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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 < 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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