Unit 17: Failure-Time
Regression Analysis
Ramón V. León
Notes largely based on
“Statistical Methods for Reliability Data”
by W.Q. Meeker and L. A. Escobar,
Wiley, 1998 and on their class notes.
11/14/2010
Stat 567: Unit 17 - Ramón León
1
Unit 17 Objectives
• Describe applications of failure-time regression
• Describe graphical methods for displaying
censored regression data
• Introduce time-scale transformation functions
• Describe simple regression models to relate life
to explanatory variables
• Illustrate the use of likelihood methods for
censored regression data
• Explain the importance of model diagnostics
• Describe and illustrate extensions to
nonstandard multiple regression models
11/14/2010
Stat 567: Unit 17 - Ramón León
2
Computer Program Execution Time
Versus System Load
• Time to complete a computationally
intensive task
• Information from the Unix uptime
command
• Predictions needed for scheduling
subsequent steps in a multi-step
computational process
11/14/2010
Stat 567: Unit 17 - Ramón León
3
Scatter Plot of Computer Program
Execution Time Versus System Load
11/14/2010
Stat 567: Unit 17 - Ramón León
4
Explanatory Variables for Failure Times
Useful explanatory variables explain/predict why some units
fail quickly and some survive a long time
• Continuous variables like stress, temperature,
voltage, and pressure
• Discrete variables like number of hardening
treatments or number of simultaneous users of a
system
• Categorical variables like manufacturer, design, and
location
11/14/2010
Stat 567: Unit 17 - Ramón León
5
Failure-Time Regression Analysis
11/14/2010
Stat 567: Unit 17 - Ramón León
6
Nickel-Base Super-alloy Fatigue Data
26 Observations in Total, 4 Censored
(Nelson 1984, 1990)
• Thousands of cycles to failure as a
function of pseudo-stress in ksi
• 26 units tested; 4 units did not fail.
Objective: Explore models that might be used
to describe the relationship between life length
and the amount of pseudo-stress applied to the
tested specimens.
11/14/2010
Stat 567: Unit 17 - Ramón León
7
Nickel-Based Super-Alloy Fatigue Data
(Nelson 1984, 1990)
11/14/2010
Stat 567: Unit 17 - Ramón León
8
Scale Accelerated Failure Time (SAFT) Models
Illustration of
Acceleration and
Deceleration
11/14/2010
Stat 567: Unit 17 - Ramón León
9
Scale Accelerated Failure Time (SAFT) Model
11/14/2010
Stat 567: Unit 17 - Ramón León
10
The SAFT Transformation Models
and Acceleration
In a time transformation plot of T(x0) vs. T(x)
• The SAFT models are straight lines through
the origin
• Accelerating SAFT models are straight lines
below the diagonal
• Decelerating SAFT models are straight lines
above the diagonal
11/14/2010
Stat 567: Unit 17 - Ramón León
11
Some Properties of SAFT Models
F  t ; x   P T ( x)  t 
 T ( x0 )

 P
t
 AF ( x) 
 P T ( x0 )  t AF ( x) 
 F  AF ( x)t , x0 
11/14/2010
Stat 567: Unit 17 - Ramón León
12
Some Properties of SAFT Models
p  F  t p ( x), x   F  AF ( x)t p ( x), x0 
and
p  F  t p ( x0 ), x0 
 t p ( x0 )  AF ( x)t p ( x)
11/14/2010
Stat 567: Unit 17 - Ramón León
13
Some Properties of SAFT Models
11/14/2010
Stat 567: Unit 17 - Ramón León
14
Weibull Probability Plot of Two Members from an SAFT
Model with a Baseline Lognormal Distribution
p
t p ( x)
11/14/2010
t p ( x0 )
Stat 567: Unit 17 - Ramón León
15
 log t p ( x)   ( x) 
since p   nor 




11/14/2010
Stat 567: Unit 17 - Ramón León
16
1
since log t p ( x)    0  1 x   nor
( p)  1 x  log t p (0) 
11/14/2010
Stat 567: Unit 17 - Ramón León
17
Also log T   ( x)   where 
N(0,1) since
P T  t   P  log T  log t 
 P   ( x)    log t 
log t   ( x) 

