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Let's say I have a sample of "events" done by a certain number of subjects, and some (although not most) of these subjects have underwent more than one event. I'd like to fit a logit model to these data to find out which characteristics contribute to a subject, over the course of an arbitrary period of time, undergoing another event within an arbitrary interval after a preceding event -- let's say three days.

The way I see it, I can set my data up in one of two ways:

  • take all of the events underwent by each subject, and define the dependent/indicator variable to be yes/no to "did this subject have another event within 3 days of a preceding event, over the course of the last year (for example)?" In this case, each row of the data for the model would correspond to every subject and be an aggregate of their event history.

  • partition each subject's event history into pairs and define the dependent variable to be yes/no to "did this subject have another event within 3 days of this particular event?" In this case, each row would correspond to every event in the data. However, there are three outcomes: 1) the subject has another event within 3 days; 2) subject has another event within greater than 3 days; and 3) subject doesn't have any further events. Could I collapse 2) and 3) into one outcome, or would I be better off considering using a multinomial logit model instead?

I'm leaning towards going with the latter option, but I'm worried that those subjects who undergo relatively more events will be over-represented in the final model (how would I deal with that, if I should?). However, I do think I gain significantly more information that way. Anyway, with that said, I'd love to hear insights on the pros/cons of each approach to setting up these particular data for a logit model.

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  • $\begingroup$ What are the characteristics you want to fit here - and how many days/different types of outcomes are there? It would be reasonably straightforward to use GEE for repeated measures (outcome yes/no, for different types of outcomes on different days clustered among different people; a working assumption is that outcomes nearby in time are more correlated than those far apart) but it depends on what the characteristics are. $\endgroup$ – guest Feb 16 '12 at 1:35
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At first I thought I'd lean to your second option too. It seems a slightly more orderly approach to the data and makes the most of the fact that subjects may have several pairs of close events.

However, the argument for the first option would be that it better represents your research question, which as I understand it is about the characteristics of subjects that lead to them experiencing events in close succession. A natural way of going about this is to identify the subjects who experience such an issue and look at their characteristics - which is your option 1. An extension of option 1 would be to actually produce a count for each subject "how many times has this subject experienced a second event close after a first event?". This count variable would then be the response in a generalized linear model with a poisson response. Once I'd thought of this, I decided I prefer option 1.

If you go for option 2, whether you can collapse your outcomes 2 and 3 together depends a bit on your research question and the underlying theory you want to test. How arbitrary is the three days limit for example? If you don't collapse them however you don't want a multinomial logit model, you want ordinal logistic regression. A multinomial model would be if you have three outcomes with no real ordering to them - in your case, clearly there is a natural order between "another event within three days", "another event before observations finished" and "no other event".

One argument in favour of collapsing your outcomes 2 and 3 will be that if you don't you have a problem similar to many survival studies. That is, the chances of getting the second outcome rather than the third depend on how late in the study the first event happened, purely due to the fact that you observe the post-event environments earlier in the experiment for a longer time. Collapsing outcomes 2 and 3 fixes that problem at least.

To control for subjects under option 2 you must introduce a level of randomness for subject. Otherwise you certainly stuff up your research question (and would be better off with option 1). You can do this with a generalized linear mixed effects model, for which there are implementations in various software applications.

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  • $\begingroup$ Thanks, Peter. I'll consider a GLMM, although I would like to capture other characteristics of the situations within which these recurrences happen, so perhaps that would call for a HLM. $\endgroup$ – tetragrammaton Feb 17 '12 at 0:59
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@PeterEllis has made some good suggestions. I like his idea of using Poisson regression (which I suspect should be supplemented by an offset), and his point that ordinal logistic regression is more appropriate than multinomial.

However, I want to suggest that you look seriously at survival analysis. As best I can follow your study, and what you hope to find out, survival analysis is appropriate, most likely the Cox Proportional Hazards model. In particular, the fact that you explicitly state that your window of 3 days is arbitrary, and the fact that it appears very often a second event does not occur within that interval, imply that results from other analyses may be badly biased.

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  • $\begingroup$ Thanks, gung. I can't upvote your answer yet, but I'll consider a CoxPH model as well. However, I'm concerned that some of the assumptions that such a model makes wouldn't apply here. $\endgroup$ – tetragrammaton Feb 17 '12 at 0:59

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