1. Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015

2. Up to now… General Linear Model (lm) – y i = α + β 11 +... + β + ε with ε ~ N(0, σ i 2 ) Generalized Linear Model (glm) – Non-normal error distributions for response variable; link function Generalized Additive Model (gam) – Identify smoothed lines of best fit for non-linear relationships Generalized Least Squares (gls) – Altered variance structures of Normal distribution

3. Residuals are normally distributed Histogram or Q-Q plot Residuals are “homogenous” or “homoscedastic” (constant variance) – No autocorrelation between observations plot residuals No colinearity between independent variables Pairs plot in R The model is not biased by unduly influential observations “Cook’s Distance” and leverage Independent observations

4. Nested data? Blocking? Repeated measures? Split-plot designs? Spatial or temporal autocorrelation? But what about…

5. Use of mixed models “Mixed effects models or multilevel models and are used when the data have a hierarchical form…which can have both fixed and random coefficients together with multiple error terms.” 1 Zuur et al. 2007. Analyzing Ecological Data. Pg 127

6. Rob Thomas

8. GLS LM, GAM, GLM Model Structure LMM, GLMM, GAMM

9. Parameter estimation ML = Maximum Likelihood – Common with GLM REML = Restricted Maximum Likelihood – Corrects ML estimation for the number of fixed covariates – Less influenced by outliers than ML estimates – Common with LMM

10. R packages library(nlme) = Non-Linear Mixed Effects –lme = Linear Mixed Effects –gls = Generalised Least Squares –model<- lme (y ~ fixed, random = ~1|random, data) library ( lme4) = Linear Mixed Effects v.4 –lmer = Linear Mixed Effects REML –model <- lmer(y ~ fixed + (1|random), family = gaussian (link = “identity”), data)

11. Nested Design Example Are there any differences between the NAP- richness relationship at these 9 beaches? – NAP = tidal height, predictive variable – Species richness = response variable From Zuur 2009

12. richness values for beach i, i = 1,…,9 Fixed Term Richness-NAP fixed effect across all beaches Random Term Richness-NAP random effect for each beach Mixed Effects Model Structure From Zuur 2009

14. Model 1: Constant slope/intercept y i = α + β i + ε with ε ~ N(0, σ i 2 ) Assumes that the richness-NAP relationship is the same at all beaches model1 <- gls (richness ~ 1 + NAP, method = “REML”, data ) From Zuur 2007 and Rob Thomas A model fitted using the REML method, but containing no random effects at all–basically a REML-fitted linear regression

15. i.e. a model that fits the same slope for each level of the random factor (fitted by REML by default) Model 2: varying intercept, same slope y ij = α+ β ij + a j + ε j where a j ~ N(0, σ a 2 ) and ε j ~ N(0, σ 2 ) model2 <- lme (richness ~ NAP, random = ~1|beach, method = “REML”, data) From Zuur 2007 and Rob Thomas

16. Model 3: varying slope, varying intercept y ij = α + β ij + a j + b j x ij + ε j where a j ~ N(0, σ a 2 ), b j ~ N(0, σ b 2 ), and ε j ~ N(0, σ 2 ) model3 <-- lme (richness ~ NAP, random = ~NAP | beach, method = “REML”, data) i.e. a model that fits a different slope for each level of the random factor (fitted by REML by default) From Zuur 2007 and Rob Thomas

17. Additional complexity Generalized Mixed models: lmer() and mgcv() GLMM and GAMM; different underlying error distributions

18. Mixed Effects Resources Mixed Effects Models and Extensions in Ecology with R (2009). Zuur, Ieno, Walker, Saveliev and Smith. Springer

19. Model selection? Check assumptions of the model (e.g., residuals and colinearity) Compare competing models (R. Thomas recommends comparing a gls (containing no random effects) to a linear effects mixed model to assess the importance of the random effect). Compare nested models using AIC OR: stepwise model refinement

20. The End!

21. Presenting examples (how many?) – How many types of models (nested- repeated measures, split-plot designs, nonlinear, linear) – Page 102 (online) and 71 (book) of Zuur – BDRipley 271 (pdf) – Random slope vs intercept models Relate to R code / packages? – LME4 and NLME

22. General/Generalised linear model General/Generalised additive model (GAM): identify smoothed lines of best fit through a dataset. A non-parametric smoothed relationship is chosen to fit a curve. – Non-Gaussian error distributions can also be chosen as with GLM Generalised least squares (GLS): incorporates a random term that takes into account heteroskedasticity (non-homogenous variance and/or autocorrelation structures) – Can use gls function in nlme package – Multiple variance structures to pick from

23. General/Generalised linear mixed modeling – nlme; lme4; asreml – Model fitting “ML” (Maximum likelihood): common with GLM “REML” (Restricted Maximum likelihood): corrects ML estimation for the number of fixed covariates. Less influenced by outliers than ML estimates – Structure Random intercept (same autocorrelation function for all levels of fixed factors) Random intercept + slope (autocorrelation function varies across different levels of the fixed factor) Random slope and > 1 random effect Random effect only aside from the intercept (useful as a null model to evaluate the importance of fixed effects in a GLMM) Fixed factors only (not really a mixed model, but useful as a null model to evaluate the importance of random effects in a GLMM) Generalised additive mixed modeling

24. Model 5: Random effect, no fixed effect y i = α + b i + ε j with ε j ~ N(0, σ 2 ) model5 <-- lme(richness ~1, random = ~1|beach, method = “RMEL”, data) i.e. a model that fits the mean value for each level of the random factor (fitted by REML by default) y x From Rob Thomas