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

[Steven Samuels <sjhsamuels@earthlink.net>]

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.

The answer to each of your questions appears to be “No.”

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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