# Survival analysis: difference between GLM clog-log family binomial vs GLM clog-log family Poisson

I'm trying to find ways to do survival analysis on data with asynchronous interval-censored outcomes and time-varying covariates.

I know that GLM binomial with a complementary log-log link function can be used in the analysis of synchronous interval-censored cases ref. On the other hand, GLM Poisson can be used to approximate (is equivalent to) piece-wise exponential- each interval has its own hazard function ref.

From what I gather, they both require the data to be in the same format. Multiple rows per individual, where time is categorical, does not have to be in equal-sized bins but each time value has to be recorded for every individual until event/censoring. GLM Poisson takes time as an offset(log(time)), while GLM binomial can, but doesn't have to have to be a log offset (just include it as any other covariate).

Could someone please explain how those models differ and when is it appropriate to use each?

Binomial regression of event probabilities over time with a complementary log-log (cloglog) link is sometimes called a "grouped proportional hazards (PH)" survival model. It represents an underlying continuous-time PH survival model in which survival and censoring times are grouped into bins too wide to be considered continuous in time. Like a continuous-time Cox PH model, there is no assumption of a form for the baseline hazard, around which the covariate hazard ratios are associated with survival. The intercept for each time bin is the estimate of the difference in the empirical cumulative baseline hazard between the beginning and the end of the time bin.

Binomial regression with a cloglog link is used when PH is assumed to hold but you have panel data with the same individuals observed repeatedly at wide and common intervals, for example each year. You know whether an event occurred during that time interval for each individual, but not exactly when within it--hence the "grouped" PH description. Technically, the survival times are interval-censored.

The piecewise-exponential model you cite is a type of continuous-time analysis that models the baseline hazard directly. That's different from a Cox PH model, in which the non-parametric baseline hazard doesn't enter the regression directly and is only inferred after the model is fit.

The time axis is broken into a set of time periods; within each time period, the baseline hazard is assumed to be constant. A constant hazard means an exponential survival model, with events thus assumed to be Poisson-distributed (conditional upon covariate values) within that time period. The Poisson/exponential modeling within each time period requires individual event/censoring times on a continuous scale; this is not a "grouped" model in the sense of panel data analyzed by a binomial model with cloglog link and it isn't intended for interval-censored survival times.

A piecewise-exponential model can be considered an intermediate way to estimate the baseline hazard between a completely non-parametric Cox PH model and a standard parametric model like a Weibull model. It can provide more flexibility than a standard parametric model while still providing a closed form (unlike a Cox model) for the baseline hazard and thus for making predictions from the model.

Offsets

An offset term in a regression model is a predictor whose regression coefficient is set to be exactly 1. For example, with count data when observations are over different extents of time, you need to account for the different observation times if you want to model the underlying event rate. A Poisson model with a log link for such data then would use log(time) as an offset. The log comes from the log link. If you used an identity link, time itself could be an offset for modeling rates. Alternatively, you might choose to model counts instead of rates and use time or a function of time as a covariate. This page and its links discuss these issues extensively.

Offsets aren't required for either of the types of models in this question.

The piecewise exponential model directly incorporates time in the calculation within each time period, so there's no need for an offset.

Even if panel data time periods have different durations, the cloglog model will work fine. The intercepts incorporate the duration information, as they are differences in cumulative baseline hazards between the ends of the individual time intervals. Other things being equal, a longer time interval will tend to have a larger intercept in the model.

The Tutz and Schmid book on "Modeling Discrete Time-to-Event Data" doesn't seem to show any examples of using an offset for time, although it does mention offsets in other contexts. You might want to incorporate (a function of) time as a predictor in some survival models, but there isn't much point in forcing the associated regression coefficient to be 1 with an offset term.

• again, thanks so much for the answer! Could you make it a bit more detailed by explaining when/why use offset(time) or offset(log(time)) in either model? Oct 20, 2022 at 14:07
• @Wojty I added a few paragraphs. Not much need for offset terms with coefficients required to be equal to 1 in survival models, although including time (or a function of time) as a covariate directly can sometimes be useful.
– EdM
Oct 20, 2022 at 14:59