[R-sig-ME] Characterizing correlation between binomial effects

[David Duffy]

On Wed, 5 Sep 2012, Mike Lawrence wrote:

> I have data from multiple human participants who each completed two
> tasks. In each task there were two conditions, and in each condition
> there are multiple trials that yield a binomial (1/0) outcome. I would
> like to evaluate whether there is a correlation between Ss' condition
> effect in task 1 and their condition effect in task 2.
>
> One idea I had was to combine the two task data frames to a single
> data frame, code task as a variable, and fit a full glmm model:
>
>    fit = glmer(
>        family = binomial
>        , data = both_tasks
>        , formula = response ~ task*condition + (1+task*condition|participant)
>    )

If I understand this correctly, which I probably don't, you want something 
like a P-value for the likelihood ratio test comparing 
(1+task*condition|participant) with (1 + condition + task|participant). Or
should that be (1|condition) + (1 + task|participant)?