186.21 158.20
180 I .000 198,16 151.52 197.48 161.65
190
.999 209.52 160.21 208.77 165.05
200
.999 220.90 168.91 220.07 168.39
210
.999 232.29 177.62 231.37 171.66
220
.998 243.66 186.32 242.65 174.86
230
.998 255.02 195.00 253.91 178.00
240
.997 266.35 203.67 265.13 181.06
250
.996 277.64 212.30 276.30 184.06
260
.995 288.87 220.89 287.41 186.98
270
,994 300.03 229.42 298.44 189.84
280
.993 311.12 237.90 309.38 192.61
290
.992 322.11 246.30 320.22 195.31
300
.990 332.99 254.62 330.95 197.94
*STOP* OY
E(T)
10.71
21.46
32.27
43.14
54.07
65.08
76.17
87.35
98.63
110.00
121.49
133.09
144.82
156.67
168.65
180.76
192.99
205.36
217.84
230.43
243.13
255.92
268.79
281.73
294.72
307.75
320.79
333.83
346.86
359.85
MSE(M)
393
809
1249
1717 ■
2213
2740
3302
3900
4538
5219
5945
6718
7540
8412
9333
10304
11321
12381
13480
14613
15774
16955
18150
19349
20545
21729
22894
24030
25131
26189
MSE(C)
232
485
757
1050
1364
1702
2063
2449
2862
3302
3770
4267
4794
5350
5936
6550
7191
7857
8546
9255
9981
10720
11470
12227
12987
13747
14505
15257
16000
16735
MSE(F)
391
806
1244
1709
2201
2725
3282
3875
4508
5182
5900
6666
7478
8340
9250
10206
11207
12249
13327
14436
15569
16719
17878
19038
20191
21327
22439
23519
24559
25552
MSE(S)
7622
6674
5798
4994
4261
3599
3008
2487
2038
1661
1356
1125
969
890
888
966
1125
1368
1696
2113
2619
3219
3915
4710
5607
6609
7720
8942
10279
11735
MSE(T)
395
816
1268
1755
2280
2849
3469
4147
4891
5708
6607
7596
8683
9876
11180
12598
14134
15787
17556
19436
21422
23505
25675
27921
30230
32587
34978
37387
39799
42199
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00334
UHAT IS THE DILUTION FACTOR?
?2.0
INPUT STARTING L, ENDING Lr JUMP SIZE
?10r300rl0
L
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
T
1.000
1.000
1,000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.999
.999
.999
.999
.998
.998
.997
.996
.995
.994
.993
.992
.990
TABLE C
appendix
V
*STOP* OY
E(M)
10.83
21.70
32.63
43.60
54 i63
65.72
76.86
88.08
99.35
110.70
122.12
133,60
145.16
156.78
168.47
180.21
192.01
203.85
215.72
227.62
239.53
251.44
263.33
275.19
287.01
298.77
310.45
322.05
333.55
344.94
E(C)
8.28
16.59
24.95
33.34
41.77
50.25
58.77
67.35
75.97
84.65
93.38
102.16
110.99
119.88
128.82
137.80
146.82
155.87
164.95
174.05
183.16
192.26
201.35
210,42
219.46
228.45
237.39
246.26
255.05
263.76
E(F)
10.81
21.64
32.52
43.43
54.39
65.39
76.45
87.56
98.72
109.94
12,1.21
132.55
143.94
155.39
166.89
178,43
190.01
201,63
213.26
224.90
236.54
248.16
259.75
271.30
282.79
294.21
305.54
316.77
327.89
338.89
E(S)
60.11
65.32
70.63
76.04
81.52
87.06
92.66
98.28
103.93
109.59
115.25
120.89
126.52
132.11
137.67
143.17
148.62
154.01
159.32
164.56
169.71
174.78
179.75
184.63
189.40
194.07
198.63
203.08
207.43
211.66
E(T)
10.85
21.78
32.79
43.90
55.10
66.41
77.84
89.40
101.08
112.90
124.87
136.99
149-. 25
161.67
174.24
186.95
199.81
212.80
225.90
239.12
252.43
265.82
279,27
292.77
306.29
319.82
333.33
346.82
360.25
373.61
MSE(M)
347
724
1134
1580
2067
2598
3179
3814
4509
5269
6097
6999
7976
9030
10163
11371
12652
14002
15414
16881
18394
19944
21520
23111
24707
26296
27867
29411
30916
32374
MSE(C)
205
433
684
961
1263
1595
. 1957
2352
2782
3249
3755
4302
4890
5520
6191
6902
7652
8436
9253
10097
10965
11852
12753
13663
14578
15491
16400
17300
18188
19060
MSE(F)
344
716
1120
1557
2033
2552
3118
3737
4413
5152
5958
6834
7782
8805
9902
11070
12306
13606
14963
16368
17813
19288
20782
22285
23786
25273
26737
28166
29553
30888
MSE(S)
2589
2222
1923
1685
1504
1377
1301
1272
1288
1346
1446
1586
1766
1984
2240
2535
2869
3243
3659
4117
4620
5170
5769
6421
7128
7894
8723
9618
10584
11624
MSE(T)
350
735
1161
1634
2162
2752
3415
4159
4995
5936
6990
8169
9481
10933
12531
14276
16168
18205
20382
22690
25119
27655
30286
32995
35764
38578
41416
44262
47098
49906
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
?.00359
WHAT IS THE DILUTION FACTOR?
?2.5
INPUT STARTING Lr ENDING Lr JUMP SIZE
?10r300r10
E(T)
.HE(F)
E(S)
T
E(M)
E(C)
11.01
10 1.000
10.97
8.39
10.93 41.29
47.09
22 r14
20 1.000
22.02
16.84
21.91
33.41
53.12
30 1.000
33.15
25.35
32.94
44.82
59.36
40 1.000
44.36
33.92
44.04
56.39
50 1.000
65.77
55.67
42.56
55.20
68.12
60 1.000
72.33
67.06
51.28
66.43
80,01
70 1.000
79.01
78.55
60.07
77.73
92.06
80 1.000
85.78
68.93
89.11
90.14
92.62 104.28
90 1.000 101.83
77.87 100.56
99.50 116.67
100 1.000 113.62
86.88 112.09
H O 1.000 125.51
95.97 123.70 106.40 129.23
MSE(M)
302
644
1031
1467
1962
2521
3153
3866
4669
5568
6570
MSE(C)
179
384
618
884
1183
1521
1899
2323
2795
3319
3897
MSE(F)
298
632
1006
1426
1900
2435
3040
3721
4489
5349
6309
141.94
154.80
167.81
180.95
194.20
207.56,
220.99
234.50
248.05
261.63
275.22
288.79
302.33
315.82
329.24
342.58
355.80
368.91
381.89
7680
8901
10234
11676
13224
14872
16612
18434
20327
22277
24271
26296
28337
30380
32411
34417
36385
38305
40166
4532
5224
5973
6778
7635
8543
9495
10487
11513
12566
13640
14729
15825
16924
18017
19101
20171
21222
22250
7372
8540
9815
11195
12674
14246
15903
17636
19431
21277
23160
25066
26982
28895
30790
32655
34479
36251
37962
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
*STnp*
I
FL
1.000
1.000
1.000
1.000
I .OOO
1.000
.999
.999
.999
.998
.998
.997
.997
.996
.995
.994
.993
.992
.990
nv
137.50
149.58
161.74
173.99
186.30
198.66
211.07
223.50
235.95
248.39
260.81
273.20
285.53
297.80
309.98
322.06
334.04
345.88
357.60
105.14
114.38
123.68
133.04
142.45
151.91
161.39
170.90
180.42
189.93
199,43
208.90
218.33
227.71
237.03
246.27
255.42
264.48
273.44
135.39
147.14
158.96
17.0.84
182.76
194.72
206.71
218.71
230.70
242.67
254.62
266.51
278.34
290.09
301.75
313.30
324.74
336.05
347.22
113.31
120.20
127.06
133.87
140.63
147.31
153.92
160,43
166,84
173.14
179.33
185.41
191.35
197.17
202.86
208.41
213.84
219.12
224,27
MSE(S)
1059
918
848
841
891
991
1136
1322
1545
1802
2088
2402
2741 .
3104
3490
3897
4324
4773
5242
5734
6249
6789
7356
7953
8581
9245
9948
10692
11483
12325
MSE(T)
306
660
1071
1547
2099
2739
3477
4327
5298
6403
7650
9047
10598
12306
14170
16186
18348
20646
23067
25599
28224
30926
33687
36488
39311
42137
44949
47729
50462
53133
WHAT IS N?
?3
TABLE E
appendix
V.
WHAT IS THE MIDDLE DILUTION?
?«00389.
WHAT IS THE DILUTION FACTOR?
