对于模拟研究，我将不同LMEM的功效与重复测量结果进行对比。我想指定一个允许随机截距和斜率相关的模型和一个不允许它的模型。但是，当我比较两个模型时，看起来它们完全相同。当我在两个模型上运行anova时，没有相关性的模型甚至具有一个更大的自由度，而它应该是相反的。
# correlation allowed
ml1 <- lmer(rt ~ treatment + (1+treatment|subject),data=df)
Linear mixed model fit by REML ['lmerMod']
Formula: rt ~ treatment + (1 + treatment | subject)
Data: df
REML criterion at convergence: 10910.52
Random effects:
Groups Name Std.Dev. Corr
subject (Intercept) 72.87
treatment2 37.59 -0.17
treatment3 72.32 -0.65 0.73
Residual 98.22
Number of obs: 900, groups: subject, 30
Fixed Effects:
(Intercept) treatment2 treatment3
602.537 -8.971 33.803
# no correlation
ml2 <- lmer(rt ~ treatment + (1|subject) + (0+treatment|subject) ,data=df)
Linear mixed model fit by REML ['lmerMod']
Formula: rt ~ treatment + (1 | subject) + (0 + treatment | subject)
Data: df
REML criterion at convergence: 10910.52
Random effects:
Groups Name Std.Dev. Corr
subject (Intercept) 29.34
subject.1 treatment1 66.70
treatment2 70.31 0.85
treatment3 52.95 0.29 0.68
Residual 98.22
Number of obs: 900, groups: subject, 30
Fixed Effects:
(Intercept) treatment2 treatment3
602.537 -8.971 33.803
# anova
anova(ml1,ml2)
refitting model(s) with ML (instead of REML)
Data: df
Models:
ml1: rt ~ treatment + (1 + treatment | subject)
ml2: rt ~ treatment + (1 | subject) + (0 + treatment | subject)
Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
ml1 10 10950 10998 -5465.2 10930
ml2 11 10952 11005 -5465.2 10930 0 1 1
数据：
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,
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440.029552147731)), .Names = c("subject", "treatment", "rt"
), class = "data.frame", row.names = c(NA, -900L))
答案 0 :(得分：2)
source("lmer_nocorr_dat.R") ## get data
library(lme4)
ml1 <- lmer(rt ~ treatment + (1+treatment|subject),data=dd)
汇总函数，用于获取我们目前感兴趣的输出位。 （打印RE差异而非标准偏差，因为它们是相加的，因此更容易分辨哪些术语被组合或分离。）
sumfun <- function(x) {
cat("REML criterion:",REMLcrit(x),"\n")
invisible(print(VarCorr(x),comp="Variance"))
}
## REML criterion: 10910.52
## Groups Name Variance Corr
## subject (Intercept) 5309.9
## treatment2 1412.8 -0.168
## treatment3 5230.4 -0.652 0.733
## Residual 9648.1
对角线模型的原始尝试：
ml2 <- lmer(rt ~ treatment + (1|subject)+(0+treatment|subject),data=dd)
这会发出警告
Model failed to converge: degenerate Hessian with 1 negative eigenvalues
并且摘要输出表明我们仍然适合使用模型 相关性：
## REML criterion: 10910.52
## Groups Name Variance Corr
## subject (Intercept) 860.71
## subject.1 treatment1 4449.14
## treatment2 4942.74 0.851
## treatment3 2803.64 0.286 0.683
## Residual 9648.07
事实上，前两个术语的总和等于前一个模型中的第一个术语，进一步证明了不可识别性（我不太确定为什么其他两个方差是不同的，但我猜随机效应的总方差没有改变。）
有几种方法可以解决这个问题。
dd <- cbind(dd,model.matrix(~treatment,data=dd))
可能首选不同的参数化（例如~treatment-1），但如果相关性设置为零，则不对应相同的模型......
ml3 <- lmer(rt ~ treatment +
(0+`(Intercept)`|subject)+
(0+treatment2|subject)+
(0+treatment3|subject),
data=dd)
## REML criterion: 10926.79
## Groups Name Variance
## subject `(Intercept)` 4366.25
## subject.1 treatment2 860.97
## subject.2 treatment3 3233.28
## Residual 9761.63
REML标准已经上升（正如我们所预期的那样，对于灵活性较低的模型），尽管残差方差仍然相同。 getME(ml3,"theta")表明参数数量确实从6减少到3。
dfun <- update(ml1,devFunOnly=TRUE)
lwr <- getME(ml1,"lower")
covpos <- which(lwr<0)
parvec <- numeric(length(lwr))
dfunwrap <- function(p) {
parvec[varpos] <- p
dfun(parvec)
}
opt <- minqa::bobyqa(par=getME(ml1,"theta")[varpos],
fn=dfunwrap,lower=0)
parvec[varpos] <- opt$par
opt$par <- parvec
opt$conv <- 0
lf <- lFormula(rt ~ treatment + (1+treatment|subject),data=dd)
ml4 <- mkMerMod(environment(dfun),opt,lf$reTrms,lf$fr)
## REML criterion: 10926.79
## Groups Name Variance Corr
## subject (Intercept) 4366.25
## treatment2 860.98 0.000
## treatment3 3233.28 0.000 0.000
## Residual 9761.63
lf$reTrms$theta <- c(1,1,1) ## only 3 theta parameters
lf$reTrms$lower <- rep(0,3)
lf$reTrms$Lind <- rep(1:3,30) ## correspondence of theta vector
## with non-zero elements in Lambdat
## there should be an easier way to create a sparse diagonal matrix
## of class 'dgCMatrix' ....
lf$reTrms$Lambdat <- sparseMatrix(i=1:90,j=1:90,dims=c(90,90),x=1)
devfun <- do.call(mkLmerDevfun,lf)
opt <- optimizeLmer(devfun)
ml5 <- mkMerMod(environment(devfun), opt, lf$reTrms, fr=lf$fr)
这给出了与之前方法相同的答案，但它打破了 输出方法......
ml5@pp$theta
## [1] 0.6687950 0.2969846 0.5755197
ml4@pp$theta
## [1] 0.6687949 0.0000000 0.0000000 0.2969847 0.0000000 0.5755197
最终我们希望在flexlambda分支中构建这类东西，但是我们必须对输出方法做一些工作......