fitting LMEMs for repeated measures with no correlation between intercept and slope

For a simulation study, I contrast the power of different LMEMs for repeated measures. To get p-values, I use likelihood ratio tests where I compare a model including a fixed treatment effect with one that has the same random effects structure but without having the fixed treatment effect. I want to specify a model in which random intercept and slope are allowed to correlate and one in which it is not allowed. But, when I extract the p-values of both models, it appears that they are exactly the same. Why? I am not an expert for mixed models but that seems weird. Here's my code using lmer syntax:

# correlation allowed
ml3 <- lmer(rt ~ treatment + (1+treatment|subject),data=df)
ml0 <- lmer(rt ~ 1 + (1+treatment|subject),data=df)
lrt <- anova(ml3,ml0)
pVal <- lrt$"Pr(>Chisq)"[2] # no correlation ml2 <- lmer(rt ~ treatment + (1|subject) + (0+treatment|subject) ,data=df) ml0 <- lmer(rt ~ 1 + (1|subject) + (0 + treatment|subject),data=df) lrt <- anova(ml2,ml0) pVal <- lrt$"Pr(>Chisq)"[2]


Data:

dput(DF)
structure(list(subject = structure(c(1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L,
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 7L, 7L,
7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L,
7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 8L, 8L, 8L, 8L,
8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L,
8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 9L, 9L, 9L, 9L, 9L, 9L,
9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L,
9L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 10L, 10L, 10L, 10L, 10L, 10L,
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L,
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 11L, 11L,
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L,
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L,
11L, 11L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L,
12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L, 12L,
12L, 12L, 12L, 12L, 12L, 12L, 13L, 13L, 13L, 13L, 13L, 13L, 13L,
13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L,
13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 13L, 14L, 14L, 14L,
14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L,
14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L, 14L,
14L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L,
15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L, 15L,
15L, 15L, 15L, 15L, 15L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L,
16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L,
16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 16L, 17L, 17L, 17L, 17L,
17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L,
17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L, 17L,
18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L,
18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L, 18L,
18L, 18L, 18L, 18L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L,
19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L,
19L, 19L, 19L, 19L, 19L, 19L, 19L, 19L, 20L, 20L, 20L, 20L, 20L,
20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L,
20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 20L, 21L,
21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L,
21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L, 21L,
21L, 21L, 21L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L,
22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L, 22L,
22L, 22L, 22L, 22L, 22L, 22L, 22L, 23L, 23L, 23L, 23L, 23L, 23L,
23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L,
23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 23L, 24L, 24L,
24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L,
24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L, 24L,
24L, 24L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L,
25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L, 25L,
25L, 25L, 25L, 25L, 25L, 25L, 26L, 26L, 26L, 26L, 26L, 26L, 26L,
26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L,
26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 26L, 27L, 27L, 27L,
27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L,
27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L, 27L,
27L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L,
28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L, 28L,
28L, 28L, 28L, 28L, 28L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L,
29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L,
29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 29L, 30L, 30L, 30L, 30L,
30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L,
30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L
), .Label = c("1", "2", "3", "4", "5", "6", "7", "8", "9", "10",
"11", "12", "13", "14", "15", "16", "17", "18", "19", "20", "21",
"22", "23", "24", "25", "26", "27", "28", "29", "30"), class = "factor"),
treatment = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L), .Label = c("1", "2", "3"), class = "factor"),
rt = c(551.798792586772, 693.014255128461, 715.599061613616,
670.119831344829, 777.748610260388, 736.018489208224, 636.791011800404,
864.593711496912, 604.529352905588, 596.673178487122, 860.858066937491,
717.975814131377, 531.672833100059, 571.150454430927, 644.315598150879,
601.914697283216, 583.92746647402, 702.714068138085, 660.346853172676,
541.292786332608, 608.233103066463, 740.593415976325, 686.059921551164,
706.522723402261, 567.648255604935, 596.111352599386, 625.779084220279,
752.776987343973, 922.314285125596, 720.736074757203, 768.585671134519,
539.657760625667, 431.193030969184, 739.341430343149, 581.505474510558,
485.905153431116, 524.085545405872, 876.566370460358, 631.259754679943,
587.887351105621, 624.365050240473, 642.528438460209, 440.661792577731,
517.142782023978, 705.840003729944, 557.122142924839, 645.711579229236,
477.292943229673, 578.522058679457, 623.879658296107, 480.855063147831,
622.295733392922, 611.623490658329, 594.974733982977, 546.239019853272,
551.638287622872, 567.791819285002, 539.239628365136, 541.328446070423,
609.931976806498, 549.492601324081, 479.862984098331, 592.411150981731,
466.224011597179, 489.388878789762, 565.187127159354, 806.196199699478,
565.001071713299, 449.529036961143, 446.824243314547, 357.993777663337,
370.054045045062, 546.443479822161, 473.894296409884, 335.821704077378,
370.498649523398, 486.052525038318, 436.53292033153, 359.637460079864,
333.146018287273, 597.894114487158, 551.993792800734, 518.563432886515,
513.629383189428, 572.062676720248, 366.611317255576, 353.934207291842,
492.273303938824, 414.632984933654, 456.987565377718, 585.524844348671,
453.577328112778, 665.072248688078, 459.204631254183, 452.028605442515,
422.731506299078, 522.84363619892, 771.586286956136, 478.422495080758,
530.925498291748, 457.030618882822, 446.313635696342, 537.708665959068,
815.929601138346, 460.420404065423, 603.027278932425, 538.526470664698,
571.491835856551, 567.234631106499, 450.878624452358, 650.340961680322,
598.829083718722, 620.85411026516, 573.498196791879, 519.953442801483,
1143.14393274202, 505.028670926264, 685.875665196364, 605.316954852204,
645.269429082978, 678.192056130499, 605.671978651269, 650.564580984954,
641.331733928499, 687.164180542278, 613.873849203194, 789.829709495785,
560.793473918547, 707.169378961089, 680.753196215641, 659.262985906231,
600.712008959484, 662.275074291484, 661.346206480403, 568.31000899618,
661.439508442242, 761.227769640367, 699.901658463283, 631.422448673388,
734.257735977184, 585.776345181453, 714.587957176744, 893.931699334816,
586.343993838929, 664.205207596859, 828.003782888565, 906.448165648461,
584.196768113385, 747.575564348236, 687.698668648395, 398.221092516595,
490.332613905338, 502.683377386602, 451.168200674477, 620.606534311108,
369.820458042713, 429.483129392912, 628.153937257066, 476.31856841443,
608.016880363378, 402.588700424079, 460.302916138576, 341.209753425326,
600.165531243784, 454.003777748405, 589.089266888531, 504.033320854066,
399.871492203846, 421.426563579218, 423.375093277487, 587.614013919312,
689.18637317583, 652.10069672704, 553.995320740249, 570.86533170596,
636.399559100471, 801.517490092303, 653.425223465164, 684.914139340214,
639.418654954543, 555.718100869331, 677.062768399616, 579.433200999322,
561.757369869387, 672.316124102021, 701.108131071079, 588.129426947175,
438.090900053591, 562.520435558598, 610.177372103278, 564.672192806652,
552.305226838045, 586.912866128373, 872.521433158083, 654.971253063189,
575.068762782096, 784.098108527601, 648.265348029739, 590.541840435637,
552.569131260877, 554.840084955354, 582.798864712891, 573.196470707737,
512.123960183202, 579.838037093289, 710.216216611067, 779.786949219207,
615.995650564573, 549.096392807351, 600.781394864656, 415.016144351654,
765.924387691343, 401.541419432177, 436.050367769487, 536.508634405116,
445.112952169149, 478.003493101049, 509.819044087032, 490.265270275681,
594.667876389766, 781.844411855516, 827.832351086729, 379.401116898897,
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That is quite simple: For your data, estimating the correlation, doesn't improve the fit at all. Hence, the p-values need to be identically.

