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

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

Mixed Effects Models

Rebecca Atkins and Rachel SmithMarch 30, 2015

Page 2: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Up to now…• General Linear Model (lm)

– yi = α + β1𝑥1 + ... + β𝑝𝑥𝑝𝑖 + ε𝑖 with ε𝑖 ~ N(0, σi2)

• 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

Page 3: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

• 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

Page 4: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Nested data?Blocking?

Repeated measures? Split-plot designs?

Spatial or temporal autocorrelation?

But what about…

Page 5: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

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

Page 6: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Rob Thomas

Page 7: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Rob Thomas

Page 8: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

GLS

LM, GAM, GLM

Model Structure

LMM, GLMM, GAMM

Rachel S Smith
Just "Model Structure"
Page 9: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

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

Page 10: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

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)

Page 11: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

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

Page 12: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

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

Rachel S Smith
Is this the right way to say this?
Rachel S Smith
How does slope/intercept relate to this model?
Page 13: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.
Page 14: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Model 1: Constant slope/intercept• yi = α + β𝑥i + ε𝑖 with ε𝑖 ~ N(0, σi

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

Rachel S Smith
Which notation do we want to do?
Rachel S Smith
Is gls() the right thing to use here?
Page 15: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

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

• yij = α+ β𝑥ij + aj + ε𝑖j where aj ~ N(0, σa2)

and ε𝑖j ~ N(0, σ2)• model2 <- lme (richness ~ NAP, random = ~1|

beach, method = “REML”, data)

From Zuur 2007 and Rob Thomas

Rachel S Smith
Want to include REML vs. ML methods in these; left in for now
Rachel S Smith
Change to have fitted line from Zuur 2009 book instead?
Page 16: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Model 3: varying slope, varying intercept

• yij = α + β𝑥ij + aj + bjxij + ε𝑖j where aj ~ N(0, σa2),

bj ~ N(0, σb2), and ε𝑖j ~ N(0, σ2)

• 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

Rachel S Smith
Page 17: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Additional complexity

• Generalized Mixed models: lmer() and mgcv()• GLMM and GAMM; different underlying error

distributions

Rachel S Smith
Do we have questions for Dr.Drake? What do we say when he asks "what else we want the class to take away"?
Rachel S Smith
Page 18: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Mixed Effects Resources• Mixed Effects Models and Extensions in

Ecology with R (2009). Zuur, Ieno, Walker, Saveliev and Smith. Springer

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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

Page 20: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

The End!

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• 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

Page 22: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

• 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

Page 23: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

• 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

Page 24: Mixed Effects Models Rebecca Atkins and Rachel Smith March 30, 2015.

Model 5: Random effect, no fixed effect

• yi= α + bi+ ε𝑖j with ε𝑖j ~ N(0, σ2)

• 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

xFrom Rob Thomas