[R-sig-ME] Random vs. fixed effects

[Gabor Grothendieck]

Here is a simulation of 10k cases using 4 and 50 level factors for the
random effect.  With 4 levels there are numerical problems and the
accuracy of the random effect is terrible.  With 50 levels there are
no numerical problems and the accuracy is much better.

> library(lme4)
> set.seed(1)
> n <- 10000
> k <- 4
> f <- function(n, k) {
+ set.seed(1)
+ x <- 1:n
+ fac <- gl(k, 1, n)
+ fac.eff <- rnorm(k, 0, 4)[fac]
+ e <- rnorm(n)
+ y <- 1 + 2 * x + fac.eff + e
+ lmer(y ~ x + (1|fac))
+ }

> # simulation with 4 level random effect
> f(n, 4)
Linear mixed model fit by REML
Formula: y ~ x + (1 | fac)
   AIC   BIC logLik deviance REMLdev
 28733 28762 -14363    28702   28725
Random effects:
 Groups   Name        Variance Std.Dev.
 fac      (Intercept) 1.1162   1.0565
 Residual             1.0298   1.0148
Number of obs: 10000, groups: fac, 4

Fixed effects:
             Estimate Std. Error t value
(Intercept) 1.313e+00  5.286e-01       2
x           2.000e+00  3.515e-06  568923

Correlation of Fixed Effects:
  (Intr)
x -0.033
Warning message:
In mer_finalize(ans) : false convergence (8)

> # simulation with 50 level random effect
> f(n, 50)
Linear mixed model fit by REML
Formula: y ~ x + (1 | fac)
   AIC   BIC logLik deviance REMLdev
 29040 29069 -14516    29009   29032
Random effects:
 Groups   Name        Variance Std.Dev.
 fac      (Intercept) 11.2016  3.3469
 Residual              1.0251  1.0125
Number of obs: 10000, groups: fac, 50

Fixed effects:
             Estimate Std. Error t value
(Intercept) 1.396e+00  4.738e-01       3
x           2.000e+00  3.507e-06  570242

Correlation of Fixed Effects:
  (Intr)
x -0.037




On Fri, Apr 23, 2010 at 9:38 AM, Schultz, Mark R. <Mark.Schultz2 at va.gov> wrote:
> I just read a post by Andrew Dolman suggesting that a factor with only 3
> levels should be treated as a fixed effect. This seems to be a perennial
> question with mixed models. I'd really like to hear opinions from
> several experts as to whether there is a consensus on the topic. It
> really makes me uncomfortable that such an important modeling decision
> is made with an "ad hoc" heuristic.
>
>
>
> Thanks,
>
> Mark Schultz, Ph.D.
>
> Bedford VA Hospital
>
> Bedford, Ma.
>
>
>        [[alternative HTML version deleted]]
>
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