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

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Semiparametric Proportional Hazards Regression (Part III)

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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)

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Three Tests

They are all likelihood-based tests:

Likelihood Ratio (LR) Test Wald Test Score Test

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Three Tests

Asymptotically equivalent Approximately low-order Taylor series

expansion of each other LR test considered most reliable and

Wald test the least

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Global Tests

Overall test for a model containing p covariates

H0: p

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Global Tests

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Global Tests

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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

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Local Tests

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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)

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Example: PBC

Consider the following models:

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

conclusion?

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Estimation of Survival Function

To estimate S(y|X), the baseline survival function S0(y) must be estimated first.

Two estimates:Breslow estimateKalbfleisch-Prentice estimate

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Breslow Estimate

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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

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Example: PBC

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Estimation of the Median Survival Time

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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).

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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

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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)

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Cox-Snell Residuals

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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.

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Martingale Residuals

Martingale residuals are a transformation of Cox-Snell residuals.

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Martingale Residuals

Martingale residuals are useful for exploring the correct functional form for the effect of a (ordinal) covariate.

Example: PBC

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Martingale Residuals

1. Fit a full model.

2. Plot the martingale residuals against each ordinal covariate separately.

3. Superimpose a scatterplot smooth (such as LOESS) to see the functional form for the covariate.

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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.

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Deviance Residuals

Martingale residuals are a transformation of Cox-Snell residuals

Deviance residuals are a transformation of martingale residuals.

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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

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Example: PBC

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Schoenfeld Residuals

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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

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Scaled schoenfeld residuals

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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.

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Example: PBC

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Example: PBC

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Example: PBC

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Strategies for Non-proportionality

Stratify the covariates with non-proportional effectsNo test for the effect of a stratification

factorHow to categorize a numerical covariate?

Partition the time axis Use a different model (such as AFT

model)

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The End

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