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Estimation of coefficient of
Cox regression by iteration

Whilst we would not recommend that maximum likelihood estimation is routinely done manually, it is nevertheless informative to carry out one such estimation so you can see what is happening.

Probability
of each death
Possible values for
hazard ratio (expβ)
Null
model
0.050.100.15expβ
= 1.00
p1 = 1/(5+5expβ)0.19050.18180.17390.1000
p2 = 1/(4+5expβ)0.23530.22220.21050.1111
p3 = 1/(3+5expβ)0.30770.28570.26670.1250
p4 = expβ/(2+5expβ) 0.02220.04000.05450.1429
p5 = 1/(2+4expβ)0.45450.41670.38460.1667
p6 = 1/(1+4expβ)0.83330.71430.62500.2000
p7 = expβ/4expβ 0.25000.25000.25000.2500
p8 = expβ/3expβ 0.33330.33330.33330.3333
p9 = expβ/2expβ 0.50000.50000.50000.5000
p10 = expβ/1expβ 1.00001.00001.0001.0000
Σ(lnp):-12.2393-12.0705-12.1414-15.1044
Look first at the probabilities in the blue column for the 'null model'. In this model we assume there is no treatment effect - in other words the hazard ratio is equal to one, so the regression coefficient β is equal to zero. The probabilities depend only on the number in the risk set so p1 = 1/10 = 0.1; p2 = 1/9 = 0.1111 and so on. The sum of the logs of these probabilities (the log likelihood function) for this null model is -15.10.

The probabilities in the yellow/brown columns are worked out assuming there is a difference between the two groups with hazard ratios ranging from 0.05 to 0.15. The sums of the logs of the probabilities are all larger than was the case for the null model. The largest of these - that with maximum likelihood - is given by a value for the hazard ratio of 0.1. This provides our first estimate. Further iterations with values close to 0.1 give the best estimate of the hazard ratio as 0.10035, or β = -2.3.