R^2 meaning in linear mixed-effects model

The linear mixed-effect model class provides the Rsq property (ordinary and adjusted) which captures the proportion of variability in the response explained by the model. Is that the variability explained by fixed effects only or both by fixed and random effects? From the documentation I get the feeling that it's fixed effects only. How would I find the proportion of variability explained by the random effects?

[Rik]

This is not a question about Matlab, but about statistics.

[Katharina]

If MATLAB offers the Rsq property, it should be specified in the documentation what the Rsq they provide stands for.

Answer 1

[Katharina]

Ordinary Rsq = 1 - SSE / SST

SST is SSR + SSE

SSR = sum((F – mean(F)).^2)

SSE = sum((y – F).^2); SSR = sum((F – mean(F)).^2), where F is the fitted conditional response of the linear mixed-effects model. The conditional model has contributions from both fixed and random effects.

Therefore, MATLAB's Rsq calculation for linear mixed-effect model does take both fixed effects and random effects into account.

Answer 2

[Rik]

The information you seek should be available on the Wikipedia page for the R².

This is one of the most basic goodness-of-fit parameters. It is so basic even Excel inculdes it when you plot a trendline.

[Katharina]

I am not interested in the general calculation of R^2, I want to know how specifically MATLAB calculates this value for the linear mixed-effect model. Does it make the direct use of the estimated variaces, or does MATLAB use the residual sum of squares in analogy to lin. regression, or some other measure of explained randomness based on information gain. Statistical software, and I count MATLAB Statistical toolbox as such, should list the method so the results are interpretable.

[Rik]

As far as I'm aware, there is only one way to calculate the R2 (or the multiple ways are equivalent). If you were asking about the confidence interval, you would have more of a point.

[Katharina]

MATLAB's support is looking into this issue, I'll post their answer

[Michael]

Estimating an R^2 for a linear mixed effects model is non-trivial and is certainly not basic statistics - suitable measures have only relatively recently been developed. In SPSS, the Nakagawa pseudo-R^2 is calculated.

Refs:

Nakagawa, S & Schielzeth, H, 2013. A general and simple method for obtaining R2 from generalized linear mixed-effects models. Methods in Ecology and Evolution, 4(2), 133-142.

Johnson, PCD, 2014. Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models. Methods in Ecology and Evolution, 5(9), 944-946.

Nakagawa, S, Johnson, PCD & Schielzeth, H, 2017. The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. Journal of the Royal Society Interface, 14, 20170213.

[Katharina]

Thank you so much! I'll check these out!