# [R] covariate selection in cox model (counting process)

Thomas Lumley tlumley at u.washington.edu
Wed Jul 28 16:59:57 CEST 2004

```On Wed, 28 Jul 2004, Mayeul KAUFFMANN wrote:

> >No, I mean recurrent events.  With counting process notation but no
> >recurrent revents the partial likelihood is still valid, and the approach
> >of treating it as a real likelihood for AIC (and presumably BIC) makes
> >sense.
> >
> >Roughly speaking, you can't tell there is dependence until you see
> >multiple events.
>
> Thanks a lot, I got it (well, I hope so)!
>
>
> I've read in several places that events in the Andersen-Gill model must be
> "conditionnaly independent", which is sometimes more precisely written as
> "conditionnaly independent given the covariates"
>
> or even more precisely:
>
> "the Andersen-Gill (AG) model assumes that each [individual] has a
> multi-event counting process with independent increments. The observed
> increments must be conditionally independent given the history of all
> observable information up to the event times."
> (http://www.stat.umu.se/egna/danardono/licdd.pdf)

More precisely still, for the criterion function in coxph() to be a
partial likelihood the estimating function must be a martingale. This
is actually a slightly weaker assumption than independent increments.

The proportional rates model doesn't require this assumption, and is also
sometimes called the Andersen-Gill model.  The criterion function isn't a
likelihood but it still gives valid estimators.

>
> Then, there is still another option. In fact, I already modelled
> explicitely the influence of past events with a "proximity of last event"
> covariate, assuming the dependence on the last event decreases at a
> constant rate (for instance, the proximity covariate varies from 1 to 0.5
> in the first 10 years after an event, then from 0.5 to 0.25 in the next
> ten years, etc).
>
> With a well chosen modelisation of the dependence effect, the events
> become conditionnaly independent, I do not need a +cluster(id) term, and I
> can use fit\$loglik to make a covariate selection based on BIC, right?

If you can get the conditional independence (martingaleness) then, yes,
BIC is fine.

One way to check might be to see how similar the standard errors are with
and without the cluster(id) term.

-thomas

> Thanks a lot again for your time.
>
> Mayeul KAUFFMANN
> Univ. Pierre Mendes France
> Grenoble - France
>
> PS: I wrongly concluded from the R statement "(Note: the likelihood ratio
> and score tests assume independence of observations within a cluster, the
> Wald and robust score tests do not). " that it meant independence between
> two consecutive observations (without any event). It made sense to  me
> because when only one covariate changes for a given individual, and with a
> small change, there is a new observation, with a risk very simlar to the
> risk for the previous observation. But there is still independence with
> respect to the question of recurrent event. Maybe the warning should be
> rewritten saying "assume *conditionnal* independence of *events* (given
> the covariates)"
>
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