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7. Topics in Cox Regression
7.1 Time-Dependent Covariates
It is often the eccentric that the values of some explanatory variables in a
survival abridgment change over time.
For example, in Rossi et al.s recidivism data, information on exinmates employment status was collected on a periodical basis.
In contrast, other covariates in this data set, such as race and the
provision of ?nancial aid, are time-constant.
The Cox-regression model with time-dependent covariates takes the
form
loge hi(t) = ?(t) + ? 1xi1(t) + ? 2xi2(t) + · · · + ? k xik (t)
= ?(t) + xi(t)?
? Of course, not all of the covariates have vary with time: If covariate
xj is time-constant, consequently xij (t) = xij .
Sociology 761
Introduction to Survival Analysis
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Copyright c 2003 by stern Fox
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? For example, the complete initial data record for the ?rst subject is
as follows:
Rossi[1,]
week arrest fin age race wexp bollix up paro prio educ emp1
20
1
0 27
1
0
0
1
3
3
0
emp2 emp3 emp4 emp5 emp6 emp7 emp8 emp9 emp10 emp11
0
0
1
1
0
0
0
0
0
0
emp12 emp13 emp14 emp15 emp16 emp17 emp18 emp19 emp20
0
0
0
0
0
0
0
0
0
emp21 emp22 emp23 emp24 emp25 emp26 emp27 emp28 emp29
NA
NA
NA
NA
NA
NA
NA
NA
NA
emp30 emp31 emp32 emp33 emp34 emp35 emp36 emp37 emp38
NA
NA
NA
NA
NA
NA
NA
NA
NA
emp39 emp40 emp41 emp42 emp43 emp44 emp45 emp46 emp47
NA
NA
NA
NA
NA
NA
NA
NA
NA
emp48 emp49 emp50 emp51 emp52
NA
NA
NA
NA
NA
Sociology 761
Introduction to Survival Analysis
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Although the inclusion of time-dependent covariates can usher in nontrivial data-management issues (that is, the construction of a suitable
data set can be tricky), the Cox-regression model can easily handle
such covariates.
Both the data-management and conceptual treatment of timedependent covariates is facilitated by the so-called counting-process
representation of survival data.
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