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EPI 5344:Survival Analysis in

EpidemiologyEstimating S(t) from Cox models

April 1, 2014

Dr. N. Birkett,Department of Epidemiology & Community

Medicine,University of Ottawa

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Objectives

• Theory of estimating S(t)• SAS methods

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• Cox methods do not require an equation for h(t)– do not produce direct estimates of h(t), H(t) or S(t).

• If we could estimate any of these, we could get the others:

Introduction (1)

Assuming h(t) is constant in interval

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• Since hazards are proportional, all we need to know is h(t) or S(t) for the baseline group: h0(t) or S0(t)

Introduction (2)

So, estimate S0(t)and we can get S(t)for any ‘x’s’

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• Must use methods outside Cox regression.• Two common approaches are used:

– Generalize the Kaplan-Meier method to estimate S0(t)– Generalize Nelson-Aalen method to estimate H0(t)

• Implemented by using the BASELINE statement.

• First, a quick review of the technical background

Introduction (3)

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A bit of technical stuff

• We assume a piecewise constant hazard model– It keeps coming up, doesn’t it

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We don’t care about the origin of the next formulae.Assume there are no ties

Where:‘l’ is subject having event at tj

‘k’ is subject whose survival curve is needed

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• Non-RCT study of therapy• Hypernephroma (type of kidney cancer)• All patients treated with

– chemotherapy and – Immunotherapy

• Some also had their affected kidney removed.• Questions

– Does having a nephrectomy affect survival? – Does age affect survival in patients having a

nephrectomy?

Example (1)

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• Answer is YES

A2i = 1 if age is 60-70

= 0 otherwiseA3i = 1 if age is 70+

= 0 otherwiseNi = 1 if had nephrectomy

= 0 if no nephrectomy

• A3 & N are statistically significant.• What do survival curves look like?

Example (2)

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Could try smoothing the h(t) curve

Estimated baseline hazard curve

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• How did we produce these curves?• Using the formulae given earlier.• SAS uses the BASELINE statement

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Example (3)

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• SAS implements these methods in Phreg using the BASELINE statement

• By default, this statement produces S(t) at a pre-defined set of covariate values:– The reference level of any variable mentioned

in a class statement– The mean value of each other variable

SAS Methods (1)

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ODS GRAPHICS ON;ODS RTF;

PROC PHREG DATA=allison.recid PLOTS=S; MODEL week*arrest(0)=fin age prio / TIES=EFRON; BASELINE OUT=a SURVIVAL=s LOWER=lcl UPPER=ucl;RUN;

proc print data=a;run;

ODS RTF close;ODS GRAPHICS OFF;

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ODS GRAPHICS ON;ODS RTF;

PROC PHREG DATA=allison.recid PLOTS=S; Class fin; MODEL week*arrest(0)=fin age prio / TIES=EFRON; BASELINE OUT=a SURVIVAL=s LOWER=lcl UPPER=ucl;RUN;

proc print data=a;run;

ODS RTF close;ODS GRAPHICS OFF;

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• What if you want S(t) for some other set of covariates?– Interested in specific target group– Contrast extremes of the range of variables

• Use COVARIATES statement in SAS

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DATA covals; INPUT fin age prio; DATALINES;0 40 3;run;

PROC PHREG DATA=allison.recid; MODEL week*arrest(0)=fin age prio / TIES=EFRON; BASELINE OUT=a COVARIATES=covals SURVIVAL=s LOWER=lcl UPPER=ucl;run;

proc print data=a;run;

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• What if you want to display the results of multiple covariate sets on the same graph?– Method #1

• Run several of previous models, for different covariates.

• Combine the output datasets into on dataset• Plot using SAS Graph, etc.

– Method #2• list more than one set of covariates in the covariate

data set

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DATA covals; INPUT fin age prio; DATALINES;0 40 31 50 2;run;

ODS graphics ON;

PROC PHREG DATA=allison.recid PLOTS(OVERLAY)=S; MODEL week*arrest(0)=fin age prio / TIES=EFRON; BASELINE OUT=a COVARIATES=covals SURVIVAL=s LOWER=lcl UPPER=ucl;run;

ODS graphics off;

proc print data=a;run;

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