 P  




 log t   ( x) 
  nor 




11/14/2010
Stat 567: Unit 17 - Ramón León
18
1
log tˆp  x    ˆ0  ˆ1 x   nor
 p  ˆ
11/14/2010
Stat 567: Unit 17 - Ramón León
19
Likelihood for Lognormal Distribution Simple
Regression Model with Right Censored Data
11/14/2010
Stat 567: Unit 17 - Ramón León
20
Estimated Parameter VarianceCovariance Matrix
11/14/2010
Stat 567: Unit 17 - Ramón León
21
Standard Errors and Confidence
Intervals for Parameters
11/14/2010
Stat 567: Unit 17 - Ramón León
22
11/14/2010
Stat 567: Unit 17 - Ramón León
23
11/14/2010
Stat 567: Unit 17 - Ramón León
24
11/14/2010
Stat 567: Unit 17 - Ramón León
25
11/14/2010
Stat 567: Unit 17 - Ramón León
26
SPLIDA Output
11/14/2010
Stat 567: Unit 17 - Ramón León
27
Recall
cov(aX  bY , Z )  a cov( X , Z )  b cov(Y , Z )
Var(aX  bY )  a VarX  b VarY  2ab cov( X ,Y )
2
11/14/2010
2
Stat 567: Unit 17 - Ramón León
28
Standard Errors and Confidence Intervals
for Quantities at Specific Explanatory
Variable Conditions
11/14/2010
Stat 567: Unit 17 - Ramón León
29
11/14/2010
Stat 567: Unit 17 - Ramón León
30
11/14/2010
Stat 567: Unit 17 - Ramón León
31
Likelihood for Weibull Distribution Quadratic
Regression Model with Right Censored Data
11/14/2010
Stat 567: Unit 17 - Ramón León
32
11/14/2010
Stat 567: Unit 17 - Ramón León
33
11/14/2010
Stat 567: Unit 17 - Ramón León
34
11/14/2010
Stat 567: Unit 17 - Ramón León
35
JMP Loss Function
11/14/2010
Stat 567: Unit 17 - Ramón León
36
Stanford Heart Transplant Data
Quadratic Log-Mean Lognormal Regression Model
11/14/2010
Stat 567: Unit 17 - Ramón León
37
Stanford Heart Transplant Data
Quadratic Log-Location Weibull Regression Model
11/14/2010
Stat 567: Unit 17 - Ramón León
38
Extrapolation and Empirical Models
• Empirical models can be useful, providing a
smooth curve to describe a population or a
process
• Should not be used to extrapolate outside of the
range of one’s data
• There are many different kinds of extrapolation
– In an explanatory variable like stress
– To the upper tail of a distribution
– To the lowest tail of a distribution
• Need to get the right curve to extrapolate: look
toward physical or other process theory
11/14/2010
Stat 567: Unit 17 - Ramón León
39
Checking Model Assumptions
• Graphical checks using generalizations of usual
diagnostics (including residual analysis)
–
–
–
–
Residuals versus fitted values
Probability plot of residuals
Other residual plots
Influence analysis (or sensitivity) analysis. (See
Escobar and Meeker 1992 Biometrics)
• Most analytical test can be suitably generalized,
at least approximately, for censored data
(especially using likelihood ratio tests).
11/14/2010
Stat 567: Unit 17 - Ramón León
40
Definition of Residuals
11/14/2010
Stat 567: Unit 17 - Ramón León
41
Plot of Standardized Residuals Versus Fitted Values for
the Log-Quadratic Weibull Regression Model Fit to the
Super Alloy Data on Log-Log Axes
11/14/2010
Stat 567: Unit 17 - Ramón León
42
Probability Plot of the Standardized Residuals for
the Log-Quadratic Weibull Regression Model Fit to
the Super Alloy Data
11/14/2010
Stat 567: Unit 17 - Ramón León
43
Empirical Regression Models and
Sensitivity Analysis Objectives
• Describe a class of regression models that can
be used to describe the relationship between
failure time and explanatory variables
• Describe and illustrate the use of empirical
regression models
• Illustrate the need for sensitivity analysis
• Show how to conduct a sensitivity analysis
11/14/2010
Stat 567: Unit 17 - Ramón León
44
Picciotto Data (Picciotto 1970)
• Cycles to failure on specimens of yarn of
different lengths
• Subset of data at particular level of stress and
particular specimen lengths (30mm, 60mm, and
90mm)
• Original data were uncensored. For the
purposes of illustration, the data censored at
100 thousand cycles
• Suppose that the goal is to estimate life for 10
mm units.
11/14/2010
Stat 567: Unit 17 - Ramón León
45
Piccioto Data Showing Fraction
Failing at Each Length
11/14/2010
Stat 567: Unit 17 - Ramón León
46
Picciotto Data
Multiple Weibull Probability Plot
with Weibull ML Estimates
11/14/2010
Stat 567: Unit 17 - Ramón León
47
Picciotto Data
Multiple Lognormal Probability Plot
with Lognormal ML Estimates
11/14/2010
Stat 567: Unit 17 - Ramón León
48
Suggested Strategy for Fitting Empirical
Regression Models and Sensitivity Analysis
• Use data and previous experience to chose a base-line
model:
– Distribution at individual conditions
– Relationship between explanatory variables and distributions at
individual conditions
• Fit models
• Use diagnostics (e.g., residual analysis) to check models
• Assess uncertainty
– Confidence intervals quantify statistical uncertainty
– Perturb and otherwise change the model and reanalyze
(sensitivity analysis) to assess model uncertainty
11/14/2010
Stat 567: Unit 17 - Ramón León
49
Picciotto Data Multiple Lognormal Probability Plot with
Lognormal ML Log-Linear Regression Model Estimates
11/14/2010
Stat 567: Unit 17 - Ramón León
50
 Model: log T   
11/14/2010
0
 1 log  Length    , 
Stat 567: Unit 17 - Ramón León
N(0,1) 
51
Picciotto Data
Log-Linear Model Residual vs. Fitted Values Plot
11/14/2010
Stat 567: Unit 17 - Ramón León
52
Picciotto Data
Multiple Lognormal Probability Plot with Lognormal ML
Square Root-Linear Regression Model Estimates
11/14/2010
Stat 567: Unit 17 - Ramón León
53
 Model: log T  
11/14/2010
0
 1 log Length   Z , Z~N(0,1)
Stat 567: Unit 17 - Ramón León

