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Statistical Methods to Select Kinetics Models for 

Analysis of Pesticide Environmental Fate Data 

Jeremy Aldworth, RTI International

Wenlin Chen, Syngenta Crop Protection, LLC

Scott Jackson, BASF Corporation

Lauren Padilla, Stone Environmental, Inc.

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EMPM Meeting, April 28, 2015
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Outline

 Brief discussion of environmental fate model 

selection methods: 

– From statistical perspective

– Will suggest statistical corrections in guidance docs

Background: Statistician’s Perspective 

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Outline

 Generic statistical model: 

Data = Signal + Noise

– Noise (error) – critical for statistical inference

 Generic error structure:

Statistical Model

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(;) ii i y hx = + θ ε

() 0 E i ε =

2

cov( , ) i

i j

i j

i j

i j

σ

ε ε

σ σ

 =

= 

 ≠ 

<= Heteroskedasticity (unequal errors)

<= Correlated errors
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Outline

 F and t tests useful for selecting models 

 Requirements: 

– Errors have normal distribution

– Error structure:

Model Selection Tests: F and t

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() 0 E i ε =

2

cov( , ) 0 i j

i j

i j

σ

ε ε

 =

= 

 ≠

<= Homoskedasticity (equal errors)

<= Uncorrelated errors
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Outline

 Need to check error structure

– If normality/homoskedasticity requirements not met: 

 Need to transform data so that requirements are met (so F test valid) 

 How do we check error structure?

– Easy if data replicates exist at each time point – next slide...

– Not so easy for unreplicated data – discuss later on...

Residual Structure

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Outline

 If replicates at time points – can define full model

with Time as a categorical variable: 

– Categorical ANOVA model – not continuous (can’t 

interpolate between time points)

– Number of parameters = number of distinct time points

– Extracts more info from data than any other model

– Residuals represent error structure of pure measurement 

error (noise) – uncontaminated with model error

Replicated Data – Full Model

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Example: Fit Full Model and Check Residuals Outline

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Original Scale Residuals:

“Trumpet” shape - heteroskedastic

Log Scale Residuals:

Less heteroskedastic

- Select Log Scale
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Outline

 Define:

 Single First Order (SFO) kinetics (signal):

– Assumes constant dissipation rate k (homogeneous medium)

– Problem if medium heterogeneous and dissipation rate not 

constant

Kinetics: SFO 

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0

 concentration at time 

 initial concentration

C t t

C

=

=

0 exp( ) C C kt t

= −

ln ln (C C kt t ) = − ( 0 )

<= Original scale

<= Natural log scale

Signal component:

mathematically equivalent

Statistical models: different 

due to added noise term – use 

error structure to select scale
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Outline

 First Order Multi-Compartment (FOMC) kinetics (signal):

– Assumes medium spatially heterogeneous and dissipation rate k 

varies according to continuous Gamma distribution

 Notes: 

– As , FOMC degenerates into SFO (= special case of FOMC)

– FOMC exactly equivalent to IORE model (different 

parameterization)

Kinetics: FOMC

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0 exp(1 ) CC t t

α

= − β

ln ln ln(1 ) (CC t t ) = −+ ( 0 ) α β

β → 0
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Gamma Distribution: 3 Examples Outline

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

k

Probability

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

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shape parameter, scale parameter

1,

 

    

 

 

    1, 1

    2, .5

    10, .1
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Outline

 Gamma distribution very flexible, but has limitations

 Couple of examples not covered by Gamma:

– 2 compartments, with k constant within each compartment, 

but different between compartments 

 Need Double First Order in Parallel (DFOP) model

– 2 compartments, with k varying continuously within each 

compartment (i.e., different Gamma distribution within each 

compartment – mixed distribution bimodal) 

 Need to develop Double First Order Multi-Compartment in Parallel 

(DFOMCP) model

Limitations of Gamma Distribution

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Outline

 If error structure reasonably normal and homoskedastic then F 

and t tests valid

 F test useful for testing: 

– If regression model differs significantly from Null Model 

– If more general model fits better than special-case model (e.g., 

FOMC v. SFO) – extra-sum-of-squares principle

 If general model has only one extra parameter – F test is same as t test

– If selected model fits as well as the Full Model (need replicates) –

lack-of-fit principle 

F Tests for Model Selection

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Outline

 Use parsimony principle: 

– Select simplest model that adequately fits the data

Illustration: Model Selection Strategy

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 SFO v. Null Model test: p < 0.0001** 

 SFO v. Full Model test: p = 0.0011**

 Decision: SFO does NOT provide adequate fit
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 FOMC v. SFO test: p < 0.0001** 

 FOMC v. Full Model test: p = 0.9650

 Decision: FOMC DOES provide adequate fit
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Outline

 NAFTA test:

– Create 50% confidence region (CR) around FOMC (or IORE) measure (SSR), 

and reject SFO in favor of FOMC if SFO measure (SSR) lies outside CR 

 Problems:

– Use CRs to simultaneously test vector of parameters (> 1) against null hyp.

 Ex: FOMC (alternative hyp.) against Null Model (null hyp.) – simultaneously test 2 parameters –

equivalent to F test of FOMC against Null Model

 But test of FOMC (alternative hyp.) against SFO (null hyp.) is a one parameter test – use 1 

dimensional confidence interval (CI), not CR

– Use approach above for testing against fixed null hypothesis value! – not

the SFO measure, which is random and has its own CI! 

– Also, FOMC and SFO fitted values are correlated!

NAFTA Test: FOMC v. SFO

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Outline

 F Test between FOMC and SFO based on difference in SSRs

– If two measures are random and correlated, difference accounts for 

variance in both variables and correlation between them

– Can create equivalent test based on CI of difference – if CI does not contain 

zero then difference significant

NAFTA Test: FOMC v. SFO (2)

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Outline

 Don’t mess with alpha!

– NAFTA: alpha = 50%

– Alpha = Pr(False positive|H0 true)

– Usually set at 5% (or lower) – almost never higher

 To narrow CI, rather increase n!

– Power = Pr(True positive|H0 false)

– Power calculation: Fix alpha = 5%, but increase n until power = 80% (or 

90%)

NAFTA Test: FOMC v. SFO (3)

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Outline

 No replicates => no Full Model 

– No direct way to get pure measurement error

 Approximate measurement error – use best-fitting model to 

minimize model error – use those residuals

 Fit model in original and log scales and see which residuals 

have required structure

– No Full Model to compare selected model against!

Data without Replicates

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Outline

 FOCUS EU Guidance formula 6-1, page 87:

 where (formula 6-2, page 89):

 Do the math:

 Which explains: 

FOCUS Chi-squared Calculated Value

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2 2 χ χ ≡ tab
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Outline

 FOCUS test based on err (formula 6-2, page 89)

 Called “Error level Chi2 test” in spreadsheet

 Properties of err:

– Higher values indicate poor fit – lower values better fit 

– Not sure of exact statistical properties

– How is it calibrated?

 How high before not acceptable?

FOCUS Chi-squared “Test”

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Outline

 Moral: 

– Get replicates!

Data without Replicates

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Outline

 Contact info:

– jaldworth@rti.org

– wenlin.chen@syngenta.com

– scott.jackson@basf.com

– lpadilla@stone-env.com

Questions?

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