> deviance(ml3)
[1] 10910.52
> deviance(ml2)
[1] 10910.52


btw, when using anova on two lmer objects which are fitted with REML (instead of ML), anova per default refits the models. To already fit the models with ML use lmer(..., REML = FALSE) or to compare the two REML models (which is not recommended when the fixed effects structure varies), use anova(..., refit = FALSE).

• thanks, but why does estimating the correlation not improve the fit for my data? if, say, there is no correlation, then the model not assuming one should have better fit, shouldn't it? – beginneR Jun 13 '14 at 7:35
• No, you are wrong. Adding a parameter cannot decrease the fit (never)! The fit index just measures the distance of the predictions from the actually observed data. If adding a parameter does not decrease the distance of predictions and data, the fit index just remains the same. Which happens in your case. Only p-values based on the $\chi^2$ distribution or measures based on complexity such as AIC or BIC change as they take fit and the number of parameters into account. – Henrik Jun 13 '14 at 9:19
• So, what would you recommend? Just use one of both models or use another criterion? – beginneR Jun 13 '14 at 9:50
• I do not fully remember the details of your simulation, but if you do not simulate data with correlation, it should not make sense to add or test it as a parameter. – Henrik Jun 13 '14 at 10:26
• ok thanks, I'll adjust my code so that there is a correlation between both. – beginneR Jun 13 '14 at 11:51