54
Picciotto Data
Square Root – Linear Model Residuals vs.
Fitted Values Plot
11/14/2010
Stat 567: Unit 17 - Ramón León
55
Examples of Power Transformations
11/14/2010
Stat 567: Unit 17 - Ramón León
56
Box-Cox Transformation
11/14/2010
Stat 567: Unit 17 - Ramón León
57
Picciotto Data
Relationship Sensitivity Analysis for the .1 Quantile at Length 30 mm.
11/14/2010
Stat 567: Unit 17 - Ramón León
58
Picciotto Data
Relationship Sensitivity Analysis for the .1 Quantile at Length 10 mm.
11/14/2010
Stat 567: Unit 17 - Ramón León
59
Picciotto Data
Profile Plot for Different Box-Cox
Parameters for the Length Relationship
11/14/2010
Stat 567: Unit 17 - Ramón León
60
Picciotto Data
Relationship Sensitivity Analysis .1, .5, .9
Quantiles at 10 mm Length
11/14/2010
Stat 567: Unit 17 - Ramón León
61
Picciotto Data
Relationship/Distribution Sensitivity Analysis .1
Quantile at 10 mm Length
11/14/2010
Stat 567: Unit 17 - Ramón León
62
Glass Capacitor Failure Data
• Experiment designed to determine the
effect of voltage and temperature on
capacitor life.
• 2 x 4 factorial, 8 units at each combination.
• Test at each combination run until 4 of 8
units failed (Type II Censoring).
• Original Data from Zelen (1959)
11/14/2010
Stat 567: Unit 17 - Ramón León
63
Scatter Plot the Effect of Voltage and Temperature
on Glass Capacitor Life (Zelen 1959)
11/14/2010
Stat 567: Unit 17 - Ramón León
64
Weibull Probability Plots of Glass Capacitor Life Test Results at
Individual Temperature and Voltage Test Conditions
11/14/2010
Stat 567: Unit 17 - Ramón León
65
Likelihood Ratio Test
Let L( H ) be the maximized loglikelihood under model H .
To test H 0 versus H 1 where H 0  H 1 use
2   L( H 1 )  L( H o )    2log L  H 0     2log L  H 1  
= Loss H 0  Loss H 1
which has a Chi Square distribution under the null hypothesis H o
with  dim  H 1   dim  H 0   degrees of fredom.
11/14/2010
Stat 567: Unit 17 - Ramón León
66
Two-Variable Regression Models
11/14/2010
Stat 567: Unit 17 - Ramón León
67
-31.8
-31.7
-30.2
-30.3
-24.8
-28.4
-26.0
-28.4
-231.6
Sum of
maximized
loglikelihoods
is -231.6
11/14/2010
Stat 567: Unit 17 - Ramón León
68
Test of Additive Model Versus
Independent Models at each
Experimental Condition
Test statistic:
2   231.6   244.24  25.28
2
2
.95,16


