Workshop 9.1: Mixed effects models

Transcription

1 -1- Workshop 91: Mixed effects models Murray Logan October 10, 2016 Table of contents 1 Non-independence - part Non-independence - part 2 11 Linear models Homogeneity of variance σ y i = β 0 + β 1 x }{{} i +ε i ε i N (0,σ 2 0 σ ) V = cov = 2 }{{} σ 2 Linearity Normality 0 σ 2 Zero covariance (=independence) How maximize power? 12 Linear models 1 Homogeneity of variance σ y i = β 0 + β 1 x }{{} i +ε i ε i N (0,σ 2 0 σ ) V = cov = 2 }{{} σ 2 Linearity Normality 0 σ 2 Zero covariance (=independence) How maximize power? increase replication 1

2 -2- add covariates (account for conditions) block (control conditions) 13 Hierarchical models Q5 Q6 To increase power - without more sites (replicates) 14 Hierarchical models Subreplicates - yet not independent

3 -3-15 Hierarchical models 151 Nested design Treatment Site Quadrat 16 Hierarchical models 161 Nested design

4 -4- Treatment (F) Site (R) Quadrat (R) 17 Hierarchical models 171 Nested design Treatment (F) Site (R) Quadrat (R)

5 -5-18 Hierarchical models Q7 Q6 0 4 Q8 5 Q5 Q To increase power 19 Hierarchical models Block treatments together - yet not independent 110 Hierarchical models

6 Randomized complete block Block Treatment Quadrat 111 Hierarchical models 1111 Randomized complete block

7 -7- Block (R) Treatment (F) Quadrat (R) 112 Hierarchical models 1121 Randomized complete block Block (R) Treatment (F) Quadrat (R) 113 Linear modelling assumptions Normality Homogeneity of Variance Linearity Independence

8 -8- Homogeneity of variance σ y i = β 0 + β 1 x }{{} i +ε i ε i N (0,σ 2 0 σ ) V = cov = 2 }{{} σ 2 Linearity Normality 0 σ 2 Zero covariance (=independence) 114 Non-independence one response is triggered by another temporal/spatial autocorrelation nested (hierarchical) design structures 1 Homogeneity of variance σ y i = β 0 + β 1 x }{{} i +ε i ε i N (0,σ 2 0 σ ) V = cov = 2 }{{} σ 2 Linearity Normality 0 σ 2 Zero covariance (=independence) 115 Hierarchical models linear model with separate covariance structure per block fixed and random factors (effects) 116 Example 1 > datarcb <- readcsv('/data/datarcbcsv') > head(datarcb) y x block Block Block Block Block Block Block1

9 Example > library(ggplot2) > ggplot(datarcb, aes(y=y, x=x)) + geom_point() + geom_smooth(method='lm') x y 118 Example > library(ggplot2) > ggplot(datarcb, aes(y=y, x=x,color=block))+geom_point()+ + geom_smooth(method='lm')

10 y block Block1 Block2 Block3 Block4 Block5 Block x 119 Example Simple linear regression - wrong > datarcblm <- lm(y~x, datarcb) Generalized least squares - more correct > library(nlme) > datarcbgls <- gls(y~x, datarcb, method='reml') 120 Example Model validation > plot(datarcbgls)

11 -11- Fitted values Standardized residuals Example > plot(residuals(datarcbgls, type='normalized') ~ + datarcb$block)

12 -12- residuals(datarcbgls, type = "normalized") Block1 Block2 Block3 Block4 Block5 Block6 datarcb$block - So what about ANCOVA 122 Example > library(ggplot2) > ggplot(datarcb, aes(y=y, x=x, color=block))+ + geom_smooth(method="lm")+geom_point()+theme_classic() 350 y block Block1 Block2 Block3 Block4 Block5 Block x

13 Example What if we add block as a predictor? (like ANCOVA) > library(nlme) > datarcbgls1 <- gls(y~x+block, datarcb, method='reml') > plot(datarcbgls) Standardized residuals Fitted values 124 Example > plot(residuals(datarcbgls1, type='normalized') ~ + datarcb$block)

14 -14- residuals(datarcbgls1, type = "normalized") Block1 Block2 Block3 Block4 Block5 Block6 datarcb$block 125 Example Looks good, but for INDEPENDENCE Can we deal with that with correlation structure? 126 Example Model in dependency structure > library(nlme) > datarcbgls2<-gls(y~x,datarcb, + correlation=corcompsymm(form=~1 block), + method="reml") > plot(residuals(datarcbgls2, type='normalized') ~ + fitted(datarcbgls2)) σ 2 ρ ρ Variance-covariance per Block:V = ρ σ 2 σ 2 ρ σ 2

15 fitted(datarcbgls2) residuals(datarcbgls2, type = "normalized") 127 Example > plot(residuals(datarcbgls2, type='normalized') ~ + datarcb$block)

16 -16- residuals(datarcbgls2, type = "normalized") Block1 Block3 Block5 datarcb$block 128 Example > summary(datarcbgls2) Generalized least squares fit by REML Model: y ~ x Data: datarcb AIC BIC loglik Correlation Structure: Compound symmetry Formula: ~1 block Parameter estimate(s): Rho Coefficients: Value StdError t-value p-value (Intercept) x Correlation: (Intr) x Standardized residuals:

