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.
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.