Statistics for Laboratory Scientists ( 140.615 )
Linear Mixed Effects - Longitudinal Data Analysis
The data used in the following examples are in the SPH.140.615 package.
library(SPH.140.615)Use the function install.packages() if you haven’t installed the following packages, i.e. use install.packages(“lme4”) etc in the code chunk. Alternatively, hit the ‘Knit’ button, and you should be prompted about installing the packages.
library(lattice)
library(Matrix)
library(lme4)Example 1
A figure of the data.
xyplot(y ~ x|id, data=dat.ex1, type=c("p","r"), layout=c(6,2), as.table=TRUE, xlab="Time", ylab="Response")
We can think of the data being described by a “random intercept” model, i.e.
Yij = b0 + Bi + b1xj + εij,
where i referes to the subject and xj to the jth time point.
lme.fit <- lmer(y ~ x + (1|id), data=dat.ex1)
summary(lme.fit)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (1 | id)
## Data: dat.ex1
##
## REML criterion at convergence: 13
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.25586 -0.67363 0.06082 0.65787 2.17048
##
## Random effects:
## Groups Name Variance Std.Dev.
## id (Intercept) 1.03232 1.0160
## Residual 0.03483 0.1866
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.049705 0.295603 67.83
## x -0.204096 0.005931 -34.41
##
## Correlation of Fixed Effects:
## (Intr)
## x -0.110How to extract some of the statistics.
fixef(lme.fit)## (Intercept) x
## 20.0497051 -0.2040963vcov(lme.fit)## 2 x 2 Matrix of class "dpoMatrix"
## (Intercept) x
## (Intercept) 0.0873809165 -0.0001934779
## x -0.0001934779 0.0000351778diag(vcov(lme.fit))## (Intercept) x
## 0.0873809165 0.0000351778sqrt(diag(vcov(lme.fit)))## (Intercept) x
## 0.295602633 0.005931088fixef(lme.fit)/sqrt(diag(vcov(lme.fit)))## (Intercept) x
## 67.82654 -34.41128Testing the regression, fitting a reduced model with slope zero.
lme.fit.red <- lmer(y ~ 1 + (1|id), dat.ex1)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ 1 + (1 | id)
## Data: dat.ex1
##
## REML criterion at convergence: 273.6
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.03318 -0.81320 0.07943 0.76815 1.91393
##
## Random effects:
## Groups Name Variance Std.Dev.
## id (Intercept) 0.9942 0.9971
## Residual 0.4163 0.6452
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 18.9272 0.2938 64.42anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex1
## Models:
## lme.fit.red: y ~ 1 + (1 | id)
## lme.fit: y ~ x + (1 | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 3 278.920 287.282 -136.460 272.920
## lme.fit 4 11.951 23.101 -1.976 3.951 268.97 1 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Note that the ‘lmer’ function does not return p-values for the fixed effects in the output. See the end of this markdown how to use the ‘lmerTest’ package to add those p-values.
Example 2
A figure of the data.
xyplot(y ~ x|id, data=dat.ex2, type=c("p","r"), layout=c(6,2), as.table=TRUE, xlab="Time", ylab="Response")
We can think of the data being described by a model with a random intercept and a random slope, i.e.
Yij = b0 + Bi + b1xj + Cixj + εij,
where i referes to the subject and xj to the jth time point.
lme.fit <- lmer(y ~ x + (x|id), dat.ex2)
summary(lme.fit)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (x | id)
## Data: dat.ex2
##
## REML criterion at convergence: 33.1
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.08758 -0.50793 -0.01624 0.54409 2.27667
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.880584 0.93839
## x 0.003622 0.06018 -0.25
## Residual 0.034004 0.18440
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.00217 0.27332 73.182
## x -0.17860 0.01834 -9.741
##
## Correlation of Fixed Effects:
## (Intr)
## x -0.270Do we need the random slope?
lme.fit.red <- lmer(y ~ x + (1|id), dat.ex2)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (1 | id)
## Data: dat.ex2
##
## REML criterion at convergence: 77
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.12097 -0.60125 0.01364 0.51411 2.38342
##
## Random effects:
## Groups Name Variance Std.Dev.
## id (Intercept) 0.83364 0.9130
## Residual 0.06472 0.2544
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.002171 0.268304 74.55
## x -0.178603 0.008086 -22.09
##
## Correlation of Fixed Effects:
## (Intr)
## x -0.166anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex2
## Models:
## lme.fit.red: y ~ x + (1 | id)
## lme.fit: y ~ x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 4 76.375 87.525 -34.188 68.375
## lme.fit 6 38.068 54.793 -13.034 26.068 42.307 2 6.504e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Note the degrees of freedom - there is also a correlation parameter between the random effects in the full model.
