Using Non-Linear Mixed Models for Agricultural...

Introduction Barley N response Statistical Models Application to Meta-analysis

Using Non-Linear Mixed Models for AgriculturalData

Fernando E. Miguez

Energy Biosciences InstituteCrop Sciences

University of Illinois, [email protected]

Oct 8th, 2008


Outline

1 Introduction

2 Barley N response

3 Statistical Models

4 Application to Meta-analysis


Objectives of Statistical Modeling

Objectives

1 Develop the simplest model which still captures the structureof the data

2 Interpret the model (give meaning to the parameters)

3 Generate predictions (validation)


Non-Linear and Mixed Models

Non-Linear Models

1 Parsimony

2 Interpretability

3 Model the mean structure

Mixed Models

1 Flexibility

2 Hierarchy

3 Model the error structure


Outline

1 Introduction

2 Barley N response




Barley N response trialsAril Vold (1998). A generalization of ordinary yield response functions. EcologicalApplications. 108:227-236.

Details

19 years of data, Norway

N rates (0, 3.38, 7.76 and11.69 g N m−2) raised by20% in 1978

Agronomic Questions

1 How does it respond to N?

2 How does it vary amongyears?


N fertilizer (g/m2)

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Outline

1 Introduction

2 Barley N response




Basics of Statistical Models

y = f(x, θ) + ε

where,y = observedf = mean structurex = inputθ = parametersε = error


Basics of Statistical Models

y = f(x, θ) + ε

D =M+ Ewhere,y = observedf = mean structurex = inputθ = parametersε = error


Choosing the Mean Structure

Asymptotic RegressionModel

y = θ1 + (θ2 − θ1)×exp(− exp(θ3)× x)

where,θ1 is the maximum value of yθ2 is the value of y for x = 0.θ3 is the growth rate of y

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Barley N response trialsNon-linear regression with years combined

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Barley N response trialsBox-plots of residuals for each year

Residuals

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One single regression to all the data

Wide confidence intervalsIgnores the structure of the data

Fitting one function for each separate year

Over-parameterized model3 parms × 19 y = 57 parms


Barley N response trialsConfidence Intervals for Non-linear regressions for each year

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lrc


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Non-Linear Mixed Model

Asymptotic regression with random effects

yij = (θ1+b1i)+((θ2+b2i)−(θ1+b1i))×exp(− exp(θ3+b3i)×xij)+εij

i = the year (or experimental unit)j = the N rate

bi ∼ N (0, Ψ), εij ∼ N (0, σ2)

Ψ =

σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33


Random EffectsDot plot for the random effects

Random effects

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Random EffectsScatter plot matrix for the random effects

Scatter Plot Matrix

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Non-Linear Mixed ModelFixed and BLUP

N fertilizer (g/m2)

Yie

ld (

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2)

100200300400500

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fixedBLUP


Comparison of NLS and NLME

Estimate, and 95% confidence intervals for the three parameters ofthe asymptotic regression model (NLS) and the mixed-effectsmodel (NLME).

Fixed term Estimate Lower Upperθ1 NLS 381 335 507θ1 NLME 390 337 443

θ2 NLS 133 101 166θ2 NLME 132 107 157

lrc NLS -1.7 -2.7 -1.1lrc NLME -1.7 -1.9 -1.4

σ̂ NLS 71.2σ̂ NLME 18.8 13.8 25.6


Summary: Using NLME

NLME are able to accomodate the mean anderror structure

NLME produce a parsimonious and easy tointerpret model

The NLME estimates are more accurate and theconfidence intervals are narrower


Outline

1 Introduction

2 Barley N response




Application to Meta-analysisMeta-analysis of the effects of management factors on Miscanthus x giganteus growthand biomass production. Miguez et al (2008) Agricultural and Forest Meteorology.148:1280-1292.

R Code and DataE-mail: [email protected]

Website: https://netfiles.uiuc.edu/miguez/www


Questions?

Using Non-Linear Mixed Models for Agricultural...

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