[R-lang] Re: Investigating random slope variance

[T. Florian Jaeger]

and to follow-up on my email, which immediately raised some questions ;),
here's what I meant when I said that "as far as i know, shrinkage does not
enforce / bias towards normality of blups":

shrinkage itself depends on the (estimated) marginal and by-grouping level
(e.g., by-item) means and variances. These variance (estimates) are derived
under the assumption of normality. But the shrinkage itself makes no
reference to normality. to simplify somewhat, shrinkage means that we
describe the posterior estimates of the by-grouping level means (e.g., the
by-item means) are described as a weighted sum of the marginal (overall)
mean and the by-grouping level mean. the weights of each of these
components depends (in the simplest case) on the relation between the
residual variance and the between-grouping level variance AND the amount of
data available for that grouping level.

see

Kliegl, R., Masson, M. E. J., & Richter, E. M. (2010). A linear mixed model
analysis of masked repetition priming. Visual Cognition, 18(5), 655-681.
doi:10.1080/13506280902986058

footnote 3

for further details.

Florian


On Fri, Apr 4, 2014 at 2:40 PM, T. Florian Jaeger
<tiflo@csli.stanford.edu>wrote:

> Hi Titus,
>
> shrinkage has larger effects on cells with
>
> a) means further away from the predicted marginal mean
> b) fewer cells counts.
>
> A good paper to see that is Kliegl et al 2010. I also have some
> demonstration of this effect in my lmer intro slides (see a recent blog
> post on the HLP lab blog). Note that you might see deviation away from the
> marginal mean, because of correlations between the grouping identity (e.g.,
> item) and other fixed effects in the model. I don't recall whether you had
> other fixed effects predictors in your model? If so, that could also be the
> reason for estimates of the random effect correlations.
>
> As far as I know shrinkage does not enforce / bias towards normality of
> BLUPs.
>
> I hope of this is helpful.
>
> florian
>
>
> On Fri, Apr 4, 2014 at 11:20 AM, Titus von der Malsburg <
> malsburg@posteo.de> wrote:
>
>>
>> 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
>>
>
>
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