Contrasts for interactions in lmer

There is a bit of evidence for an interaction of COND and PCU:
COND1:PCU ? ? ? ?48.309 ? ? 29.850 ? 1.618
If the t-value were larger it would indicate that slopes for the
regression of RRT on PCU differ between the two condition.

There is no statistical support for the the interaction of DIR and PCU
PCU:DIR1 ? ? ? ?-26.835 ? ? 29.814 ?-0.900
Now to some of your questions relating to correlations:
(1) The Fixed Effects correlations are probably not what you are
after. For example, in a perfectly balanced design, these correlations
will be zero.

(2) I suspect what you might be after are effect correlations related
to subjects or items. Assuming cond and verb bias are within-subject
effects, you could get an estimate of the parameter for the covariance
component with the following specification.
RRT ~ COND * PCU * DIR + (1 + COND + DIR  | SUBJECT) + (1 | ITEM)

You should check whether adding these variance components to the model
improves the goodness fo fit, for example with an ANOVA..

(3) You used a sum contrast specification for the two factors (COND
and DIR). This is fine. For two-level factors there is no point in
specifying Helmert contrasts. So it is unclear what you referring to
in this context.

Finally, it is generally a bad idea to specify models with
interactions terms leaving out the factors contributing to the
interactions. If you do so, you need to have very good theoretical
reasons.

Reinhold Kliegl
Dear all.

I am currently analyzing eye-tracking data and am interested in a main effect of condition (COND) plus its interaction with subjects' operation span (PCU) and the direction of a verb bias (1 or 2). The contrasts are:

contrasts(COND)
?[,1]
a ? -1
b ? ?1
and

contrasts(DIR)
?[,1]
1 ? -1
2 ? ?1
PCU is a continuous predictor which I centered by subtracting the mean (the problem does, however, persist when I split the sample into extreme groups and work with a categorial predictor). With the following model, I don't get a correlation between the fixed effects:

Linear mixed model fit by REML
Formula: RRT ~ COND * PCU * DIR + (1 | SUBJECT) + (1 | ITEM)
? ?Data: fm3
? ?AIC ? BIC logLik deviance REMLdev
?46733 46801 -23355 ? ?46768 ? 46711
Random effects:
?Groups ? Name ? ? ? ?Variance Std.Dev.
?SUBJECT ?(Intercept) ?8918.29 ?94.437
?ITEM ? ? (Intercept) ? 404.85 ?20.121
?Residual ? ? ? ? ? ? 34881.69 186.766
Number of obs: 3503, groups: SUBJECT, 59; ITEM, 59

Fixed effects:
? ? ? ? ? ? ? ?Estimate Std. Error t value
(Intercept) ? ? 122.900 ? ? 12.963 ? 9.481
COND1 ? ? ? ? ? ?15.924 ? ? ?3.165 ? 5.031
PCU ? ? ? ? ? ? 139.411 ? ?120.025 ? 1.162
DIR1 ? ? ? ? ? ? -7.746 ? ? ?4.107 ?-1.886
COND1:PCU ? ? ? ?48.309 ? ? 29.850 ? 1.618
COND1:DIR1 ? ? ? -3.396 ? ? ?3.164 ?-1.073
PCU:DIR1 ? ? ? ?-26.835 ? ? 29.814 ?-0.900
COND1:PCU:DIR1 ? -8.069 ? ? 29.838 ?-0.270

Correlation of Fixed Effects:
? ? ? ? ? ? (Intr) COND1 ?PCU ? ?DIR1 ? COND1:PCU COND1:D PCU:DI
COND1 ? ? ? ?0.002
PCU ? ? ? ? ?0.004 -0.001
DIR1 ? ? ? ? 0.002 -0.004 ?0.004
COND1:PCU ? -0.001 -0.001 ?0.003 ?0.000
COND1:DIR1 ?-0.001 ?0.000 ?0.000 ?0.007 ?0.021
PCU:DIR1 ? ? 0.005 ?0.000 -0.003 ?0.000 -0.009 ? ?-0.005
COND1:PCU:D ?0.000 ?0.021 -0.002 -0.004 -0.009 ? ?-0.001 ? 0.011
But, since I'm mainly interested in the interactions and not so much the main effects of PCU and DIR, I changed the model to the following:

Linear mixed model fit by REML
Formula: RRT ~ COND + COND:PCU + COND:DIR + (1 | SUBJECT) + (1 | ITEM)
? ?Data: fm3
? ?AIC ? BIC logLik deviance REMLdev
?46744 46800 -23363 ? ?46769 ? 46726
Random effects:
?Groups ? Name ? ? ? ?Variance Std.Dev.
?SUBJECT ?(Intercept) ?8911.15 ?94.399
?ITEM ? ? (Intercept) ? 406.16 ?20.153
?Residual ? ? ? ? ? ? 34869.91 186.735
Number of obs: 3503, groups: SUBJECT, 59; ITEM, 59

Fixed effects:
? ? ? ? ? ? Estimate Std. Error t value
(Intercept) ?122.962 ? ? 12.959 ? 9.489
COND1 ? ? ? ? 15.941 ? ? ?3.164 ? 5.039
CONDa:PCU ? ? 91.049 ? ?123.553 ? 0.737
CONDb:PCU ? ?187.055 ? ?123.714 ? 1.512
CONDa:DIR1 ? ?-4.340 ? ? ?5.168 ?-0.840
CONDb:DIR1 ? -11.160 ? ? ?5.204 ?-2.144

Correlation of Fixed Effects:
? ? ? ? ? ?(Intr) COND1 ?CONDa:PCU CONDb:PCU CONDa:DIR1
COND1 ? ? ? 0.002
CONDa:PCU ? 0.004 -0.001
CONDb:PCU ? 0.004 -0.001 ?0.883
CONDa:DIR1 ?0.002 -0.003 ?0.006 ? ? 0.000
CONDb:DIR1 ?0.001 -0.003 ?0.000 ? ? 0.006 ? ? 0.256
Not I do get a considerable correlation between the interactions. From the output (CONDa:?, CONDb:?), I infer that the model didn't always use helmert coding for condition but applied something else for the interactions. Is that right? When I code COND numerically as -1 and 1, the correlations turn out fine, which supports my conclusion. I would be very grateful for suggestions.

Thanks,
Paul

---
Paul Metzner

Humboldt-Universit?t zu Berlin
Philosophische Fakult?t II
Institut f?r deutsche Sprache und Linguistik

Post: Unter den Linden 6 | 10099 Berlin | Deutschland
Besuch: Dorotheenstra?e 24 | 10117 Berlin | Deutschland

+49-(0)30-2093-9726
paul.metzner at gmail.com
http://amor.rz.hu-berlin.de/~metznerp/

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Contrasts for interactions in lmer

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