# proportional hazards model with fixed interval censoring = cloglog GLM with fixed effect of time?

Consider a survival analysis with time-constant coefficients, interval-censored, where the observation intervals are consistent across all individuals (e.g. each individual is observed at the end of every time period). I think I remember seeing it asserted somewhere that in this case a Cox proportional hazards model is equivalent to a binomial (Bernoulli) GLM with a complementary log-log link and a fixed effect of time for every observation period (this corresponds to the baseline hazard that is factored out of the Cox PH likelihood). Is this known to be true/false, and can someone provide a supporting argument or pointers to supporting references?

If true, this provides a very convenient way to avoid the computational/technical difficulties of fitting interval-censored Cox models in the specific case where the interval censoring dates are completely (or mostly) identical across individuals (e.g. see this question and more generally these questions ...)

• it looks like this might be covered here ... also, I realized my question looks a lot like this one ... (I don't think it can be closed as duplicate until there's an answer to one or the other) (The link from that Q is broken, on wayback machine here: web.archive.org/web/20141121080306/http://www.ics.uci.edu/… looks to be substantially similar ... Sep 30 '19 at 21:12

This particular situation is explained in Section 3.3.2 of Tutz and Schmidt, Modeling Discrete Time-to-Event Data, Springer, 2016, adapted slightly below.

Say that there is an underlying continuous-time proportional hazards (PH) model, but failures are only coded within intervals of time, e.g. the event is coded as happening at $$T=t$$ if is occurs within $$[t-1,t)$$. Then with covariate vector $$x$$ and regression coefficients $$\beta$$ the discrete hazard $$\lambda(t|x)=P(T=t|T\ge t,x)$$ can be written as

$$\lambda(t|x)= 1- \exp(-\exp (\gamma_{0t}+ x^T \beta)),$$

where the fixed time coefficient $$\gamma_{0t}=\log (\Lambda_0(t)-\Lambda_0(t-1))$$ and $$\Lambda_0(t)$$ is the cumulative baseline hazard function of the underlying continuous-time PH process. This is the form of a complementary log-log (Gompertz) model, as so nicely described here. This is called a "grouped proportional hazards model."

How many "computational/technical difficulties" are avoided, however, isn't so clear. Unlike a continuous-time Cox model, in which the baseline hazard effectively disappears from the maximum-partial-likelihood calculations, this fixed-time-effect discrete approach requires estimation of the baseline hazard (as parameterized in $$\gamma_{0t}$$) at each interval end-point along with estimation of the regression coefficient values. That's not a big problem with the type of panel data envisioned in this question. Except for the link function, it's not different from standard discrete-time survival models that use logistic regression for panel data with time modeled as a fixed effect.

"Computational/technical difficulties" arise with this approach when each individual has her own interval-censoring times, as in Finkelstein's original adaptation of PH models to interval censoring (Biometrics 42: 845-854, 1986). This occurs e.g. with modeling of disease progression in cancer, when between-visit progression is detected at clinical appointment times that aren't uniform among patients. In that situation with $$N$$ patients you might need to estimate as many as $$2N$$ nuisance values of a baseline hazard function (one for both ends of each patient's event-containing interval) in addition to the regression coefficient values of primary interest. That's why tools like those in the icenReg package can be needed.

• @CliffAB I wanted to clear up this long-standing question but parts of it are outside my specific expertise. When you get a chance, could you see if I fairly described why the approach of the OP in its simple situation doesn't avoid "computational/technical difficulties" with semi-parametric models in a broader context?
– EdM
Mar 31 at 20:29
• thanks, this seems like what I was looking for! Mar 31 at 21:07