Example 1 continued
The same data as above, but now with a treatment and a control group.
xyplot(y~x|group:id,data=dat.ex1,type=c("p","r"),layout=c(6,2),as.table=TRUE,
xlab="Time",ylab="Response",strip=strip.custom(var.name=c("id","group")))
We can again think of the data being described by a random intercept model, with possible differences in the average intercept between treatment and control groups:
Yij = b0 + c0I{subject i is in treatment} + Bi + b1xj + εij,
where i referes to the subject and xj to the jth time point, and I{subject i is in treatment} indicates whether subject i is in the treatment or control arm.
lme.fit <- lmer(y ~ group + x + (1|id), data=dat.ex1)
summary(lme.fit)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ group + x + (1 | id)
## Data: dat.ex1
##
## REML criterion at convergence: 10.3
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.26018 -0.66584 0.06588 0.66638 2.17797
##
## Random effects:
## Groups Name Variance Std.Dev.
## id (Intercept) 0.93697 0.9680
## Residual 0.03483 0.1866
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 19.642547 0.397247 49.447
## grouptreatment 0.814317 0.559895 1.454
## x -0.204096 0.005931 -34.411
##
## Correlation of Fixed Effects:
## (Intr) grptrt
## grouptrtmnt -0.705
## x -0.082 0.000Testing for treatment effect:
lme.fit.red <- lmer(y ~ x + (1|id), dat.ex1)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (1 | id)
## Data: dat.ex1
##
## REML criterion at convergence: 13
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.25586 -0.67363 0.06082 0.65787 2.17048
##
## Random effects:
## Groups Name Variance Std.Dev.
## id (Intercept) 1.03232 1.0160
## Residual 0.03483 0.1866
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.049705 0.295603 67.83
## x -0.204096 0.005931 -34.41
##
## Correlation of Fixed Effects:
## (Intr)
## x -0.110anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex1
## Models:
## lme.fit.red: y ~ x + (1 | id)
## lme.fit: y ~ group + x + (1 | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 4 11.951 23.101 -1.9757 3.9514
## lme.fit 5 11.649 25.586 -0.8244 1.6488 2.3026 1 0.1292Example 2 continued
The same data as above, but now with a treatment and a control group.
xyplot(y ~ x|group:id, data=dat.ex2, type=c("p","r"), layout=c(6,2), as.table=TRUE,
xlab="Time", ylab="Response", strip=strip.custom(var.name=c("id","group")))
We can think of the data being described by a model with a random intercept and a random slope, with possible differences in the average intercept and slope between treatment and control groups.
Yij = b0 + c0I{subject i is in treatment} + Bi + b1xj + c1I{subject i is in treatment}xj + Cixj + εij,
where i referes to the subject and xj to the jth time point, and I{subject i is in treatment} indicates whether subject i is in the treatment or control arm.
lme.fit <- lmer(y ~ group*x + (x|id), dat.ex2)
summary(lme.fit)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ group * x + (x | id)
## Data: dat.ex2
##
## REML criterion at convergence: 34.6
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.04693 -0.51271 -0.02664 0.59745 2.25953
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.803667 0.89648
## x 0.003463 0.05885 -0.11
## Residual 0.034005 0.18440
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 19.62970 0.36958 53.114
## grouptreatment 0.74493 0.52267 1.425
## x -0.15696 0.02542 -6.176
## grouptreatment:x -0.04329 0.03594 -1.204
##
## Correlation of Fixed Effects:
## (Intr) grptrt x
## grouptrtmnt -0.707
## x -0.145 0.102
## grptrtmnt:x 0.102 -0.145 -0.707Testing for differences in intercepts:
lme.fit.red <- lmer(y ~ x + group:x + (x|id), dat.ex2)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + group:x + (x | id)
## Data: dat.ex2
##
## REML criterion at convergence: 37.1
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.05343 -0.51777 -0.02044 0.58514 2.26374
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.880517 0.93836
## x 0.003471 0.05891 -0.12
## Residual 0.034005 0.18440
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.00217 0.27331 73.185
## x -0.16066 0.02529 -6.352
## x:grouptreatment -0.03588 0.03556 -1.009
##
## Correlation of Fixed Effects:
## (Intr) x
## x -0.108
## x:grptrtmnt 0.000 -0.703anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex2
## Models:
## lme.fit.red: y ~ x + group:x + (x | id)
## lme.fit: y ~ group * x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 7 39.094 58.606 -12.547 25.094
## lme.fit 8 38.875 61.175 -11.437 22.875 2.2192 1 0.1363Testing for differences in slopes:
lme.fit.red <- lmer(y ~ group + x + (x|id), dat.ex2)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ group + x + (x | id)
## Data: dat.ex2
##
## REML criterion at convergence: 31.2
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.07811 -0.51819 -0.01486 0.55877 2.28594
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.804338 0.89685
## x 0.003622 0.06018 -0.12
## Residual 0.034005 0.18440
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 19.67526 0.36771 53.507
## grouptreatment 0.65382 0.51715 1.264
## x -0.17860 0.01834 -9.741
##
## Correlation of Fixed Effects:
## (Intr) grptrt
## grouptrtmnt -0.703
## x -0.105 0.000anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex2
## Models:
## lme.fit.red: y ~ group + x + (x | id)
## lme.fit: y ~ group * x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 7 38.500 58.013 -12.250 24.500
## lme.fit 8 38.875 61.175 -11.437 22.875 1.6256 1 0.2023Testing jointly for differences in intercepts and slopes:
lme.fit.red <- lmer(y ~ x + (x|id), dat.ex2)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (x | id)
