l'v\
\
ANNOTATED COMPUTER OUPUT FOR SPLIT PLOT DESIGN: GENSTAT ANOVA
by
W.T. Federer, Z.D. Feng, and N.J. Miles-McDermott
Mathematical Sciences Institute, Cornell University, Ithaca, NY
ABSTRACT
In order to provide an understanding of covariance
split plot
without
designs,
the covariate.
performed.
the
.e
analyses
conducted on
an analysis
with
the
a data set
covariate is
The third example has a different experiment design for
whole plots and
within
Then
are first
analyses for
a whole
the covariate is constant for all split plots
plot.
Computer packages may only give portions of
the analysis correctly.
to obtain a portion
Some packages require two procedural calls
of the correct results.
GENSTAT requires only
one procedural call.
INTRODUCTION
This is
part of a continuing
project that produces annotated
computer output for the analysis of balanced split plot experiments
with covariates. The complete project will involve processing three
examples
on SAS/GLM,
SYSTAT/MGLH.
BMDP/2V,
SPSS-X/MANOVA,
GENSTAT/ANOVA, and
Only univariate results are considered.
We show here
the results from GENSTAT ANOVA.
For
Example 1, the
data are artificial
and were constructed
for ease of computation; the experiment design
for the whole plots
is a randomized
plot treatments are
complete block
and the split
randomly allocated to the split plot experimental units within each
whole
plot.
Example 2 is
the same
as
Example 1
except
that a
covariate
varies from
split plot to
split
plot.
The
data
for
Example 3 come from an experiment wherein the whole plot treatments
were laid out in a
treatments were
units
completely randomized design and the split plot
randomly
within each
from whole
allotted to
whole plot.
plot to whole
the split plot experimental
The value
of the covariate varies
plot but is constant for all split plots
within a whole plot treatment.
We
present
the
elementary
steps.
computational
Simple
hypothetical data are used for the first two examples so that it is
easy to
number
provide all
is obtained.
computations.
The
detailed
Some
third
computations to illustrate how each
readers may
wish to skip
example comes
from
Winer
the detailed
(1971).
The
detailed computations are given in his book (p. 803).
Data SP-1
Split plot data with whole plots arranged in
randomized complete block design
(hypothetical data)
Block
1
2
3
Total
Whole plot treatment
W2
W1
split plot treatment
split plot treatment
Total
s3
s2
s1
s2
s4
s1
s3
s4
3
6
6
15
4
10
10
24
7
1
4
12
6
11
4
21
20
28
24
72
2
3
8
10
21
2
8
8
18
1
2
9
12
14
18
13
45
Total
20
36
40
96
e
Total and Means
1
S(split plot)
(6 observations)
Total
Mean
36
6
42
7
8
24
4
Grand Total
168
66
11
Grand Mean
7
2
64
8
3
64
Model:
W2
Y, , k
l.J
=
'j.L
96
+
p,
J
1-L
= mean
p.
= effect of block j
J
0 ..
l.J
Eijk
--
W(whole plots)
(12 observations)
Total
Mean
W1
72
6
Blocks
(8 observations)
Total
Mean
40
5
=
=
8
+ T,l. + 0 l.J
.. + ak + (aT) , k +
l.
€, , k
l.J
T, = effect of whole plot i
l.
ak = effect of split plot k
(aT) ik = effect of interaction of
whole plot i and split plot k
error (a)
error (b)
Analysis of Variance
Source
B (Blocks)
w (whole plot treatments)
BxW (error (a))
s (split plot treatments)
sxw (interaction of S and W)
(**} SxB:W (error (b))
Total (Corrected for mean)
Mean
Total (Uncorrected for mean}
=
=
=
=
=
=
=
=
=
(*)
R(p 'j.L 1 T 1 a,aT)
R(T 'j.L,p,a,aT)
R(o 'j.L 1 p 1 T,a 1 aT)
R(a 'j.L,p,T,aT)
R(aTI!-L,a,T,p)
R(Eiu.a TaT o)
R(p,T,o,a,aT,EI!-L)
R(l.l)
R(u o.T o a,aT E)
df
2
1
2
3
3
12
23
1
24
ss
48
24
16
156
84
112
440
1176
1616
(*)Notation follows that of Searle(1971); since the design is balanced,
R(PIJ.l,T,a,aT) = R(pjJ.l), etc. The simpler notation is used later.
