What to do if a specific parametric survival model does not converge but most other model do (implications for decision analysis)?

Question

Please consider the following:

In (health) decision modelling, an often-used approach is to extrapolate observed survival data with parametric functions.

The NICE Technical Support Unit summarised this methodology here and in a one-page summary here.

Usually, several parametric models are estimated. However, depending on the data used, it can happen that one (or more) specific model is ‘not converging’.

In R, the package flexsurvreg will give a warning message if this occurs:

Warning message: In flexsurvreg(Surv(times_start_ELd, times_end_ELd, type = "interval2") ~ : Optimisation has probably not converged to the maximum likelihood - Hessian is not positive definite.

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In a report, I have seen colleagues addressing this issue as follows:

“[…] model did not converge due to the limited number of events over the long-term.”

and

“[…] model could not converge […]. No parametric function were estimated using these models.”

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This post suggests a couple of solutions to avoid this error.

However, here I am rather interested in the implications of a non-converging model when other (alternative) models do give estimates.

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My questions are:

1. How can you determine that the non-convergence is due to the limited number of events and not due to other causes (like massive censoring)?

2. Am I correct in assuming that since the problem of non-convergence often happens in only one or a few models using the same data set, this is not per se a problem of the data but rather the parametric function? (What I mean here is that I do not need to investigate the scaling of the variables used for the parametric estimation as this works fine with the majority of the models. Hence, I would not re-scale the data but rather accept the model cannot be properly estimated.)

3. If this non-convergence happens with one (or a few) model(s) only, is it common to ignore this model?

Answer 1

It may be the case that the model assumptions are far from resembling your data, thus making the algorythm that calculates the model parameters unable to converge.

Another possibility is that one of the parameters tends towards infinity, as would be the case if you tried logistic regression on the following data (explaining_variable, health_state):

(1, healthy); (1, healthy); (-1, sick); (-1, sick)

Here a logistic regression model would try to make the effect of the explanatory as big as possible, with the optimization algorithm that fits the model never converging (it can always keep making the parameter bigger and bigger)