Symposium: Advances in Dose-response Methodology Applied to the Science of Weed Control Presenters:...
-
date post
20-Dec-2015 -
Category
Documents
-
view
215 -
download
1
Transcript of Symposium: Advances in Dose-response Methodology Applied to the Science of Weed Control Presenters:...
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 Stomachache
What did hunter gathers do?
One Several Dinner
Tasty Tasty Filling
Tasty Tasty Stomachache
Stomach Dead Still Deadache
General principle
0
20
40
60
80
100
1 10 100 1000
Dose
Res
po
nse
(%
of
Co
ntr
ol)
Response curve
0
20
40
60
80
100
120
0 500 1000 1500
Dose
Res
po
nse
(%
of
Co
ntr
ol) Assumptions:
1. Small dose increases at some threshold result in very large responses and 2. susceptibility to dose is normally distributed
Linear regression
0
20
40
60
80
100
120
0 500 1000 1500
Dose
Res
po
nse
(%
of
Co
ntr
ol) Initially can determine least
squares, but is it useful for estimating anything other than dose resulting in 50 % response?
Remember least squares?
With 4 bivariate optionsX Y XY X2
1 2 2 1
2 1 2 2
3 3 9 9
4 3 12 16
Total 10 9 25 30
Sums of SquaresX Observed
YPrediction Ŷ=1+.5X
Error (Yi-Ŷi)2
Total (Yi-Yi)2
1 2 1.5 0.25 .0625
2 1 2.0 1.00 1.5625
3 3 2.5 0.25 .5625
4 3 3.0 0.00 .5625
9 ESS = 1.5
TSS = 2.75
X=10/4=2.5 Y=9/4=2.25b=(25-4(2.5)(2.25))/((30-(4(2.5) 2)=0.5a=2.25-0.5(2.5)=1Line equation is y=1 + 0.5x
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, andeventually 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
y=C+
D = Upper limitC = Lower limitb = Related to slope
I = Dose giving 50% response
D-C1+exp[b(log(x)-log(I ))]50
50
Seefeldt et al. 1995
Log transformation of dose
0
20
40
60
80
100
120
0 1 10 100 1000
Dose
Res
po
nse
(%
of
Co
ntr
ol) More or less symmetric
sigmoidal curve that expands the critical dose range where response occurs
Dose-response curve
0.01 0.1 1 10 1000
20
40
60
80
100
Herbicide Dose
Perc
en
t of
con
trol Upper limit (D=100)
I50Slope (b=
2)
Lower limit (C=4)
Treatment comparison
0.01 0.1 1 10 1000
20
40
60
80
100
Herbicide Dose
Perc
en
t of
con
trol Upper limit (D=100)
I50Slope (b=
2)
Lower limit (C=4)I50
Treatment comparison
0.01 0.1 1 10 1000
20
40
60
80
100
Herbicide Dose
Perc
en
t of
con
trol Upper limit (D=100)
I50Slope (b=
2)
Lower limit (C=4)
I50
Slope (b=1.2)
Comparing crop (pale blue) to weed (yellow)
0.01 0.1 1 10 1000
20
40
60
80
100
Herbicide Dose
Perc
en
t of
con
trol
I95
I5
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.