(**) SxB:W means SxB within W.
Calculations of sums of squares:
N
=
2•3•4
=
24
R(p,T,o,a,aT,eiJ.l)
y = 7
=
1616- 1176
=
440
3
2 +64 2 +64 2 )
8
- - 1176
R(pl~)
=
R(~,p)
- R(~)
= -( 40
R(TI~)
=
R(~ 1 T)
- R(~)
= -(72 12+96
2
R(o!~ 1 p 1 T) = R(o,~ 1 p 1 T)
=
R(aTI~,a,T)
2)
- - 1176
R(aT 1 ~ 1 a 1 T)
= 1440 R(el~,p,o,a,T,aT)
=
1224 - 1176
48
= 1200- 1176 = 24
- R(~ 1 P) - R(T,~) + R(~)
1264 - 1224 - 1200 + 1176
=
=
=
16
- R(~,a) - R(~ 1 T) + R(~)
1332 - 1200 + 1176 = 84
=
R(e,~,a,p,o,T,aT)- R(~ 1 p,T,5)- R(~,a,T,aT)
=
1616 - 1264 - 1440 + 1200
=
+
112
Data 5P-2
Data 5P-2:
Data 5P-1 with the following covariate Z
which varies with split plot
Covariate (Z)
51
52
whole plot
W1
Total
53 54
B1
1
2
1
2
6
2
0
2
4
8
B2
2
2
0
4
8
4
1
3
4
12
B3
3
5
2
0
10
3
2
4
7
16
Total
6
9
3
6
24
9
3
9
15
36
4
51
52
W2
53 54
Total
R(T,~)
Totals and Means
W (whole plot)
(12 observations)
Total
Mean
24
2.0
1
2
36
3.0
blocks
(8 observations)
Total
Mean
1
14
14/8
2
20
20/8
3
26
26/8
Grand
2.5
Total 60
s (split plot)
(6 observations)
Total
Mean
15
2. 5
12
2. 0
12
2. 0
21
3. 5
1
2
3
4
. . +ak+(aT) J..k+f3 1 (z l.J.
.. -z • • • )+f3 2 (z l.J
.. k-z l.J.
.. )+c..l.J'k
Model: Yijk = 11 +p.+T.+o
J
J.
l.J
P· = effect of jth block
J
ak = effect of kth split plot
/32 = split plot regression slope
T. = effect of ith whole plot
J.
/31 = whole plot regression slope
0 .. = error a
c.ijk = error b
l.J
Table of sum of squares and products
Source
B
.e
yy
48
24
16
156
84
112
1176
1616
df
2
1
2
3
3
12
1
24
w
BxW (error a)
s
sxw
SxB:W (error b)
Mean
Total
YZ
18
12
4
33
33
17
420
537
zz
9
6
1
9
21
20
150
216
yy column is the same as in SP-1, zz column is computed in the same
fashion.
Thus, only computations for YZ column are illustrated.
2
Totalyz
Meanyz
= NY
Byz
e
3
= };
4
};
}; y .. k. z .. k
i=1 j=1 k=1 l.J
l.J
= 3(1)+ 6(2)+···+ 14(4)+ 18(4)+ 13(7) = 537
...z ...
