Trend analysis with three time points in a repeated measures design Assume that there are 15 subjects and that a quantitative feature is measured for all subjects at three equally-spaced times points. The research of interest is to find out wether there is a linear trend across time. Admittedly the number of time points is quite limited, but what are the options for trend analysis? I can think of two approaches:
1) The traditional trend analysis is usually performed with weights assigned based on orthogonal polynomials. With three time points, this would lead to the weights of -1, 0, and 1. The problem with this approach is that the data at the midpoint is essentially ignored (under the equal space assumption), leading to a contrast between the two end time-points.
2) Alternatively I can try linear mixed-effects modeling with lme from R package nlme using the three time points as an explanatory variable (1, 2 and 3?). Would this approach be reasonable?
Any suggestions? Thanks!
 A: If you have a constant term in your model, the midpoint is not being ignored, even though it is given a value of zero in this coding. The value equal to the mean of the two other values indicates to the software that the time spacings are equal. You can code your time as 1, 2 and 3, but then it will be collinear with the constant/intercept, resulting in somewhat less efficient estimates, while what you have is mean-centered coding. You can even do a one d.f. test for linearity by including a quadratic term.
The two approaches you are contemplating are nested within one another: by restricting the variance of the random effects to zero, you will obtain the simple trend model. With 15 subjects, however, your power is likely to be very low, and the estimates of the variance components, very imprecise.
Trying both approaches is a matter of three lines:
    lm(outcome ~ time)
    library(lme4)
    lmer(outcome ~ time + (1|subject))
    lmer(outcome ~ time + (time|subject))

Why don't you just do this and post the results? May be we'd see clearly what's going on, and whether it is worth bothering working with the mixed effects model.
A: Sorry to disappoint, but I would NOT try to categorize this as trend analysis, at least not in the usual time series sense.
That said, you can still do productive modeling.  You look at testing the hypothesis that the difference between (t1,t2) is non-zero (i.e. test against the null hypothesis).  You could do the same for (t2,t3).
Where this gets tricky is that the intervals (t2 - t1) and (t3 - t2) are not well defined in your question.  There may be a constant interval, but is the response believed to have a linear relationship with the interval?  Try examining students' math scores in January, May, and September, and you may see that the second interval drops a lot relative to the gain made in the first interval.  Students' math scores croak in the summer.
So, I would first begin with two separate tests.
Second, what I would try is sign-testing of the deltas for the two periods.  For this, you can do simple binomial hypothesis testing.  (NB: If all values change monotonically, regardless of the factor being studied, then de-trending will be a little hard, but you could center with the sample mean at each timepoint.)
