[R-sig-ME] random effects for a second-order polynomial

[John Kingston]

I am trying to model response times in an experiment in which listeners
hear one of five synthesized syllables, one which is clearly "da", another
which is clearly "ga", or one of three steps in between the clear "da" and
clear "ga", which differ in how ambiguous they are between the clear "da"
or "ga". Listeners must identify each syllable as "da" or "ga". As is
typical in such experiments, response times are much faster to the clear
instances of each syllable than to the ambiguous steps in between them, and
a plot of response times against the steps along this series of syllables,
da 2 3 4 ga, has roughly the shape of an inverted parabola. So, I naturally
modeled the effect of the series with poly(step, 2), and I obtained an
expected negative estimate for the second-order term. My question concerns
the random effect structure. When I include only a first-order effect of
step in the random effects, like this (scDgRsp = whether the response was
"da" or "ga"):

dgRTStep0 <- lmer(logRT ~ scDgRsp +
                    poly(scStep, 2)  +
                (1 | participant) +
                (0 + scDgRsp | participant) +
                (0 + scStep | participant),
                    control = lmerControl(optimizer="bobyqa",
                                           optCtrl=list(maxfun=2e5)),
              data = dgTestTrim)

I get this as the random effects:

Random effects:
 Groups        Name        Variance  Std.Dev.
 participant   (Intercept) 0.0051506 0.07177
 participant.1 scDgRsp     0.0005302 0.02303
 participant.2 scStep      0.0002087 0.01445
 Residual                  0.0194742 0.13955
Number of obs: 13272, groups:  participant, 28


but when I include the second-order term, like this:


dgRTStep1 <- lmer(logRT ~ scDgRsp +
                    poly(scStep, 2)  +
                (1 | participant) +
                (0 + scDgRsp | participant) +
                (0 + poly(scStep, 2) | participant),
                    control = lmerControl(optimizer="bobyqa",
                                           optCtrl=list(maxfun=2e5)),
              data = dgTestTrim)


I get this instead:


Random effects:
 Groups        Name             Variance  Std.Dev. Corr
 participant   (Intercept)      0.0051593 0.07183
 participant.1 scDgRsp          0.0005799 0.02408
 participant.2 poly(scStep, 2)1 3.5037792 1.87184
               poly(scStep, 2)2 1.8811428 1.37155  -0.57
 Residual                       0.0193518 0.13911
Number of obs: 13272, groups:  participant, 28


I'm not so puzzled by the correlation as the sizes of the randow
effects, which four orders of magnitude larger. Is this sensible,
interpretable?


The fixed effects estimates differ little between the two models:


poly(scStep, 2)1  1.077e+00  3.841e-01  2.825e+01   2.804 0.009033 **
poly(scStep, 2)2 -2.969e+00  1.502e-01  1.300e+04 -19.768  < 2e-16 ***


versus


poly(scStep, 2)1  1.102060   0.418011 27.340208   2.636 0.013648 *
poly(scStep, 2)2 -2.968686   0.299938 27.103765  -9.898  1.7e-10 ***


except in the standard errors and degrees of freedom for the
second-order estimate.


Thanks!


Best,

John Kingston
Professor
Linguistics Department
University of Massachusetts
N434 Integrative Learning Center
650 N. Pleasant Street
Amherst, MA 01003
1-413-545-6833, fax -2792
jkingstn using umass.edu
https://blogs.umass.edu/jkingstn
<https://blogs.umass.edu/jkingstn/wp-admin/>

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