I am wondering how one should properly analyse longitudinal data from simple experiments (not observational studies). Imagine an experiment where subjects are first measured at baseline, randomised to different treatments and then followed for a number of time periods. We are interested in differences (typically in means, but perhaps also variances) between the the effect of treatments at each time period. We only look at a single outcome measure.
The simple(st) case
Let $Y_{ij}$ be the measurement of subject i at time period j, $i=1,\dotsc,n$, $j=0,\dotsc,k$, where $j=0$ corresponds to baseline (i.e., pre-intervention) measurements. If we have two treatments and only $k=1$ follow-up time period (and are interested in means), it’s easy to analyse the data. We can either look at the change scores $Y_{i1}-Y_{i0}$ and compare these between the two treatment groups using a t-test, or do an ANCOVA where we adjust for baseline, $Y_{i1}=\alpha + \beta_1 Y_{i0} +\beta_2 T_i+\epsilon_i$, where $T_i$ is either 0 or 1, depending on the treatment subject i received. Both methods of analysis are valid, but the ANCOVA is more powerful (and may give us ‘better’ answers if the treatment groups happened to be unbalanced at baseline?).
(Note that all this for an experiment; for observational studies, other analyses may be more appropriate.)
The more interesting/difficult case
But how should we properly analyse data like this when we have measurements at more than one follow-up time? It seems to be that things quickly gets much more complicated. The simplest example is an experiment with two follow-up times. Let’s make it more concrete, and imagine that we are looking at weight-loss, and are investigating the effect of an intervention (diet, exercise or nutritional advice) on overweight people. So we gather a bunch of overweight people, measure their weight (or BMI), randomise them to either intervention or control, perform the intervention, and then measure them again after, say, 6 months and 2 years.
I see several ways to analyse this:
- Ignore the baseline values and analyse only the follow up-data.
- Compare change scores like we did in the simpler case. We now have two changes scores $Y_{i1}-Y_{i0}$ and $Y_{i2}-Y_{i0}$.
- Model the three measurements as correlated random variables, and estimate the parameters using generalised least squares with an unstructured covariance matrix.
- Do an ANCOVA-style analysis at each time point.
Here’s my understanding of these methods:
This seems like a really bad idea, as we’re throwing away valuable data. :-( But it is a valid method of analysis.
This also seems valid, but I guess it too wastes power. And we really need to jointly model to two change scores. How should we do this?
This method of analysis initially sounded perfect (to me). We get nice estimates for both the means at each time point and the difference in means between the two groups at each time point. We can easily add the restriction that the mean at baseline is equal for the two groups (to gain some power), but one problem is the distributional assumptions. Even if we are prepared to assume approximately normally distributed errors at the two follow-up times, for the baseline measurements the errors are guaranteed not to be normally distributed, as the inclusion criterion was that the people participating should be overweight (e.g., weight > some fixed constant). So if weight was normally distributed in the population, it would have a truncated normal distribution for the baseline measurements for our subjects (at follow-up, this would probably be less of a problem).
A similar issue would of course arise if the inclusion criteria didn’t exactly correspond to limits on baseline values, but only were (cor)related with them (e.g., where the inclusion criteria specifies a BMI > a constant, and we are measuring triglycerides). Because of this, it doesn’t feel right treating the baseline measurements on the same footing as the follow-up measurements; they are fundamentally different.
This I’m actually not sure how one should go about doing. I guess we could create a linear model $Y_{ij}=\beta_j + \beta_{1j} Y_{i0} +\beta_{2j} T_{i} + \epsilon_{ij}$, $j=1, 2$, and fit this using generalised least squares (with an unstructured covariance matrix). Are there any conceptual or technical problems with doing this? Will a similar method work just as well even if we have many different treatments and/or time points (as long as we have plenty of observations)? Or if we have some missing data (MCAR or MAR)?
Basically, my question is: For longitudinal data in experiments like the one described above, which of the methods 1, 2, 3 and 4 (or other methods) can be considered valid methods of analyses, and which of them are preferred (e.g., have the greatest power, minimum variance of estimators, least bias …)? I know many different methods of analysis have been used in the past, but what are the current recommendations?