?3.0
INPUT STARTING L, ENDING L, JUMP SIZE
?10,300,10
L
T
E(M)
E(C)
E(F)
E(S)
10 1.000
11.11
8.50
30,41
11.05
20 1.000
22.34
22.17
36.61
17.08
30 1.000
33.71
25.77
43.20
33.38
40 1.000
45.20
34.57 44.68
50.12
50 1.000 .56,84
57.30
43.46
56.08
60 1.000
68.62
52.47
67.58
64.70
70 1.000
72.26
80.54
61.58
79.18
80 1.000
70\ 80 90.88
79.94
92.59
90 1.000 104.78
80.12 102.68 •87.70
100 1.000 117.09
95.50
89.54 114.57
H O 1.000 129.52
99.04 126.55 103.32
120 1,000 142.06 108.63 138.61 111.12
130 1.000 154.69 118.29 150.73 118,89
140 1.000 167.40 128.00 162.92 126,60
150 1.000 180.17 137.77 175.15 134.23
160 1.000 192.99 147,57 187.40 141.78
170
.999 205.84 157.40 199.68 149.22
180
.999 218.70 167.23 211.95 156.55
190
.999 231.55 177.05 224,21 163.76
200
.999 244.37 186.86 236.43 170.84
210
.998 257.15 196.63 248.61 177.79
220
.998 269.87 206.35 260.73 184.60
230
.997 282.51 216.02 272.77 191.26
240
.996 295.06 225.62 284.72 197.78
250
.996 307.51 235.14 296.57 204.15
260
.995 319.84 244.56 308.31 210.38
270
.994 332.04 253.89 319.92 216.46
280
,993 344.10 263.11 331.40 222.39
290
.991 356.01 272.22 342.74 228.18
300
.990 367.77 281.22 353.93 233.82
*STOP* OY
E(T)
11.17
22.53
34.11
45.91
57.92
70.15
82.58
95.20
108.00
120.94
134,02
147.22
160.51
.173.87
187.28
200.71
214.16
■227.58
240.98
254.31
267.58
280.76
293.83
306.78
319.60
332.28
344.80
357.16
369.35
381.36
MSE(M)
264
578
9.51
1394
1918
2533
3253
4089
5050
6146
7383
8765
10290
11957
13759
15688
17732
19878
22111
24416
' 26776
29175
■ 31594
34020
36435
38825
41177
43478
45717
47883
MSE(C)
155
343
566
829
1136
1494
1908
2382
2922
3532
4214
4969
5797
6696
7662
8690
9775
10910
12087
13298
14535
15790
17056
18325
19590
20846
22085
23304
24498
25664
MSE(F)
258
560
914
1332
1824
2403
3080
3868
4777
5815
6988
8299
9747
11330
13040
14870
16809
18842
20955
23134
25361
27621
29898
32176
34440
36676
38872
41015
43095
45102
MSE(S)
493
465
512
626
799
. 1022
1292
1601
1946
2321
2724
3151
3599
4066
4549
5046
5557
6081
6617
7165
7727
8303
8895
9504
10133
10785
11463
12169
12907
13680
MSE(T)
269
601
1009
1505
2103
2814
3649
4618
5729
6988
8398
9961
■11673
13529
15520
17636
19863
22186
24590
27057
29569
32111
34663
37211
39738
42229
44670
47050
49357
51581
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00422
WHAT IS THE DILUTION FACTOR?
?3.5
INPUT STARTING Lr
710,300,10
T
L
E(M)
10 1.000
11.24
20 1.000
22.67
30 1.000
34.31
40 1.000
46.14
50 1.000
58.18
60 1.000
70.40
70 1.000
82.80
80 1.000
95.37
90 1,000 108.08
100 1.000 120.91
H O 1.000 133.85
120 1.000 146.87
130 1.000 159.95
140 1.000 173.07
150 1.000 186.21
160 1.000 199.34
170
.999 212.46
180
.999 225.54
190
.999 238.56
200
.998 251.51
210
.998 264.38
220
.998 277.16
230
.997 289.82
240
.996 302.38
250
.995 314.80
260
.995 327.10
270
.994 339.25
280
.993 351.26
290
.991 363.12
300
.990 374.83
*STOP* OY
>
X
H
W
Ph
ENDING Lr JUMP SIZE
E(C)
8.60
17.34
26.23
35.28
44.48
53.83
63.32
72.92
82.64
92.45
102.35
112.30
122.31
132.34
142.38
152.43
162,46
172.46
182.42
192.32
202.16
211.93
221.62
231.21
240.72
250.12
259.41
268.59
277.66
286.61
E(F)
11.16
22.43
33.84
45.37
57.05
68.85
80.78
92.83
104.98
117.23
129.55
141.93
154,35
166.81
179.27
191.73
204.18
216.59
228.95
241.26
253.49
265.65
277.71
289.67
301.51
313.24
324.84
336.31
347.64
358.82
E(T)
E(S)
23.68
11.32
30.26
22.95
34.90
37.38
47.15
44.93
52.81
59.69
72.46
60.94
69.24
85.43
98.56
77.65
86.12 111.80
94.60.125.11
103.06 138.46
111.45 151.82
119.77 165.16
127.99 178.45
136.10 191.67
144.08 204.81
151.93 217.83
159.63 230.74
167.20 243.52
174.61 256.15
181.88 268,64
188.99 280.97
195.96 293.13
202.77 305.14
209.44 316.97
215.95 328.62
222.32 340.11
228.55 351.41
234.63 362.54
240.57 373.49
MSE(M)
232
528
901
1368
1945
2647
3490
4487
5648
6981
8488
10168
12016
14023
16177
18463
20867
23370
25954
28600
31291
34008
36734
39452
42148
44807
47415
49962
52436
54828
MSE(C)
136
311
530
800
1128
1522
1989
2535
3165
3883
4689
5582
6560
7618
8750
9948
11203
12508
13854
15230
16628
18039
19456
20870
22274
23663
25031
26373
27685
28963
MSE(F)
225
504
852
1284
1817
2470
3256
4189
5280
6534
7954
9538
11282
13178
15213
17374
19646
22012
24454
26955
29497
32062
34635
37198
39738
42241
44695
47087
49409
51651
MSE(S)
259
297
414
602
852
1157
1510
1906
2340
2807
3303
3823
4365
4924
5498
6085
6683
7290
7906
8529
9160
9800
10448
11107
11778
12463
13165
13886
14628
15396
MSE(T)
239
560
979
1511
2167
2954
3881
4950
6163
7520
9014
10640
12388
14246
16201
18239
20345
22503
24698
26913
29135
31350
33544
35705
37821
39883
41882
43809
45657
47422
WHAT IS NT
?3
WHAT IS THE MIDDLE DILUTION?
?.00457
WHAT IS THE DILUTION FACTOR?
?4.0
INPUT STARTING Lr ENDING Lr JUMP SIZE
? IO r300 rIO
L
T
E(M)
E(C)
E(S)
E(F)
10 1.000
11.37
11.27
19.33
8.69
20 1.000 23.02
17.60
22.70
26.28
30 1.000 34.96
26.74
34.32
33.92
40 1.000
47.19
46.12
42,07
36.08
50 1.000 59.67
58.10
50.59
45.63
60 1.000 72.38
70.23
59.35
55.34
70 1.000 85.28 > 65.21
82.50
68.26
80 1.000
94 ..88 -77.22
98.33
75.19
90 1.000 111.50
86.18
85.26 107.34
100 1.000 124.75
95.10
95.39 119.87
H O 1.000 138.06 105.57 132.43 103.93
120 1.000 151.40 115.77 145.02 112.66
130 1.000 164.74 125.97 157.62 121.26
140 1.000 178.06 136.16 170.20 129.73
150 1.000 191.35 146,32 182.77 138.05
160
.999 204.59 156,44 195.30 146.23
170
.999 217.76 166,51 207.79 154.26
180
.999 230.87 176.53 220.22 162.13
190
.999 243.88 186.49 232.60 169.86
200
.998 256.81 196.37 244.91 177.44
210
.998 269.64 206,18 257.14 184,87
220
.997 282.37 215.92 269.29 192.16
230
.997 294.99 225.57 281.34 199.30
240
,996 307.50 235.13 293.30 206.30
250
.995 319.88 244.60 305.15 213.16
260
.994 332.15 253.98 316.89 219.89
270
.993 344.28 263.26 ,328.51 226.47
280
.992 356.28 272.43 340.01 232.92
290
.991 368.15 281.51 351.38 239.24
300
.990 379.88 290.47 362.61 245.42
2STnP*
X
H
I
Pm
<!
AY
E(T)
11.48
23,41
35.79
48.56
61.62
74.89
88.28
101.71
115.14
128.49
141.74
154.86
167.83
180.63
193.26
205.71
217.97
230.06
241.96
253.68
265.24
276.61
287.82
298.86
309.74
320.45
331.00
341.39
351.62
361.69
MSE(M)
207
494
880
1389
2041
2855
3848
5032
6415
7998
9779
11749
13898
16211
18672
21263
23965
26758
29623
32541
35493
38461
41428
44379
47298
50172
52988
55733
58399
60976
MSE(C)
122
289
511
797
1157
1601
2136
2769
3503
4339
5277
6311
7436
8646
9931
11282
12690
14144
15635
17152
18687
20230
21773
23308
24828
26328
27801
29242
30647
32012
MSE(F)
199
464
817
1281
1880
2632
3554
4657
5948
7429
9097
10945
12963
15139
17457
19901
22451
25092
27802
30566
33363
36178
38994
41796
44568
47299
49976
55588
55125
57578
MSE(S)
156
235
400
640
946
1313
1734
2202
2712
3259
3838
4445
5075
5723
6388
7064
7751
8445
9145
9850
10559
11272
11989
12711
13438
14173
14916
15670
16438
17222
MSE(T)
217
536
977
1552
2264
3115
4105
5231
6485
7862
9350
10938
12614
14364
16175
18033
19924
21836
23756
25672
27572
29446
31285
33079
34821
36504
38122
39669
41141
42534
WHAT IS N?
?3
— I
I
cn
WHAT IS THE MIDDLE DILUTION?
7.00492
• WHAT IS THE DILUTION FACTOR?
?4.5
INPUT STARTING L, ENDING L, JUMP1 SIZE
710,300,10
L
T
E(M)
E(C)
E(S)
E(F)
10 1.000
11.50
8.79
16.43
11.37
20 1.000 23.39
17,89
23.77
22.97
30 1.000
35.69
27.29
31.93
34.83
40 1.000
48.34
36.97
40.65
46.93
50 1.000
61.29
46.87
49.72
59.22
60 1.000
56.94
58.98
74.46
71.67
70 1.000
87.80
67.14
68.32
84.24
80 1.000 101.24
77.64
77.41
96.88
• 90 1.000 114.73' 87.73 109.56
86.90
100 1.000 128.24
98.06 122-. 25 -96.04
H O 1.000 141.74 108.38 134.94 105.05
120 1.000 155.20 118.67 147.61 113.91
130 1.000 168.61 128.93 160.26 122.62
140 1.000 181.96 139.14 172.87 131.18
150 1.000 195.25 149.30 185.44 139.58
160
.999 208.46 159.40 197.98 147.84
170
.999 221,60 169.44 210.46 155.96
180
.999 234* 66 179.43 222.90 163.93
190
.999 247.64 189.36 235.29 171.77
200
.998 260.55 199.23 247.61 179.47
210
.998 273.37 209.03 259.87 187.05
220
.997 286.11 218.77 272.06 194.49
230
.997 298.75 228.44 284.17 201.80
240
.996 311.30 238.04 296.19 208.98
250
.995 323.76 247.56 308.12 216.04
260
.994 336.10 257.00 319.95 222.96
270
.993 348.34 266.36 331.68 229.76
280
.992 360.45 275.62 343.29 236.44
290
.991 372.45 284.79 354.77 242.98
300
.990 384.32 293.87 366.13 249.40
3ST0P* : OY
S
I
PM
E(T)
11.64
23.93
36.79
50.07
63.59
77.21
90.79
104.24
117.51
130.56
143.36
155.91
168.21
180.27
192.11
203.73
215.14
226.37
237.42
248.31
259.03
269.60
280.02
290.29
300.42
310.41
320.26
329.97
339.53
348.95
MSE(M)
189
474
886
, 1452
2197
3141
4299
5679
7282
9105
11137
13367
15778
18354
21074
23921
26874
29913
33019
36173
39355
42549
45736
48902
52031
55108
58121
61058
63907
66660
MSE(C)
HO
275
506
817
1219
1723
2336
3063
3905
4861
5925
7093
8355
9702
11125
12614
14157
15743
17363
19007
20663
22324
23980
25624
27247
28844
30408
31934
33418
34856
MSE(F)
179
438
809
1323
2004
2874
3944
5222
6709
8402
10295
12375
14630
17044
19602
22284
25072
27949
30895
33893
36924
39972
43020
46053
49057
52017
54921
57759
60519
63194
MSE(S)
108
217
416
697
1051
1471
1951
2485
3066
3689
4347
5037
5752
6487
7240
8005
8779
9560
10345
11132
11920
12708
13495
14282
15068
15855
16645
17438
18238
19045
MSE(T)
201
527
997
1611
2366
3254
4269
5400
6638
7971
9389
10878
12427
14024
15658
17315
18987
20661
22328
23979
25604
27196
28746
30248
31696
33084
34409
35665
36851
37963
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00527
WHAT IS THE DILUTION FACTOR?
?5.0
INPUT STARTING Lr
TlOrSOOrlO
L
T
E(M)
10 1.000
11.63
20 1.000 23.80
30 1.000
36.49
40 1.000
49.58
50 1.000
62.97
60 1.000
76.54
70 1.000
90.20
80 1.000 103.88
90 1.000 117.53
100 1,000 131.12
H O 1.000 144.64
120 1.000 158.09
130 1.000 171.46
140 1.000 184.76
150 1.000 198.00
160
.999 211.18
170
.999 224.31
180
.999 237.39
190
.999 250.42
200
.998 263.39
210
,998 276.31
220
.997 289.16
230
.997 301.95
240
.996 314.66
250
.995 327.29
260
.994 339.83
270
.993 352.26
280
.992 364.58
290
.991 376.78
300
.990 388.85
ItSTOPA OY
>
X
H H
W r4
ENDING Lr JUMP SIZE
E(C)
8.89
18.20
27.90
37.91
48.15
58.53
68.97
79.43
89.87
100.26
110.60
120.88
131.11
141.28
151.40
161.48
171.52
181.52
191.48
201.40
211.28
221.11
230.89
240.61
250.27
259.85
269.36
278.78
288.10
297.33
E(F)
11.47
23,26
35.38
47.78
60.38
73.10
85.89
98.69
111.47
124.22
136.93
149.60
162.24
174.84
187.41
199,95
212.47
224.95
237.39
249.80
262.16
274.46
286.70
298.86
310,94
322.92
334.80
346.57
358.21
369.72
E(S)
14,44
22.19
30.85
40.07
49.59
59.21
68.82
78.32
87.69
96.91
105.95
114,84
123.58
132.17
140.62
148.94
157.14
165.21
173,17
181.02
188.75
196.36
203.86
211.24
218.51
225.64
232.66
239.55
246.32
252.95
E(T)
11.82
24.50
37.86
51.60
65,45
79.19
92.68
105.85
118.67
131.14
143.26
155.07
I66,60
177.87
188.92
199.77
210.44
220.95
231.31
241.53
251.63
261.59
271.43
281.14
290.72
300,18
309.51
318.70
327.75
336,67
MSE(M)
175
468
915
1550
2399
3481
4806
6375
8184
10222
12476
14931
17570
20372
23321
26395
'29574
32839
36170
39546
42950
46362
49765
53141
56474
59750
62955
66076
69102
72023
MSE(C)
102
268
514
857
1308
1877
2572
3394
4342
5410
6593
7883
9269
10741
12290
13903
15569
17277
19016
20775
22543
24312
26071
27812
29527
31210
32853
34453
36004
37503
MSE(F)
164
425
825
1400
2176
3169
4386
5829
7494
9375
. 11461
13741
16199
18820
21588
24483
27489
30587
33758
36984
40246
43528
46811
50080
53320
56516
59655
62725
65715
68614
MSE(S)
85
216
444
760
1156
1626
2162
2757
3405
4099
4833
5602
6*398
7217
8053
8903
9762
10626
11492
12358
13221
14080
14935
15783
16627
17466
18301
19134
19967
20802
MSE(T)
190
530
1027
1671
2447
3342
4345
5442
6624
7880
9199
10571
11985
13430
14897
16377
17859
19334
20793
22228
23632
24998
26318
27587
28801
29956
31047
32074
33033
33924
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00563
WHAT IS THE DILUTION FACTOR?
