B Appendix Estimation of Parameters and Baseline Hazard Function in Proportional Hazards Model B.1 Estimation of the Coefficient Among the proposed methods for estimating the parameters , those of marginal and partial likelihood are the most commonly used in practice. Since the partial likelihood method developed by Cox (1972) is considered to be the most general of the existing estimation techniques and will be used in the data analysis involved in the present study, further details on its derivation are given in this section. Consider the following two scenarios. In scenario 1 an item fails at 100 hours. In scenario 2 an item fails at 100 hours given that it has survived for 99 hours. In the first scenario we use the failure time density to evaluate the corresponding probability function. In the second, we need to use the hazard function to evaluate the conditional probability. Now let us consider the event that item i fails from the survivor or risk set R( j ) and R( j ) = {k: tk j } in which: tk is the failure time for item k and 1 < 2 < ... < n denotes the ordered failure times of the n pipes. To evaluate the corresponding probability, we first evaluate the probability that one item fails in the survivor set R( j ) which is given by: P[one item fails at t from R( j )] = P[item k fails at time j from R( j )] = =
kR(t)
P[ j t k < j + t | t k j ] h k (t )t
k
kR ( j )
Consider the probability, P[item `i' fails at time j given that it belongs to R( j ) | one item fails in R( j )] in which R( j ) ={k: tk j } which in turn can be evaluated as P[item k R( j ) fails] P[one item fails in R( j )] = = P[ j t k < j + t] P[t k t] P[one item fails in R( j )] h i ( j )
kR( j )
h k ( j )
By regarding the selection of an item from the risk set at its observed failure time as an independent event, a joint probability statement more appropriately called the likelihood function is written as:
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L(; t 1 , t 2 ,..., t n ) =
i =1
n
kR(t i )
h i (t ) h k (t)
By applying the proportional hazards model structure, h(t;z) = h 0 (t) (z ; ) , the likelihood function becomes n n (B.1.1) h 0 ( j ) exp(z i ) exp( z i ) L() = = h ( ) exp(z k ) j=1 exp( z k ) j=1 0 j
kR( j ) kR( j )
Several methods have also been proposed for dealing with the problem of ties in failure times [Cox and Oaks (1984), Kalbfleisch and Prentice (1980)]. In the approximate likelihood function in the case of ties as proposed by Breslow (1974), the partial likelihood function is written as: n exp(s i ) L() = m i i =1 exp(z j ) jR ( j ) where mi is the number of failures at i and s i is the vector sum of the covariates of the mi items. The conditional log likelihood obtained from equation (B.1.1) is given by (B.1.2) n n n = l L() = z i - log exp( z k ) i i =1 i =1 kR ( j ) i =1 In order to obtain maximum likelihood estimates the of first and second derivatives of li are first calculated. If zir denotes the value of the rth component of the explanatory variable z on the ith subject, then we have: (B.1.3) z exp( T z ) li kR (i ) = z ir - lr exp( T z k )
kR ( i )
kr
k
= z ir - A ir ()
Also: li kR (i ) = z ir - r s exp( T z k )
kR ( i ) 2
z kr exp( T z k )
+ A ir ()A is () = -C irs ()
For all risk sets i = 1, ... , k, we obtain from equation (B.1.3) the score function Ur( ) for the rth component:
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li k = z ir - A ir () r i =1 i =1 The information matrix I( ) of negative second derivatives has elements: U r () =
k
(
)
(B.1.4)
I rs () = C irs ()
i =1
k
(B.1.5)
Maximum likelihood estimates of can be obtained by iterative use of equation (B.1.4) and (B.1.5). The Newton-Raphson iterative algorithm can be applied to obtain the maximum likelihood estimates of . B.2 Estimation of the Baseline Hazard Function By referring to equation (5.1.5), it is seen that we still need to estimate the baseline hazard function h0(t). From equation (5.1.2), it is seen that the baseline hazard function represents the hazard rate that a piece of equipment would experience if all the values of the covariates are equal to zero. We can say that the baseline hazard function is the hazard rate when the covariates have no influence on the failure pattern. The baseline hazard function can be estimated by first obtaining a cumulative baseline hazard function. First, compute the cumulative hazard function as follows. From the given data we know the observed failure times, j . Then for a fixed value of t, H0(t) is estimated by (B.2.1) dj ^ H 0 (t ) = ^ < t exp( z k )
j kR ( j )
where, j : observed and ordered failure times dj : number...