Why am I getting such different hazard ratios from the Andersen-Gill model compared with the Prentice-Williams-Peterson model?

Question

I am trying to model recurrent composite events to compare differences in recurrent events between two therapy groups. I have completed time to first event Cox regression as well as recurrent event regression using the Andersen-Gill (AG) and Prentice-Williams-Peterson (PWP) models. However, the results for the two recurrent event models are quite different. Following are the hazard ratios (HR) I get for the three models:

Cox time to first event (TTE): 6.42 (p<0.001) Andersen-Gill: 10.23 (p<0.001) PWP: 6.24 (p<0.001)

For more context, the two therapy groups are time dependent covariates. All models are multivariable models adjusted for potential confounders.

All methodology papers I have encountered so far show very similar HRs for both the AG and PWP models and I am wondering why the AG HR is so inflated in my results. Is there a specific reason why this may be the case? Below is the R code I used. I can provide more information about the scenario if required.

coxmodel(TTE) <- coxph(Surv(time = tstart, time2 = tstop, 
                        event = event) ~ therapy + var1 + var2 + 
                        var3 + var4 + var5 + var6, data = mydata)

AGModel <- coxph (Surv (tstart, tstop, event) ~ therapy + var1 + 
                   var2 + var3 + var4 + var5 + var6 + 
                   cluster(ID),data=mydata)

PWPModel <- coxph (Surv (tstart, tstop, event) ~ therapy + var1 + 
                    var2 + var3 + var4 + var5 + var6 + 
                    cluster(ID) + strata(enum), data=mydata)

FYI - therapy, var1, var2, var3 are time dependent covariates.

Answer 1

Therneau and Grambsch discuss recurrent-event models in Chapter 8.

For the Andersen-Gill model: "This method is the simplest to visualize and set up, but makes the strongest assumptions" (page 185); "This model is ideally suited to the situation of mutual independence of the observations within a subject" (page 186).

For the PWP model: "It assumes that a subject cannot be at risk for event 2 until event 1 occurs; in general, a subject is not at risk for the $k^{th}$ event until he/she has experienced event $k - 1$. To accomplish this... each event is assigned to a separate stratum... The use of time-dependent strata means that the underlying intensity function may vary from event to event, unlike the AG model, which assumes that all events are identical" (page 187).

What you found is what you might expect if all events within an individual are not identical, so that the stronger assumption underlying the Andersen-Gill model isn't met.