?5.5
CO
o\
INPUT STARTING Lr
?10,300 ,10
L
T
E(M)
10 I .000
11.76
20 I .000 24.25
30 I .000 37.35
40 I .000
50.87
50 I .000
64.64
60 I .000
78.49
70 I .000
92.33
80 I .000 106.09
90 I .000 119.75
100 I .000 133.30
H O I .000 146.76
120 I .000 160,14
130 I .000 173.45
140 I .000 186.73
150
.999 199.97
160
.999 213.20
170
.999 226.40
180
.999 239.60
190
.998 252.77
200
.998 265.92
210
.998 279.04
220
.997 292.11
230
.996 305.12
240
.996 318.07
250
.995 330.93
260
.994 343.70
270
.993 356.36
280
.992 368.90
290
.991 381.31
300
.990 393.58
*STOP* OY
>
I
PL
ENDING L , JUMP SIZE
E(C)
9.00
18.54
28.56
38.90
49.43
60.02
70.60
81.12
91.57
101.93
112.22
122.45
132.63
142.78
152.91
163.02
173.12
183.21
193.28
203.34
213.37
223.36
233.31
243.21
253.05
262.81
272.49
282.08
291.57
300.95
E(F)
11.57
23.57
35.98
48.67
61.53
74.45
87.36
100.22
113,01
125.74
138.42
151.07
163.69
176.30
188.91
201.51
214.11
226.71
239.28
251.84
264.35
276.82
289,23
301.57
313.82
325.97
338.01
349.94
361.73
373.38
E(S)
13.04
21.21
30.34
39.99
49.84
59.67
69.39
78.94
88.31
97.50
106.53
115.41
124.15
132,77
141.28
149.69
157.99
166.20
174.32
182.33
190.24
198.04
205.73
213.31
220.77
228.11
235.32
242.40
249.35
256.16
E(T)
12.01
25.13
38.98
53.08
67.04
80.64
93.78
106.42
118.59
130.33
141.69
152.75
163.54
174,11
184.50
194.73
204.82
214.79
224.64
234.38
244.01
253,53
262.93
272.22
281.38
290.42
299.32
308.08
316.70
325.17
MSE(M)
165
471
961
1674
2634
3854
5337
7080
9072
11300
13750
16404
19245
22254
25411
28698
32092
35573
39120
42712
46329
49951
53558
57132
60657
64116
67495
70781
73962
77028
MSE(C)
96
267
532
911
1415
2053
2829
3743
4789
5962
7254
8655
10155
11743
13406
15134
16913
18733
20580
22444
24313
26177
28026
29851
31645
33400
35109
36768
38373
39918
MSE(F)
153
422
860
1506
2381
3494
4847
6437
8258
10301
12556
15010
17650
20459
23423
26521
29738
33052
36445
39897
43389
46903
50419
53921
57392
60816
64180
67470
70676
73786
MSE(S)
75
224
477
825
1260
1776
2364
3017
3727
4489
5294
6135
7008
7904
8819
9747
10684
11624
12565
'13503
14435
15359
16274
17180
18076
18962
19840
20711
21576
22439
MSE(T)
184
539
1059
1717
2492
3367
4331
5371
6481
7649
8868
10127
11419
12732
14059
15388
16712
18021
19307
20562
21780
22955
24080
25152
26168
27124
28019
28852
29623
30332
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00598
WHAT IS THE DILUTION FACTOR?
?6.0
APPENDIX V
TABLE K
INPUT STARTING Lr
?IO r300 rIO
L
T
E(M)
10 1.000
11.91
20 1.000
24.73
30 1.000
38.25
40 1.000
52.15
50 1.000
66.20
60 1.000
80.20
70 1.000
94.08
80 1.000 107.80
90 1.000 121.37
100 1.000 134.83
H O I .000 148.21
120 I .000 161.55
130 1.000 174.87
140 1.000 188.20
150
.999 201.54
160
.999 214.91
170
.999 228.31
180
.999 241.71
190
.998 255.12
200
.998 268.53
210
.998 281.90
220
.997 295.24
230
.996 308.52
240
.996 321.72
250
.995 334.84
260
.994 347.84
270
.993 360.72
280
.992 373.47
.991 386♦06
290
300
.990 398.50
*STOP* OY
ENDING Lr JUMP SIZE
E(C)
9.11
18.91
29.24
39.88
50.62
61.33
71.94
82.43
92.81
103.10
113.33
123.53
133,72
143.91
154.11
164.33
174.57
184.83
195.08
205.33
215.56
225.76
235.91
246.01
256.03
265.98
275.83
285.57
295.20
304.72
E(F)
11.68
23.91
36.60
49.56
62.62
75.64
88.58
101.42
114.16
126.85
139.50
152.15
164,80
177.48
190.19
202.92
215.67
228.43
241,19
253.94
266.65
279.31
291.91
304.43
316.85
329.17
341.36
353.43
365.35
377.12
E(S)
12.06
20.64
30.21
40.19
•50.23
60.15
69.87
79.38
88.70
97.84
106.84
115.72
124.50
133.20
141.81
150.35
158.81
167.18
175.47
183.67
191.77
199.77
207.65
215.42
223.06
230.58
237.96
245.21
252.32
259.28
E(T)
12.23
25.81
40.09
54.38
68.23
81.47
94.04
106.01
117.45
128.46
139.13
149.53
159.72
169,75
179.65
189.43
199.11
208.69
218.18
227.58
236.87
246.05
255.11
264.05
272.87
281.54
290.07
298.45
306*68
314.74
MSE(M)
159
482
1023
1817
2887
4237
5865
7763
9917
12315
14941
17780
20813
24021
27385
30883
34493
38193
41959
45769
49602
53434
57247
61020
64735
68377
71930
75381
78717
81928
MSE(C)
92
270
559
976
1535
2240
3094
4093
5231
6500
7893
9398
11005
12702
14475
16313
18201
20127
22079
24044
26009
27965
29902
31809
33679
35504
37279
38998
40656
42252
MSE(F)
145
428
912
1632
2603
3826
5299
7017
8973
11159
13567
-16183
18997
21991
25150
28454
31885.
35423
39045
42731
46460
50211
53966
57704
61409
65064
68654
72165
75586
78904
MSE(S)
71
235
511
889
1362
1921
2558
3266
4036
4861
5733
6645
7590
8561
9551
10554
11565
12579
13591
14597
15595
16582
17556
18517
19464
20398
21319
22230
23130
24024
MSE(T)
181
553
1087
1744
2500
3338
4250
5228
6266
7356
8489
9656
10848
12054
13266
14473
15667
16838
17979
19083
20145
21158
22120
23027
23878
24671
25407
26086
26710
27281
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00633
WHAT IS THE DILUTION FACTOR?
?6.5
appendix
V
INPUT STARTING Lr
? IO r300 rIO
L
T
E(M)
10 1.000
12.07
20 1.000
25.24
30 I ,000
39.16
40 1.000
53.38
50 1.000
67.58
60 1.000
81.60
70 1.000
95,41
80 1.000 109.02
90 1.000 122.49
100 1.000 135.87
H O 1.000 149.23
120 1.000 162.60
130 1.000 176.02
140 1.000 189.49
150
.999 203.03
160
.999 216.63
170
.999 230.28
180
.999 243.96
190
.998 257.65
200
.998 271.34
210
.997 285.00
220
.997 298.61
230
.996 312.14
240
.996 325.59
250
.995 338.92
260
.994 352.13
270
.993 365.19
280
.992 378.10
290
.991 390.84
300
.990 403.39
*STOP* OY
I
S
ENDING Lr JUMP SIZE
E(C)
9.23
19.30
29.94
40.81
51.67
62.40
72.96
83.36
93.66
103.89
114.11
124.33
134.59
144,89
155.25
165.65
176.08
186.54
197.02
207.48
217.93
228.33
238.68
248.96
259.16
269.25
279.24
289.11
298.85
308.46
E(F)
11.79
24.26
37.23
50.41
63.58
76.63
89.53
102.30
114.99
127.65
140.31
153.01
165.76
178.57
191.43
204.34
217.29
230.26
243.22
256.17
269.08
281.93
294.71
307.39
319.97
332,43
344.75
356.92
368.94
380.79
E(S)
11.37
20.36
30.29
40.50
50.62
60.52
70.18
79.61
88.86
97.97
106.97
115.90
124.77
133.58
142.33
151.03
159.66
168.22
176.70
185.09
193.37
201.54
209.60
217.53
225.33
232.99
240.51
247.89
255.11
262.19
E(T)
12.46
26.52
41.13
55.42
68.96
81.64
93.53
104.78
115.52
125.88
135.97
145.86
155.61
165.25
174.80
184.26
193.65
202.96
212.18
221.30
230.31
239.20
247.97
256.60
265.08
273.42
281.59
289.60
297.45
305.12
MSE(M)
156
501
1096
1973
3147
4616
6374
8410
10712
13266
16060
19076
22297
25703
29272
32981
36805
40721
44702
48725
52765
56800
60808
64769
68664
72476
76190
79792
83271
86616
MSE(C)
89
277
591
1049
1662
2432
3358
4436
5659
7019
8507
10111
11820
13620
15498
17440
19432
21460
23510
25569
27625
29667
31684
33667
35608
37499
39335
41110
42821
44465
MSE(F)
139
441
975
1770
2829
4150
5728
7558
9636
11955
14508
17284
20271
23452
26809
30323
33973
37735
41587
45506
49469
53454
57440
61406
65333
69206
73008
76724
80343
83853
MSE(S)
69
248
545
952
1460
2060
2744
3503
4329
5214
6150
7129
8143
9184
10245
11319
12400
13482
14561
15631
16691
17736
18766
19780
20776
21756
22720
23669
24607
25534
MSE(T)
181
567
1106
1748
2471
3264
4124
5045
6021
7046
8109
9200
10310
11427
12541
13642
14721
15770
16783
17752
18674
19546
20365
21130
21840
22498
23103
23660
24170
24639
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00668
WHAT IS THE DILUTION FACTOR?
