# When fitting a GLMM, is the predicted value for any success or all successes or what?

I am relatively new to multilevel modeling and have just been given an assignment that uses a generalized linear mixed effects model. The outcome is smoking status (1=yes, 0=no) measured at three different time points. I am using proc glimmix in SAS to get the odds ratio estimates. I am having a hard time of understanding what the ORs are measuring exactly. Is it the odds of smoking at all three time points (compared to not smoking at all)? Or is it the odds of any smoking (i.e., smoked at 1 or more of the three time points where smoking was assessed)? It is just a a bit more confusing since there are not just two options of smoking or non smoking. There are 8 different combinations that each participant could have regarding the smoking status at three different measurements. Also, is there a way to get the correlation amongst smoking status from the model?

• Can you clarify your question in the last sentence? Commented Apr 24, 2015 at 3:33

Typically you will have several variables in this situation. You will have a subject ID variable that indexes which subject the data came from. You will also have a time variable indicating which time point the particular observation came from. So for example, your dataset might look like this:

SS time ID
0  1    1
1  2    1
1  3    1
0  1    2
0  2    2
1  3    2
...


Your model will have a coefficient for time, and will also fit a mean and variance of the distribution of random intercepts and possibly random slopes. From those distributions you can predict a random intercept (and slope, if appropriate) for a particular subject. Then, using the fitted coefficients for the fixed effects (notably the coefficient for time) and the predicted random effects you can calculate the predicted value for smoking (actually ${\rm logit}(\hat p)$ since your example is binomial) at a particular time. That is, when you plugged values into the formula, you plugged in a particular timepoint (1, 2, or 3). The predicted value is for that timepoint.

Edit: To get an odds ratio from your logistic regression output, you exponentiate the fitted beta. Thus, you would do: $\exp(\hat\beta_\rm{time})$. This gives you the multiplicative change in the odds as you increase time by one unit. As I set up the data above, that would be going to time=2 from time=1, or going to time=3 from time=2. If your data were in months, and the waves were six months apart, it would be the multiplicative change associated with a one month passage of time (i.e., $^1/_6$ of the interval between waves). On the other hand, if you solved for a particular subject, or just got a prediction for a specified set of input values, you would get the log odds of smoking. Upon exponentiating that, you would have the odds of smoking. There is no such thing as the 'odds ratio of smoking'.

It is very likely that there will be a correlation between the timepoints. (That is, it is very common—it isn't a logical necessity.) The random effects will account for this. You can get the amount of correlation from the model by using the intraclass correlation (ICC). It is the variance of the random intercept divided by the sum of the variance of the random intercept and the residual variance.

• Okay that makes since to plug in values once SAS has estimated the formula, however, I used the odds ratio option in SAS to output the odds ratio and 95% CIs. It is very clear in regular logistic regression that the OR is the odds of event=1 given exposure compared to the unexposed group. Now that there are three repeated measurements for the event of interest its confusing me as to what the odds ratio that is output is looking at in the outcome. Is in the odds of smoking at every one of the three waves? Or is it the odds of having smoked at least once during the three measurements? Commented Apr 24, 2015 at 12:09
• Also, my question in the last sentence. What I meant was should've there be a correlation amongst the smoking status at ll three time points? For instance, if someone smoked at the first time point, is that more predictive of smoking at the later time points, or is the covariates in the model more predictive? Commented Apr 24, 2015 at 12:14