Week 4 Lecture Slides (1)

# This test statistic has a 2 distribution with q

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Unformatted text preview: l hypothesis is H0 : βp+1 = βp+2 = · · · = βp+q = 0 ie the associated covariates are not signiﬁcant. We will introduce two tests: 1 the likelihood ratio test 2 the Wald test Both rely on the properties of the MpLE. 26/45 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s The likelihood ratio test The likelihood ratio statistic is −2 [log Lp − log Lp+q ] , where two (nested) models have been ﬁtted one with p covariates (z1 , z2 , · · · , zp ) and another with p + q covariates (z1 , z2 , · · · , zp , zp+1 , · · · , zp+q ), and their associated likelihoods are Lp and Lp+q , 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 – Week 4: The 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 &lt; 0.05 and hence, 3 we reject H0 . 28/45 Actuarial Statistics – Week 4: The 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 β = (β1 , · · · , βp+q ). The test statistic of the Wald test has a χ2 distribution with q degrees of freedom under the null hypothesis for large n. In general, the Likelihood ratio test and the Wald test give very similar conclusions in practice. (Note: Here, the vectors are row vectors. In statistics literature, column vectors seem more popular.) 29/45 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s Covariance term Recall the observed information matrix of the model with p + q covariates is I (b ) = − ∂ 2 ln Lp+q (β ) |β =b ∂βi ∂βj i ,j =1,··· ,p +q Now partition (I (b ))−1 into (I (b ))−1 = I (11) (b ) I (12) (b ) I (21) (b ) I (22) (b ) where I11 (b ) is of dimension p × p , and I (22) (b ) is of dimension q × q . We have then ˜ ˜ Cov (βp+1 , · · · , βp+q ) = I (22) (b ) 30/45 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s Example (continued) Consider the problem in the last example, that is, t...
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