?7.0
INPUT STARTING Lr
?10 r300 r10
L
T
E(M)
10 1.000
12,24
20 1.000 25.79
30 1.000
40.07
40 1.000
54.49
50 1.000
68.73
60 1.000
82.68
70 1.000
96.36
80 1.000 109.84
• 90 1.000 123.23
100 1.000 136»60
H O 1.000 150.01
120 1.000 163.51
130 1,000 177.11
140 I .000 190.81
150
.999 204.62
160
.999 218.50
170
.999 232.44
180
.999 246.42
190
.998 260.41
200
.998 274.38
210
.997 288.30
220
.997 302.15
230
.996 315.91
240
.996 329.56
250
.995 343.07
260
.994 356.44
.993 369.63
270
280
.992 382.65
290
.991 395.48
300
.990 408.11
*STOP* O
H
I
FM
%
ENDING Lr JUMP SIZE
E(C)
9.36
19.72
30.64
41.67
52.56
63.22
73.68
83.99
94.23
104.45
114.71
125,03
135.43
145.91
156.46
167.08
177.74
188.43
199.12
209.80
220.45
231.04
241.56
252.00
262.33
272.55
282.64
292.60
302.40
312.06
E(F)
11.91
24.63
37.86
51.20
64.40
77.40
90.22
102.92
115.58
128.25
140.98
153.79
166.69
179.68
192.75
205.87
219.04
232.22
245.39
258.54
271.63
284.66
297.59
310.42
323.12
335.68
348.09
360.34
372.42
384.32
E(S)
•10.89
20.27
30.50
40.83
50.94
60.76
70.31
79,66
88.86
97.98
107.04
116.06
125.05
134.01
142.93
151.81
160.62
169.36
178.02
186.58
195.03
203.36
211.56
219.62
227.54
235.32
242.94
250.41
257.72
264.87
E(T)
12.72
27,24
42.06
56.16
69.19
81.21
92.39
102.96
113.09
122.93
132.58
142.10
151.53
160.90
170.21
179.45
188.62
197.71
206.70
215.58
224.34
232.97
241.45
249.77
257.94
265.94
273.76
281,41
288.88
296.16
MSE(M)
154
526
1176
2134
3405
4982
6857
9019
11459
14164
17121
20314
23722
27325
31098
35015
39049
43174
47363
51589
55827
60053
64243
68378
72439
76407
80268
84008
87617
91084
MSE(C)
88
288
629
1128
1792
2622
3616
4768
6072
7518
9096
10795
12602
14502
16480
18522
20612
22735
24877
27025
29166
31288
33380
35434
37441
39394
41288
43117
44878
46568
MSE(F)
136
462
• 1048
1912
3050
4457
6127
8060
10252
12702
15400
10338
'21502
24874
28434
32161
36030
40017
44096
48242
52432
56640
60844
65024
69160
73234
77229
811.33
84931
88614
MSE(S)
70
. 262
580
1013
1554
2193
2921
3728
4608
5551
6548
7591
8670
9278
10906
12047
13194
14340
15481
16612
17729
18830 •
19912
20976
22019
23044
24050
25039
26013
26975
MSE(T)
183
580
1113
1730
2414
3163
3976
4849
5777
6749
7755
8783
9822
10860
11887
12891
13866
14803
15698
16546
17344
18091
18786
19429
20024
20571
21074
21538
21966
22364
UHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00703
WHAT IS THE DILUTION FACTOR?
?7.5
appendix
V
INPUT STARTING L,
710,300,10
T
E(M)
.Il
10 1.000 12.42
20 1.000 26.35
30 1.000 40.93
40 1.000 55.46
50 1.000 69.64
60 1.000 83.45
70 1.000 96.98
80 1.000 110.37
90 1.000 123.74
100 1.000 137.18
H O 1.000 150.74
120 1.000 164.44
130 1.000 178.29
140 1.000 192.28
150 .999 206,38
160 .999 220.57
170 .999 234.82
180 .999 249.09
190 .998 263.35
200 .998 277.57
210 .997 291.73
220 .997 305.79
230 .996 319.73
240 .996 333.54
250 .995 347.18
260 .994 360,66
270 .993 373.94
280 .992 387,02
290 .991 399.89
300 .990 412.54
♦STOP# OY
ES
W
ENDING L, JUMP SIZE
E(C)
9.50
20.15
31.30
42.41
53.25
63.81
74.16
84.40
94 ..62
104.90
115.26
125.74
136.33
147.03
157,81
168.66
179,56
190,47
201.37
212.25
223.07
233.82
244.49
255.04
265.48
275.78
285.93
295.93
305.77
315.45
E(F)
12.03
25.02
38.47
51.88
65.05
77.96
90.69
103.35
116.02
128.77
141.62
154.60
167.69
180.90
194.18
207.53
220.91
234.31
247.68
261.00
274.26
287.44
300.50
313.43
326.22
338.86
351.33
363.62
375.73
387.65
E(S)
10.56
20.31
30.75
41.11
51.14
60.84
70.29
79.59
88.79
97.96
107.12
116.27
125.41
134.54
143.64
152.70
161.69
170.60
179.42
188.13
196.72
205.17
213.49
221.66
229.67
237.53
245.22
252.75
260.12
267.32
E(T)
13.00
27.94
42.82
56.55
68.96
80.27
90.78
100.76
110.40
119.85
129.19
138.46
147.68
156.86
165.99
175.06
184.05
192.94
201.73
210.38
218,90
227.26
235.46
243.48
251.33
258.99
266.47
273.75
280.85
287.75
MSE(M)
MSE(C)
155
87
555
301
1262
670
2296
1208
3655
1922
5329
2809
7311
3866
9593
5089
12166
6469
15021
7998
18141
9664
21510
11455
25106
13355
28903
15351
17425
32877
36997
19561
41235
21744
23958
45561
49947
26187
28418
54364
30637
58786
32833
63189
34996
67547
37116
71842
39184
76052
41195
80162
43143
84156
45022
88020 '
46831
91745
48567
95321
MSE(F)
135
487
1125
2053
3260
4742
6497
8528
10834
13415
16264
19369
22714
26281
30047
33986
38073
42280
46580
50945
55350
59768
64177
68555
72881
77137
81308
85379
89338
93173
MSE(S)
72
276
613
1073
1645
2320
3089
3943
4874
5872
6928
8032
9174
10346
11537
12741
13950
15157
16357
17544
1871619869
21001
22112
23201
24269
25316
26345
27357
28355
MSE(T)
186.
590
1109
1693
2338
3048
3823
4660
5549
6479
7437
8410
9385
10349
11293
12208
13085
13919
14707
15445
16133
16771
17361
17904
18404
18866
19293
19692
20067
20425
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00738
WHAT IS THE DILUTION FACTOR?
?8.0
INPUT STARTING Lr ENDING Lr JUMP SIZE
T l O r 300 r10
L
10
20
30
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
OO
T
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
I. OOP
1.000
1.000
1.000
1.000
1.000
.999
.999
.999
.999
.998
.998
.997
.997
.996
.996
.995
.994
.993
.992
.991
.990
*RTnP*
X!
H
I
OY
E(M)
12.62
26,92
41.73
56.26
70,30
83.94
97.36
110.72
124.16
137.75
151.52
165.48
179,63
193.93
208.34
222.83
237.37
251.91
266.41
280.86
295,21
309.43
323.51
337.43
351.16
364.70
378.02
391.12
403.99
416.62
E(C)
9.65
20.58
31.91
■ 43,02
53.76
64.19
74.45
84.66
94.94
105.33
115.86
126.54
137.35
148.29
159.31
170.39
181.50
192.62
203.71
214.76
225.73
236.61
247.38
258.02
268.52
278.87
289.05
299.07
308.91
318.57
E(F)
12.17
25.41
39,03
52.46
65,54
78.34
91.01
103.66
116.40
129.27
142.30
155,48
168.80
182.23
195.74
209.31
222.91
236.49
250.04
263.53
276.93
290.22
303.38
316.40
329.25
341.94
354.44
366.75
378.86
390.76
E(S)
10.35
20.43
31.01
41.33
51.23
60.81
70.18
79.46
88.71
97.97
107.26
11.6.57
125.88
135.19
144.47
153.69
162.85
171,91
180.87
189.70
198,41
206.96
215.37
223.61
231.69
239.60
247.34
254.90
262.30
269.51
E(T)
13.29
28.61
43,39
56.61
68.33
78.96
88.87
98.38
107.66
116.83
125.96
135.06
144.13
153.18
162.17
171.09
179.91
188.62
197.20
205.63
213.89
221.99
229.90
237.62
245.15
252.48
259.61
266.54
273.27
279.80
MSE(M)
158
588
1350
2455
3893
5657
7740
10139
12845
15848
19133
22678
26459
30449
34617
38933
43365
47882
52452
57048
61640
66204
70715
75154
79499
83735
87847
91822
95650
99322
MSE(C)
88
316
713
1290
2049
2990
4108
5398
6852
8462
10213
12092
14084
16171
18337
20563
22834
25132
27442
29750
32043
34308
36535
38716
40842
42906
44904
46832
48686
50466
MSE(F)
136
517
1203
2188
3456
5006
6842
8971
11394
14112
17115
20392
23924
27688
31660
35810
40111
44531
49043
53615
58222
62836
67433
71991
76490
80910
85237
89456
93554
97523
MSE(S)
74
291
646
1130
1731
2441
3249
4149
5129
6181
7294
8456
9659
10891
12143
13406
14674
15937
17192
18433
19656
20858
22038 "
23195
24328
25438
26526
27594
28644
29679
MSE(T)
190
596
1093
1643
2253
2932
3678
4487
5345
6240
7155
8076
8989
9884
10750
11578
12364
13104
13794
14435
15027
15573
16075
16538
16966
17365
17741
18099
18447
18792
WHAT IS N?
