4- Semi parametric survival analysis: Cox modeSummary: How to perform a Cox analysis1- univariate analysis - enter all covariates individually into a Cox model, - check that the proportional hazards assumption holds.2- multivariate analysis - introduce all (or part of) covariates into the model, - identify covariates that are independently associated with survival. 51
4- Semi parametric survival analysis: Cox mode• To check if a covariate has an effect on survival ⇒ likelihood ratio test.Assumptions H0 : βk = 0 , for all categories k H1 : at least 1 parameter is different from 0La = log-likelihood for the model with the covariate to be testedLs = log-likelihood for the model without the covariate to be tested2(Ls − La ) ~ χ2 ⇒ using this test, we can say whether the k -1 covariate has a significant effect on survival. k = number of categories for the covariate. 52
4- Semi parametric survival analysis: Cox mode stcox list_covariate0 estimates store m0 stcox liste_covariate1 estimates store m1 lrtest m0 m1• The likelihood ratio test is useful to: - identify covariates significantly associated with survival: starting from the full model, eliminate non-significant covariates. - regroup categories for one covariate. 53
4- Semi parametric survival analysis: Cox mode• Example : incidence of undernutrition Has the covariate age (in categories) a significant effect? xi: stcox educa i.age_cat estimates store LL0 xi: stcox educa if age_cat!=. estimates store LL1 lrtest LL0 LL1 54
4- Semi parametric survival analysis: Cox modexi: stcox educa i.age_catNo. of subjects = 6238 Number of obs = 6238No. of failures = 1143------------------------------------------------------------------------------_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------educa | 1.464797 .1143098 4.89 0.000 1.257047 1.706882_Iage_cat_1 | .8828032 .0544445 -2.02 0.043 .7822911 .9962295_Iage_cat_3 | 2.249075 .2887221 6.31 0.000 1.748767 2.892516------------------------------------------------------------------------------estimates store LL0xi: stcox educaNo. of subjects = 6260 Number of obs = 6260No. of failures = 1147------------------------------------------------------------------------------_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------educa | 1.490285 .1155236 5.15 0.000 1.280225 1.734813------------------------------------------------------------------------------estimates store LL1 Messages in red usually indicate errors !!!!lrtest LL0 LL1observations differ: 6238 vs. 6260 55
4- Semi parametric survival analysis: Cox modexi: stcox educa i.age_catNo. of subjects = 6238 Number of obs = 6238No. of failures = 1143------------------------------------------------------------------------------_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------educa | 1.464797 .1143098 4.89 0.000 1.257047 1.706882_Iage_cat_1 | .8828032 .0544445 -2.02 0.043 .7822911 .9962295_Iage_cat_3 | 2.249075 .2887221 6.31 0.000 1.748767 2.892516------------------------------------------------------------------------------estimates store LL0xi: stcox educa if age_cat !=.No. of subjects = 6238 Number of obs = 6238No. of failures = 1143------------------------------------------------------------------------------_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------educa | 1.478473 .1153281 5.01 0.000 1.268865 1.722706------------------------------------------------------------------------------estimates store LL1lrtest LL0 LL1 LR chi2(2) = 44.07 Prob > chi2 = 0.0000Likelihood-ratio test(Assumption: LL1 nested in LL0) 56
4- Semi parametric survival analysis: Cox modeWald’s parametric testTo test whether categories k et l have the same effect on survival. xi: stcox i.covariate testparm category to be tested 57
4- Semi parametric survival analysis: Cox modexi: stcox educa i.age_catNo. of subjects = 6238 Number of obs = 6238No. of failures = 1143Time at risk = 30483.00891------------------------------------------------------------------------------_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------educa | 1.464797 .1143098 4.89 0.000 1.257047 1.706882_Iage_cat_1 | .8828032 .0544445 -2.02 0.043 .7822911 .9962295_Iage_cat_3 | 2.249075 .2887221 6.31 0.000 1.748767 2.892516------------------------------------------------------------------------------testparm educa 23.93 0.0000( 1) educa = 0 chi2( 1) = Prob > chi2 =xi: testparm i.age_cat( 1) _Iage_cat_1 = 0( 2) _Iage_cat_3 = 0 chi2( 2) = 55.09Prob > chi2 = 0.0000 58
Summary of a survival analysis :1. Describe the mortality using non-parametric methods :- mortality rate,- Kaplan-Meier survival curves (+ logrank test)these methods only describe the data.2. For further understanding, model the survival data - Cox Model, if the proportional hazard assumption is valid for themain covariates.Univariate and multivariate analysis to identify factors independentlyassociated with mortality. 59
References• Yoann Madec , Survival data analysis , The Capacity Building in Statistics and Cohort Data Analysis Expertise Mission Conference, 2013 60
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