coxphfit+matlab,Cox Proportional Hazards Model

Partial Likelihood Function for Tied Events

When you have tied events, coxphfit approximates

the partial likelihood of the model by either Breslow’s method

(default) or Efron’s method, instead of computing the exact

partial likelihood. Computing the exact partial likelihood requires

a large amount of computation, which involves an entire permutation

of the risk sets for the tied event times.

The simplest approximation method is Breslow’s method.

This method uses the same denominator for each tied set.

L=∏i=1d∏j∈DiHR(Xj)∑k∈RiHR(Xk),

where d is

the number of distinct event times, and Di is

the index set of all subjects whose event time is equal to the ith

event time.

Efron’s method is more accurate than Breslow’s

method, yet simple. This method adjusts the denominator of the tied

events as follows:

L=∏i=1d∏j∈DiHR(Xj)∑k∈RiHR(Xk)−j−1di∑k∈DiHR(Xk),

where di is

the number of indexes in Di.

For an example, assume that the first two events are tied, that

is, t1 = t2 and t2

In Breslow’s method, the denominators of the first two terms

are the same:

L=HR(X1)∑j=1nHR(Xj)×HR(X2)∑j=1nHR(Xj)×HR(X3)∑j=3nHR(Xj)×HR(X4)∑j=4nHR(Xj)×⋅⋅⋅×HR(Xn)HR(Xn).

Efron’s method

adjusts the denominator of the second term:

L=HR(X1)∑j=1nHR(Xj)×HR(X2)0.5HR(X1)+0.5HR(X2)+∑j=3nHR(Xj)×HR(X3)∑j=3nHR(Xj)×HR(X4)∑j=4nHR(Xj)×⋅⋅⋅×HR(Xn,tn)HR(Xn,tn).

You can specify an approximation method by using the name-value

pair 'Ties' in coxphfit.

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