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A friend of mine approached me to help her to interpret her multinomial logistic regression model. They had measured people as 1 of 2 states at 2 time periods. So, each person can have 1 of 4 configurations: start at state 1, end at state 1; start at state 1, end at state 2; start state 2, end state 1; start state 2, end state2. They had performed an analysis where the outcome had 4 levels corresponding to each of these configurations. This struck me as a somewhat unnatural way to do this. I feel that this is a situation similar to if you had a continuous outcome, measured at 2 time points. If you were interested in the change from baseline, you can model the end value and adjust for the baseline and other covariates.

Does this translate to logistic regression? Can you model the end state, and adjust for the beginning state with other covariates? I couldn't think of a good reason why this would not work, but I also had this problem. Is this better than modeling the outcome as 4 possible categories?

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Does this translate to logistic regression?

Sure. As you said, you can model the binary end state (either 1 or 2) as a function of the beginning state.

We can look at the simplest case where we only have beginning state and ending state.

In this model, you would have two parameters on the log odds scale. That is, the intercept and the coefficient for beginning state. Assuming that we are modeling the probability of ending in state 2, and that the beginning state of 1 is the reference state, then a positive coefficient for beginning state means a higher probability of ending in state 2.

Is this better than modeling the outcome as 4 possible categories?

It is a different model. With four possible categories, this model will estimate three parameters. Change on the log odds scale would not have to be the same in that case.

How you interpret the models would be different. In the first model, you are asking questions like "How does a particular beginning state predict the probability of the end state?" For example, maybe you could set the beginning state, and are interested in what happens to the ending state. Or, perhaps people arrive in the beginning state, and we want to predict their ending state.

In the second model you are asking questions like "What is the probability of various sequences of states?" This would represent the situation where you are interested in the sequence itself as the unit of analysis.

The situation gets more complex with other factors or covariates in the model. But, the underlying idea of looking at the ending state versus the sequence of states will still hold.

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