[R-lang] Re: Investigating random slope variance

[Titus von der Malsburg]

On 2014-04-04 Fri 05:10, T. Florian Jaeger <tiflo@csli.stanford.edu> wrote:
> I would be careful making anything out of this. The BLUP estimates of the
> random effects (and, I assume, their distribution) are affected by
> shrinkage, which is often a desirable (conservative) feature, although it
> will make differences appear smaller. So, it's not surprising that the
> fixed effect model mirrors the empirical means more closely. That doesn't
> mean though that it's the better model to draw conclusion from (about those
> differences).

Florian, your comment is spot on.  Here is a plot showing the effect of
shrinkage in my data set:

    http://users.ox.ac.uk/~sjoh3968/R/effect_of_shrinkage.png

Unfilled circles show the empirical mean reading times and differences
between conditions, one circle for each item.  The dots show the BLUP
estimates for each item.

The difference is fairly dramatic.  I assumed that shrinkage would pull
all data points to the mean with the same force (I have the same amount
of data for all items).  If that were the case, the ordering of items
would be preserved.  However, shrinkage affects the individual items in
quite different ways, and some items are even pushed away from the
overall means (1, 5, 7, 8, 9, 10, 13, 14, 35) effectively expanding a
subset of the estimates instead of shrinking them.

I must say that I find it hard to swallow that two seemingly valid ways
to analyze the data (item as random effect or fixed effect) yield
results that are so different.

Another observation: in the BLUP estimates, the correlation of
intercepts and slopes seems to be much higher than in the raw data.  The
correlation of the estimated random intercepts and slopes is -0.86. (The
summary of the model reports -0.62.)  The correlation of the empirical
item means and differences is only -0.4.  Why does lmer believe in such
a high correlation?

  Titus