I find this post relevant. The mock data-set is constructed the same way as in the post but with a different seed to produce zero variance in the mixed model. Think this as a three way anova, with A as temperature factor, B as day factor and C as replicate. The output of fit1 gives a SD of 0.455 for subject, and anova with p-value 0.02483 suggests a significant non-zero SD. I did some research and find this chi-square test is a conservative test considering it is on the boundary, so p-value should have been even lower. My question is whenever we see zero SD for some factors (in fit1, SD for factors A and B are zero), does this imply that the variance estimates for other factors are not reliable? The residual SD is correctly estimated to be 1 though.
> set.seed(8) > d <- data.frame( + Y = rnorm(96), + subject = factor(rep(1:12, 4)), + A = factor(rep(1:2, each=24)), + B = factor(rep(rep(1:2, each=12))), + C = factor(rep(rep(1:2, each=48)))) > > fit1 <- lmer(Y ~ (1|subject) + (1|A) + (1|B), data=d) > > fit2 <- lmer(Y ~ (1|A) + (1|B), data=d) > > fit1 Linear mixed model fit by REML ['lmerModLmerTest'] Formula: Y ~ (1 | subject) + (1 | A) + (1 | B) Data: d REML criterion at convergence: 286.2724 Random effects: Groups Name Std.Dev. subject (Intercept) 0.455 A (Intercept) 0.000 B (Intercept) 0.000 Residual 1.008 Number of obs: 96, groups: subject, 12; A, 2; B, 2 Fixed Effects: (Intercept) -0.09281 > > anova(fit1,fit2) refitting model(s) with ML (instead of REML) Data: d Models: fit2: Y ~ (1 | A) + (1 | B) fit1: Y ~ (1 | subject) + (1 | A) + (1 | B) Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq) fit2 4 297.52 307.78 -144.76 289.52 fit1 5 294.48 307.31 -142.24 284.48 5.0357 1 0.02483 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1