023 with a log partial likelihood l1 195906 all four

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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 . q 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, 3 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 estimates of...
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