## Data: dat.ex2
##
## REML criterion at convergence: 33.1
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.08758 -0.50793 -0.01624 0.54409 2.27667
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.880584 0.93839
## x 0.003622 0.06018 -0.25
## Residual 0.034004 0.18440
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.00217 0.27332 73.182
## x -0.17860 0.01834 -9.741
##
## Correlation of Fixed Effects:
## (Intr)
## x -0.270anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex2
## Models:
## lme.fit.red: y ~ x + (x | id)
## lme.fit: y ~ group * x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 6 38.068 54.793 -13.034 26.068
## lme.fit 8 38.875 61.175 -11.437 22.875 3.1936 2 0.2025Example 3
An example with large differences in slopes, comparing treatment and control group.
xyplot(y ~ x|group:id, data=dat.ex3, type=c("p","r"), layout=c(6,2),as.table=TRUE,
xlab="Time" ,ylab="Response", strip=strip.custom(var.name=c("id","group")))
As above, we can think of the data being described by a model with a random intercept and a random slope, with possible differences in the average intercept and slope between treatment and control groups.
Yij = b0 + c0I{subject i is in treatment} + Bi + b1xj + c1I{subject i is in treatment}xj + Cixj + εij,
where i referes to the subject and xj to the jth time point, and I{subject i is in treatment} indicates whether subject i is in the treatment or control arm.
lme.fit <- lmer(y ~ group*x + (x|id), dat.ex3)
summary(lme.fit)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ group * x + (x | id)
## Data: dat.ex3
##
## REML criterion at convergence: 54.4
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.22100 -0.63843 0.02772 0.70797 2.01538
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.874956 0.9354
## x 0.001376 0.0371 -0.24
## Residual 0.044818 0.2117
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 19.67384 0.38641 50.915
## grouptreatment 0.76479 0.54646 1.400
## x -0.06544 0.01789 -3.658
## grouptreatment:x -0.23799 0.02530 -9.408
##
## Correlation of Fixed Effects:
## (Intr) grptrt x
## grouptrtmnt -0.707
## x -0.271 0.192
## grptrtmnt:x 0.192 -0.271 -0.707Testing for differences in intercepts:
lme.fit.red <- lmer(y ~ x + group:x + (x|id), dat.ex3)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + group:x + (x | id)
## Data: dat.ex3
##
## REML criterion at convergence: 57
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.18712 -0.63842 0.02218 0.71625 2.03571
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.953040 0.97624
## x 0.001389 0.03727 -0.25
## Residual 0.044818 0.21170
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.05623 0.28489 70.400
## x -0.07023 0.01759 -3.994
## x:grouptreatment -0.22840 0.02435 -9.380
##
## Correlation of Fixed Effects:
## (Intr) x
## x -0.203
## x:grptrtmnt 0.000 -0.692anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex3
## Models:
## lme.fit.red: y ~ x + group:x + (x | id)
## lme.fit: y ~ group * x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 7 57.506 77.019 -21.753 43.506
## lme.fit 8 57.360 79.660 -20.680 41.360 2.1464 1 0.1429Testing for differences in slopes:
lme.fit.red <- lmer(y ~ x + group + (x|id), dat.ex3)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + group + (x | id)
## Data: dat.ex3
##
## REML criterion at convergence: 74
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.07830 -0.66540 -0.03376 0.69802 2.07654
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 1.39784 1.1823
## x 0.01665 0.1290 -0.64
## Residual 0.04482 0.2117
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.37011 0.43291 47.054
## x -0.18443 0.03785 -4.873
## grouptreatment -0.62775 0.52604 -1.193
##
## Correlation of Fixed Effects:
## (Intr) x
## x -0.512
## grouptrtmnt -0.608 0.000anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex3
## Models:
## lme.fit.red: y ~ x + group + (x | id)
## lme.fit: y ~ group * x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 7 82.811 102.32 -34.405 68.811
## lme.fit 8 57.360 79.66 -20.680 41.360 27.451 1 1.611e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Testing jointly for differences in intercepts and slopes:
lme.fit.red <- lmer(y ~ x + (x|id), dat.ex3)
summary(lme.fit.red)## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (x | id)
## Data: dat.ex3
##
## REML criterion at convergence: 74.7
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.10794 -0.64942 -0.02818 0.69732 2.06154
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.95302 0.9762
## x 0.01665 0.1290 -0.45
## Residual 0.04482 0.2117
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 20.05623 0.28489 70.400
## x -0.18443 0.03785 -4.873
##
## Correlation of Fixed Effects:
## (Intr)
## x -0.463anova(lme.fit,lme.fit.red)## refitting model(s) with ML (instead of REML)## Data: dat.ex3
## Models:
## lme.fit.red: y ~ x + (x | id)
## lme.fit: y ~ group * x + (x | id)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## lme.fit.red 6 80.983 97.708 -34.491 68.983
## lme.fit 8 57.360 79.660 -20.680 41.360 27.623 2 1.004e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Linear mixed effects: longitudinal data analysis with lmerTest
The ‘lmerTest’ package overwrites the ‘lmer’ function in the ‘lme4’ package, and adds p-values to the fixed effects in the output.