=
168•60
= 420
24
3
2
4
2
4
};
( };
};
y .. k) ( };
};
j=1
i=1 k=1
l.J
i=1 k=1
z.l.J'k)
-420
40(14)+64(20)+64(26)
2•4
= 438
2
3
4
3
4
}; ( };
};
};
y .. k) ( };
z. 'k)
j=1 k=1 l.J
i=1 j=1 k=1 l.J
WYZ =
3(4)
5
-
420 = 432
8
420 = 18
-
420 = 12
-
420
2
3
4
2:
2:
( 2:
4
y, 'k) ( 2:
lJ
= i=1 j=1 k=1
z. 'k)
lJ
k=1
- 438 - 432 + 420
4
= 454 - 438 - 432 + 420 = 4
3
2
2: Y. 'k) ( 2:
i=1
1=1 j=1 lJ
2
( . 2:
4
2:
k=1
- 420 = 453 - 420 = 33
2(3)
2
4
3
2:
2:
( 2:
i=1 k=1
3
y, 'k) ( 2: z. 'k)
j=1 lJ
j=1 lJ
- 453 - 432 + 420
3
= 498 - 453 - 432 + 420 = 33
SxB:Wyz:
2
3
4
2:
2:
2:
i=1 j=1 k=1
Y .. kz. 'k- 454 - 498 + 432
lJ
lJ
= 537 - 454 - 498 + 432 = 17
Analysis of Variance and Covariance
Source
B (block)
w (whole plot treatment)
Regression (a)
= R(p ,J..L,T)
= R(T ].1. 1 p,{3 1 )
= R(f3 1 IJ..L,p,T)
BxW (error (a))
s
s
Regression (b)
16.0
= R(oi]J.,p ,T ,{3 1 )
1
0.0
= R(a!J..L,p,T,aT,{3 2 )
3
84.243
and W) = R(aTIJ..L,p,T,a,{3 2 )
= R(f3 2 !J..L,p,T,a,aT)
3
37.474
1
14.450
11
97.550
= R(ci]J.,p,a,T,aT,{3 2 )
SxB: w (error (b))
ss
48
3.4286
1
(split plot treatment)
sxw (interaction of
df
2
1
A
{3 1 = BxWYZ I BxWZZ = 411 = 4
A
{3 2 = SxB:Wyz 1 sxB:Wzz = 17120 = 0.85
6
The SS's adjusted by regression on Z are illustrated below:
R(PI~>
=
48, remains same since it is not of interest to adjust for Z
on the blocks.
=
(12 + 4) 2
-
(24 + 16) -
6
=
--
40 - 256 - 0
=
+ 1
=
40 - 256
=
3.4286
7
3.4286
7
=
42
=
16
1
=
(156 + 112) -
=
268 - 86.207
(33+17}- 2
9+20
=
= 84 + 112 -
Note:
181.793
<33 + 17 >
21+20
2
R(a,~l~,~ 2 ) and R(aT,~I~,a,T,~ 2 )
later use.
7
= 196 -
60.976 = 135.024
are intermediate steps for
sxB:Wzz
R(al~,p,T,aT,~ 2 )
=
=
=
14.450
20
R(a,cl~,p,T,aT,~ 2 )
=
135.024 - 97.55
- SS error b
=
181.793 - 97.55
=
84.243
37.474
Data SP-3
Split plot data with plots arranged in a completely
randomized design and a covariate z that is constant
within the whole plot. (Winer, 1971, p. 803)
whole plot
Subject
Split plots
B1
B2
y
z
y
Total
y
A1
1
2
3
4
10
15
20
12
8
12
14
6
3
5
8
2
18
27
34
18
5
6
7
8
Total
Mean
15
25
20
15
132
16.5
10
20
15
10
95
11.9
1
8
10
2
39
4.88
25
45
35
25
227
A2
T.
~
ak
=
=
A effect (whole plot)
B effect (split plot)
0 .•
l.J
~1
=
=
error (a)
c. 'k = error (b)
l.J
whole plot regression slope
e
8
4lt
Analysis of variance and covariance
Source
A (whole plot)
Regression
Error (a)
B (split plot)
AxB (interaction)
Error (b)
ss
df
1
1
5
1
1
6
R(T IJ.L,/31)
= R(J3 1 IJ.L,T)
= R(o I11 ,T,/3 1 )
= R(a!J.L,T,aT)
= R(TaiJ.L,T,a)
= RCc.lu,T,a,Ta)
=
44.492
166.577
61.298
85.563
0.563
6.375
Table of SS and products
Symbol
w
E(a)
s
ws
E(b)
&1
y2
ZY
12.38
163.00
0
0
0
68.06
227.88
85.563
0.563
6.375
z2
2.25
159.50
0
0
0
= 163.00 = 1.02
159.50
Since the computations are illustrated in Winer
have omitted them here.
(1971,
p.
803-5)
we
References
Federer. W.T.
(1955), Experimental Design, Theory
The Macmillan Co., New York, Chapter 16.
and Application.
Federer, W.T., and Henderson, H.V.
(1979), Covariance Analysis of
Designed Experiments X Statistical Packages: An Update, Proc.,
Comp. Sci. and Stat.: 12th Ann. Sym. on the Interface.
GENSTAT
b General
Statistical
Rothamsted Experimental Station.
Searle, S.R.,
Program
Release
(1983),
(1971), Linear Models, Wiley, N.Y., 532pp.
Searle, S.R., Hudson, G.F.S., and Federer, W.T.
(1985), Annotated
Computer Output for Covariance-Text, BU-780-M, Biometrics Unit Mimeo
Ser., Cornell University, Ithaca, NY.
Winer, B.J.,
(1971), Statistical Principles in Experimental Design,
McGraw-Hill Book Company, New York: 907pp.
9
SP-1 and SP-2:
Control Language
control Language is typed in upper case and comments are in bold face.
REFE' SPLIT ~ file name
'UNITS' $ 24 ~number of observations
'FACTOR' BLOCKS $3
: PLOTS $2
} define levels of each factor
1
: SUBPLOTS $4
READ/FORM=P, PRIN=D' BLOCKS, PLOTS, SUBPLOTS, X, Y
'RUN'
1
1
1
1
1
1
1
1
1
1
2
1
2
3
4
1
1
2
1
2
2
~
Input variables
3
4
7
6
3
1 2 2 0 2
1 2 3 2 1
1 2 4 4 14
2 1 1 2 6
2 1 2 2 10
2 1 3 0 1
2 1 4 4 11
2
2
2
2
3
3
3
3
2
2
2
2
1
1
1
1
1
2
3
4
1
2
3
4
4
1
3
4
3
5
2
0
8
8
2
18
6
10
4
4
3 2 1 3 10
3 2 2 2 8
3 2 3 4 9
3 2 4 7 13
1 EOD 1
~ tell GENSTAT that data flow ends
'BLOCKS' BLOCKS/PLOTS/SUBPLOTS ~define error terms (strata) of a
factorial model
'TREATMENTS' PLOTS*SUBPLOTS ~ treatment terms of a factorial model
1 COVARIATES 1
X ~covariate
'ANOVA' Y ~ invokes analysis of variance on Y variable
'RUN'
'STOP'
10
SP-3:
Control Language
OK, SLIST SPLIT3.0GEN
'REFE' SPLIT3
'UNITS' $ 16
'FACTOR' PLOT $2
: SUBPLOT $2
: SUBJECT $8
'READ/FORM=P, PRIN=D' SUBJECT, PLOT, SUBPLOT, Y, Z
'RUN'
1 1 1 10 3
1 1 2 8 3
2 1 1 15 5
2 1 2 12 5
3 1 1 20 8
3 1 2 14 8
4 1 1 12 2
4 1 2 6 2
5 2 1 15
5 2 2 10
6 2 1 25
6 2 2 20
7 2 1 20
7 2 2 15
8 2 1 15
8 2 2 10
1
1
8
8
10
10
2
2
'EOD'
'BLOCKS' PLOT/SUBJECT/SUBPLOT
'TREATMENT' PLOT*SUBPLOT
'COVARIATES' Z
'ANOVA' Y
'RUN'
'STOP'
11
SP-1:
SP-2:
Split plots with whole plots arranged in RCB design
Split plots with whole plots arranged in RCB with a
covariate with split plot
Analysis of variance table on covariate Z
