More on random slopes and what it means if your effect is not longer significant after the inclusion of random slopes

I thought the following snippet from a somewhat edited email I recently wrote in reply to a question about random slopes and what it means that an effect becomes insignificant might be helpful to some. I also took it as an opportunity to updated the procedure I described at http://hlplab.wordpress.com/2009/05/14/random-effect-structure/. As always, comments are welcome. What I am writing below are just suggestions.

[...] an insignificant effect in an (1 + factor|subj) model means that, after controlling for random by-subject variation in the slope/effect of factor, you find no (by-convention-significant) evidence for the effect. Like you suggest, this is due to the fact that there is between-subject variability in the slope that is sufficiently large to let us call into question the hypothesis that the ‘overall’ slope is significantly different from zero.

[...] So, what’s the rule of thumb here? If you run any of the standard simple designs (2×2, 2×3, 2x2x2,etc.) and you have the psychologist’s luxury of plenty of data (24+item, 24+ subject [...]), the full random effect structure is something you should entertain as your starting point. That’s in Clark’s spirit. That’s what F1 and F2 were meant for. [...] All of these approaches do not just capture random intercept differences by subject and item. They also aim to capture random slope differences.

[...] here’s what I’d recommend during tutorials now because it often saves time for psycholinguistic data. I am only writing down the random effects but, of course, I am assuming there are fixed effects, too, and that your design factors will remain in the model. Let’s look at a 2×2 design:

Two interesting papers on mixed models

While searching for something else, I just came across two papers that should be of interest to folks working with mixed models.

• Schielzeth, H. and Forstmeier, W. 2009. Conclusions beyond support: overconfident estimates in mixed models. Behavioral Ecology Volume 20, Issue 2, 416-420.  I have seen the same point being made in several papers under review and at a recent CUNY (e.g. Doug Roland’s 2009? CUNY poster). On the one hand, it should be absolutely clear that random intercepts alone are often insufficient to account for violations of independence (this is a point, I make every time I am teaching a tutorial). On the other hand, I have reviewed quite a number of papers, where this mistake was made. So, here you go. Black on white. The moral is (once again) that no statistical procedure does what you think it should do if you don’t use it the way it was intended to.
• The second paper takes on a more advanced issue, but one that is becoming more and more relevant. How can we test whether a random effect is essentially non-necessary – i.e. that it has a variance of 0? Currently, most people conduct model comparison (following Baayen, Davidson and Bates, 2008).  But this approach is not recommended (and neither do Baayen et al recommend it) if we want to test whether all random effects can be completely removed from the model (cf. the very useful R FAQ list, which states “do not compare lmer models with the corresponding lm fits, or glmer/glm; the log-likelihoods [...] include different additive terms”). This issue is taken on in Scheipl, F., Grevena, S. and Küchenhoff, H. 2008. Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models. Computational Statistics & Data Analysis.Volume 52, Issue 7, 3283-3299. They present power comparisons of various tests.

Random effect: Should I stay or should I go?

One of the more common questions I get about mixed models is whether there are any standards regarding the removal of random effects from the model. When should a random effect be included in the model? This was also one of the questions we had hope to answer for our field (psycholinguistics) in the pre-CUNY Workshop on Ordinary and Multilevel Models (WOMM), but I don’t think we got anywhere close to a “standard” (see Harald Baayen’s presentation on understanding random effect correlations though for a very insightful discussion).

That being said, I find most of us would probably agree on a set of rules of thumb, at least for factorial analyses of balanced data: