You're misinterpreting these results, which is easy to do as with mixed models there's more than one type of 'fitted value' and the documentation of lmer
isn't as clear as it might be. Try using fixed.effects()
in place of fitted()
and you should get correlations which makes more intuitive sense if you're interested in the contribution of the fixed effects.
The fitted()
function of lmer
is documented as giving the 'conditional means'. I had to check the Theory.pdf vignette to work out that these include the predictions of the modelled random effects. Your modelled random effect variances are, overall, smaller in the model including the fixed effect. But smaller random effects mean less shrinkage, i.e. the predicted random effect is closer to the observed residual. When calculating the correlation, it seems that in your case this smaller shrinkage just overcomes the improvement from the fixed effect.
The interpretation of $R^2$ as 'proportion of variance explained' gets more complex with mixed models, as it depends whether you think of random effects as 'explaining' variance. Probably not, in most cases.
I can provide some references:
Xu, R. (2003). Measuring explained variation in linear mixed effects models. Statistics in Medicine, 22, 3527-3541. DOI:10.1002/sim.1572
Edwards, L. J., Muller, K. E., Wolfinger, R. D., Qaqish, B. F., & Schabenberger, O. (2008). An $R^2$ statistic for fixed effects in the linear mixed model. Statistics in Medicine, 27, 6137-6157. DOI:10.1002/sim.3429
Hössjer, O. (2008). On the coefficient of determination for mixed regression models. Journal of Statistical Planning and Inference, 138, 3022-3038. DOI:10.1016/j.jspi.2007.11.010
Nakagawa, S., & Schielzeth, H. (2013). A general and simple method for obtaining $R^2$ from generalized linear mixed-effects models. Methods in Ecology and Evolution, 4, 133-142. DOI:10.1111/j.2041-210x.2012.00261.x
Happy reading!
Best Answer
This is actually a very complex question. Defining what the proportion of explained variance means in these models is a non-trivial exercise. I would start with chapter 7 of Tom Snijders & Roel Bosker (1999) "Multilevel Analysis: An introduction to basic and advanced multilevel modeling" Thousand Oaks: Sage.