***** ANALYSIS OF VARIANCE *****
VARIATE: Z
DF
ss
SS%
MS
BLOCKS STRATUM
2
9.000
13.64
4.500
BLOCKS.PLOTS STRATUM
PLOTS
RESIDUAL
TOTAL
1
2
3
6.000
1.000
7.000
9.09
10.61
6.000
0.500
2.333
12.000
1. 52
BLOCKS.PLOTS.SUBPLOTS STRATUM
SUBPLOTS
3
9.000
PLOTS.SUBPLOTS
3
21.000
RESIDUAL
12
20.000
TOTAL
50.000
18
13.64
31.82
30.30
75.76
3.000
7.000
1. 667
2.778
1. 800
4.200
SOURCE OF VARIATION
GRAND TOTAL
23
66.000
GRAND MEAN
2.50
TOTAL NUMBER OF OBSERVATIONS
100.00
24
12
VR
ANOVA for Y variable without covariate Z
***** ANALYSIS OF VARIANCE *****
VARIATE: Y
SOURCE OF VARIATION
ss
DF
MS
SS%
VR
(F-statistics)
10.91
BLOCKS STRATUM
2 R(pi~,T,a,aT)
48.000
24.000
BLOCKS.PLOTS STRATUM
PLOTS
1 R(TI~,p,a,aT)
24.000
5.45
24.000
3.000
16.000
8.000
40.000
13.333
3.64
RESIDUAL
TOTAL
2 R(ol~,p,T,a,aT)
3
BLOCKS.PLOTS.SUBPLOTS STRATUM
SUBPLOTS
3 R(al~,p,T,aT,O)
PLOTS.SUBPLOTS
RESIDUAL
TOTAL
GRAND TOTAL
156.000
52.000
84.000
28.000
12 R(EI~,a,T,aT,p,o)112.000
9.333
352.000
18
19.556
23
440.000
GRAND MEAN
7.00
TOTAL NUMBER OF OBSERVATIONS
24
13
9.09
35.45
5.571
19.09
3.000
25.45
80.00
100.00
=
24.00
8.00
~****
ANALYSIS OF VARIANcr
(ADJUSTED FOR COVARIATE)
*****
VARIATE: Y
SOURcr OF VARIATION
ss
DF
MS
SS'/.
VR
OOV EF
= covariance
efficiency factor
(F-statislic)
BLOCKS STRATUM
COVARIATE
RESIDUAL
TOTAL
1
(*) pooled {36.000
1
12.000
18.000
2 R(p!Jl,T)
BLOCKS.PLOTS STRATUM
PLOTS
COVARIATE
3. 429
16.000
R(T j,J, p./1 1 )
R{l1l j,t,p,T)
o.ooo
R(ol~.p.T,/1 1 )
RESIDUAL
TOTAL
3
BLOCKS.PLOTS.SUBPLOTS STRATUM
SUBPLOTS
PLOTS.SUBPLOTS
COVARIATE
19.129
RESIDUAL
TOTAL
3 R(aljJ,p,T,aT,/12 )
3 R(aTijJ,p,T,a,Jl2 )
1 R{I12 11J,p,T,a,aT)
11 R(EhJ,p,a,T,ar,{}2 )
18
GRAND TOTAL
23
GRAND MEAN
TOTAL NUMBER OF OBSERVATIONS
(*)
301. 146
7.00
24
Block is just a nuisance factor; hence only
R(pj,t,T)
= 18
is used in Analysis of variance
81.243
37.174
14.150
97.550
233.717
8.18
2.73
10.91
36.000
12.000
21.000
0.78
3.64
0.00
4.12
3.129
16.000
0.000
6.176
19.15
8.52
3.28
22.17
53.12
28.081
12.491
11.150
3.000
2.000
0.113
3.166
1.409
1.629
8.868
12.984
0.870
0.711
1.052
68.14
NOTE:
low value of COY EF (usually close to unity)
indicates high correlation between treatment
and covariate e.g. COY EF for plots is .113;
indicating correlation between
wi
and Zi(Wl
= G,
w2 = s.
