Proportional hazard model

In this model the conditional hazard functions are assumed homothetic and the parametric term exp(x,0) is introduced as the coefficient of proportionality:

X( Уі 1 xi; ^ f0) = exp(x0)X 0( y4 (21.22)

where X 0 is an unconstrained baseline hazard function. It is defined up to a multiplicative scalar. The term proportional hazard indicates that the hazards for two individuals with regressor vectors x1 and x2 are in the same ratio. For
example, the exponential model with X(y | x, 0) = exp(x;0) is a proportional hazard model with X 0 = 1.

The parameter 0 can be consistently estimated by the partial maximum likeli­hood introduced by Cox (1975). The approach is the following. Let us first rank the duration data by increasing values ym < уй < … < y{N), where we implicitly assume that all observed durations are different. Then we consider the sub­population at risk just before the exit of the ith individual:

R(i) = {j : У(і) > У(о).

The probability that the first individual escaping from this subpopulation R^ is individual (i) is given by:

Подпись:

image24

Х(У( i )|x( i), 0, Xo)

^ jR i) X[y( j )|x( j); 0, Xo] exp(X(i)0)

j (i)exp(x( j )0).

Подпись: 0 = argmax £ log p(i )(0, X o)

Подпись: (21.23)

image518
Подпись: exp(x(i)0)
Подпись: ^j^R, i)exp(x( j )0)
Подпись: i=1

It no longer depends on the baseline distribution. The partial maximum likelihood estimation of 0 is defined by:

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