[R-sig-ME] Longitudinal covariation parameter estimate does not match average association over time

[Matthew Boden]

Hello,

I am thoroughly perplexed and could greatly benefit from your feedback
(thanks in advance!).

I am examining the longitudinal covariation between two variables (N, P)
measured each month for 26 months among 140 subjects. I am interested in
determining the average relation between these two variables when
accounting for dependencies due to repeated measures. Thus, I am interested
in between-subject variation more so than within-subject variation. Yet,
there exists considerable variation in both trajectories and intercepts for
individual subjects.

The issue is that the average association between N and P at each time
point is negative (e.g., r = -.29).  Yet, in most LMM models I run, the
fixed effects estimate for P predicting N is positive.

For example, including random effects for both intercept and slope (to
account for within subject variation in each) using the following code
yields a positive estimate for P.  This is also true if I include only a
random effect for the intercept or a random effect for the slope.

long <- lmer (N ~ P + (1 + P | ID), data = lip)

Fixed effect estimate (SE), Z
Intercept = 46.9 (4.38), 10.68
P = 2.99 (.57), 5.19

The only way I obtain a negative estimate for P is when I include
duration/time in the model and a random effect for duration/time, but
exclude the random effect for P AND exclude the random intercept.

long <- lmer (N ~ P + time +  (1 + time | ID), data = lip)

Fixed effect estimate (SE), Z
Intercept = 73.99 (2.17), 34.09
Time = .18 (.06), 2.84
P = -1.01 (.28), -3.51

Besides the fact that I'm not really interested in the structure of N over
time, and thus seemingly do not need a duration/time parameter, there is
substantial variation in the intercept and slope for N by P for which
random effects would be needed.

It is my understanding, perhaps wrong, that the fixed effect parameter
estimate for P should be akin to the average association between N and P
across time. Thus, this parameter estimate should be negative, regardless
of whether or not duration/time is included in the model. Indeed, plotting
N by P without consideration of duration/time reveals a negative average
regression slope.

I'm at a loss and could use some help.

Thank you,

Matt

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