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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

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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