Contrasts for interactions in lmer

I saw that now you use log-transformed DVs and that in my
experience is a good choice for durations collected in eye tracking.
Nevertheless you should check the distribution of model residuals to
back up this decision. Anyway, the log-transformation of RRT may have
lifted the t-value for the PCU  X COND interaction. So I am curious
whether it did or not?

Maybe I responded to quickly ...

First, I guess for a two-level factor a sum contrast can also be
called a Helmert contrast; it is a bit unusual, I think.

Second, the story about the fixed-effect correlations is complicated.
What I wrote for balanced designs returning a zero correlation matrix
of fixed effects assumes also that contrasts for the fixed effects are
orthogonal and that the variance components are specified only for the
intercepts, as you had set up your first model. If you specify a
non-orthogonal set of treatment contrasts, the fixed-effects
correlations will  be 0.5. Thus, these correlations inform about the
correlations of the predictors in the model matrix.
       Moreover, the story changes again  if you estimate values for
parameters representing (co-)-variance components for random effects
in a balanced design, the fixed-effect correlations return values that
(sometimes?) are close to within-subject correlations (i.e.,
correlations unadjusted for shrinkage); maybe for balanced designs
with orthogonal predictors there is actually a specification under
which they are actually identical with them.  This would be cool.

Third, I think the fixed-effect part of the model you give now looks
fine; it is defensible (and sometimes necessary) to exclude
non-significant higher-order interactions.  I still don't think you
need the variance components associated with COND and DIR for subjects
and you may be communicating the wrong thing, but opinions may differ
on this, because non-significance of these components is a thorny
issue.
        In this case, as far as I can see, these correlations are not
really related to the COND x PCU interaction  you are interested in.
Significant effect correlations  can be mapped onto a different kind
of interaction (e.g.,, subjects with a large COND effect may tend to
have a larger DIR effect than subjects with a small COND effect), but
this does not bear on your PCU covariate, at least not directly. (This
could happen independent of a DIR x COND interaction in the
fixed-effect part of the model.)
       I saw that now you use log-transformed DVs and that in my
experience is a good choice for durations collected in eye tracking.
Nevertheless you should check the distribution of model residuals to
back up this decision. Anyway, the log-transformation of RRT may have
lifted the t-value for the PCU  X COND interaction. So I am curious
whether it did or not?

Reinhold Kliegl


On Fri, Aug 13, 2010 at 10:20 AM, Paul Metzner <paul.metzner at gmail.com> wrote:

Thank you for the quick answer!

(1) The Fixed Effects correlations are probably not what you are
after. For example, in a perfectly balanced design, these correlations
will be zero.

They are not, but like you suggested, I wanted them to be at least close to zero. When I changed the model like mentioned before, I noticed an increase in fixed effects correlations and a curious change in contrast coding (see below), that I couldn't explain.
My main interest are the fixed effects interactions. My hypothesis is that subjects with a higher PCU will be affected more strongly by the condition manipulation. Also, in some studies only one kind of verbs (DIR) has been shown to evoke the effect, hence the desired interaction of COND and DIR. But, because I really don't want individual differences over and above what is explained by PCU, I implemented the random effect term like you suggested and re-included the factors contributing to the interactions. My model now looks like this:

lmer(log(RRT)~COND + PCU + COND:PCU + DIR + COND:DIR + (1+COND+DIR|SUBJECT) + (1|ITEM), data=fm3)

Although including the covariance component did not improve model fit, I decided to leave it in the model for the reasons mentioned above. I did, however, exclude the three-way interaction COND:DIR:PCU.

(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.

Being a novice to contrast coding, I thought it was the same. Coincidentally, that seems to be the case for two-level factors. Thanks again for the suggestions!

Paul


On 12 Aug 2010, at 11:24, Reinhold Kliegl wrote:

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


On Thu, Aug 12, 2010 at 10:44 AM, Paul Metzner <paul.metzner at gmail.com> wrote:

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/

---
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/