Love does not come by demanding from others, but it is a self initiation. Survival Analysis

Semiparametric Proportional Hazards Regression (Part III) Survival Analysis Semiparametric Proportional Hazards Regression (Part III) Survival Analysis

Hypothesis Tests for the Regression Coefficients Does the entire set of variables contribute significantly to the prediction of survivorship? (global test) Does the addition of a group variables contribute significantly to the prediction of survivorship over and above that achieved by other variables? (local test) Survival Analysis

Three Tests They are all likelihood-based tests: Likelihood Ratio (LR) Test Wald Test Score Test Survival Analysis

Three Tests Asymptotically equivalent Approximately low-order Taylor series expansion of each other LR test considered most reliable and Wald test the least Survival Analysis

Global Tests Overall test for a model containing p covariates H0: b1 = b2 = ... = bp = 0 Survival Analysis

Global Tests Survival Analysis

Global Tests Survival Analysis

Local Tests Tests for the additional contribution of a group of covariates Suppose X1,…,Xp are included in the model already and Xp+1,…,Xq are yet included Survival Analysis

Local Tests Survival Analysis

Local Tests Only one: likelihood ratio test The statistics -2logPLn(MPLE) is a measure of “amount” of collected information; the smaller the better. It sometimes inappropriately referred to as a deviance; it does not measure deviation from the saturated model (the model which is prefect fit to the data) Survival Analysis

Survival Analysis

Example: PBC Consider the following models: LR test stat = 2.027; DF = 2; p-value =0.3630  conclusion? Survival Analysis

Estimation of Survival Function To estimate S(y|X), the baseline survival function S0(y) must be estimated first. Two estimates: Breslow estimate Kalbfleisch-Prentice estimate Survival Analysis

Breslow Estimate Survival Analysis

Kalbfleisch-Prentice Estimate An estimate of h0(y) was derived by Kalbfleisch and Prentice using an approach based on the method of maximum likelihood. Reference: Kalbfleisc, J.D. and Prentice, R.L. (1973). Marginal likelihoods based on Cox’s regression and life model. Biometrika, 60, 267-278 Survival Analysis

Example: PBC Survival Analysis

Estimation of the Median Survival Time Survival Analysis

Survival Analysis

Example: PBC The estimated median survival time for 60-year-old males treated with DPCA is 2105 days (=5.76 years) with an approximate 95% C. I. (970.86,3239.14). The estimated median survival time for 40-year-old males treated with DPCA is 3584 days (=9.81 years) with an approximate 95% C. I. (2492.109, 4675.891). Survival Analysis

Assessment of Model Adequacy Model-based inferences depend completely on the fitted statistical model  validity of these inferences depends on the adequacy of the model The evaluation of model adequacy are often based on quantities known as residuals Survival Analysis

Residuals for Cox Models Four major residuals: Cox-Snell residuals (to assess overall fitting) Martingale residuals (to explore the functional form of each covariate) Deviance residuals (to assess overall fitting and identify outliers) Schoenfeld residuals (to assess PH assumption) Survival Analysis

Cox-Snell Residuals Survival Analysis

Survival Analysis

Limitations Do not indicate the type of departure when the plot is not linear. The exponential distribution for the residuals holds only when the actual parameter values are used. Crowley & Storer (1983, JASA 78, 277-281) showed empirically that the plot is ineffective at assessing overall model adequacy. Survival Analysis

Martingale Residuals Martingale residuals are a transformation of Cox-Snell residuals. Survival Analysis

Martingale Residuals Martingale residuals are useful for exploring the correct functional form for the effect of a (ordinal) covariate. Example: PBC Survival Analysis

Martingale Residuals Fit a full model. Plot the martingale residuals against each ordinal covariate separately. Superimpose a scatterplot smooth (such as LOESS) to see the functional form for the covariate. Survival Analysis

Survival Analysis

Survival Analysis

Martingale Residuals Example: PBC The covariates are now modified to be: Age, log(bili), and other categorical variables. The simple method may fail when covariates are correlated. Survival Analysis

Deviance Residuals Martingale residuals are a transformation of Cox-Snell residuals Deviance residuals are a transformation of martingale residuals. Survival Analysis

Deviance Residuals Deviance residuals can be used like residuals from OLS regression: They follow approximately the standard normal distribution when censoring is light (<25%) Can help to identify outliers (subjects with poor fit): Large positive value  died too soon Large negative value  lived too long Survival Analysis

Example: PBC Survival Analysis

Schoenfeld Residuals Survival Analysis

Assessing the Proportional Hazards Assumption The main function of Schoenfeld residuals is to detect possible departures from the proportional hazards (PH) assumption. The plot of Schoenfeld residual against survival time (or its rank) should show a random scatter of points centered on 0 A time-dependent pattern is evidence against the PH assumption. Ref: Schoenfeld, D. (1982). Partial residauls for the proportional hazards regression model. Biometrika, Vol. 69, P. 239-241 Survival Analysis

Scaled schoenfeld residuals Survival Analysis

Assessing the Proportional Hazards Assumption Scaled Schoenfeld residuals is popular than the un-scaled ones to detect possible departures from the proportional hazards (PH) assumption. (SAS uses this.) A time-dependent pattern is evidence against the PH assumption. Most of tests for PH are tests for zero slopes in a linear regression of scaled Sch. residuals on chosen functions of times. Survival Analysis

Example: PBC Survival Analysis

Example: PBC Survival Analysis

Example: PBC Survival Analysis

Strategies for Non-proportionality Stratify the covariates with non-proportional effects No test for the effect of a stratification factor How to categorize a numerical covariate? Partition the time axis Use a different model (such as AFT model) Survival Analysis

The End Good Luck for Finals!! Survival Analysis