Correct interpretation of Lmer output

Question

I have produced the following model:

>lmer(TotalPayoff~PgvnD*Type+Type*Asym+PgvnD*Asym+Game*Type+Game*PgvnD+Game*Asym+
                   (1|Subject)+(1|Pairing),REML=FALSE,data=table1)->m1

PgvnD=A percentage (numeric)
Asym= a factor 0 or 1
Type=a factor 1 or 2
Game= a factor 1 or 2

from this model the terms Type, Game and PgvnD:Asym were shown to be significant by removal from the model. PgvnD and Asym on there own were not significant but were left in the model because the interaction between them was. The summary of this model is as follows;

> m7
Linear mixed model fit by maximum likelihood 
Formula: TotalPayoff ~ Type + PgvnD * Asym + Game + (1 | Subject) + (1 |Pairing) 
   Data: table1 
  AIC  BIC logLik deviance REMLdev
 1014 1038 -497.8    995.6   964.4
Random effects:
 Groups   Name        Variance Std.Dev.
 Subject  (Intercept)   0.000   0.0000 
 Pairing  (Intercept) 716.101  26.7601 
 Residual              89.364   9.4533 
Number of obs: 113, groups: Subject, 73; Pairing, 61

Fixed effects:
            Estimate Std. Error t value
(Intercept)   81.727      6.332  12.907
Type2          7.926      2.852   2.779
PgvnD         -8.466      7.554  -1.121
Asym1        -12.167      7.583  -1.604
Game2         15.374      7.147   2.151
PgvnD:Asym1   26.618      9.710   2.741

Correlation of Fixed Effects:
            (Intr) Type2  PgvnD  Asym1  Game2 
Type2       -0.188                            
PgvnD       -0.218 -0.038                     
Asym1       -0.620  0.081  0.189              
Game2       -0.483  0.009 -0.010 -0.015       
PgvnD:Asym1  0.233 -0.267 -0.766 -0.328 -0.011

Am I interpreting these results correctly?

• TotalPayoff is higher when Type=1 than in Type=2, it is also higher when game=2 than when game=1.
• Also TotalPayoff increases significantly with PgvnD if Asym=1 but not if ASym=0 (indicated by significant interaction term but non-significant single terms).

Also I notice that the Subject random effect has SD and variance of 0. Can this then be removed from the model? What does this really mean?

Answer 1

I address your interpretations 1 and 2 in order:

1) How you interpret factors depends on which level of the factor is the reference category. The fact that the model calls it Type2 suggests to me that Type1 is the reference, and that the parameter represents how the estimate changes when Type == 2. Thus, I disagree with your interpretation. I would say TotalPayoff is higher when Type == 2 because the parameter is positive and significant (assuming alpha == .05).

2) I think your interpretation basically makes sense. I prefer to say it like this: The slope for PgvnD changes by the amount estimated as the parameter for the interaction term when Asym == 1 (i.e. when Asym is not equal to the reference category). So the PgvnD parameter is its main effect estimate plus the interaction estimate when Asym == 1. This would be -8.466 + 26.618. Keep in mind, though, if you want to make an estimate of TotalPayoff you must also account for the main effect of Asym. Bottom line, the interaction parameter tells you how much the main effects change under the conditions specified by the interaction (value of PgvnD and the Asym == 1).

Alternatively, the interaction allows you to say that the effect of Asym==1 on TotalPayoff changes positively along with changes in PgvnD by the amount estimated as the interaction parameter.

A quick example: ignoring all but the two discussed main effects which I now refer to as $A$ and $P$, and the interaction $AP$,

$$ y = \beta_{A}A + \beta_{P}P + \beta_{AP}AP $$

Clearly, if $A$ is $0$ (i.e. reference category), then neither the $AP$ interaction nor the main effect for $A$ contributes anything to $y$. But $\beta_PP$ still does so long as $P \ne 0$.

If $A = 1$ (i.e. probably meaning Asym is true, or not reference), and $P = 1$, then

$$y = \beta_{A}(1) + \beta_{P}(1) + \beta_{AP}(1 \times 1)$$

$$y = -12.167 + -8.466 + 26.618$$

Finally, I think it is probably safe to remove the variance component that was estimated 0 from the model. It might be worth it to explore the data a little to make sure that it seems like a reasonable estimate and not an artifact of a misspecified model or other oddity.