zl = 2,
z2 = 3)
14
-
-
-
***** COVARIANCE REGRESSIONS *****
SE
COEFFICIENT
COVARIATE (Z)
BLOCKS STRATUM
({3 0 is not of interest)
z
A
1.15
{30 = 2.0
Byz
18
= -- =
9
Bzz
BLOCKS.PLOTS STRATUM
"
{31 = 4
z
0
BLOCKS.PLOTS.SUBPLOTS STRATUM
A
z
0.666
{32 = 0.85
***** TABLES OF MEANS *****
(ADJUSTED FOR COVARIATE)
--
VARIATE: Y
GRAND MEAN
7.00
PLOTS
1
8.oo
SUBPLOTS
2
A
6.oo = Y2 •• - f3 1
1
2
3
6.00
7.43
4.43
cz 2 •• - z... >
-
= 8-4.0(3-2.5)=6
4
1o.15 =
Y.. 4 - 52 cz .. 4 -
z... >
!-
= 11-0.85( 2
~~)
= 11-.85 = 10.15
SUBPLOTS
PLOTS
1
2
3
4
1
7.00
9.15
6.85
9.00
2
5.00
5.70
2.00
11.3o =
- 2.5) -
15
Y2 • 4 -5 1 cz 2 •• 15
.85(~
z... >
-3) = 11.30
***** STANDARD ERRORS OF DIFFERENCES OF MEANS *****
TABLE
PLOTS
SUBPLOTS
REP
SED
12
0.000
6
1. 844
PLOTS
SUBPLOTS
3
2.718
***** STRATUM STANDARD ERRORS AND COEFFICIENTS OF VARIATION *****
STRATUM
DF
SE
CV%
BLOCKS
BLOCKS.PLOTS
BLOCKS.PLOTS.SUBPLOTS
1
1
11
1. 225
17.5
0.0
42.5
0.000
2.978
2.978 = unadjusted standard error
Jcov EF of Residual
=
..../9.333
..../1. 052
--
NOTE:
A
A
)
S.E (Y i .. - y..
l ••
A
I
)
=
0.0
W
r
=
=
2Eb
r(w)
(1 +
{
=
{ 2{97.55}
3(2)(11)
=
{ -r-
=
{ 2{97.55}
3 ( 11)
2Eb
Wzz I (w-1)
BxW
)}1/2
S 22 / (s-1)
SxB:W
)} 1/2
zz
=
A
S.E (Y.l• k- yi•k')
where
{ 2E
r(:)
A
S.E (Y •• k - yo o k
A
=
(1 +
zz
(1 +
<1 + 9L{4-1}
20
=
1. 844
(S zz + SxW zz )/w(s-1)
)}1/2
SxB:W
zz
(1 + {9+21}L{2} {3} >}1/2
20
no. of whole plot
no. of blocks = 3
Eb = error b
)}1/2 =
=
2
97.55/11
16
=
2.718
S = no. of split plot
Ea = error a = o
=
SP-3:
Split plots with whole plot arranged in CRD with a covariate
constant within whole plot
***** ANALYSIS OF VARIANCE *****
ANOVA on Covariate Z
VARIATE: Z
ss
SS%
2.250E
1. 595E
1.618E
0
2
2
1. 39
98.61
100.00
PLOT. SUBJECT. SUBPLOT STRATUM
SUBPLOT
O.OOOE
1
PLOT . SUBPLOT
1
O.OOOE
RESIDUAL
O.OOOE
6
TOTAL
8
O.OOOE
0
0
0
0
0.00
0.00
GRAND TOTAL
2
100.00
SOURCE OF VARIATION
PLOT.SUBJECT STRATUM
PLOT
RESIDUAL
TOTAL
e
DF
1
6
7
15
1.618E
GRAND MEAN
4.88
TOTAL NUMBER OF OBSERVATIONS
o.oo
o.oo
MS
VR
2.250E
2.658E
2.311E
0
1
1
0.085
O.OOOE
O.OOOE
O.OOOE
O.OOOE
0
0
0
0
16
Notice that ss in plot.subject.subplot stratum are all zeroes
since Z is constant within whole plot.
