Longitudinal Binary Logistic Model vs. Cox Model

Question

Longitudinal logistic regression can be used to approximate the hazard ratios from a Cox model:

https://hbiostat.org/stat/binarySurv.html

In the first paragraph, Frank Harrell states:

In the special case where the response variable for a time interval is binary, the event is a terminating event, and interval event probabilities are small, longitudinal binary logistic models’ odds ratios are the same as Cox model hazard ratios.

My questions are:

1. How do we assess interval event probabilities to decide whether they are "small enough"?
2. What are the advantages of running the analysis as a longitudinal logistic regression as opposed to a Cox model? I'm guessing perhaps computation speed and to reduce assumptions of the model (e.g. to avoid the strong proportional hazards assumption in the Cox model)?
3. If the analsyis is performed using longitudinal logistic regression, should we still call the effect sizes (exponentiated coefficients) "odds ratios" or is it OK to call them "hazard ratios" in this context?

Answer 1

Question 1. In this context, a logistic regression models at each time the odds of the event, given that the event hadn't yet happened; a Cox regression models at each time the hazard, the probability of the event given that the event hadn't yet happened. So this is the question of when odds ratios and hazard ratios are close to equivalent. An example on your linked Harrell web page used an exponential model with constant hazards of 0.004 and 0.0026, where the odds and probability are essentially identical. At a hazard of 0.05 the odds are numerically 5% higher, 0.0526.

For a binomial regression, the complementary log-log (cloglog) link, $\log(-\log(1-p))$, stays closer to the probability scale for longer than the log-odds link, $\log(p/(1-p))$, of logistic regression. For example, the exponentiated cloglog is within 5% of the probability up to a probability near 0.1, as $-\log(1-0.1)=0.1054$. The complementary log-log link is appropriate for proportional hazards models when survival times are grouped into intervals.

Question 2. I don't know that there is much practical advantage to the longitudinal logistic model for continuous-time data. You then need to make multiple data rows for each individual for all at-risk times, as performed by Harrell's expandit() function at the top of that web page. That's not needed for Cox models unless you are fitting time-varying coefficients.

Tied event times don't require special handling in a discrete-time binomial model, unlike in a Cox model. Note that Harrell's examples are essentially discrete-time survival models, just with a lot more time values than you might usually see.

Logistic regression makes a proportional-odds instead of a proportional-hazards assumption. Is either assumption "stronger"?

The logistic model can allow you to model a baseline hazard as part of the process, while a Cox model can return an empirical baseline hazard.

Question 3. If you model odds ratios, you should call them "odds ratios" in your presentations. You can choose to note their near equivalence to hazard ratios (if appropriate) in discussion.