# Correct interpretation of Lmer output

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?

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.

• Hi 1)sorry yes that was my mistake Type==2 is higher than Type==1 Apr 19 '13 at 16:09
• Hi Thanks 1)sorry yes that was my mistake Type==2 is higher than Type==1. 2) You say the PgvnD parameter is -8.466 + 26.618=18.152. Does this mean that when Asym==0 the slope for PgvnD is 18.152? and so when Asym==0, TotalPayoff still increase with PgvnD just not as much as when Asym==1. Could you therefore say for Asym==0 the effect of PgvnD on TotalPayoff is positive but non-significant but with Asym==1 it is positive and significant? Apr 19 '13 at 16:18
• It's worth commenting that using factors with levels called 0 and 1, and 1 and 2 adds considerable confusion to interpretation. I recommend giving the levels of the factors meaningful names to ease interpretation. Alternatively, use dummy variables in the standard way by naming a dummy sensibly (I assume you're not using dummies, but factors which are then converted to dummies in a R-special way). Apr 19 '13 at 16:29
• I made an error in the last sentence of this comment but couldn't edit. Remade the comment instead. Re comment #2: The PgvnD slope moves to 18.152 when Asym==1 because the interaction was estimated 26.618 (26.618 - 8.466 == 18.152). When Asym==0 (the reference), then PgvnD parameter is just as it was estimated -8.466. This is because the contribution to TotalPayoff from the Asym:PgvnD parameter is effectively 0 when Asym:PgvnD is 0 (i.e. Beta * PgvnD * 0 = 0) Apr 19 '13 at 16:47