# Re: Re: st: Predicted probabilities in a competing risks model in discrete time?

 From Steven Samuels To statalist@hsphsun2.harvard.edu Subject Re: Re: st: Predicted probabilities in a competing risks model in discrete time? Date Wed, 6 Aug 2008 17:15:35 -0400

Thanks, Katharina.

I don't have either of the two references, but I do have Stephen Jenkins's book “Survival Analysis”, downloaded from his website ( http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/pdfs/ ec968lnotesv6.pdf ). The multinomial approach to competing risks that you refer to appears, I think in Chapter 9, document pages 92-97, actual pdf pages 105 to 109.

In ordinary multinomial logistic regression, there is one equation for predicting each outcome. In the competing risk set up, there is a different equation for each outcome in each time point. The multinomial probabilities are the time-specific hazards of each outcome--the probability that the outcome occurs the time point among those still at risk. Therefore none of the time-point-specific equations will provide a single predictive summary. Effectively, you are fitting an interaction of each covariate with time. Even if the coefficient for a covariate is constant over intervals, the impact, measured as a difference in probabilities can also change over time.

The notion of a “reference person” is not clear-cut for these equations. The population at risk in each interval is not constant, but consists of the survivors of prior intervals. If your model includes time-varying covariates, you would need to define a constant value for these covariates.

If there is no single summary prediction, you could plot the ensemble of predictions-the outcome specific hazards-against time for a reference set of covariates and for a set which changes one of the covariates. You must assume that your covariates are constant throughout.

Better yet would be to estimate the cumulative incidence of each event as a function of time. A plot of the cumulative incidence at reference and changed values would then display the total impact of the baseline covariate. To see how to convert the interval probabilities to cumulative ones, consult Section 2.21 (document page 17, PDF page 29) of Stephen's book. I imagine that -nlcom- can do much of the work for you.

All this work assumes that the discrete model is a good approximation to the data-generating process; that risks are independent; and that the proportional-hazards assumption holds. See Chapter 9 of Stephen's book. There are other issues in assessing the impact of covariates on competing risks; a good reference is JG Kalbfleisch and RL Prentice, Analysis of Failure Time Data, 2nd Edition, Wiley, 2002, Chapter 8, especially pp. 247-265. If you cannot find a copy, the 1st Edition (1978) covers most of the same territory.

-Steve

On Aug 6, 2008, at 1:59 PM, LachZak@web.de wrote:

Hi Steve,
I meant the following two papers:
Jenkins, S.P. (1995): Easy ways to estimate discrete time duration models, Oxford Bulletin of Economics and Statistics, 57, 129-138.
Allison, P. (1982): Discrete time methods for the analysis of event histories, pp. 61-98, in Sociological Methodology (ed. by S. Leinhardt).
These models were developed for intrinsically discrete time data, assuming a particular functional form for the destination-specific hazards in the competing risks framework., namely, hazard to destination A = exp(betaA*X)/[1+exp(betaA*X)+exp(betaB*X)] The resulting likelihood function is exactly the same as for a "standard" multinomial logit.
In Stata, estimation works as follows: Using expand, you create a dataset in person-month format and estimate it using a command as the following:
mlogit depvar regressors f(time)
My question is: Does it make any sense to interpret predicted probabilities after this estimation command, e.g. something like
prvalue, x(female=1) rest(mean) ?
Sorry for the first post, but this was my first try with Statalist...
Best, Katharina
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