[R] Re: smooth non cumulative baseline hazard in Cox model

[Mayeul KAUFFMANN]

Hello everyone,
I am searching for a covariate selection procedure in a cox model formulated
as a counting process.
I use intervals, my formula looks like coxph(Surv(start,stop,status)~
x1+x2+...+cluster(id),robust=T) where id is a country code (I study
occurence of civil wars from 1962 to 1997).
I'd like something not based on p-values, since they have several flaws for
this purpose.
I turned to other criteria but all the articles I read seems to apply to the
classical formulation of the cox model, not the counting process one (or
they apply to both but I am not aware of this)

I've tried AIC with
> step(cox.fit)
or
> stepAIC(cox.fit)

and BIC using
>step(cox.fit,k = log(n))
but there seems to be 2 theoretical problems to address:

(1) These values are based on partial loglikelihood ("loglik")
I wonder if this is correct with the cox model formulated as a *counting
process*, with many (consecutive) observations for a given

individual, and then some observation not being independent

Since "the likelihood ratio and score tests assume independence of
observations within a cluster, the Wald and robust score tests

do not" (R warning), and the likelihood ratio being based on loglik, can I
use loglik in BIC with some dependent observations?

[I have 170 individuals (namely, countries) for 36 year, some single
countries having up to 140 very short observation intervals, other having
(on

the other extreme) only 1 long interval per year. That's because I created
artificial covariates measuring the proximity since some

events: exp(-days.since.event/a.chosen.parameter). I splitted every yearly
interval for which these covariates change rapidly (i.e.

when the events are recent) yielding up to 11 intervals a year]

(2) What penalized term to used?

It seems natural to include the number of covariates, k.
What about the number of observations?

I found several definitions:
AIC= -2 loglik(b) +  2.k
Schwartz Bayesian information criteria:
SBIC= -2 loglik(b) +  k ln(n)

Frédérique Letué (author of PhD thesis "COX MODEL: ESTIMATION VIA MODEL
SELECTION AND BIVARIATE SHOCK MODEL",
http://www-lmc.imag.fr/lmc-sms/Frederique.Letue/These3.ps)
suggested me
AIC= - loglik(b) +  2.k/n
BIC= - loglik(b) +  2.ln(n).k/n, with other possible values for parameter
"2" in this case (see her thesis p.100, but this section is in French)

All these do not tell *what to take for n*. There are 3 main possibilities:

a) Taking the number of observations (including censored one) will give a
huge n (around 6000 to 8000), which may seem meaningless

since some observations are only a few days long.
With n at the denominator (Letué's criteria), the penalized term would be so
low that it's like not having it:
>log(7000)/7000
[1] 0.001264809

(where loglik from summary(cox.fit) range from -155 to -175, dependig on the
model)

b) Volinsky & Raftery "propose a revision of the penalty term in BIC so that
the penalty is defined in terms of the number of

uncensored events instead of the number of observations." (Volinsky &
Raftery , Bayesian Information Criterion for Censored

Survival Models, June 16, 1999,
http://www.research.att.com/~volinsky/papers/biocs.ps)

This could be computed with
>sum(coxph.detail(cox.interac.dsi6mois)$nevent)

Letué's BIC penalized trerm with 50 events will then be
> 2*log(50)/50
[1] 0.1564809
which will have more effects.

However, adding or removing a country which has data for the 36 years but no
event (then, it is censored) will not change this BIC.

Thus, it is not suitable to account for missing data that do not reduce the
number of event.
I'd like the criteria to take this into account, because all covariates do
not have the same missing data.
The question is: When I have the choice with adding a covariate, x10 or x11,
which have different (not nested) set of missing

values, which one is best?
Estimating all subsets of the full model (full model = all covariates) with
a dataset containing no missing data for the full model

would be a solution but would more than halve the dataset for many subsets
of the covariates.

I should mention that step(cox.fit) gives a warning and stops:
"Error in step(cox.fit) : number of rows in use has changed: remove missing
values?"

which makes me ask whether the whole procedure is OK with model of different
sample size.


c) "For discrete time event history analysis, the same choice has been made,
while the total number of exposure time units has also

been used, for consistency with logistic regresion (Raftery,Lewis,Aghajanian
and Kahn,1993;Raftery Lewisand Aghajanian,1994)"

(Raftery, Bayesian Model Selection in Social Research, 1994,
http://www.stat.washington.edu/tech.reports/bic.ps)

I am not sure what "exposure time units" mean. But since I could have used a
logit model with yearly observations [but with many

flaws...], I suggest I could use the number of years (sum of length of
intervals, in year)

> sum((fit$y)[,2]-(fit$y)[,1])/365.25
[1] 3759.537

This may still be too high.

Since I have datas from 1962 to 1997 (36 years), I have the folowing numbers
of "complete cases"-equivalent:
>sum((fit$y)[,2]-(fit$y)[,1])/ (365.25*36)
[1] 104.4316
This seems more resonable and would account for NAs in different models.

However, it might be to high, because some countries are not at risk during
the all period: some did not existed because they gain

independence near the end of the period (E.G. ex-USRR countries in arly
1990's) or because they were experiencing an event (new

wars in countries already experiencing a war are not taken into account).

I may take the *proportion* of available data to time at risk to adjust for
this: a country at risk during 1 year and for which

data are available for this entire year will increase n by 1, not by 1/36.
If the data frame "dataset" contains all countries at risk (including some
with NA), assuming id == id[complete.cases(dataset)]

(all countries have at least one complete observation) this will be computed
by

>attach(dataset)
>sum(tapply(stop[complete.cases(dataset)]-start[complete.cases(dataset)],CCO
DE,sum) /tapply(stop-start,CCODE,sum))

But this would be a rather empirical adjustment, maybe with no theoretical
basis.
And I don't think I can enter this as argument k to step()....


Thank you for having read this. Hope I was not too long.
And thank you a lot for any help, comment, etc.

(sorry for mistakes in English as I'm a non native English speaker)

Mayeul KAUFFMANN
Univ. Pierre Mendes France
Grenoble - France