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I am using the BaSTA package in R to estimate sex differences in survival parameters for a capture-recapture dataset. When I use a simple Gompertz model the parameters are very interpretable (b0 represents the baseline hazard and b1 represents age-dependent mortality rate, or senescence). However, when I look at a simple Weibull model, I'm not sure how to biologically interpret the differences in survival parameters. For my dataset, females have a higher shape parameter (b0) than males, but a lower rate parameter (b1) (see model output and plots). These differences are substantial enough to interpet according to the KLDC (0.88 and 0.99, respectively), but how would you interpret these differences biologically?

The best I can tell is that the shape parameter suggests that females show a later increase in age-specific mortality than males do. The higher rate parameter for males seems like it should mean males are senescing faster than females, but that isn't consistent with the visualization of the results. If males were senescing earlier and faster than females, I would expect them to have higher mortality rates than females at the oldest age classes, but instead it appears that the mortality hazard estimate is converging. Am I misinterpreting the parameters?

Thanks!

Coefficient estimates from a simple Weibull model in BaSTA

plots of survival parameters from a simple Weibull model in BaSTA

momce
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  • "How to interpret it biologically" sounds like a biology question rather than a statistics question. However, the plots $S(x)$ and $\mu(x)$ seems clear taken together (presuming by mortality you mean force of mortality $-$ i.e. the hazard function). Initially there's higher mortality for males, making their survival proportion dip more quickly but as age increases, the hazard rates become very similar (the historical gap in survival from the mortality difference at younger x remains), then higher female mortality at later ages results in very similar survival curves near the end. – Glen_b Mar 14 '23 at 21:39
  • Thanks! That is what I thought was happening: males have higher early life mortality, and then female mortality hazard increases relative to male, closing the gap. I guess I was wondering if the parameters in the Weibull function correspond with biologically meaningful measures (as they do in Gompertz functions), or if the best course of action is just to interpret the survival and mortality plots for inference. – momce Mar 14 '23 at 22:25
  • To clarify my earlier (but now deleted) second comment: The change from lower Female mortality to higher could be related back to the different values of the two parameters. Specifically, if you consider the posterior mean (say) of the log of the hazard-ratio, it will directly relate to the parameters in a simple way. Again, if you want biological meaning, keep in mind that you're not talking to a biologist (I'm not, nor are most people here). Even if I could relate it to biological meanginfulness, I don't know exactly which interpretations you'd say were biological and which ones not. – Glen_b Mar 14 '23 at 22:31
  • "males have higher early life mortality, and then female mortality hazard increases relative to male, closing the gap" -- beware; this is potentially misleading. If you're referring to the gap in mortality, $\mu(x)$, that's closed by the middle ages (~30-40 ish) and then female mortality is higher. If you're referring to gap in survival, that requires a period of higher female mortality before that gap will close, which is why it doesn't occur until the latest ages. Mortality is instantaneous, but the ratio of survival curves shows effects of the whole past experience. – Glen_b Mar 14 '23 at 22:36

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With so many parameterizations of Weibull models in survival analysis, you first need to identify the parameterization that has been used.

According to the BaSTA manual, its Weibull parameterization is what Wikipedia calls the second alternative. The BaSTA $b_0$ is the Wikipedia shape parameter $k$ and its $b_1$ is Wikipedia's rate parameter $\beta$. The rate parameter is the inverse of the scale parameter $\lambda$ in Wikipedia's "standard parameterization."

The survival function in terms of $b_0$ and $b_1$ is:

$$S(x) = \exp(-(b_1 x)^{b_0}).$$

The discussion of Parametric Survival Models by Germán Rodríguez puts this into a form that might be more readily interpreted. For a Weibull model parameterized this way, the distribution of log-survival times can be written:

$$\log X = -\log b_1 + \frac{W}{b_0} ,$$

where $W$ is a standard minimum extreme value distribution. In that form, $b_1$ is related to a location of the distribution in log time, and $b_0$ is related to the inverse of the width of the distribution.

For your results, that location in time is earlier for males, but the width of the distribution is nominally wider for males ($b_0$ is smaller). Eventually, with that wider distribution, the tail of the distribution for males overlaps the distribution for females.

That said, I'd be careful in interpreting these results too closely. The standard errors of the $b_0$ values for males and females overlap, so one might argue that there isn't much evidence for a difference in distribution widths. I suspect that there is also covariance in the estimates of $b_1$ and $b_0$, complicating their separate interpretation.

Although it's possible to model $b_0$ as a function of covariate values (male/female), a frequent practice is only to model $-\log b_1$ as a function of covariates and assume a shared $b_0$ value for all cases. That might explain your data just as well, leading to a simple interpretation in terms of either accelerated failure times or proportional hazards for a Weibull model.

EdM
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