[R-sig-ME] Interpreting the coefficients of a main effect

[Ista Zahn]

Hi Sverre,
I hope you'll forgive me if I get this wrong, but I don't think the
interpretation of the fixed effects differs much from regular
regression. See below.

On Mon, Mar 7, 2011 at 11:00 PM, Sverre Stausland
<johnsen at fas.harvard.edu> wrote:
> Hi mixed-models users,
>
> I am trying to interpret the coefficients I get from lmer models. In
> model A, I have these variables:
>
> Dependent variable:
> Correct response (0=incorrect, 1=correct)
>
> Independent variables:
> Trial (continuous)
> ReactionTime (continuous)
> Stimulus (binary: 'same' or 'different')
>
> Here is the output:
>
> Fixed effects:
>                           Estimate Std. Error z value Pr(>|z|)
> (Intercept)                3.756272   1.344013   2.795  0.00519 **
> Trial                      0.019962   0.007494   2.664  0.00773 **
> ReactionTime              -2.354231   1.236785  -1.904  0.05697 .
> Stimulussame              -5.364448   1.203831  -4.456 8.34e-06 ***
> Trial:ReactionTime        -0.014469   0.006982  -2.072  0.03822 *
> Trial:Stimulussame        -0.010312   0.002525  -4.085 4.41e-05 ***
> ReactionTime:Stimulussame  4.838763   1.105341   4.378 1.20e-05 ***
>
> I am curious about the overall effect of ReactionTime. If I understand
> it correctly, I should add the all the coefficients with ReactionTime
> in it to see that, i.e. (-2.25)+(-0.02)+4.84 = 2.57. In other words,
> slower responses give more accurate responses.

Describing an "overall" effect is bound to be misleading in the
presence of such a strong interaction. Your data are telling you that
the effect of ReactiontTime differs depending on the Stimulus. Why do
you insist that there must be a single number for an effect that is
clearly variable?

>
> Question 1:
> Is that the correct way to see the overall effect of ReactionTime?

No I don't think so. If you _really_ want average effects (and again,
I'm hard pressed to imagine why) then center your continuous
predictors and contrast-code (not dummy code!) your factors and re-run
the model including the interactions or interpret the value from model
B below.


>
> In model B, I do the same as above, but I leave out interactions with
> ReactionTime. Now I get this:
>
> Fixed effects:
>                       Estimate Std. Error z value Pr(>|z|)
> (Intercept)            1.992515   0.598379   3.330 0.000869 ***
> Trial                  0.006496   0.002061   3.152 0.001622 **
> ReactionTime          -0.739240   0.452896  -1.632 0.102626
> Stimulussame          -0.331084   0.353614  -0.936 0.349126
> Trial:Stimulussame    -0.011652   0.002440  -4.776 1.79e-06 ***
>
> Question 2:
> How do I interpret the fact that the overall effect of ReactionTime in
> model A with interactions is positive, whereas the overall effect in
> model B without interactions is negative? Is the coefficient in model
> B simply not trustworthy as a result of leaving out significant
> interactions?

The ReactionTime coefficient is not a trustworthy estimate of the
"overall" effect in either model, given the strong interaction.

>
> Question 3:
> Normally I do model comparisons with anova to determine the effect of
> a variable. In model A, where I am curious about the significance of
> ReactionTime in predicting the dependent variable, should I compare
> model A (= the superset model) with a subset model where I remove the
> main effect ReactionTime _as well as_ all the interactions it is a
> part of?

I think that's a reasonable test.

>
> Thank you
> Sverre
>
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>



-- 
Ista Zahn
Graduate student
University of Rochester
Department of Clinical and Social Psychology
http://yourpsyche.org