fitting an exponential decay model to binary data (repeated measures)

[mjwenger]

I have a data set for a smoking cessation trial in which participants were sampled on the day they first attempted to quit and then at a number of time points after that date. At each time point, they were classified as being either abstinent (1) or non-abstinent (0). If you plot the proportion of participants who were abstinent as a function of time the data look very cleanly like an exponential decay to a lower limit. Participants were also measured for their iron status. Preliminary analyses have suggested that higher levels of two of the iron measures are associated with a higher likelihood of being abstinent at any given time point. I'd like to be able to estimate an exponential decay model in which two predictors increase the probability of being abstinent and time decreases it. Does this seem reasonable and, if so, how would I go about it? My thanks in advance.

[StatDave]

If you think of the classic S shape of a logistic curve, the lower half of it probably is quite similar to what you see in your data. So, a logistic model might fit well. Since the data consist of repeated measures, you could use either a Generalized Estimating Equations model with PROC GEE or a random effects model with PROC GLIMMIX, depending on whether you want a population-averaged model or a subject-specific model. In the model you could include both time and your iron measures as predictors.

[mjwenger]

Thanks, Dave -- I'm assuming that you mean a function defined as 

y(t) = a / (1 + b c^(-t))

with 0 < c < 1, a estimating the upper asymptote, and b estimating the lower asymptote. So if c specifies the decay rate as a function of time, how would you specify the effects of the two (iron) predictors that would attenuate the decay? Apologies if I'm missing something obvious and thanks again for the help.

-Michael

[StatDave · Accepted solution]

No, I'm referring to an ordinary logistic regression model fit to binary response data: log(p/(1-p)) = Xβ, where p=Pr(Y=1), 

as can be fit by PROC LOGISTIC or, since this is repeated measures analysis, by PROC GEE or PROC GLIMMIX - that is, a generalized linear model with binomial distribution and logit link function. Examples can be found in the documentation of all of those procedures. 

[mjwenger]

Excellent, and thank you.