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

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What did hunter gathers do?

One Several Dinner

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General principle

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Response curve

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1. Small dose increases at some threshold result in very large responses and 2. susceptibility to dose is normally distributed

Linear regression

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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

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sigmoidal curve that expands the critical dose range where response occurs

Dose-response curve

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Treatment comparison

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Treatment comparison

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Comparing crop (pale blue) to weed (yellow)

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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.