Rasch with lme4
Daniel Ezra Johnson wrote:
What happens in practice when you compare the two approaches of [Subject] as a fixed versus as a random effect? Consider: M1 = lmer(Reaction ~ Days + (1|Subject), sleepstudy) M2 = lm(Reaction ~ Days + factor(Subject), sleepstudy) The slope estimates for Days for are practically identical, the mean intercepts differ...
One important difference arises if you have a fixed effect that, unlike Days, is "between subject", like Gender for example. In that case you MUST use the M1 (mixed-model) approach with Subject as a random effect. You CANNOT have Gender as a fixed effect AND Subject as a fixed effect.
They're still similar for the example I tried. (lmer vs. lm)
Intercept: 250.0442 (SE = 10.0442) vs 293.6628 (SE = 10.8062)
Days: 10.4498 (SE = 0.8067) vs 10.4511 (SE = 0.8067)
SexMale: 2.5910 (SE = 4.6849) vs 2.4017 (SE = 4.6864)
(See below.) The largest difference is in intercepts.
So I still need a good counterexample. I feel a term like "shrinkage"
will be involved in an explanation.
A
P.S. I guess lm still is a mixed effects approach - the residuals are a
random effect at the level of observations? :-)
-------------------------------------------------------------------
> M1 = lmer(Reaction ~ Days + (1|Subject), sleepstudy)
>
> # Use the random intercept to make up a Male/Female IV
> sleepstudy$Sex =
cut(ranef(M1)$Subject$"(Intercept)",2,labels=c("Female","Male"))
>
> # Now again: the same models:
>
> M1 = lmer(Reaction ~ Days + Sex + (1|Subject), sleepstudy)
> summary(M1)
...
Estimate Std. Error t value
(Intercept) 250.0442 10.0442 24.894
Days 10.4498 0.8067 12.954
SexMale 2.5910 4.6849 0.553
>
> M2 = lm(Reaction ~ Days + Sex + factor(Subject), sleepstudy)
> summary(M2)
...
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 293.6628 10.8062 27.176 < 2e-16 ***
Days 10.4511 0.8067 12.956 < 2e-16 ***
SexMale 2.4017 4.6864 0.512 0.609020
...