(generalized) mixed-effects models – (g)mems

(Generalized) Mixed-Effects Models – (G)MEMs

Yann Hautier, NutNet meeting 16th Aug 2011

Maximum Entropy Models (Grace 2011)

MEM

Maximum Entropy Models (Grace 2011)Muddled Eric Models (Seabloom 2011)

MEM

Maximum Entropy Models (Grace 2011)Muddled Eric Models (Seabloom 2011)Mixed-Effects Models (Fischer 1918)

MEM

Books

LM – Classical least-square

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MEM – Maximum Mikelihood

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Given a set of dataand a chosen model which parameters of the model produce the best model fit and make the data most likely to be observed

Fixed and Random EffectsFixed effects:• Experimentally manipulated factors• Influence only the mean of a response• BLUEs (Best Linear Unbiased Estimates)• Unknown constants to be estimated from data• Limited to the range of the fixed effect examined

Random effects• Blocks, Plots, Sites, etc.; random subset of larger

population• Influence the variance/covariance structure• BLUPs (Best Linear Unbiased Predictions)• ‘Prediction’ of the population of random effects

Fixed and Random EffectsFixed effects:• Experimentally manipulated factors• Influence only the mean of a response• BLUEs (Best Linear Unbiased Estimates)• Unknown constants to be estimated from data• Limited to the range of the fixed effect examined

Random effects• Blocks, Plots, Sites, etc.; random subset of larger

population• Influence the variance/covariance structure• BLUPs (Best Linear Unbiased Predictions)• ‘Prediction’ of the population of random effects

Not inequivocal – grey area in between

Why use mixed-effects models?

• To properly account for covariance structure in grouped data

• To treat fixed and random effects appropriately

• Fixed effects: Estimate and test

• Random effects: Predict and test

Why use Modern Mixed-effects Models (in particular)?

• Modern methods (ML, REML etc.) can give unbiased predictions of variance components for unbalanced data

• More efficient use of degrees of freedom than traditional approach because fixed x random interactions are random terms and only require 1 DF to estimate VC

• Estimating variance components by ML-based method avoids the mixed-model debate over correct error term since the VCs are estimated directly by REML etc.

• To get shrinkage estimates and reduce risk of over-fitting• To properly account for covariance structure in grouped

data

Generalized Linear Mixed Models (GLMMs)

• Combine the properties of two statistical frameworks

• Linear mixed models (incorporation random effects)

• Generalized linear models (handling non normal data by using link functions and exponential family)

lmer(X ~ Y + (1 + logS | site) + (1| Block ) + (1| mix), family = " ", data)

lme(X ~ Y, random = list(Site = ~1 + sr.log2, Block = ~1, mix = ~1), data)

lm, lme, lmer syntax

Family:binomial(link = "logit")gaussian(link = "identity") Gamma(link = "inverse") inverse.gaussian(link = "1/mu^2") poisson(link = "log") quasi(link = "identity", variance = "constant") quasibinomial(link = "logit") quasipoisson(link = "log")

lm(X ~ Y, data)

ls1Machine <- lme( score ~ Machine, data = Machines)

Example – MachinesBalanced vs. Unbalanced data

delete selected rows from the Machines dataMachinesUnbal <- Machines[ -c(2,3,6,8,9,12,19,20,27,33), ]

ls2Machine <- lme( score ~ Machine, data = MachinesUnbal)

(Intercept) MachineB MachineC 52.355556 7.966667 13.916667

(Intercept) MachineB MachineC 52.32102 8.00837 13.95120

fm1Machine <- lme( score ~ Machine, data = Machines, random = ~ 1 | Worker / Machine)

Example – MachinesBalanced vs. Unbalanced data

(Intercept) MachineB MachineC 52.355556 7.966667 13.916667

delete selected rows from the Machines dataMachinesUnbal <- Machines[ -c(2,3,6,8,9,12,19,20,27,33), ]

fm2Machine <- lme( score ~ Machine, data = MachinesUnbal , random = ~ 1 | Worker / Machine)

(Intercept) MachineB MachineC 52.354000 7.962446 13.918222

Example – BIODEPTH Manipulation of diversity:

Species richness

Species composition

Hector et al. 1999, Science

ls1 <- lm(terms(ANPP~ Site+Block+SR.log2+Site:SR.log2+Mix+Site:Mix, keep.order= TRUE), data= Biodepth)