?3
APPENDIX V
TABLE P
WHAT IS THE MIDDLE DILUTION?
7.00774
WHAT IS THE DILUTION FACTOR?
?8.5
INPUT STARTING Lr ENDING Lr JUMP' SIZE
? IO r300 rIO
' L
E(M)
E(S)
T
E(C)
E(F)
10 1.000
12.83
9.81
10.23
12.31
20 1.000
27.48
21.01
25.81
20.59
30 1.000
42.43
32.44
39.54
31.24
40 1.000
56.88
41.45
43.49
52.91
50 1.000
70.74
54.09
65.87
51.21
60 1.000
84.23
64.40
78.58
60.68
70 1.000
97.58
74.61
91.22
70.01
80 1.000 110.99
84.87 103.93
79.32
90 1.000 124.57
95.25 116.79
88»66
100 1.000 138.37 105.81 129.83
98.05
H O 1.000 152.42 116.55 143.06 107.49
120 1.000 166.69 127.46 156.47 116.97
130 1.000 181.15 138.51 170.02 126.46
140 1.000 195.76 149.69 183.69 135.95
150
.999 210.48 160.94 197.43 145.39
160
.999 225.26 172.24 211.21 154.77
.999 240.06 183.56 225.00 164.07
170
180
.999 254.83 194.86 238.76 173.26
190
.998 269.55 206.11 252.46 182.33
200
.998 284.17 217.29 266.08 191.26
210
.997 298.66 228.37 279.59 200.05
220
.997 313.01 239.34 292.97 208.67
230
.996 327.18 250.18 306.20 217.13
240
.995 341.16 260.87 319.27 225.42
250
.995 354.93 271.40 332.16 233.53
260
.994 368.48 281.76 344.85 241.47
270
.993 381.80 291.94 357.35 249.22
280
.992 394.87 301.94 369.65 256.79
290
.991 407.69 311.74 381.73 264.18
300
.990 420.26 321.35 393.60 271.39
*STOP* OA
E(T)
13.61
29.22
43.74
56.33
67.35
77.35
86.78
95.92
104.95
113,94
122.93
131.91
140.88
149.81
158.68
167.45
176.11
184.63
192.99
201.19
209.20
217.02
224.63
232.05
239.25
246.24
253.02
259.60
265.96
272.13
MSE(M)
161
623
1438
2607
4118
5966
8148
10663
13503
16657
20105
23824
27786
31961
36315
40815
45427
50117
54853
59606
64347
69050
73691
78249
82704
87042
91247
95308
99215
102961
MSE(C)
89
332
756
1370
2173
3164
4341
5697
7224
8912
10746
12711
14789
16963
19214
21524
23875
26249
28632
31008
33365
35689
37972
40203
42376
44485
46525
48492
50384
52200
MSE(F)
139
549
1281
2315
3637
5252
7169
9399
11944
14804
17968
21421
25141
29102
33276
37631
42136
46757
51465
56228
61016
65804
70566
75279
79923
84480
88934
93272
97482
101555
MSE(S)
77
305
678
1184
1813
2555
3403
4346
5374
6477
7644
8862
10121
11409
12717
14036
15356
16672
17977
19266
20535
21782
23005
24203
25376
26524
27650
28755
29841
30912
MSE(T)
194
596
1067
1584
2165
2819
3545
4331
5163
6024
6897
7766
8619
9444
10233
10979
11679
12329
12931
13484
13993
14460
14891
15291
15666
16023
16368
16709
17053
17408
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
7.00809
WHAT IS THE DILUTION FACTOR?
?9.0
INPUT STARTING L, ENDING L, JUMP SIZE
710,300,10
L
T
E(C)
E(F)
E(M)
E(S)
10 1.000
13.05
9.98
12.46
10.17
20 1.000
21.43
26.19
20.78
28.02
30 1.000
43.02
32.90
39.97
31.41
40 1.000
41.49
57.32
43.83
53.25
50 1.000 70.99
54.29
66 *08 51.11
60 1.000
84.37
64.51
60.50
78.72
70 1.000
91.38
69.84
97.73
74.73
80 1.000 111.25
85.07 104.19
79.22
90 1.000 125.03
88.68
95.61 117.21
100 1.000 139.10 106.37 130.46
98.22
H O 1.000 153.45 117.34 143.92 107.83
120 1.000 168.03 128.49 157.56 117.49
130 1.000 182.81 139.79 171.35 127.15
140
.999 197.72 151.19 185.24 136.80
150
.999 212.73 162.66 199.19 146.39
160
.999 227.77 174.16 213.17 155.91
170
.999 242.80 185.65 227.13 165,33
180
.998 257.77 197.11 241.04 174.63
190
.998 272.66 208.49 254.88 183.80
200
.998 287.42 219.78 268.61 192.81
210
.997 302.03 230.95 282.21 201.66
220
.997 316.46 241.99 295.66 210.33
230
.996 330.69 252.87 308.94 218.83
240
.995 344.71 263,58 322.04 227.15
250
.995 358.49 274.12 334.94 235.28
260
.994 372.02 284.47 347.64 243.22
270
.993 385.31 294.62 360.13 250.97
280
.992 398.33 304.58 372.40 258.53
290
.991 411.08 314.34 384.44 265.91
300
.990 423.57 323.89 396.26 273.10
*STOP* OY
W
a
E(T)
13.94
29.75
43.87
55.79
66*15
75.63
84.68
93.57
102.42
111.30
120.20
,129.10
137.98
146.81
155.55
164.17
172.66
180.99
189.14
197.10
204.86
212.40
219.73
226.84
233.73
240.39
246.84
253.07
259.10
264.91
MSE(M)
166
660
1524
2751
4330
6259
8540
11173
14149
17454
21066
24958
29098
33453
37988
42664
47448
52304
57199
62101
66982
71816
76579
81251
85811
90246
94541
98686
102671
106489 '
MSE(C)
92
350
800
1448
2292
3332
4566
5987
7587
9351
11267
13315
15479
17737
20071
22461
24889
27338
29790
32232
34650
37033
39370
41652
43873
46027
48110
50119
52051
53906
MSE(F)
142
583
1356
2432
3805
5484
7485
9820
12493
15500
18830
22462
26371
30529
34903
39460
44163
48980
53876
58820
63782
68734
73651
78510
83291
87976
92549
96999
101313
105484
MSE(S)
80
319
708
1236
1891
2666
3551
4537
5613
6767
7986
9259
10572
11916
13278
14650
16022
17388
18742
20078
21393
22685
23951
25191
26405
27594 '
28759
29903
31027
32136
MSE(T)
199
592
1034
1523
2081
2719
3430
4199
5007
5837
6669
7488
8283
9043
9760
10430
11052
11624
12149
12630
13071
13477
13855
14212
14554
14889
15225
15570
15931
16316
WHAT IS N?
?3
WHAT IS THE MIDDLE DILUTION?
?.00844
WHAT IS THE DILUTION FACTOR?
