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Love does not come by demanding from others, but it is a self initiation.
Survival Analysis
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Survival Analysis
Semiparametric Proportional
Hazards Regression (Part III)
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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)
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Survival Analysis
Three Tests
They are all likelihood-based tests:
Likelihood Ratio (LR) Test
Wald Test
Score Test
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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
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Survival Analysis
Global Tests
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Overall test for a model containing p
covariates
H0: b1 = b2 = ... = bp = 0
Survival Analysis
Global Tests
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Survival Analysis
Global Tests
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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
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Survival Analysis
Local Tests
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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)
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Survival Analysis
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Survival Analysis
Example: PBC
Consider the following models:
LR test stat = 2.027; DF = 2; p-value =0.3630
conclusion?
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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
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Survival Analysis
Breslow Estimate
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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
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Survival Analysis
Example: PBC
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Survival Analysis
Estimation of the Median
Survival Time
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Survival Analysis
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Survival Analysis
Example: PBC
The estimated median survival time for 60year-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 40year-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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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
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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)
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Survival Analysis
Cox-Snell Residuals
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Survival Analysis
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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.
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Survival Analysis
Martingale Residuals
Martingale residuals are a transformation of Cox-Snell residuals.
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Survival Analysis
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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Survival Analysis
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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Survival Analysis
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Survival Analysis
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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.
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Survival Analysis
Deviance Residuals
Martingale residuals are a
transformation of Cox-Snell residuals
Deviance residuals are a
transformation of martingale residuals.
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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):
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Large positive value died too soon
Large negative value lived too long
Survival Analysis
Example: PBC
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Survival Analysis
Schoenfeld Residuals
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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
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Survival Analysis
Scaled schoenfeld residuals
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Survival Analysis
Assessing the Proportional
Hazards Assumption
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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
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Survival Analysis
Example: PBC
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Survival Analysis
Example: PBC
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Survival Analysis
Strategies for Non-proportionality
Stratify the covariates with nonproportional 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)
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Survival Analysis
The End
Good Luck for Finals!!
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Survival Analysis