Using Non-Linear Mixed Models for Agricultural Data

Transcription

1 Using Non-Linear Mixed Models for Agricultural Data Fernando E. Miguez Energy Biosciences Institute Crop Sciences University of Illinois, Urbana-Champaign Oct 8 th, 8

2 Outline 1 Introduction 2 Barley N response 3 Statistical Models 4 Application to Meta-analysis

3 Objectives of Statistical Modeling Objectives 1 Develop the simplest model which still captures the structure of the data 2 Interpret the model (give meaning to the parameters) 3 Generate predictions (validation)

4 Objectives of Statistical Modeling Objectives 1 Develop the simplest model which still captures the structure of the data 2 Interpret the model (give meaning to the parameters) 3 Generate predictions (validation)

5 Objectives of Statistical Modeling Objectives 1 Develop the simplest model which still captures the structure of the data 2 Interpret the model (give meaning to the parameters) 3 Generate predictions (validation)

6 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

7 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

8 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

9 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

10 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

11 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

12 Outline 1 Introduction 2 Barley N response 3 Statistical Models 4 Application to Meta-analysis

13 Barley N response trials Aril Vold (1998). A generalization of ordinary yield response functions. Ecological Applications. 108: Details 19 years of data, Norway N rates (0, 3.38, 7.76 and g N m 2 ) raised by 20% in 1978 Agronomic Questions 1 How does it respond to N? 2 How does it vary among years?

14 Yield (g/m2) N fertilizer (g/m2)

15 Outline 1 Introduction 2 Barley N response 3 Statistical Models 4 Application to Meta-analysis

16 Basics of Statistical Models where, y = observed f = mean structure x = input θ = parameters ɛ = error y = f(x, θ) + ɛ

17 Basics of Statistical Models y = f(x, θ) + ɛ where, y = observed f = mean structure x = input θ = parameters ɛ = error D = M + E

18 Choosing the Mean Structure Asymptotic Regression Model y = θ 1 + (θ 2 θ 1 ) exp( exp(θ 3 ) x) Yield (g/m2) where, θ 1 is the maximum value of y θ 2 is the value of y for x = 0. θ 3 is the growth rate of y N fertilizer (g/m2)

19 Barley N response trials Non-linear regression with years combined N fertilizer (g/m2) Yield (g/m2) 500

20 Barley N response trials Box-plots of residuals for each year year Residuals

21 Barley N response trials Yield (g/m2) 500 N fertilizer (g/m2) One single regression to all the data Wide confidence intervals Ignores the structure of the data Yield (g/m2) Fitting one function for each separate year Over-parameterized model 3 parms 19 y = 57 parms 500 N fertilizer (g/m2)

22 Barley N response trials Confidence Intervals for Non-linear regressions for each year year th th lrc

23 Yield (g/m2) N fertilizer (g/m2)

24 Non-Linear Mixed Model Asymptotic regression with random effects y ij = (θ 1 +b 1i )+((θ 2 +b 2i ) (θ 1 +b 1i )) exp( exp(θ 3 +b 3i ) x ij )+ɛ ij i = the year (or experimental unit) j = the N rate b i N (0, Ψ), ɛ ij N (0, σ 2 ) Ψ = σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33

25 Random Effects Dot plot for the random effects th1 th2 lrc year Random effects

26 Random Effects Scatter plot matrix for the random effects lrc th th1 0 0 Scatter Plot Matrix

27 Non-Linear Mixed Model Fixed and BLUP fixed BLUP Yield (g/m2) N fertilizer (g/m2)

28 Comparison of NLS and NLME Estimate, and 95% confidence intervals for the three parameters of the asymptotic regression model (NLS) and the mixed-effects model (NLME). Fixed term Estimate Lower Upper θ 1 NLS θ 1 NLME θ 2 NLS θ 2 NLME lrc NLS lrc NLME ˆσ NLS 71.2 ˆσ NLME

29 Summary: Using NLME NLME are able to accomodate the mean and error structure NLME produce a parsimonious and easy to interpret model The NLME estimates are more accurate and the confidence intervals are narrower

30 Summary: Using NLME NLME are able to accomodate the mean and error structure NLME produce a parsimonious and easy to interpret model The NLME estimates are more accurate and the confidence intervals are narrower

31 Summary: Using NLME NLME are able to accomodate the mean and error structure NLME produce a parsimonious and easy to interpret model The NLME estimates are more accurate and the confidence intervals are narrower

32 Outline 1 Introduction 2 Barley N response 3 Statistical Models 4 Application to Meta-analysis

33 Application to Meta-analysis Meta-analysis of the effects of management factors on Miscanthus x giganteus growth and biomass production. Miguez et al (8) Agricultural and Forest Meteorology. 148: R Code and Data miguez@illinois.edu Website:

34 Questions?