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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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survival analysis

Survival Analysis

Semiparametric Proportional Hazards Regression (Part III)

hypothesis tests for the regression coefficients
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
Three Tests

They are all likelihood-based tests:

  • Likelihood Ratio (LR) Test
  • Wald Test
  • Score Test
three tests5
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
Global Tests
  • Overall test for a model containing p covariates
  • H0: b1 = b2 = ... = bp = 0
local tests
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 tests11
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
Example: PBC
  • Consider the following models:

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

 conclusion?

estimation of survival function
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
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 pbc20
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).
assessment of model adequacy
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
residuals for cox models
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
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

Martingale residuals are a transformation of Cox-Snell residuals.

martingale residuals27
Martingale Residuals
  • Martingale residuals are useful for exploring the correct functional form for the effect of a (ordinal) covariate.
  • Example: PBC
martingale residuals28
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 residuals31
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
Deviance Residuals
  • Martingale residuals are a transformation of Cox-Snell residuals
  • Deviance residuals are a transformation of martingale residuals.
deviance residuals33
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
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 assumption38
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
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
The End

Good Luck for Finals!!