Example – BIODEPTHDegrees of freedom

Df Sum Sq Mean Sq F value Pr(>F) Site 7 14287153 2041022 126.5168 < 2.2e-16 ***Block 7 273541 39077 2.4223 0.02072 * SR.log2 1 5033290 5033290 311.9986 < 2.2e-16 ***Site:SR.log2 7 1118255 159751 9.9025 9.316e-11 ***Mix 189 17096361 90457 5.6072 < 2.2e-16 ***Site:Mix 28 1665935 59498 3.6881 2.252e-08 ***Residuals 224 3613660 16132

mem1 <- lmer(ANPP~ SR.log2 +(1|Site)+(1|Block)+(1|Mix)+(1|Site:Mix), data= Biodepth)

Df Sum Sq Mean Sq F valueSR.log2 1 772347 772347 47.544

ls1 <- lm(terms(ANPP~ Site+Block+SR.log2+Site:SR.log2+Mix+Site:Mix, keep.order= TRUE), data= Biodepth)

Example – BIODEPTHDegrees of freedom

Df Sum Sq Mean Sq F value Pr(>F) Site 7 14287153 2041022 126.5168 < 2.2e-16 ***Block 7 273541 39077 2.4223 0.02072 * SR.log2 1 5033290 5033290 311.9986 < 2.2e-16 ***Site:SR.log2 7 1118255 159751 9.9025 9.316e-11 ***Mix 189 17096361 90457 5.6072 < 2.2e-16 ***Site:Mix 28 1665935 59498 3.6881 2.252e-08 ***Residuals 224 3613660 16132

mem1 <- lmer(ANPP~ SR.log2 +(1|Site)+(1|Block)+(1|Mix), data= Biodepth) mem2 <- lmer(ANPP~ SR.log2 +(1|Site)+(1|Block), data= Biodepth)

Df AIC BIC logLik Chisq Chi Df Pr(>Chisq) mem2 5 6394.7 6415.4 -3192.4 mem1 6 6277.3 6302.1 -3132.6 119.41 1 < 2.2e-16 ***

Uses only 6 DF instead of 7+7+1+7+189+28=239!!!

Example – BIODEPTHError terms

Spehn et al. 2005

Example – BIODEPTHError terms

Spehn et al. 2005

The Great Mixed Model Muddle

F tests for fixed-effects, random-effects and mixed-effects models

Source A and B fixed A and B randomA fixed, B randomRestricted version

A fixed, B randomUnrestricted version

A MSA

MSResidual

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B MSB

MSResidual

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AB MSAB

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Example – BIODEPTHError terms

mem4 <- lme (ANPP~ SR.log2+Block, random=list(Site = ~1, Mix = ~1), data= Biodepth, na.action=na.omit)

numDF denDF F-value p-value(Intercept) 1 413 1781.8981 <.0001SR.log2 1 413 94.3233 <.0001Site 7 7 33.4856 1e-04

mem3 <- lme (ANPP~ SR.log2+Site, random=list(Block = ~1, Mix = ~1), data= Biodepth, na.action=na.omit)

numDF denDF F-value p-value(Intercept) 1 224 457.6593 <.0001SR.log2 1 224 57.2133 <.0001Block 7 217 6.0297 <.0001

Crossed-random effects – to be considered with care. Not handled by lme, lmer would do the job, but no P-values!!!

Crossed random effects

Block

Species compossition

ShrinkageNo pooling (lmList(ANPP~ SR.log2|Site, data= Biodepth))

ShrinkageNo pooling (lmList(ANPP~ SR.log2|Site, data= Biodepth))

Complete pooling (lm(ANPP~ SR.log2, data= Biodepth))

ShrinkageNo pooling (lmList(ANPP~ SR.log2|Site, data= Biodepth))

Complete pooling (lm(ANPP~ SR.log2, data= Biodepth))

Mixed-effects models (lmer(ANPP~ SR.log2 +(SR.log2|Site)+(1|Block)+(1|Mix)+(1|Site:Mix), data= Biodepth))

Mixed-effects model analysis: Overview

• Estimate and test the fixed effects just as in fixed-effects analysis but it makes no sense to estimate and test the random effects.

• Instead the random effects are treated as in variance components analysis: we predict and test the variance components (often expressed in standard deviation form so that they are back on the original scale of the response).

• The covariances (or correlations) between random effects can also be estimated – between repeated measurement locations or times for example

• Fixed effects and random effects are tested differently.