Symposium:
Advances in Dose-response
Methodology Applied to the
Science of Weed Control
• Presenters:
– Dr. Steven Seefeldt
– Dr. Bahman Shafii
– Dr. William Price
Historical development of
dose-response relationships
Steven Seefeldt, ARS, Fairbanks, AK
Bahman Shafii, Univ. of ID, Moscow, ID
William Price, Univ. of ID, Moscow, ID
Before the scientific method and
hypothesis testing
What did hunter gathers do?
One
Several
Dinner
Tasty
Tasty
Filling
What did hunter gathers do?
One
Several
Dinner
Tasty
Tasty
Filling
Tasty
Tasty
Stomach
ache
What did hunter gathers do?
One
Several
Dinner
Tasty
Tasty
Filling
Tasty
Tasty
Stomach
ache
Stomach
ache
Dead
Still Dead
Response (% of Control)
General principle
100
80
60
40
20
0
1
10
100
Dose
1000
Response (% of Control)
Response curve
120
Assumptions:
1. Small dose increases at
some threshold result in
very large responses and
2. susceptibility to dose is
normally distributed
100
80
60
40
20
0
0
500
1000
Dose
1500
Response (% of Control)
Linear regression
120
Initially can determine least
squares, but is it useful for
estimating anything other
than dose resulting in 50 %
response?
100
80
60
40
20
0
0
500
1000
Dose
1500
Remember least squares?
With 4 bivariate options
Total
Sums of Squares
X
Y
XY
X2
X
Observed
Y
Prediction
Ŷ=1+.5X
Error
(Yi-Ŷi)2
Total
(Yi-Yi)2
1
2
2
1
1
2
1.5
0.25
.0625
2
1
2
2
2
1
2.0
1.00
1.5625
3
3
9
9
3
3
2.5
0.25
.5625
4
3
12
16
4
3
3.0
0.00
.5625
10
9
25
30
ESS =
1.5
TSS =
2.75
X=10/4=2.5 Y=9/4=2.25
b=(25-4(2.5)(2.25))/((30-(4(2.5) 2)=0.5
a=2.25-0.5(2.5)=1
Line equation is y=1 + 0.5x
9
R2 = 1-ESS/TSS=1-(1.5/2.75)=0.455
Early work on response curves
• Pearl and Reed. 1920. Proceed.
Nat. Acad. of Sci. V6#6:275-288.
• Mathematical representation of
US population growth.
• Improved on Pritchett’s 1891
model (a third order parabola) on
US population growth.
• Made it binomial and logarithmic
(y = a + bx + cx2 + d log x)
Early work on response curves
• They recognized that equation would not predict US
population into the future so, assuming that resources
would limit populations, they postulated:
y = b/(e-ax + c) for x > 0, y = b/c
point of inflection is x = -(1/a)log e and y = b/2c
slope at point of inflection is ab/4c
Early work on response curves
• They recognized that equation would not predict US
population into the future so, assuming that resources
would limit populations, they postulated:
y = b/(e-ax + c) for x > 0, y = b/c
point of inflection is x = -(1/a)log e and y = b/2c
slope at point of inflection is ab/4c
• Their inflection point was April 1, 1914 at a population
of 98,637,000 and a population limit of 197,274,000
Early work on response curves
• They recognized that equation would not predict US
population into the future so, assuming that resources
would limit populations, they postulated:
y = b/(e-ax + c) for x > 0, y = b/c
point of inflection is x = -(1/a)log e and y = b/2c
slope at point of inflection is ab/4c
• Their inflection point was April 1, 1914 at a population
of 98,637,000 and a population limit of 197,274,000
• They recognized 2 problems
• Location of the point of inflection
• Symmetry
Early work on response curves
• Pearl in 1927 published “The Biology of Superiority”,
which disproved basic assumptions of eugenics and
went on to a career in Mendelian genetics.
• Reed in 1926 became the second chair of Biostatistics
at John Hopkins and by 1953 was president of the
university.
Early work on response curves
• Pearl in 1927 published “The Biology of Superiority”,
which disproved basic assumptions of eugenics and
went on to a career in Mendelian genetics.
• Reed in 1926 became the second chair of Biostatistics
at John Hopkins and by 1953 was president of the
university.
• In 1929 Reed and Joseph Berkson published “The
Application of the Logistic Function to Experimental
Data” in an attempt to correct rampant misuse.
• “in almost all cases, the mathematical phases of
the treatment have been faulty, with consequent
cost to precision and validity of the conclusions”
Early work on response curves
• They made the recommendation that this curve be
referred to as logistic instead of autocatalytic because
the curve was being used where “the concept of
autocatalysis has no place”.
Early work on response curves
• They made the recommendation that this curve be
referred to as logistic instead of autocatalytic because
the curve was being used where “the concept of
autocatalysis has no place”.
• Later they state that “the method of least squares,
when certain assumptions regarding the distribution of
errors can be made, is mathematically the most
proper”.
Early work on response curves
• They made the recommendation that this curve be
referred to as logistic instead of autocatalytic because
the curve was being used where “the concept of
autocatalysis has no place”.
• Later they state that “the method of least squares,
when certain assumptions regarding the distribution of
errors can be made, is mathematically the most
proper”.
• After acknowledging the computational difficulties,
they consider other techniques to determine the
parameters: Logarithmic Graphic Method; Function of
(r, y, t) vs. y; Slope of the Logarithmic Function vs. y;
and Parameters of the Hyperbola.
Early work on response curves
• All these methods involved graphing, fitting a line by
eye, and in some cases changing the multiplier and
repeating the process until better linearity results.
