Dear all,

Here is a question regarding the difference between using poly() and rcs().

If you fit a model using poly(), as shown below, the summary returns
statistics for the term "linear" term, poly(TrialContinuous,2,raw=TRUE)1,
and statistics for the quadratic, poly(TrialContinuous,2,raw=TRUE)2. Here,
the summary tells us that the "linear" and quadratic terms are significant,
that the interaction between the linear term and the factor variable
FreqGroup is significant, but that the interaction between the quadratic and
FreqGroup is not significant.

> m1=lmer(LogRT~poly(TrialContinuous,2,raw=TRUE)*FreqGroup+(1|Subject),data=
dat.li)
> print(m1,corr=F)
Linear mixed model fit by REML
Formula: LogRT ~ poly(TrialContinuous, 2, raw = TRUE) * FreqGroup + (1 |
 Subject)
   Data: dat.li
   AIC   BIC     logLik deviance REMLdev
 17199 17266  -8592    17056   17183
Random effects:
 Groups   Name        Variance   Std.Dev.
 Subject  (Intercept)   0.014825  0.12176
 Residual                  0.101560  0.31868
Number of obs: 30742, groups: Subject, 25

Fixed effects:

   Estimate    Std. Error    t value
(Intercept)
6.307e+00  2.456e-02    256.85
poly(TrialContinuous, 2, raw = TRUE)1                       -1.570e-04
4.639e-06   -33.84
poly(TrialContinuous, 2, raw = TRUE)2                        9.779e-08
1.120e-08    8.73
FreqGroupLow
3.299e-02   6.085e-03    5.42
poly(TrialContinuous, 2, raw = TRUE)1:FreqGroupLow  1.769e-05   8.736e-06
 2.03
poly(TrialContinuous, 2, raw = TRUE)2:FreqGroupLow -1.168e-08   2.123e-08
-0.55

Now, if you use rcs() instead, the summary return statistics for the first
and second splines, as shown below. We see that the 2 splines are
significant, but there is no interaction between any of the splines and
FreqGroup. This reflects what we see in the model fitted with poly(), where
the the linear and quadratic terms were significant but the quadratic by
FreqGroup interaction was not significant. What's missing in the model
fitted with rcs() is the linear term.

> m2=lmer(LogRT~rcs(TrialContinuous,3)*FreqGroup+(1|Subject),data=dat.li)
> print(m2,corr=F)
Linear mixed model fit by REML
Formula: LogRT ~ rcs(TrialContinuous, 3) * FreqGroup + (1 | Subject)
   Data: dat.li
   AIC   BIC     logLik deviance REMLdev
 17180 17247  -8582    17065   17164
Random effects:
 Groups   Name        Variance   Std.Dev.
 Subject  (Intercept)  0.014826   0.12176
 Residual                 0.101592   0.31874
Number of obs: 30742, groups: Subject, 25

Fixed effects:

   Estimate    Std. Error    t value
(Intercept)
6.286e+00  2.498e-02    251.62
rcs(TrialContinuous, 3)TrialContinuous                        -2.538e-04
 1.253e-05   -20.26
rcs(TrialContinuous, 3)TrialContinuous'                        1.285e-04
1.547e-05    8.31
FreqGroupLow
3.599e-02   1.061e-02    3.39
rcs(TrialContinuous, 3)TrialContinuous:FreqGroupLow   3.018e-05   2.379e-05
   1.27
rcs(TrialContinuous, 3)TrialContinuous':FreqGroupLow -1.660e-05    2.936e-05
  -0.57

I tried adding a linear term "TrialContinuous" to the model fitted with
rcs(), as shown below, but that doesn't work, the model is not positive
definite.

> m3=lmer(LogRT ~ (TrialContinuous + rcs(TrialContinuous,3)) * FreqGroup +
(1|Subject), data = dat.li)
> Error in mer_finalize(ans) : Downdated X'X is not positive definite, 7.

The question is where is the linear term (if there is one)?
Is it hidden somewhere? Or is it simply not a question to ask when using
rcs() to fit models?
Should I use poly() rather than rcs()?
Are there situations where you would want to use poly() over rcs() and vice
versa?

Additionally, how should we interpret the statistics for the splines?
Does their significance mean that the slopes in the splines are
significantly different than 0?
In the case of the second spline, does it's significance mean that it is
significantly different than the first spline?

Thank you very much for your time and efforts. Your help is greatly
appreciated.
Sincerely
-- 
Antoine Tremblay
Department of Neuroscience
Georgetown University
Washington DC

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