# [S] [s} Variation on Cox Proportional Hazards

John Maindonald (john.maindonald@anu.edu.au)
Thu, 15 Oct 1998 16:54:05 +1000 (EST)

Recently I put up the message
>I have data where, rather than proportional hazards, it seems more
>plausible that the relative hazards change linearly with time. Is it
>possible to fit such models using S-PLUS4.5 survival analysis, or
>available library routines? While cox.zph() is able to do something
>of this type for testing purposes, I cannot see how one can do the
>corresponding estimation problem.

Kenneth Hess sent tbe following reply
========================================================================
While it is possible to do this in S+, it requires constructing a new
dataset with additional rows so that an Anderson-Gill type model can be
constructed. In effect, you use the same sort of data structure as if you
had a time-dependent covariate. The advantage to using S+ is that cox.zph
works for the extended Cox model as well and this is the best way to check
that the time-dependence in indeed linear. If all you want to do is fit the
extended Cox model, then I would suggest you try SAS or BMDP or one of the
other packages that does not require transforming the data. If you insist
on fitting the model in S+, we have functions that will transform your
dataset in the appropriate fashion. Ordinarily, you would need to construct
new rows for each patient such that the patient had a row for each unique
event time not greater than the patient's follow-up time. These datasets
can be rather large. It is possible to form a grid of times that covers the
observed time range but uses fewer points - then construct rows using these
grid points rather than the unique event times. This yields an approximate
model that will compute faster. It is also possible to fit coefficients
which are nonlinear functions of time. It is my experience that most
violations of the PH assumption are nonlinear in nature. Note that by
'extended' Cox model, I mean a model of the form h(t,x) = h0(t)exp(b1*x +
b2*x*f(t))

[Ken also provided me with an S+ function to expand the dataset to fit
the model using survfit.]

--------------------------------------------------------------
Kenneth R. Hess Phone: 713-792-7251
Section of Biostatistics FAX: 713-792-4262
Department of Biomathematics e-mail: khess@odin.mdacc.tmc.edu
M. D. Anderson Cancer Center
1515 Holcombe Blvd, Box 237
Houston, Texas 77030-4095 USA
========================================================================

Frank Harrell suggested working with a parametric model. Thanks also
to Terry Therneau for expressing interest in my inquiry.

At 05:20 PM 9/24/98 +1000, you wrote:
>Dear ken
>
>You wrote
>> While it is possible to do this in S+, it requires constructing a new
>> dataset with additional rows so that an Anderson-Gill type model can be
>> constructed. In effect, you use the same sort of data structure as if you
>> had a time-dependent covariate. The advantage to using S+ is that cox.zph
>> works for the extended Cox model as well and this is the best way to check
>> that the time-dependence in indeed linear. If all you want to do is fit the
>> extended Cox model, then I would suggest you try SAS or BMDP or one of the
>> other packages that does not require transforming the data. If you insist
>> on fitting the model in S+, we have functions that will transform your
>> dataset in the appropriate fashion. Ordinarily, you would need to construct
>> new rows for each patient such that the patient had a row for each unique
>> event time not greater than the patient's follow-up time. These datasets
>> can be rather large. It is possible to form a grid of times that covers the
>> observed time range but uses fewer points - then construct rows using these
>> grid points rather than the unique event times. This yields an approximate
>> model that will compute faster. It is also possible to fit coefficients
>> which are nonlinear functions of time. It is my experience that most
>> violations of the PH assumption are nonlinear in nature. Note that by
>> 'extended' Cox model, I mean a model of the form h(t,x) = h0(t)exp(b1*x +
>> b2*x*f(t))
>
>Thank you for your complete reply. I've copied it to one other person
>(at the University of Queensland) who expressed an interest in any
>answers I might get. Assuming that it is ok by you, I'll include
>your reply along with any others that seem illuminating in a summary
>to the list.
>
>I'll probably try the fit with S+. I am not at all at ease with SAS!
>Actually, the diagnostic plots suggests a nonlinear function of time
>here also. A linear approximation may however be all that is useful
>in my rather small data set.
>
>John Maindonald email : john.maindonald@anu.edu.au
>Statistical Consulting Unit, phone : (6249)3998
>c/o CMA, SMS, fax : (6249)5549
>John Dedman Mathematical Sciences Building
>Australian National University
>Canberra ACT 0200
>Australia
>
>

John Maindonald email : john.maindonald@anu.edu.au
Statistical Consulting Unit, phone : (6249)3998
c/o CMA, SMS, fax : (6249)5549
John Dedman Mathematical Sciences Building
Australian National University
Canberra ACT 0200
Australia
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