4
.95,12  5.226
11/14/2010
Stat 567: Unit 17 - Ramón León
69
Weibull Probability Plots with Weibull Regression Model ML
Estimates of F(t) at each Set of Conditions for the Glass
Capacitor Data: Model With Interaction
11/14/2010
Stat 567: Unit 17 - Ramón León
70
Estimates of Weibull t.5 Plotted for each Combination of the
Glass Capacitor Test Conditions
11/14/2010
Stat 567: Unit 17 - Ramón León
71
Superalloy JMP Loss Function
Let’s think
of some
interesting
tests?
11/14/2010
Stat 567: Unit 17 - Ramón León
72
Review of the Weibull Plot
For the Weibull the p quantile is
t p     log(1  p )
1
or
log t p  log  log   log(1  p )
1
1
log t p  log    log   log(1  p )
 
So a plot of log   log 1  F (t )    log   log  S (t  
against t will plot as straight line if the distribution is Weibull
Proportional Hazard Failure Time Model
h(t; x )   ( x )h(t; x0 )

 log 1  F (t; x )   H (t; x )
t
t
0
0
  h( s; x )ds  =  ( x )  h( s; x0 )ds   ( x ) H (t , x0 )
  ( x )   log 1  F (t , x0 )  

 log 1  F (t; x )    log 1  F (t , x0 ) 
 (x)

1  F (t; x )  1  F (t , x0 ) 
 ( x)
11/14/2010
 S (t; x )  S (t , x0 ) ( x )
Stat 567: Unit 17 - Ramón León
74
S (t , x )  S (t , x0 ) ( x )
 log S (t , x )   ( x )log S (t , x0 )
log   log S (t , x )   log   ( x )log S (t , x0 ) 
 log ( x )  log   log S (t , x0 ) 

log   log S (t , x )   log   log S (t , x0 )   log ( x )
Thus in a Weibull probability scale F ( t ; x ) and F (t ; x0 )
are equidistant. In particular, Weibull plots of F (t ; x ) and F (t ; x0 )
are translations of each other along the probability axis.
Weibull Probability Plot of Two Members from a
PH Model with Baseline Lognormal Distribution
11/14/2010
Stat 567: Unit 17 - Ramón León
76
Weibull Probability Plot of Two Members from a
PH/SAFT Model with Baseline Weibull Distribution
11/14/2010
Stat 567: Unit 17 - Ramón León
77
Contrasting SAFT and PH Models
11/14/2010
Stat 567: Unit 17 - Ramón León
78
Interpreting PH Models as
Failure Time Transformation
11/14/2010
Stat 567: Unit 17 - Ramón León
79
Proportional Hazard Model (Lognormal
Baseline) as a Time Transformation
Amount of acceleration
(deceleration) depends
on the position in time.
11/14/2010
Stat 567: Unit 17 - Ramón León
80
General Failure Time Transformation Model
11/14/2010
Stat 567: Unit 17 - Ramón León
81
General Failure Time Transformation Graph
11/14/2010
Stat 567: Unit 17 - Ramón León
82
General Failure Time Transformation Model
11/14/2010
Stat 567: Unit 17 - Ramón León
83
Some Comments on the Proportional
Hazard Failure Time Model
11/14/2010
Stat 567: Unit 17 - Ramón León
84
Cox’s Semiparametric
Proportional Hazard rates Model
11/14/2010
Stat 567: Unit 17 - Ramón León
85
Simple Example
h(t , x)  h0 (t )e  x
(x is a scalar in this example)
Consider the following data:
11/14/2010
Stat 567: Unit 17 - Ramón León
86
Motivation
Using the partial likelihood to estimate  is equivalent to estimating
 by the method of maximum likelihood knowing only
-that the units under stress x = 100 were the 1st and 4th to fail
-that the units under stress x = 200 were the 2nd and 3rd to fail
-between what failures the censoring occurred
Cox’s partial likelihood estimation method corresponds to the
method of maximum likelihood when only the ranks of the failures
times (and the location between the failure ranks of the censoring)
is considered.
11/14/2010
Stat 567: Unit 17 - Ramón León
87
Partial Likelihood
The partial likelihood can be used just as the likelihood to:
- To estimate  maximize the partial likelihood
- To estimate the variance-covariance matrix of the estimates
invert the matrix of negatives of second partial derivatives
of the log partial likelihood
-To obtain confidence intervals use the (partial) likelihood
ratio method or the standard error method
- Compare nested models using partial likelihood ratios
11/14/2010
Stat 567: Unit 17 - Ramón León
88
Risk Sets
11/14/2010
Stat 567: Unit 17 - Ramón León
89
Partial Likelihood Formula


