# Is it worth reporting small fixed-effect $R^2$ (marginal $R^2$), large model $R^2$ (conditional $R^2$)?

In a mixed model analysis (lme4 + lmerTest for R), I want to analyse the effect of 3 predictors, say A, B and C. Since it is a mixed model, there are two random effects Ran1 and Ran2.

I first built a random intercept model with Ran1 and Ran2, but without fixed terms:

mod.0 <- lmer ( outcome ~ 1 + (1|Ran1) + (1|Ran2), data = mydata)


The result (fixed part) is the following:

Fixed effects:
Estimate Std. Error       df t value Pr(>|t|)
(Intercept)  2.92381    0.07787 35.28000   37.55   <2e-16 ***


I built a random intercept mixed effect model to account for Ran1 and Ran2:

mod.1 <- lmer ( outcome ~ A + B + C + (1|Ran1) + (1|Ran2), data = mydata)


The result (fixed part) is the following:

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)            3.255e+00  8.476e-02  5.000e+01  38.400  < 2e-16 ***
A                     -1.482e-01  2.639e-02  5.671e+04  -5.617 1.95e-08 ***
B                      3.495e-01  2.462e-02  5.971e+04  14.195  < 2e-16 ***
C                     -2.083e-01  1.873e-02  3.942e+04 -11.124  < 2e-16 ***


With the following method to compute $R^2$ for models:

r2.mer <- function(m)
{
lmfit <-  lm(model.response(model.frame(m)) ~ fitted(m))
summary(lmfit)$r.squared }  mod.0 has$R^2=$0.6187513 and mod.1 has$R^2=$0.6251295. We can see that by adding the fixed terms, the model$R^2$does not change much. I also use a detailed$R^2$computation method to compare the two model and to compare the marginal and conditional$R^2$(https://github.com/jslefche/rsquared.glmer). By running the following command: rsquared.glmm(list(mod.0, mod.1,))  The result is the following:  Class Family Link Marginal Conditional AIC 1 merModLmerTest gaussian identity 0.00000000 0.5814522 300654.6 2 merModLmerTest gaussian identity 0.00555211 0.5691487 129177.1  The result is in line with the previous, i.e. for mod.1, the variances in the fixed terms only account for 0.00555 of the total variance (marginal$R^2)$. As I said at the beginning, I am interested to analyse the effect of A, B, C. As you see, the effects are significant, although the effect size (Beta values) are small, due to large number of observations. In this case, does it make sense to report that A and C have negative effects (Beta = -0.14 and -0.21), B has positive effect (Beta = 0.345), even the$R^2$values of these fixed terms are really small? Do you have a better interpretation of the results? • B seems to have a less-than-tiny effect size at least. I don't see any harm in reporting these effects if they're of interest. With a sample size this tremendous, you can get some pretty sharp confidence intervals on those estimates. That's gotta be worth something, right? – Nick Stauner Apr 2 '14 at 9:46 • @NickStauner Hi, the thing that significance is of no surprise with such a large sample. So I think maybe r-squared is a better measure to gauge the effect, although it is a different thing. The marginal r-square is so small, which indicates that even with mixed model, the observations vary so much compared to the fitted lines. The effects show the difference in means, but small marginal r-square indicates that the fix-effects are not that convincing, right? That's why I am not that confident to report them. – nan Apr 2 '14 at 11:23 • Sure, it's no surprise they're significantly different from zero, but if you report 95% confidence intervals, you might be surprised at how precise your effect size estimates really are. Whether they're meaningful is another matter, but I'd think you could at least be confident that your estimates are pretty accurate. – Nick Stauner Apr 2 '14 at 11:26 • Report, and let others decide what to make of them. In many paradigms, a zero effect is not inherently more or less interesting than a non-zero effect. At the very least, others reading your work will have an estimate of what probably won't turn out to be an important effect later on. – jona May 3 '14 at 15:44 • Make sure you aren't using the REML estimates (REML=FALSE) when comparing the AIC of the two models. – Andrew M Dec 15 '14 at 23:53 ## 2 Answers See Nakagawa and Schielzeth (2013) (or this blog entry) for more information and a discussion on$R^2$for mixed models. First of all, as Nagakawa and Schielzeth notice, using$R^2$from OLS linear model for mixed models is misleading and should not be used. Generally, there are multiple ideas how to compute$R^2$for mixed models, but there's still no consensus on it. As for me, the Nagakawa and Schielzeth's ideas are most interesting, however you should always remember that$R^2$for mixed models is not the same "variance explained" as for linear models and it is just an approximation. Among other approaches you could also check Snijders and Bosker (1994) for comparison. Statistical significance is also problematic for mixed models (and problematic in general) so I wouldn't pay much attention to it. As for your question, I'd recommend Gelman and Hill's (2007) book. First of all, they suggest to compare "effect sizes" for mixed models. In your case marginal$R^2\$ is very small, but you should also look at the "Betas": if they are small compared to variance of your data, i.e. including this effects in the model does not change anything for estimation, you probably could abandon those effects. On another hand, in regression and mixed models literature there are multiple examples for leaving "non-significant" effects in the model and it is almost never a decision based on a simple rule of thumb. For example, in your case the AIC changed quite dramatically what suggests that the second model has a better fit. So I do not see a simple answer in here, not yet.

It seems strange that the conditional R^2 goes down when you add the fixed effects. I know Nagakawa and Schielzeth talk about this kind of thing, but it seems less than useful if adding significant predictors decreases the conditional (i.e. fixed + random) R^2...

• for the decreases, I think it is due to something I forgot to mention in my post. A, B and C all contains NA values, so when I added them to the model, the number of observations decrease from 149884 to 62213. This might have caused the decrease of the conditional r-squared. Anyway, mod.0 is here because I want to show the r-squared computed with the r2.mer methods. If we take a look at mod.1 only, do you think it makes sense to report the effect of A,B and C, even though the marginal r-squared are so small? With such large sample, significance is actually not of surprise. – nan Apr 2 '14 at 11:16
• btw, MuMIn package implements Nakagawa's R^2 so you could check also this package for estimation. – Tim Oct 25 '14 at 22:03