Symposium: Advances in Dose-response Methodology Applied to the Science of Weed Control

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Symposium Advances in Dose-response
Methodology Applied to the Science of Weed Control

• Presenters
• Dr. Steven Seefeldt
• Dr. Bahman Shafii
• Dr. William Price

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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

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Before the scientific method and hypothesis
testing
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What did hunter gathers do?
One Several Dinner Tasty Tasty Filling
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What did hunter gathers do?
One Several Dinner Tasty Tasty Filling Tas
ty Tasty Stomach ache
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What did hunter gathers do?
One Several Dinner Tasty Tasty Filling Tas
ty Tasty Stomach ache Stomach
Dead Still Dead ache
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General principle
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Response curve
Assumptions 1. Small dose increases at some
threshold result in very large responses and 2.
susceptibility to dose is normally distributed
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Linear regression
Initially can determine least squares, but is it
useful for estimating anything other than dose
resulting in 50 response?
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Remember least squares?

• With 4 bivariate options

• Sums of Squares

X Y XY X2
1 2 2 1
2 1 2 2
3 3 9 9
4 3 12 16
Total 10 9 25 30
X Observed Y Prediction Y1.5X Error (Yi-Yi)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
X10/42.5 Y9/42.25 b(25-4(2.5)(2.25))/((30-(4
(2.5) 2)0.5 a2.25-0.5(2.5)1 Line equation is
y1 0.5x
R2 1-ESS/TSS1-(1.5/2.75)0.455
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Early work on response curves

• Pearl and Reed. 1920. Proceed. Nat. Acad. of Sci.
  V66275-288.
• Mathematical representation of US population
  growth.
• Improved on Pritchetts 1891 model (a third order
  parabola) on US population growth.
• Made it binomial and logarithmic (y a bx
  cx2 d log x)

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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 gt 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

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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 gt 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

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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 gt 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

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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.

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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

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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.

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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.

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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.

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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

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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.

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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

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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.

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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
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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

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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

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Working with a transformation

• Special graph paper was designed

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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

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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

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Early dose-response, a primer

• Preliminary ANOVA
• Logistic equation
• Dose-response curve
• Treatment comparison
• Model comparison
• Practical use

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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

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Log-logisitic equation
D-C
yC
1expb(log(x)-log(I ))
50
D Upper limit
C Lower limit
b Related to slope
I Dose giving 50 response
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Seefeldt et al. 1995
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Log transformation of dose
More or less symmetric sigmoidal curve that
expands the critical dose range where response
occurs
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Treatment comparison
100
Upper limit (D100)
80
60
Percent of control
Slope (b2)
40
I50
I50
20
Lower limit (C4)
0
0.01
0.1
1
10
100
Herbicide Dose
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Treatment comparison
100
Upper limit (D100)
80
60
Slope (b1.2)
Percent of control
Slope (b2)
40
I50
I50
20
Lower limit (C4)
0
0.01
0.1
1
10
100
Herbicide Dose
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Comparing crop (pale blue) to weed (yellow)
100
I5
80
60
Percent of control
40
20
I95
0
0.01
0.1
1
10
100
Herbicide Dose
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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

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References

• Bliss, C. I. 1934. The method of probits.
  Science, 792037, 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.
  33760-779.
• Seefeldt, S.S., J. E. Jensen, and P. Fuerst.
  1995. Log-logistic analysis of herbicide
  dose-response relationships. Weed Technol.
  9218-227.