r
r
i
 exp  x (i )   

L(  )   

i 1  exp  xl  
i 1 ci
 lR

 i

where x(i ) is the vector of regressor variables associated
with units observed to fail at time t(i ) , i  1, 2,..., r
11/14/2010
Stat 567: Unit 17 - Ramón León
90
Partial Likelihood Calculation
11/14/2010
Stat 567: Unit 17 - Ramón León
91
Partial Likelihood Calculation
11/14/2010
Stat 567: Unit 17 - Ramón León
92
A Game
Three players throw three balls in a paper basket. We have the
following probabilities:
Player
P(Player makes basket)
1
p1
2
p2
3
p3
Suppose exactly one ball landed in the basket, what is the
probability that player i was the one that made the basket?
pi
p1  p2  p3
11/14/2010
Stat 567: Unit 17 - Ramón León
93
One Multiplicand of the Partial Likelihood
h(t(3) , 200)t
h(t(3) , 200)t  h(t(3) ,100)t  h(t(3) , 200)t
11/14/2010
Stat 567: Unit 17 - Ramón León
94
One Multiplicand of the Partial Likelihood
h(t(3) , 200)t
h(t(3) , 200)t  h(t(3) ,100)t  h(t(3) , 200)t

h0 (t(3) )e 200  t
h0 (t(3) )e
200 
200 
e
200 
100 
t  h0 (t(3) )e
t  h0 (t(3) )e
200 
t

x( 3) 
e
e

xl 
100 
200 
e
e
e
lR3
11/14/2010
Stat 567: Unit 17 - Ramón León
95
Remark
Cox’s partial likelihood estimation method ignores the actual
times of the failures. This is reasonable since without knowing
h0(t) the actual failure times provide little or no information on
the 's
11/14/2010
Stat 567: Unit 17 - Ramón León
96
Comments on the Cox PH Model Likelihood
11/14/2010
Stat 567: Unit 17 - Ramón León
97
Picciotto Cox PH Model Survival Estimates
(Linear Axes)
11/14/2010
Stat 567: Unit 17 - Ramón León
98
Picciotto Cox PH Model Survival Estimates (on
Weibull Probability Scales)
11/14/2010
Stat 567: Unit 17 - Ramón León
99
Cox Proportional Hazards Model in JMP
We analyze the failure stresses of a single carbon fiber as a function
of length x. (Data available at the course home page.)
2
.01
.05
.1
.2
1
log(-log(Surv))
0
.4
.6
-1
.8
-2
.9
-3
.95
-4
.99
-5
-6
This curves should
be vertical
displacements of
each other not
necessarily straight
lines
-7
1
2
3
4
5
6
7
Stress
11/14/2010
Stat 567: Unit 17 - Ramón León
100
JMP Analysis
11/14/2010
Stat 567: Unit 17 - Ramón León
101
Effect Likelihood Ratio Tests
Source
Nparm
Length
1
11/14/2010
DF L-R ChiSquare Prob>ChiSq
1
140.962733
Stat 567: Unit 17 - Ramón León
0.0000
102
Parameter Estimates
Term
Estimate
Lower CL
Upper CL
Length 0.04923859 0.0040304 0.0413392
0.05717
h(t , x)  h0 (t )e
Std Error
 ( x  xu )
 h0 (t )e
0.4923675( x  20.25)
where xu  20.25 is the unweighted mean
of the values of the regressors (lengths):
(1+10+20+50)/4 =20.25
11/14/2010
Stat 567: Unit 17 - Ramón León
103
11/14/2010
Stat 567: Unit 17 - Ramón León
104
JMP Analysis
S0 (t ) = Survival at xu  20.25
S (t , x)   S0 (t ) 
 ( x)
exp  ( x  xu )
  S0 (t ) 
exp 0.4923675( x  20.25)
  S0 (t ) 
Can be used to calculate Shoenfeld residuals. More
about this latter.
11/14/2010
Stat 567: Unit 17 - Ramón León
105
Some Mathematics
4
h(t ,1)  h(t ,10)  h(t , 20)  h(t ,50) 
4
4
 h(t , x )
i
i 1
4
4
  xu
 ( xi  xu )
h
(
t
)
e