17 -17- Min Med Max Residual standard error: Degrees of freedom: 60 total; 58 residual 129 Linear mixed effects model > datarcblme <- lme(y~x, random=~1 block, datarcb, + method='reml') > plot(datarcblme) 2 Standardized residuals Fitted values > plot(residuals(datarcblme, type='normalized') ~ fitted(datarcblme)) residuals(datarcblme, type = "normalized") fitted(datarcblme) > plot(residuals(datarcblme, type='normalized') ~ datarcb$block) residuals(datarcblme, type = "normalized") Block1 Block3 Block5 datarcb$block 130 Linear mixed effects model > summary(datarcblme) Linear mixed-effects model fit by REML Data: datarcb AIC BIC loglik Random effects: Formula: ~1 block (Intercept) Residual StdDev: Fixed effects: y ~ x Value StdError DF t-value p-value (Intercept)

18 -18- x Correlation: (Intr) x Standardized Within-Group Residuals: Min Med Max Number of Observations: 60 Number of Groups: Linear mixed effects model > anova(datarcblme) numdf dendf F-value p-value (Intercept) <0001 x <0001 > intervals(datarcblme) Approximate 95% confidence intervals Fixed effects: lower est upper (Intercept) x attr(,"label") [1] "Fixed effects:" Random Effects: Level: block lower est upper sd((intercept)) Within-group standard error: lower est upper Linear mixed effects model > vc<-asnumeric(asmatrix(varcorr(datarcblme))[,1]) > vc/sum(vc) [1] Linear mixed effects model > library(effects) > plot(alleffects(datarcblme, partialresiduals=true))

19 -19- x effect plot 360 y x 134 Linear mixed effects model > predict(datarcblme, newdata=dataframe(x=30:40),level=0) [1] [11] attr(,"label") [1] "Predicted values" 135 Linear mixed effects model > predict(datarcblme, newdata=dataframe(x=30:40, + block='block1'),level=1) Block1 Block1 Block1 Block1 Block1 Block1 Block1 Block1 Block1 Block1 Block attr(,"label") [1] "Predicted values" 136 Linear mixed effects model 1361 Summary figure Step 1 gather model coefficients > coefs <- fixef(datarcblme) > coefs (Intercept) x

20 Linear mixed effects model 1371 Summary figure Step 2 generate prediction model matrix > xs <- seq(min(datarcb$x), max(datarcb$x), l=100) > Xmat <- modelmatrix(~x, dataframe(x=xs)) > head(xmat) (Intercept) x Linear mixed effects model 1381 Summary figure Step 3 calculate predicted y > ys <- t(coefs %*% t(xmat)) > head(ys) [,1] Linear mixed effects model 1391 Summary figure Step 3 calculate confidence interval > SE <- sqrt(diag(xmat %*% vcov(datarcblme) %*% t(xmat))) > CI <- 2*SE > #OR > CI <- qt(0975,length(datarcb$x)-2)*se > datarcbpred <- dataframe(x=xs, fit=ys, se=se, + lower=ys-ci, upper=ys+ci) > head(datarcbpred) x fit se lower upper

21 Linear mixed effects model 1401 Summary figure Step 4 plot it > library(ggplot2) > ggplot(datarcbpred, aes(y=fit, x=x)) + + geom_ribbon(aes(ymin=lower,ymax=upper),fill='blue',alpha=02)+ + geom_line()+ + scale_y_continuous('y')+ + theme_classic()+ + theme(axistitlex=element_text(size=rel(125), vjust=-2), + axistitley=element_text(size=rel(125), vjust=2), + plotmargin=unit(c(01,01,2,2),'lines')) > > ## plot(fit~x, data=datarcbpred,type='n',axes=f, ann=f) > ## points(y~x, data=datarcb, pch=16, col='grey') > ## with(datarcbpred, polygon(c(x,rev(x)), c(lower, rev(upper)), > ## col="#0000ff50",border=false)) > ## lines(fit~x,data=datarcbpred) > ## lines(lower~x,data=datarcbpred, lty=2) > ## lines(upper~x,data=datarcbpred, lty=2) > ## axis(1) > ## mtext('x',1,line=3) > ## axis(2,las=1) > ## mtext('y',2,line=3) > ## box(bty='l') Linear mixed effects model 141 Linear mixed effects model 1411 Summary figure 320 Y x 142 Linear mixed effects model

22 Summary figure Step 4 plot it (with partial observed values) > datarcb$res <- predict(datarcblme, level=1)+ + residuals(datarcblme) > > library(ggplot2) > ggplot(datarcbpred, aes(y=fit, x=x)) + + geom_point(data=datarcb, aes(y=res))+ + geom_ribbon(aes(ymin=lower,ymax=upper),fill='blue',alpha=02)+ + geom_line()+ + scale_y_continuous('y')+ + theme_classic()+ + theme(axistitlex=element_text(size=rel(125), vjust=-2), + axistitley=element_text(size=rel(125), vjust=2), + plotmargin=unit(c(01,01,2,2),'lines')) 143 Linear mixed effects model 1431 Summary figure Y x