• They note that “One attempts in doing this to choose
a line that minimizes the total deviations.” and that
“The inexactness that might appear in such a method
is not as serious as sometimes supposed”
• Also, “Hand calculations of non-linear statistical
estimations are labor intensive and prone to error”
• And “Iterative procedures result in greater
expenditures for labor and more opportunities for
calculation error”
Working with a transformation
• Once the line was drawn (fitted) through the
data points the slope (2.30259 x m) and
intercept (log-1 a) are determined (Reed and
Berkson 1929)
• Expected and observed outcomes could then
be compared.
More linear transformations
• Integral of the normal curve (Gaddum 1933)
– Widely used to represent the distribution of
biological traits
– Direct experimental evidence for a normal
distribution of susceptibility (tolerance distribution)
• Gaddum was an English pharmacologist who
wrote classic text Gaddum's Pharmacology
More linear transformations
•Probit (C. I. Bliss 1934)
•Observation that in many physiologic processes
equal increments in response are produced when
dose is increased by a constant proportion of the
given dosage, rather than by constant amount.
•Chester Bliss was largely self
Taught, worked with Fisher, and
eventually settled at Yale.
Working with a transformation
• Tables with transformations were
prepared
% kill
probits % kill
probits % kill
probits % kill
probits
1
2.674
40
4.747
52
5.050
80
5.842
5
3.355
44
4.849
54
5.100
90
6.282
10
3.718
46
4.900
56
5.151
95
6.645
20
4.158
48
4.950
60
5.253
99
7.326
30
4.476
50
5.000
70
5.524
99.9
8.090
More linear transformations
•Logistic function applied to bioassy (Berkson
1944) and ED50
• Biologically relevant
– Autocatalysis of ethyl acetate by acetic acid
– Oxidation-reduction reaction
– Bimolecular reaction of methyl bromide and sodium
thiosulfate
– Hydrolysis of sucrose by invertase
– Hemolysis of erythrocytes by NaOH
More linear transformations
•Logistic function applied to bioassy (Berkson
1944) and ED50
• Biologically relevant
– Autocatalysis of ethyl acetate by acetic acid
– Oxidation-reduction reaction
– Bimolecular reaction of methyl bromide and sodium
thiosulfate
– Hydrolysis of sucrose by invertase
– Hemolysis of erythrocytes by NaOH
• Berkson of the Mayo clinic sadly stated in 1957 that it
was “very doubtful that smoking causes cancer of the
lung”
Working with a transformation
• Special graph paper was designed
Statistical analyses
• Least squares vs Maximum likelihood
– Berkson (1956) revived the debate started by
Fisher in 1922.
– Because of lack of computational power the point
was all but moot
– There was general agreement that maximum
likelihood was best
Computers
• By 1990, increased computational
speed and accuracy and the
development of analysis software meant
that analyses of dose-response
relationships could be conducted using
iterative least squares estimation
procedures
Early dose-response, a primer
•
•
•
•
•
•
Preliminary ANOVA
Logistic equation
Dose-response curve
Treatment comparison
Model comparison
Practical use
Preliminary ANOVA
• Determines if herbicide dose has an
effect on plant response
• Provides the basis for a lack-of-fit test of
the subsequent nonlinear analysis
• Provides the basis for assessing the
potential transformation of response
variables
Log-logisitic equation
D-C
y=C+ 1+exp[b(log(x)-log(I ))]
50
D = Upper limit
C = Lower limit
b = Related to slope
I = Dose giving 50% response
50
Seefeldt et al. 1995
Response (% of Control)
Log transformation of dose
120
More or less symmetric
sigmoidal curve that
expands the critical dose
range where response
occurs
100
80
60
40
20
0
0
1
10
Dose
100
1000
Dose-response curve
Percent of control
100
Upper limit (D=100)
80
60
40
I50
20
Lower limit (C=4)
0
0.01
0.1
1
Herbicide Dose
10
100
Treatment comparison
Percent of control
100
Upper limit (D=100)
80
60
40
I50
20
I50
Lower limit (C=4)
0
0.01
0.1
1
Herbicide Dose
10
100
Treatment comparison
Percent of control
100
Upper limit (D=100)
80
60
40
I50
20
I50
Lower limit (C=4)
0
0.01
0.1
1
Herbicide Dose
10
100
Comparing crop (pale blue) to
weed (yellow)
100
Percent of control
I5
80
60
40
20
0
0.01
I95
0.1
1
Herbicide Dose
10
100
Usefulness
•
•
•
•
Biologically meaningful parameters
Least squares summary statistics
Confidence intervals
Better estimates of response at high and low
doses
• Tests for differences in I50 or slope
• Still errors at extremes of doses
References
•
Bliss, C. I. 1934. The method of probits. Science, 79:2037, 38-39.
• Berkson, J. 1944. Application of the Logistic function to bio-assay. J. Amer. Stat.
Assoc. 39: 357-65.
• Berkson, J. 1955. Estimation by least squares and by maximum likelihood. Third
Berkeley Symposium p1-11.
• Gaddum, J. H. 1933. Methods of biological assay depending on a Quantal response.
Medical Res. Council Special Report. Series No. 183.
• Reed, L.J., and Berkson, J. 1929. The application of the logistic function to
experimental data. J. Physical Chem. 33:760-779.
• Seefeldt, S.S., J. E. Jensen, and P. Fuerst. 1995. Log-logistic analysis of herbicide
dose-response relationships. Weed Technol. 9:218-227.