17.6.2.2 Algorithms (Cox Proportional Hazard Regression)PHM-Cox-Algorithm
Let /math-b4a096d9d2236359a73b7386af0c9ab6.png?v=0 "t_i\,\!") ,for i = 1, 2, ?, n, be the failure time or censored time for the ith observation with the vector of p covariates /math-e9b2a0e549182ce07a674b99d3bdd987.png?v=0 "Z_j(j=1,2,\ldots ,p)") . It is assumed that the failure and censored mechanisms are independent. The hazard function, /math-b4edeac500164ed31e65913d1bf90051.png?v=0 "\lambda (z,t)\,\!") , is the probability that an individual with covariates z fails at time t given that the individual survived up to time t. In the Cox proportional model is of the form:
/math-7fd25fca62158bf07a508af6b3048824.png?v=0 "\lambda (z,t)=\lambda _0(t)\exp (z^{T}\beta +\omega )\,\!")
where /math-3edc1ca1e622708969948a2514e800f2.png?v=0 "\lambda _0\,\!") is the base-line hazard function, an unspecified function of time, /math-5b320b6d3d3254d936c752ae308dbfd8.png?v=0 "\beta \,\!") is a vector of unknown parameters and /math-f2f1b18a419c6fddc9ad79a13a5946e3.png?v=0 "\omega\,\!") is a known offset.
Assuming there are ties in the failure time giving /math-99af4576b28fa0e44da190e42ca9fdfb.png?v=0 "n_d < n\,\!") distinct failure times, /math-afdb5212c7209b37b90b950f8e435844.png?v=0 "t_{(1)} < t_{(2)} < ?< t_{(nd)}") , such that /math-40b4081d7e04c18f0529a2e70a11e290.png?v=0 "d_i\,\!") individuals fail at /math-659eb06b406d2e114b139d0dd37ba959.png?v=0 "t_{(i)}\,\!") , it follows that the marginal likelihood for /math-b0603860fcffe94e5b8eec59ed813421.png?v=0 "\beta") is well approximated by:
where /math-df637c345145e684f09c3e2bf73732cf.png?v=0 "s_i\,\!") is the sum of covariate of individuals observed fail at and /math-ee91b0d4f1eaf7196ca8b86c689bce4c.png?v=0 "R(t_{(i)})\,\!") is the set of individuals at risk just prior to , that is it is all the individuals that fail or censored at time /math-f82bfe760d9b9727abfa6b9f205b4291.png?v=0 "t_{(i)}") along with all individuals survived beyond the time . The MLE (maximum likelihood estimates) of /math-8d2bf18dffee8005942fdde98a69f3a1.png?v=0 "\beta\,\!") , given by/math-950a1c15a15d3e439fc0c39729c41abe.png?v=0 "\hat \beta\,\!") , are obtained by maximizing (1) using a Newton-Raphson iteration technique that includes step having and utilizes the first and second partial derivatives of (1) which are given by (2) and (3) below:
for j = 1, 2, ?, p, where /math-c187e0049b758f53e5e0d9889f652f49.png?v=0 "s_{ji}\,\!") is the jth element in the vector and
/math-a28a80d6e7722fccef4fc3249adc4ca2.png?v=0 "\alpha _{ji}(\beta )=\frac{\sum_{l\in R(t_{(1)})}z_{jl}\exp (z_l^{T}\beta +\omega _l)}{\sum_{l\in R(t_{(1)})}\exp (z_l^{T}\beta +\omega _l)}")
Similarly,
where /math-b9afb94ffa7c4b399e7da601d15f214c.png?v=0 "\gamma _{hji}=\frac{\sum_{l\in R(t_{(1)})}z_{hl}z_{jl}\exp (z_l^{T}\beta +\omega _l)}{\sum_{l\in R(t_{(1)})}\exp (z_l^{T}\beta +\omega _l)}-\alpha _{hi}(\beta )\alpha _{ji}(\beta )") h, j = 1, ? p.
/math-94b61be7dbb72780127ba83b9a55c19f.png?v=0 "U_j(\beta )\,\!") is the jth component of a score vector /math-1a14fac8fed1c86899f6f323acabceea.png?v=0 "I_{hi}(\beta )\,\!") is the (h, j) element of the observed information matrix /math-a139862e5a9fa67a8d14878476236d55.png?v=0 "I(\beta )\,\!") whose inverse /math-3d35c360006c6b692bb24cfaa0162d4b.png?v=0 "I(\beta )^{-1}=I_{hi}(\beta )^{-1}\,\!") gives the variance-covariance matrix of .
It should be noted that if a covariate or a linear combination of covariates is monotonically increasing or decreasing with time ,then one or more of the /math-35ddbb0efc4d0c98d6f19a3b84c1db78.png?v=0 "\beta _j^{\prime }s") will be infinite.
If /math-8883b73ab26908c75308092259b6d141.png?v=0 "\lambda _0(t)\,\!") varies across /math-3b0619e986d2ebf9acd80c494cb421a4.png?v=0 "\nu\,\!") strata, where the number of individuals in the kth stratum is /math-3da18b12eaeaccf089fe39883448b765.png?v=0 "n_k\,\!") , k = 1, ?, , with /math-ff8c77ee62f8208a4dbab23d6a511dcc.png?v=0 "n=\sum_{k=1}^\nu n_k") , then rather than maximizing (1) to obtain , the following marginal likelihood is maximized:
where /math-39209f726b762faf7442e04a446ad41f.png?v=0 "L_k\,\!") is the contribution to likelihood for the observations in the kth stratum treated as a single sample in (1). When strata are concluded the covariate coefficients are constant across strata but there is a different base-line hazard function .
The base-line survival function associated with a failure time , is estimated as
/math-0b8df9b2daec8e7514a7c2cd894f8faf.png?v=0 "exp(-\hat H(t_{(i)}))") ,
where /math-ca2e935da3c4ec9f78523b3db2309234.png?v=0 "\hat H(t_{(i)})=\sum_{t(j)\leq t(i)}(\frac{d_i}{\sum_{l\in R(t_{(j)})}\exp (z_l^T\hat \beta +\omega _l)})")
and is the number of failures at time . The residual of the lth observation is computed as:
/math-a5cc248e45947fea428f4b34caff7093.png?v=0 "r(t_l)=\hat H(t_l)\exp (-z_l^T\hat \beta +\omega _l)")
where /math-b77e639a3484da642fbbd419c433f190.png?v=0 "\hat H(t_l)=\hat H(t_{(i)}),t_{(i)}\leq t_l<t_{(i+1)}") .
The deviance is defined as /math-22a372fe7aefbb5295da3f13a354ea31.png?v=0 "-2^*\,\!") (logarithm of marginal likelihood). There are two ways to test whether individual covariates are significant: the differences between the deviances of nested models can be compared with appropriate /math-c15009ca5cbb034d023f145a4ed23b8d.png?v=0 "\chi ^2\,\!") -distribution; or, the asymptotic normality of the parameter estimates can be used to form z-test by dividing the estimates by their standard errors or the score function for the model under the null hypothesis can be used to form z-test.
|