library(lmerTest)##
## Attaching package: 'lmerTest'## The following object is masked from 'package:lme4':
##
## lmer## The following object is masked from 'package:stats':
##
## stepExample 1
xyplot(y ~ x|group:id, data=dat.ex1, type=c("p","r"), layout=c(6,2), as.table=TRUE,
xlab="Time", ylab="Response", strip=strip.custom(var.name=c("id","group")))
lme.fit <- lmer(y ~ group + x + (1|id), data=dat.ex1)
summary(lme.fit)## Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
## Formula: y ~ group + x + (1 | id)
## Data: dat.ex1
##
## REML criterion at convergence: 10.3
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.26018 -0.66584 0.06588 0.66638 2.17797
##
## Random effects:
## Groups Name Variance Std.Dev.
## id (Intercept) 0.93697 0.9680
## Residual 0.03483 0.1866
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 19.642547 0.397247 10.136199 49.447 2.02e-13 ***
## grouptreatment 0.814317 0.559895 10.000000 1.454 0.176
## x -0.204096 0.005931 107.000000 -34.411 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) grptrt
## grouptrtmnt -0.705
## x -0.082 0.000Example 2
xyplot(y ~ x|group:id, data=dat.ex2, type=c("p","r"), layout=c(6,2), as.table=TRUE,
xlab="Time", ylab="Response", strip=strip.custom(var.name=c("id","group")))
lme.fit <- lmer(y ~ group*x + (x|id), dat.ex2)
summary(lme.fit)## Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
## Formula: y ~ group * x + (x | id)
## Data: dat.ex2
##
## REML criterion at convergence: 34.6
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.04693 -0.51271 -0.02664 0.59745 2.25953
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.803667 0.89648
## x 0.003463 0.05885 -0.11
## Residual 0.034005 0.18440
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 19.62970 0.36958 10.00011 53.114 1.35e-13 ***
## grouptreatment 0.74493 0.52267 10.00011 1.425 0.184538
## x -0.15696 0.02542 9.99973 -6.176 0.000105 ***
## grouptreatment:x -0.04329 0.03594 9.99973 -1.204 0.256145
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) grptrt x
## grouptrtmnt -0.707
## x -0.145 0.102
## grptrtmnt:x 0.102 -0.145 -0.707Example 3
xyplot(y ~ x|group:id, data=dat.ex3, type=c("p","r"), layout=c(6,2), as.table=TRUE,
xlab="Time", ylab="Response", strip=strip.custom(var.name=c("id","group")))
lme.fit <- lmer(y ~ group*x + (x|id), dat.ex3)
summary(lme.fit)## Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
## Formula: y ~ group * x + (x | id)
## Data: dat.ex3
##
## REML criterion at convergence: 54.4
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.22100 -0.63843 0.02772 0.70797 2.01538
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 0.874956 0.9354
## x 0.001376 0.0371 -0.24
## Residual 0.044818 0.2117
## Number of obs: 120, groups: id, 12
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 19.67384 0.38641 10.00024 50.915 2.06e-13 ***
## grouptreatment 0.76479 0.54646 10.00024 1.400 0.1919
## x -0.06544 0.01789 9.99992 -3.658 0.0044 **
## grouptreatment:x -0.23799 0.02530 9.99992 -9.408 2.77e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) grptrt x
## grouptrtmnt -0.707
## x -0.271 0.192
## grptrtmnt:x 0.192 -0.271 -0.707