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Unformatted text preview: respectively.
This test statistic has a χ2 distribution with q degrees of
freedom under the null hypothesis for large n.
Reject the null hypothesis H0 at α% (eg 5%) signiﬁcance level
if the value of the test statistic is greater than the upper α%
point of χ2 .
27/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model
Hypothesis tests on the β ’s Example
Consider a model with four covariates z1 , z2 , z3 and z4 , where the
ﬁrst covariate is the patient’s age at diagnosis, and the last three
are indicators stage II, III, and IV disease, respectively.
Assume that ﬁtting a Cox model to the data with
the single covariate z1 (age) leads to b1 = 0.023 with a log
partial likelihood L1 = −195.906
all four covariates, leads to b = (0.0189, 0.1386, 0.6383,
1.6931) with a log partial likelihood L4 = −188.179.
The likelihood ratio statistic is
−2[L1 − L4 ] = −2[(−195.906) − (−188.179)] = 15.454
The p -value=Pr (χ2 ≥ 15.454) = 0.0015 < 0.05 and hence,
we reject H0 .
28/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model
Hypothesis tests on the β ’s The Wald test
The Wald statistic is
(bp+1 , · · · , bp+q ) [Cov (βp+1 , · · · , βp+q )]−1 (bp+1 , · · · , bp+q )T
where b = (b1 , · · · , bp+q ) denotes the partial maximum likelihood
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