h
(
t
)
e
0
0
i 4
4
h0 (t )e   xu e

4
4
 xi
e
 
i 1
4
 xi
i 1
 h0 (t )e   xu 4 e  4 xu  h0 (t )e   xu e  xu  h0 (t )
h0(t) is the geometric mean of the hazard rates at the four loads
(lengths)
11/14/2010
Stat 567: Unit 17 - Ramón León
106
Risk Ratios
Term
Risk Ratio Lower CL
Length
1.050471
Upper CL
1.042206 1.058836
For simplicity assume that x1  x2
 ( x2  xu )
h(t , x2 ) h0 (t )e


 ( x1  xu )
h(t , x1 ) h0 (t )e
 x2
e
x2  x1
 x2  x1
0.049 x2  x1
 e 
 e 
 1.05 
 x1
e
11/14/2010
Stat 567: Unit 17 - Ramón León
107
Categorical JMP Analysis
Whole Model
Number of Events
Number of Censorings
Total Number
Model
Difference
Full
Reduced
250
0
250
-LogLikelihood ChiSquare
105.449
1028.625
1134.075
210.8989
DF Prob>Chisq
3
<.0001
Effect Likelihood Ratio Tests
11/14/2010
Source
Nparm
Length
3
DF L-R ChiSquare Prob>ChiSq
3
210.898899
Stat 567: Unit 17 - Ramón León
0.0000
108
Parameter Estimates
Term
Estimate
Std Error
Lower CL
Upper CL
-1.9230939 0.163213 -2.250564 -1.610452
Length[1]
-0.1355
Length[10] -0.3619084 0.1178567 -0.598202
Length[20] 0.89469015 0.1231856 0.6508217 1.1345373
h(t , x)  h0 (t ) exp  1.92 x1  0.36 x2  0.89 x3 
1

x1   -1
0

1

length=50 x2   -1
0
o.w.

length=1
1

length=50 x3   -1
0
o.w.

length=10
length=20
length=50
o.w.
h(t ,1)  h0 (t )e 1.92 , h(t ,10)  h0 (t )e 0.36
h(t , 20)  h0 (t )e0.89 , h(t ,50)  h0 (t )e1.39
11/14/2010
Stat 567: Unit 17 - Ramón León
109
Categorical JMP Analysis
What mean if
regressor
is categorical?
Baseline Survival at mean
1.0
0.9
Baseline Surv
0.8
0.7
h0 (t )  geometric
0.6
0.5
mean of the hazard
0.4
0.3
rates at the four
0.2
0.1
loads
0.0
1
2
3
4
5
6
Stress
11/14/2010
Stat 567: Unit 17 - Ramón León
110
Risk Ratios
Term
Risk Ratio Lower CL
Length[1]
Length[10]
Length[20]
0.146154
0.696346
2.446578
h(t ,1)  h0 (t )e
1.92
h(t ,10)  h0 (t )e
 0.146h0 (t )
 0.696h0 (t )
0.89
 2.447 h0 (t )
1.39
 4.015h0 (t )
h(t ,50)  h0 (t )e
11/14/2010
0.10534 0.199797
0.5498 0.873279
1.917115 3.109734
0.36
h(t , 20)  h0 (t )e
Upper CL
Stat 567: Unit 17 - Ramón León
111
Shoenfeld (Imputation) Residuals
xiT 
ˆ
H 0 (ti )e
if t i is a failure time
xiT 
ˆ
H 0 (ti )e  1
if t i is a censoring time
Remark: This are uncensored residuals unlike the usual residuals
Based:
-For any positive random variable T, H(T) is an exponential
random variable with mean  = 1
-For the an exponential random variable T we have
E (T | T  c)    c
Useful for plotting against predicted values. In practice, they don’t
work too well though they are based on a very clever idea.
11/14/2010
Stat 567: Unit 17 - