How to solve a case of unbalanced repeated measures?

Question

I have animals, that could be virgin or mated (reproductive state is the fixed factor), which I've stimulated sequentially with 4 different doses of an odour (doses are the repeated measures, the same animal was blown with 4 increasing doses of the same odorant). Then, I measure the neuronal response (variable: number of spikes) of each animal to each dose of the odorant. This might be a typical case or repeated measures, however I have some missing values for the doses, I have not all the doses completed for some animals. For example, for animal 1, I missed recording 1 out of the 4 doses. what can I do? I have two statistical packages: SPSS 16 or Statistical. thanks for your help!

Answer 1

Repeated Measures

Personally I would pursue a hierarchical model where the basic observations are, for each animal, the 4 (or fewer) levels of odour and the corresponding neuronal responses. And the predictor for the per animal intercept and slope on this relationship is the animal's reproductive status. (Here I'm assuming that your interest is in the effect of reproductive status on these aspects of the response function and to what extent it is distinguishable from individual variation.) That would give you nice interpretable animal level regression parameters, e.g. moving from virgin to mated animals drops the predicted firing rate by x and increases the effect of one unit increase in odour dose by z.

Failing that, a mixed model with reproductive fixed effect would probably also work. Actually I think that's all SPSS 16 offers you anyway.

I wouldn't immediately worry about missing data in this framework. Just try it and then check for robustness of the results, as Rob suggests. The more basic problem is knowing what SPSS is telling you when you fit one of these models. For that, you'll want to read up a bit first. Other folk here may have preferred introductions to mixed models - mine are all R-oriented and therefore not so helpful.

Spikes

If you are working with spike counts they are probably conditionally Poisson distributed (and don't forget the offset, if the exposure during measurement varies). If you don't have the option to specify that fact you might need to fit the model in appropriately adjusted log counts or suchlike.

Missing Data

If you have enough animals missing a few measurements for some of them might be ok. For a lot of missing, there won't be much information as Rob also points out.

If you (or your audience) worry about the missing data, you could do multiple imputation first. If I remember right, and I don't really use SPSS for anything, 16 makes you use AMOS for multiple imputation, but later version have it built into the missing value module. So that might be an option.

Answer 2

Mixed effects analysis (available in R via the lme4 package, free as always) can handle missing data like this. My understanding is (possibly erroneous? Mixed effects modelling experts please feel free to provide correction) of how this is achieved is as follows:

Given a set of predictor variables that are combined in some way to form a matrix that I usually call the "predictor design" (for example, for a completely crossed design of 3 variables with 2 levels each, the predictor design is a 2x2x2 matrix in which you expect to have each cell filled with an observed response), and given a set of "units of observation" (ex. individual human participants in an experiment), missing data can occur if the experiment fails to obtain an observation for a given unit in a given cell of the predictor design.

If the missing data occurs in a cell of the design that involves a continuous predictor variable over which the model would typically attempt to fit a slope, the model simply goes ahead and fits the slope across those cells without missing data. I think that the model also takes into account the fact that there is missing data in determining the degree of influence that given unit's data has on inference relating to that slope; that is, slopes from units with lots of missing data will by definition be based on fewer observations and thereby are expected to be less reliable estimates, a circumstance that the model takes into account when combining that estimate with those from other units with possibly more reliable estimates.

If the slope cannot be computed (ex. one or fewer cells have data), or if the missing data occurs in a cell of the design that involves a categorical predictor variable, then that given unit will not influence inference of that particular slope/effect, but remains in the model for influence of slopes/effects for which the unit does have sufficient data. (Hm, now that I think of it, I'm not sure how this works in the context of missing data for cells that constitute the intercept level of contrasts when treatment contrasts are used...)

Note, however, that if you are missing information about the predictor variables (eg, you have a response, but you're unsure where in the predictor design it should fall), the model will not attempt to impute this information and will instead simply ignore that value.