?9.5
INPUT STARTING Lr
?10 r300 T10
L
T
E(M)
10 1.000
13.28
20 1,000 28.53
30 1.000
43.50
40 1.000 57.60
50 1.000
71.10
60 1.000
84.42
70 1.000
97.86
80 1.000 111.56
90 1.000 125.59
100 1.000 139.96
H O 1.000 154.62
120 1.000 169.52
130 1.000 184.60
140
.999 199.80
150
.999 215.06
160
.999 230.33
170
.999 245.56
180
.998 260.71
190
.998 275.73
200
.998 290.60
210
.997 305.29
220
.997 319.77
230
.996 334.02
240
.995 348.04
250
.995 361.79
260
.994 375.29
270
.993 388.51
280
.992 401.45
290
.991 414.12
300
.990 426.51
APPENDIX V
TABLE R
JtSTOP* OY
ENDING Lr JUMP SIZE
E(C)
10.16
21.81
33.26
44.04
54.37
64.56
74.83
85.30
96.*04
107.02
118.23
129.62
141.15
152.78
164.45
176.12
187.77
199.35
210,84
222.21
233,44
244.51
255.41
266.13
276.65
286.96
297.07
306.97
316»66
326.13
E(F)
12.62
26.56
40.33
53.47
66.20
78.82
91.54
104.49
117.71
131.17
144.87
158,75
172.76
186.87
201.02
215.17
229.29
243.33
257.28
271.10
284.77
298,27
3 1 1 ;58
324.69
337.59
350.27
362.73
374.95
386.94
398.70
E(S)
10.17
20.96
31.53
41.45
50.95
60.31
69.69
79.17
88.78
98.48
108.27
118.10
127.92
137.72
147.45
157.09
166.62
176.00
185.24
194.31
203.20
211.90
220,42
228.74
236.87
244.80
252.53
260,07
267.42
274.58
E(T)
14,28
30.18
43,80
55.03
64.79
73.84
82.61
91.33
100.08
108.89
117.72
126.56
135.35
144.08
152.69
161.17
169.48
177.61
185.55
193.26
200.76
208.03
215.07
221.88
228.45
234.80
240.92
246.83
252.52
258.00
MSE(M)
172
697
1607
288.7
4530
6540
8922
11674
14788
18244
22018
26080
30393
34922
39627
44471
49415
54424
59464
64502
69509
74460
79331
84101
88753
93272
97644
101860
105912
109793
MSE(C)
94
368
844
1523
2407
3495
4786
6271
7942
9783
11777
13907
16152
18491
20903
23370
25870
28387
30905
33407
35882
38317
40703
43032
45296
47491
49613
51660
53629
55520
MSE(F)
148
618
1426
2540
3961
5706
7795
10241
13047
16207
19705
23518
27617
31969
36539
41289
46183
51184
56257
61369
66490
71591
76647
81635
86536
91332
96009
100553
104956
109209
MSE(S)
83
332
737
1285
1967
2773
3695
4723
5845
7048
8319
9644
11010
12405
13819
15241
16663
18076
19476
20857
22216 '
23549
24857
26137
27391
28619
29823
31005
32168
33315
MSE(T)
203
583
997
1462
2005
2632
3331
4084
4870
5667
6459
7229
7966
8663
9312
9913
10464
10967
11426
11846
12232
12592
12932
13262
13588
13919
14263
14628
15023
15455
WHAT IS N?
?3
UHAT IS THE MIDDLE DILUTION?
7.00880
WHAT IS THE DILUTION FACTOR?
710.0
INPUT STARTING L, ENDING Lr JUMP SIZE
710,300,10
L
T
E(M)
E(C)
E(F)
E(S)
10 1.000
13.53
10.34
12.79
10.21
20 1.000
28.99
21.13
22.17
26.90
30 1.000
43.86
33.54
40.61
31.58
40 1.000
57.74
41.34
44.15
53.61
50 1.000
71.11
54.38
66.25
50.75
60 1.000
84.44
64.57
78.88
60.12
70 1.000
98.01
74.94
91.72
69.58
80 1.000 111.94
85.60 104.85
79.19
90 1.000 126.27 . 96.55 118.28
88.96
100 1.000 140.95 107.78 131.98
98.84
H O 1.000 155.92 119.23 145.91 108.80
120 1.000 171.13 130.86 160.02 118.78
130 1.000 186.50 142.61 174.26 128.76
.999 201.96 154.43 188.57 138.69
140
150
.999 217.46 166.28 202.90 148.54
160
.999 232.93 178.11 217.21 158.28
170
.999 248.32 189.88 231.46 167.89
180
.998 263.60 201.57 245,62 177.34
190
.998 278.73 213.13 259.66 186.62
200
.998 293.67 224.56 273.55 195.72
210
.997 308.40 235.82 287.26 204.63
220
.997 322.89 246.90 300,78 213.34
230
.996 337.14 257.79 314.10 221.85
240
.995 351.11 268.48 327.20 230.16
250
.995 364.81 278.96 340.08 238.26
260
.994 378.23 289.22 352.71 246.16
270
.993 391.36 299.26 365,12 253.85
280
.992 404.20 309.07 377.28 261.35
290
.991 416.75 318.67 389.20 268.65
300
.990 429.01 328.04 400.87 275.76
*STOP* or
>
%
H
CO
g M
M
kJ
SI
E(T)
14.62
30.50
43.52
54.08
63.33
72.04
80.62
89.23
97.92
106.68
115.46
124.24
132.95
141.57
150.05
158.36
166.49
174.41
182.11
189.58
196.81
203.80
210.55
217.05
223.31
229.35
235.15
240.73
246.09
251.25
MSE(M)
179
735
1686
3014
' 4720
6812
9297
12171
15423
19030
22964
27189
31668
36361
41225
46222
51312
56456
61622
66776
71889
76935
81892
86740
91461
96042
100470
104736
108834
112758
MSE(C)
97
387
886
1596
2518
3655
5001
6551
8291
10207
12279
14486
16808
19223
21708
24244
26810
29388
31961
34516
37038
39517
41943
44308
46607
48835
50988
53064
55063
56984
MSE(F)
154
652
1491
2640
4109
5923
8105
10667
13612
16928
20597
24591
28877
33418
38176
43110
48181
53351
58583
63845
69103
74332
79504
84598
89595
94477
99232
103848
108315
112627
MSE(S)
87
346
765
1333
2039
2876
3835
4904
6071
7321
8640
10014
11429
12873
14334
15802
17268
18725
20166
21587
22985
24357
25701
27018
28308
29572
30813
32031
33230
34413
MSE(T)
206
570
957
1403
1936
2555
3245
3983
4744
5508
6257
6977
7657
8292
8877
9412
9898
10339
10742
11111
11454
11780
12097
12413
12738
13080
13448
13850
14295
14789
103
APPENDIX VI
C
C
C
C
C
C
C
C
C
C
PROGRAM TO GENERATE EXACT SAMPLING DISTRIBUTIONS FOR LAMBDA
FROM I TO 100 FOR N TUBES AT EACH OF 3 DILUTIONS FOR THE
FINITE POPULATION MODEL (I.E., N(Z1+Z2+Z3).LE.l IS NECESSARY)
A(*,J) IS THE BINOMIAL COEFFCIIENT FOR X(J) ITEMS * AT A TIME
D(A) IS THE BINOMIAL COEFFCIEINT FOR N ITEMS * AT A TIME
X IS THE NUMBER OF FERTILE TUBES
Z IS THE DILUTION
AMAT SAVES THE VALUES FOR OUTPUT A PAGE AT A TIME (10 LAMBDA
VALUES BY 64 TUBE COMBINATIONS)
DIMENSION A(0:10,3),D(O=IO),X(3),Z(3),AMAT(66,40)
OUTPUT 'INPUT THE 3 Z VALUES'
INPUT Z(1),Z(2),Z(3)
OUTPUT 'INPUT N (// TUBES PER Z)'
INPUT N
S=N* (Z (I)+Z (2)+Z (3) )
D(O)=I.0
DO 5 I=IsN-I
5 D(I+1)=D(I)*(N-I)/(1+1)
L=I
9 DO 60 IM=I,40,4
IF (L>100) GO TO 99
AMAT(IsIM)=L
C=O
KOUNT=Z
DO 40 Xl=OsN
DO 40 XZ=OsN
DO 40 X3=0,N
X(I)=Xl
X(Z)=XZ
X(3)=X3
SX=X1+X2+X3
F=O
IF (SX>L) GO TO 35
CN = D(X1)*D(X2)*D(X3)
DO 10 1=1,3
A(OsI)=I.0
IF (X(I).EQ.O) GO TO 10
A(OsI)=I.0
DO 10 J=OsX(I)-I
A(J+1,1)=A(JsI) * (X(I)-J)/(J+l)
10 CONTINUE
104
APPENDIX VI (CONTINUED)
30
35
40
60
200
70
. 300
400
99
DO 30 I=O5X(I)
DO 30 I=O5X(Z)
DO 30 I=O5X(S)
B = (-!)**(SX-I-J-K)*A(I,I)*A(J ,2)*A(K,3)
C
*(1-S+I*Z(1)+J*Z(2)+K*Z(3))**L
F=F+B
F=CNaF
AMAT(KOUNT5IM)=Xl
AMAT(KOUNT,IM+1)=X2
AMAT (KOUNT,IMf 2) =X3
AMAT(KOUNT,IM+3)=F
