Biomedical Engineering Reference
In-Depth Information
to additive changes in the log of the hazard. Mathematically, the hazard function
reflecting the proportional hazards model is defined as
(tjX;;) = 0 (tj) exp(X);
(11.1)
where X is a vector of concomitant, covariate, or regressor information X =
(x 1 ;x 2 ; ;x p ), is the column vector of parameters ( 1 ; 2 ; ; p ) corresponding
to X, and 0 (tj) is the baseline hazard function with parameter vector . Based
on this structure in Equation (11.1), the hazard ratio for any two patients with
covariates X 1 and X 2 is constant over time because
(tjX 1 )
(tjX 2 ) = exp(X 1 )
exp(X 2 )
(note that 0 (t) cancels from numerator and denominator of the ratio), and therefore
the hazard for one subject is proportional to the hazard of another subject.
From the formulation in Equation (11.1), it is noted that the concomitant infor-
mation acts in a multiplicative fashion on the time-dependent-only hazard function.
Further, the term \proportional hazards" also arises by observing that if an x i is an
indicator of treatment group membership (xi i = 1 if treatment group 1; xi i = 0 if
treatment group 0), then the ratio of the hazard for treatment group 1 to the hazard
of treatment group 0 is exp(βi); i ); or the hazard for treatment group 1 is proportional
to the hazard of treatment group 0. Therefore, exp(X) sometimes is referred to as
relative risk.
Parameter estimation and attendant statistical inference are based on the par-
tial likelihood approach as presented in Cox's paper and has been shown to produce
unbiased estimates for right-censored time-to-event data. This approach has been
implemented in commonly used statistical software packages, such as SAS, R, etc.
Because of this implementation in commonly used software packages, analysts usu-
ally resort to using Cox regression for interval-censored data by defining the event
time to be the time at which the event was first observed at t 2 or utilize the mid-
point imputation of the interval, which has led to bias and erroneous conclusions.
To extend Cox's model for interval-censored time-to-event data, Pan (1999) pro-
posed a semiparametric approach to approximate the baseline cumulative distribu-
tion F 0 (tj) with piecewise constants, which leads to the iterative convex minorant
(ICM) algorithm. In this approach, the well-known proportional hazards assump-
tion in Equation (11.1) is still assumed to link the covariates X and the vector of
 
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