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Love does not come by demanding from others, but it is a self initiation. . Survival Analysis. Semiparametric Proportional Hazards Regression (Part III). Hypothesis Tests for the Regression Coefficients.

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

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### Survival Analysis

Semiparametric Proportional Hazards Regression (Part III)

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)
Three Tests

They are all likelihood-based tests:

• Likelihood Ratio (LR) Test
• Wald Test
• Score Test
Three Tests
• Asymptotically equivalent
• Approximately low-order Taylor series expansion of each other
• LR test considered most reliable and Wald test the least
Global Tests
• Overall test for a model containing p covariates
• H0: b1 = b2 = ... = bp = 0
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
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)
Example: PBC
• Consider the following models:

LR test stat = 2.027; DF = 2; p-value =0.3630

 conclusion?

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
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
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).
• 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
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)
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.
Martingale Residuals

Martingale residuals are a transformation of Cox-Snell residuals.

Martingale Residuals
• Martingale residuals are useful for exploring the correct functional form for the effect of a (ordinal) covariate.
• Example: PBC
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.
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.
Deviance Residuals
• Martingale residuals are a transformation of Cox-Snell residuals
• Deviance residuals are a transformation of martingale residuals.
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
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

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.
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)
The End

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