How to interpret the output of 'Linear mixed model fit by REML' in R?

Question

I made a mixed model to investigate the effect of 2 interventions (strength or endurance) on physical activity. Here are descriptions of my variables:

• PA = Physical Activity (measured in minutes of PA per week)
• progr = is the intervention program (1 = endurance, 2 = strength)
• time = time points of measurement (at baseline, after 6 weeks and after 12 weeks) vnr = ID-number of the subjects

I have used this code to build the model:

mix.intslo_PA_2 <- lmer(PA ~ progr + time + progr * time + (time|vnr), data_l)
summary(mix.intslo_PA_2)

Now when I get the output, I don't know how to interpret it, since I can't find a clear explanation anywhere on how to do this. What is important to interpret? The random or fixed effects and what do they mean? And how to interpret them?

This is the output:

> summary(mix.intslo_PA_2)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: PA ~ progr + time + progr * time + (time | vnr)
   Data: data_l
REML criterion at convergence: 8772.7
Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.99644 -0.77989 -0.01227  0.84138  1.88024
Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 vnr      (Intercept)  28188.4 167.89

          time           192.5  13.87   -0.54
 Residual             137102.2 370.27

Number of obs: 594, groups:  vnr, 198
Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)

(Intercept)  773.982     38.722 196.001  19.988   <2e-16 ***
progr2       -54.241     53.687 196.001  -1.010    0.314

time          -2.205      4.698 196.002  -0.469    0.639

progr2:time    2.025      6.514 196.002   0.311    0.756

Signif. codes:  0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
            (Intr) progr2 time

progr2      -0.721

time        -0.733  0.529

progr2:time  0.529 -0.733 -0.721

Answer 1

It's not clear what vnr is here, but I will answer in a general sense what your output is saying. First I start with some less important bits.

• Formula: This is just the model you specified using R syntax.
• REML criterion: This is just the likelihood estimated under REML which isn't really useful on it's own.
• Scaled residuals: These tell you some information about how your regression line is fitting the model. Generally speaking, you want the median to be somewhat near zero (here it's fine) and the minimum/maximum to be somewhat similar in absolute value (also fine here). Basically, the min/max tell you the furthest your residuals (errors) stray away from your fitted regression line, and inequality can indicate that the regression is misfit somehow, which here it isn't.

The next parts are generally the meat and potatoes that you would care about:

• Random effects (intercepts): Each part that is named (Intercept) is telling you the random intercept portion of the random effects. Here this would be your vnr variable. When looking at all of the vnr groups, they vary around the average conditional mean of the DV by about $28188.4$, and the SD of this is around $167.89$. Whether this is a lot or a little is dependent on what your DV actually measures in terms of physical activity. For example, if PA was measured in caloric deficit, then this would be a lot.
• Random effects (slopes): The slopes are usually the ones that don't have (Intercept) because they are not grouped (hence you don't see them in the group column), and have some correlation near it if the random slopes and intercepts are allowed to correlate (which here you have specified). The variance and SD here represent how much the slopes vary between groups (in other words, how much the magnitude fluctuates by grouping). The correlation shows that as the intercepts increase in value, the slopes decrease. This means that as the conditional mean for each group increases, the magnitude of time decreases.
• Residual: This part tells you the "leftover" noise in your model. Basically, there is about $370.27$ SD in PA that is left unaccounted for in your model. The rest of the output below the residuals just tell you details about the observations and groups entered into the model.
• Fixed effects: These are interpreted the same way they would a normal regression, though the $p$ values are not estimated the same way as a typical OLS regression would (they are instead estimated with Satterthwaite's method here). Reading out just one of the fixed effects, we can see that progr2 has a slope which indicates that it decreases PA with $\beta = -54.241$, in other words this group decreases PA compared to the endurance group. The rest of the information to the right (standard error (se), degrees of freedom (df) and t-value) are used to estimate the $p$ value. Here it is not statistically significant. Whether or not that is something to care about is up to how much you are married to the NHST framework of decision making around $p$ values. Those terms which include $:$ are interaction terms and have similar interpretation. The rest of this output just tells you how to read the significance terms.

As to the final part:

• Correlation of fixed effects. Just ignore this section. Many have actually asked for this output to be removed because it is confusing and leads to erroneous interpretation. More on that here.

Just as an aside, it appears you have some redundant code. If you enter progr * time, it already estimates the main effect and the interaction of time, so you don't need to enter time again as a main effect (though functionally lme4 will just ignore this redundancy).