ANOVA on Y without covariate Z
***** ANALYSIS OF VARIANCE *****
VARIATE: y
SOURCE OF VARIATION
PLOT.SUBJECT STRATUM
PLOT
RESIDUAL
TOTAL
DF
ss
SS%
MS
VR
1
6
7
68.063
227.875
295.938
17.52
58.66
76.19
68.063
37.979
42.277
1. 792
85.563
0.563
6.375
92.500
22.03
0.14
1. 64
23.81
85.563
0.563
1.063
11.563
80.529
0.529
388.438
100.00
PLOT. SUBJECT. SUBPLOT STRATUM
SUBPLOT
1
PLOT.SUBPLOT
1
RESIDUAL
6
TOTAL
8
GRAND TOTAL
e
15
GRAND MEAN
14.19
TOTAL NUI-1:BER OF OBSERVATIONS
16
17
Covariance Analysis
*****
ANALYSIS OF VARIANCE
(ADJUSTED FOR COVARIATE)
*~***
VARIATE: Y
ss
SS"/.
MS
VR
COV EF
272.367
14.492
166.577
12.260
38.910
3.629
13.587
7
11.45
12.88
15.78
70.12
0.986
1 R(P 1 1~.r)
5 R(o!Jl,T,/}l)
44.492
166.577
61.298
1 R(alrl,T,crr)
85.563
1 R(Tal~.r.a)
0.563
6 R{Ej~.T.a,Ta)
6.375
92.500
8
22.03
0.11
1.61
23.81
85.563
0.563
1.063
11.563
80.529
0.529
364.867
93.93
OF
SOURCE OF VARIATION
PLOT. SUBJECf STRATUM
PLOT
COVARIATE
1
RESIDUAL
TOTAL
PLOT. SUBJECf. SUBPLOT STRATUM
SUBPLOT
PLOT. SUBPLOT
RF.SIDUAL
TOTAL
GRAND TOTAL
15
GRAND MEAN
TOTAL NUMBER OF OBSERVATIONS
*****
R(ri~.P 1 )
COVARIANCE REGRESSIONS
1.000
1.000
1.000
11.19
16
~**
COEFFICIENT
COVARIATE
3.098
SE
PLOT. SUBJECf STRATUM
A
z
/}1 =
1.02
0.277
18
e
e
-
***** TABLES OF MEANS *****
(ADJUSTED FOR COVARIATE)
VARIATE: Y
GRAND MEAN
PLOT
14.19
2
1
12.51
15.87 = Y2 •
•
-
~ 1 cz 2·
tJ
- z•• > = 16.25-1.02(5.25-4.88)
= 15.87
SUBPLOT
.e
1
16.50
2
SUBPLOT
PLOT
1
2
1
14.63
10.38
2
18.37
13.37 =
11.88
same as unadjusted mean
Y2 • 2 -~tJ 1 cz 2 •
-
z
• •
>
=
13.75-1.02(5.25-4.88)
=
13.37
***** STANDARD ERRORS OF DIFFERENCES OF MEANS *****
TABLE
PLOT
SUBPLOT
PLOT
SUBPLOT
REP
8
8
4
1.763
0.515
1.829
SED
EXCEPT WHEN COMPARING MEANS WITH SAME LEVEL(S) OF:
PLOT
0.731
***** STRATUM STANDARD ERRORS AND COEFFICIENTS OF VARIATION *****
STRATUM
DF
SE
CV%
PLOT.SUBJECT
5
2.476
17.5
PLOT. SUBJECT. SUBPLOT
6
1. 031
= v'l. 063
7.3
v'l. 000
=
[ residual MS of unadjusted )1/2
cov EF of adjusted residual
19