KOUNT=KOUNT+l
C=C+F
CONTINUE
AMAT(66,IM)=C
L=L+1
CONTINUE
WRITE (108,200) AMAT(I5I),AMAT(I5S),AMAT(I5O)5
C AMAT(1,13) , A M A T d 517) ,AMAT(I5Zl) ,AMAT(I5ZS) ,
C AMAT(1529) 5AMAT(1533),AMAT(1537)
FORMAT (T75I3,9(9X5I3))
DO 70 1=2,65
WRITE (108,300)
(AMAT(I5J),1=1,40)
FORMAT (X510(2X53I1,F7.4))
WRITE (108,400) AMAT(66,1),AMAT(66,5),AMAT(66,9),
C AMAT(66,13)5AMAT(66,17)5AMAT(66,21)5AMAT(66,25),
C AMAT(66,29),AMAT(66,33),AMAT(66,37)
FORMAT (T7,F6.4,9(6X,F6.4))
GO TO 9
END
105
APPENDIX VII
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
PROGRAM TO GIVE THE POSTERIOR FUNCTION OF LAMBDA GIVEN
X1,X2,X3 UNDER AN IMPROPER UNIFORM PRIOR
THE PROGRAM CALCULATES THE EXACT SUM OF THE POSTERIOR
PROBABILITIES FROM (X1+X2+X3) TO INFINITY AND THE EXACT MEAN
OF THE POSTERIOR DISTRIBUTION
FOR EACH VALUE OF LAMBDA, THE PROGRAM OUTPUTS THE PROBABILITY,
THE CUMULATIVE PROBABILITY AND THE CUMULATIVE RELATIVE
PROBABILITY FOR QUICK DETERMINATION OF CONFIDENCE INTERVAL
ENDPOINTS FOR ANY ALPHA
THIS PROGRAM IS FOR THE FINITE POPULATION MODEL **********
A(*,J) IS THE BINOMIAL COEFFICIENT FOR X(J) ITEMS * AT A TIME
D(*) IS THE BINOMIAL COEFFICIENT FOR N ITEMS * AT A TIME
X IS THE NUMBER OF FERTILE TUBES OBSERVED
Z IS THE DILUTION
2
150
C
C
C
C
DIMENSION A(0:10,3),D(OrlO),X(3),Z(3)
WRITE (1,150)
FORMAT (IHl)
THE PRECEDING 2 LINES START THE OUTPUT FOR EACH INPUT X1,X2,X3
ON A NEW PAGE
5
REAL M
OUTPUT 'INPUT THE 3 Z VALUES (DILUTIONS)'
INPUT Z(1),Z(2),Z(3)
OUTPUT
'INPUT N (// TUBES PER Z)'
INPUT N
OUTPUT 'INPUT THE 3 X VALUES (# FERTILE RESPONSES PER Z)'
INPUT X(1),X(2),X(3)
S=N*(Z(1)+Z(2)+Z(3))
SX=X(1)+X(2)+X(3)
D(O)=I.0
DO 5 1=0,N-I
D(I+1)=D(I)* (N-I)/(1+1)
CN=D(X(I))*D(X(2))*D(X(3))
DO 10 1=1,3
A(O1I)=I.0
IF (X(I).EQ.O) GO TO 10
DO 10 J=I1X(I)
A(J1I)=I-O
106
APPENDIX VII (CONTINUED)
DO 10 K=O5J-I
P=(X(I)-K)/(K+l)
A(J5I)=A(J5I)AP
10 CONTINUE
OUTPUT 'TOTAL L-FN IS'
T=O
DO 20 I=O5X(I)
DO 20 J=0,X(2)
. DO 20 K=O5X(J)
B=(-1)AA(SX-I-J-K)AA(I5D *A(J ,2)AA(K5J)
C *(1-S+I*Z(1)+J*Z(2)+K*Z(J))*ASX
C /(S-I*Z(I)-JAZ(2)-RAZ(J))
20 T=TJB
T=CNaT
OUTPUT T
OUTPUT 'MEAN IS'
M=O
DO 25 I=O5X(I)
DO 25 J=O5X(Z)
DO 25 K=O5X(J)
B=(-1)A*(sx-I-J-K)*A(I,I)*A(J ,2)AA(K5J)
C *(1-S+I*Z(I)+JAZ(2)+K*Z(J))**(SX-1)
C /(S-I*Z(1)-J*Z(2)-K*Z(J))**2
25 M=MJB
• M=CN*M
M=SXJM/T
OUTPUT M
■ OUTPUT 'LAMBDA
L-FN
CUM L-FN
C=O
•■ R=O
'
■ DO 40 L=SX5400
F=O
DO JO I=O5X(I)
DO JO J=O5X(Z)
DO JO K=O5X(J)
B=(-1)AA(SX-I-J-K)*A(I51)*A(J,2)*A(K,J)
C *(1-SJI*Z(1)JJ*Z(2)JK*Z(J))*AL
JO F=FJB
F=CNAF
C=CJF
R=C/T
WRITE (108,100) L 5F 5C 5R
CUM REL
107
APPENDIX VII (CONTINUED)
40
100
200
IF (R>0.995) GO TO 200
CONTINUE
FORMAT (I4,5X,E12.5,4X,E12.5,4X,E12.5)
GO TO 2
END
108
APPENDIX VIII
To verify statements (i) and (ii) of section 5.2, one needs the
following fact.
(I)
L
L
j_(l_x)_ dx = £ (^) (1-x)k + ln(x) + c
x
k=l k
proof:
y i W
dx =
-/(1-x)^ ^dx + /-—
.L-I
X-^-- dx
-/(1-x)^ ^dx - /(1-x)^ ^dx -...+ / ~ ~ ~ " dx
-/(1-x)^ ^dx - /(1-x)^ ^dx -...+ / ( " I ) (1-x)dx
-/(1-x)^ ^dx - /(1-x)^ ^dx
/(1-x)dx + /~dx
- /dx
=
2
(1-x)^ + ln(x) - x + C1
Jj
= Z (y1) (l-x)k + ln(x) + c
k=l k
In the serial dilution problem with z^=(l/d)^ ^z1, the z's will
be equally spaced on a log scale.
Letting F(z) = P(sterile response
at dilution z), which is F(z)=l-(l-z)^ for the finite population
model and F(z) = 1-e
for the
infinite population model, the serial
dilution problem may be illustrated as follows.
ln(z)
ln(d)
109
APPENDIX VIII
(CONTINUED)
The Spearman (1908) estimate for the mean of F(x) is
(2)
y =
I n ( Z 1)
+ [ln(d)][.5-EX/n],
and the exact mean is given by
(3)
= /0 [ln(z)]d[F(z)]
= /q [ln(z)]X(l-z)^ 1dz
= lim{/^[ln(z)]X(l-z)^ ^dz)
a-^-0 3
X
]dz}
lim{-[ln(z)] [I-Zix I1 + f1
a+0
a
*
lim{-[ln(z)] [l-z]^| ^
a-)-0
X
(l-z)k |^ + ln(z) |^
by (I)
■
= -Y-In(X)
, for large X
Combining (2) and (3), one obtains
^ = e -y-ln(z1)+[ln(d)] [(ZX/n)-.5]
the same value obtained by Johnson and Brown (1971) for the infinite
population model. This verifies statement (i) of section 5.2.
To verify statement (ii) of section 5.2, consider a single
dilution z of the finite population model and let X=EV^, where V^=I
if the Ith sample is fertile and V.=0 otherwise. Note that V is a
1
X
1
binomial random variable with n=l and p=P(V^=I)=1-(1-z) . If there are
n samples at dilution z, E(X) = E(EV^) = n[l-(l-z) ].
no.
A P P E N D I X VIII
(CONTINUED)
For several dilutions,
E(X.X.) = E(EV1ZV1) = EEE(V1V 1)
ij
i 1J 3
ij
1 J
= Hi^CP(Vi=I) ■+ P(Vj=I) + P (Vi=O and Vj=O) -l]
= n . n . C l - d - z J ^ - d - z j V d - z -z )^]
1 J
1
J
1 J
and one observes that
COV(X1X 1) = E(X1X 1) - E(X1)E(X1)
1 J
1
1 J
1
J
= n.n.Cl-d-z. )^-(l-z .)^+d-z -z )^]
1 J
1
J
- HiCl-(I-Zi)
1
J
[I-(1-zJ
= n.n.C(I-Z1- Z 1)^-C(I-Z)(I-Z)]^).
1 J
1
J
1
J
Since (I-Z1)(I-Z1) = l-z.-z.+z.z. > l-z.-z., the correlation
1
J
1
J
1 J
between Xi and Xj is, as expected, negative.
1
J
Since z^zy is very close
to zero when Zi and Zj are small, however* Xi and Xj are essentially
independent whenever only a small portion of the population is sampled.
This approximate independence allows one to mimic Johnson and
Brown's derivation of a variance for y. Letting p=X/n for each
dilution,
p = ln(zi)+Cln(d) ]'C .5-Ep]
and
Var(p) = Cln(d)I2Var(Ep)
= Cln(d)]2EVar(p)
, if the p's are independent
= Cln(d)]2Ep(1-p)/n
=
^-Ep(l-p)ln(d)
^ M d l / 0 [ F(z)][1_F (,z)]dCln(z)]
n
I
Ill
APPENDIX VIII (CONTINUED)
^ ^ - / ^ [ l - ( l - z ) X][ (l-z)X]^dz
n
U
z
2A
ln(d) J rI r O ^ l ]dz _
n .
z
]dz}
2A
i£? i{^ 1(E) a 'z>k|o +
X
I
2A
" UEl(|) (I-Z)k I2, - I-(Z) IJ)
I
^-kiPE)+ k:i(E)]
ln(d)
[-In(A) + ln(2A)]
, for large A
[ln(d)][ln(2)]
Finally, note that when A is large and the portion of the
population sampled is small,
E(%) = E(e-7-0)
E [e-(Y+y)-(y-y)]
AE[e"(°-y)]
-
i A[l + E(P-U) +
AU +
[ln(d)][ln(2)]
}
, since y is unbiased
This gives [2n]/{2n+[ln(d)][ln(2)]} as the appropriate multiplicative
bias correction for X and verifies statement (ii) of section 5.2.
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