love does not come by demanding from others, but it is a self initiation
Post on 05-Jan-2016
17 Views
Preview:
DESCRIPTION
TRANSCRIPT
Survival Analysis 1
Love does not come by demanding from others, but it is a self initiation.
Survival Analysis2
Survival Analysis
Semiparametric Proportional Hazards Regression (Part III)
Survival Analysis3
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 Analysis4
Three Tests
They are all likelihood-based tests:
Likelihood Ratio (LR) Test Wald Test Score Test
Survival Analysis5
Three Tests
Asymptotically equivalent Approximately low-order Taylor series
expansion of each other LR test considered most reliable and
Wald test the least
Survival Analysis6
Global Tests
Overall test for a model containing p covariates
H0: p
Survival Analysis7
Global Tests
Survival Analysis8
Global Tests
Survival Analysis9
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 Analysis10
Local Tests
Survival Analysis11
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 Analysis12
Survival Analysis13
Example: PBC
Consider the following models:
LR test stat = 2.027; DF = 2; p-value =0.3630
conclusion?
Survival Analysis14
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
Survival Analysis15
Breslow Estimate
Survival Analysis16
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 Analysis17
Example: PBC
Survival Analysis18
Estimation of the Median Survival Time
Survival Analysis19
Survival Analysis20
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 Analysis21
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 Analysis22
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 Analysis23
Cox-Snell Residuals
Survival Analysis24
Survival Analysis25
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 Analysis26
Martingale Residuals
Martingale residuals are a transformation of Cox-Snell residuals.
Survival Analysis27
Martingale Residuals
Martingale residuals are useful for exploring the correct functional form for the effect of a (ordinal) covariate.
Example: PBC
Survival Analysis28
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.
Survival Analysis29
Survival Analysis30
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 Analysis31
Survival Analysis32
Deviance Residuals
Martingale residuals are a transformation of Cox-Snell residuals
Deviance residuals are a transformation of martingale residuals.
Survival Analysis33
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 Analysis34
Example: PBC
Survival Analysis35
Schoenfeld Residuals
Survival Analysis36
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 Analysis37
Scaled schoenfeld residuals
Survival Analysis38
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 Analysis39
Example: PBC
Survival Analysis40
Example: PBC
Survival Analysis41
Example: PBC
Survival Analysis42
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)
Survival Analysis43
The End
Good Luck for Finals!!
top related