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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, 2015RTI International 

# Outline

 

## Brief discussion of environmental fate model selection methods:

– 

### From statistical perspective

– 

### Will suggest statistical corrections in guidance docs 

# Background: Statistician’s Perspective

2RTI International 

# 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 i 

## E ε = 2 

## cov( , ) i i j i j 

## i j i j σ ε ε σ σ  = 
=  ≠   <= Heteroskedasticity (unequal errors) <= Correlated errorsRTI International 

# Outline

 

## F and t tests useful for selecting models

 

## Requirements:

– 

### E rrors have normal distribution

– 

### Error structure: 

# Model Selection Tests: F and t

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# () 0 i 

# E ε = 2 

# cov( , ) 0 i j 

# i j i j σ ε ε  = =  ≠  <= Homoskedasticity (equal errors) <= Uncorrelated errorsRTI International 

# Outline

 

## Need to check error structure

– 

### If normality/ homoskedasticity requirements not met:

 N eed 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:

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### Categorical ANOVA model – not continuous (can’t interpolate between time points)

– 

### Number of parameters = number of distinct time points

– 

### E xtracts 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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# Outline Example: Fit Full Model and Check Residuals

7 Original Scale Residuals: “Trumpet” shape
- heteroskedastic Log Scale Residuals: Less heteroskedastic

- Select Log ScaleRTI International 

# 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

8 0 

### concentration at time initial concentration t 

### C t C =

=

0 

### exp( ) t 

### C C kt = − 

# ( ) ( ) 0 

### ln ln t 

### C C kt = − <= Original scale <= Natural log scale Signal component: mathematically equivalent Statistical models: different due to added noise term

– use error structure to select scaleRTI International 

# 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

9 0 

### exp(1 ) t 

### CC t α 

### β = − 

# ( ) ( ) 0 

### ln ln ln(1 ) t 

### CC t α β = −+



### 0 β →RTI International 

# Outline Gamma Distribution: 3 Examples

10 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4

kProbability

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4 2 

## shape parameter, scale parameter 1,         

 

1, 1    

2, .5     10, .1    RTI International 

# Outline

 

## Gamma distribution very flexible, but has limitations

 

## Couple of examples not covered by Gamma:

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### 2 compartments, with **k** constant within each compartment, but different between compartments

 Need Double First Order in Paralle l (DFOP) model

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### 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:

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### S elect 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 fitRTI International



###  FOMC v. SFO test: p < 0.0001**

 FOMC v. Full Model test: p = 0.9650

 Decision: FOMC DOES provide adequate fitRTI International 

# 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|H 0 true)

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

 

### To narrow CI, rather increase n!

– Power = Pr (True positive|H 0 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

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### No direct way to get pure measurement error

 Approximate measurement error – use best - fitting model to minimize model error – use those residuals

 F it model in original and log scales and see which residuals have required structure

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### 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

20 2 2 tab 

## χ χ ≡RTI International 

# 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:

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### Get replicates! 

# Data without Replicates

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# Outline

 

## Contact info:

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### jaldworth@rti.org

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### wenlin.chen@syngenta.com

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### scott.jackson@basf.com

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### lpadilla@stone - env.com 

# Questions?

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