Hypothesis testing in mixed linear models with a factorial design

Published

September 30, 2024

dat <- expand_grid(time = factor(1:3), group = factor(1:2), ID = 1:5) |> 
  mutate(hh = paste('time', time, 'group', group, sep = '_'))

X <- model.matrix(~ hh + factor(ID), data = dat)

dat <- dat |> 
  mutate(y = X %*% rnorm(10) + rnorm(30))


fm_1 <- lmer(y ~ time * group + (1 | ID), data = dat)
fm_2 <- lmer(y ~ hh - 1 + (1 | ID), data = dat)

fm_1

Linear mixed model fit by REML ['lmerMod']
Formula: y ~ time * group + (1 | ID)
   Data: dat
REML criterion at convergence: 92.9467
Random effects:
 Groups   Name        Std.Dev.
 ID       (Intercept) 0.9389  
 Residual             1.2076  
Number of obs: 30, groups:  ID, 5
Fixed Effects:
 (Intercept)         time2         time3        group2  time2:group2  
    -0.08427       2.12602      -0.21546      -0.15532      -3.74877  
time3:group2  
    -0.65262

fm_2

Linear mixed model fit by REML ['lmerMod']
Formula: y ~ hh - 1 + (1 | ID)
   Data: dat
REML criterion at convergence: 92.9467
Random effects:
 Groups   Name        Std.Dev.
 ID       (Intercept) 0.9389  
 Residual             1.2076  
Number of obs: 30, groups:  ID, 5
Fixed Effects:
hhtime_1_group_1  hhtime_1_group_2  hhtime_2_group_1  hhtime_2_group_2  
        -0.08427          -0.23959           2.04174          -1.86234  
hhtime_3_group_1  hhtime_3_group_2  
        -0.29973          -1.10767

AIC(fm_1)

[1] 108.9467

AIC(fm_2)

[1] 108.9467

summary(fm_1)

Linear mixed model fit by REML ['lmerMod']
Formula: y ~ time * group + (1 | ID)
   Data: dat

REML criterion at convergence: 92.9

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.7375 -0.5930  0.2371  0.6309  1.1804 

Random effects:
 Groups   Name        Variance Std.Dev.
 ID       (Intercept) 0.8816   0.9389  
 Residual             1.4582   1.2076  
Number of obs: 30, groups:  ID, 5

Fixed effects:
             Estimate Std. Error t value
(Intercept)  -0.08427    0.68408  -0.123
time2         2.12602    0.76373   2.784
time3        -0.21546    0.76373  -0.282
group2       -0.15532    0.76373  -0.203
time2:group2 -3.74877    1.08007  -3.471
time3:group2 -0.65262    1.08007  -0.604

Correlation of Fixed Effects:
            (Intr) time2  time3  group2 tm2:g2
time2       -0.558                            
time3       -0.558  0.500                     
group2      -0.558  0.500  0.500              
time2:grop2  0.395 -0.707 -0.354 -0.707       
time3:grop2  0.395 -0.354 -0.707 -0.707  0.500

summary(fm_2)

Linear mixed model fit by REML ['lmerMod']
Formula: y ~ hh - 1 + (1 | ID)
   Data: dat

REML criterion at convergence: 92.9

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.7375 -0.5930  0.2371  0.6309  1.1804 

Random effects:
 Groups   Name        Variance Std.Dev.
 ID       (Intercept) 0.8816   0.9389  
 Residual             1.4582   1.2076  
Number of obs: 30, groups:  ID, 5

Fixed effects:
                 Estimate Std. Error t value
hhtime_1_group_1 -0.08427    0.68408  -0.123
hhtime_1_group_2 -0.23959    0.68408  -0.350
hhtime_2_group_1  2.04174    0.68408   2.985
hhtime_2_group_2 -1.86234    0.68408  -2.722
hhtime_3_group_1 -0.29973    0.68408  -0.438
hhtime_3_group_2 -1.10767    0.68408  -1.619

Correlation of Fixed Effects:
            h_1__1 h_1__2 h_2__1 h_2__2 h_3__1
hhtm_1_gr_2 0.377                             
hhtm_2_gr_1 0.377  0.377                      
hhtm_2_gr_2 0.377  0.377  0.377               
hhtm_3_gr_1 0.377  0.377  0.377  0.377        
hhtm_3_gr_2 0.377  0.377  0.377  0.377  0.377

fixef(fm_1)

 (Intercept)        time2        time3       group2 time2:group2 time3:group2 
 -0.08427219   2.12601590  -0.21545931  -0.15531723  -3.74876543  -0.65261665

fixef(fm_2)

hhtime_1_group_1 hhtime_1_group_2 hhtime_2_group_1 hhtime_2_group_2 
     -0.08427219      -0.23958942       2.04174371      -1.86233895 
hhtime_3_group_1 hhtime_3_group_2 
     -0.29973150      -1.10766538

glht(fm_1, c('time2 == 0',
             'time3 == 0',
             'time2 + group2 + time2:group2 == 0',
             'time3 + group2 + time3:group2 == 0'))


     General Linear Hypotheses

Linear Hypotheses:
                                   Estimate
time2 == 0                           2.1260
time3 == 0                          -0.2155
time2 + group2 + time2:group2 == 0  -1.7781
time3 + group2 + time3:group2 == 0  -1.0234

glht(fm_2, c('hhtime_2_group_1 - hhtime_1_group_1 == 0',
             'hhtime_3_group_1 - hhtime_1_group_1 == 0',
             'hhtime_2_group_2 - hhtime_1_group_2 == 0',
             'hhtime_3_group_2 - hhtime_1_group_2 == 0'))


     General Linear Hypotheses

Linear Hypotheses:
                                         Estimate
hhtime_2_group_1 - hhtime_1_group_1 == 0   2.1260
hhtime_3_group_1 - hhtime_1_group_1 == 0  -0.2155
hhtime_2_group_2 - hhtime_1_group_2 == 0  -1.6227
hhtime_3_group_2 - hhtime_1_group_2 == 0  -0.8681