Page 54 - Contributed Paper Session (CPS) - Volume 7

CPS2028 Ayon M.
where H(t) is known as the integrated hazard or the cumulative hazard. It can
therefore be seen that for the distribution of T to be proper i.e, for its density
to integrate to one, () → ∞. If this is not true, the implication is that the
individual may never die; though in some contexts this may not be an
unreasonable approximating assumption ( as, for example, where children
may be cured of a childhood tumour and live ”indefinitely” in relation to the
time scale of the study). Normally, statisticians would want to insist on the
distribution of T to be proper.
The hazard function gives the event rate at a given time t, conditional on
having survived to time t. If the hazard rate is increasing, the risk of death (or
failure) is also increasing with time because the ratio of the hazard rates will
be the same as the ratio of the risk functions.
The Lehmann family, also known as the Proportional Hazard family is an
important family of distributions in modelling surival times. If ξ is an arbitrary
constant, the form of the Lehmann Family can be generated by:

This model can clearly be used to model the log hazard and is the basis of
the important Proportional Hazard models where the covariates act additively
on the logarithm of the hazard function. The exponential distribution and the
Weibull distribution belongs to the Lehmann family.
Sir David Cox in the year 1972 had effectively used this concept to provide
a semi-parametric approach for modelling time to event data where the
survival experience of patients in different groups can be compared after
adjusting for the effects of other variables which has a significant effect on the
patients’ survival responses. His approach had been extremely popular and the
paper in 1972 about the Cox proportional Hazard model has become the most
cited paper in the statistical literature. Unlike the Accelerated Life models
which assume a particular parametric distribution for the survival time of the
patients, the Cox proportional hazard model does not make any strong
assumption about the functional form of the survival times but make a lighter
assumption about the hazard ratio between two individuals at a particular time
point being constant. Since the model makes no assumption about the
functional form of the survival times, the parameter estimates are not based
on the the probability of the observed outcomes given the parameter values.
Instead of attempting to construct a full likelihood, Sir David Cox considered
the conditional probability that, given that exactly one individual in the risk set
, with covariate vector , dies at time , it is the individual that does
ℎ

ℎ
so. Let denote the treatment indicator for the patient such that = 1, if

the patient is assigned to treatment A, and = 0, if the patient is assigned to

treatment B. Associated with patient = 1, . . . . , is a ( + 1) vector of
baseline covariates = (1, , . . . . . . . . , ) and a risk set which is

1

41 | I S I W S C 2 0 1 9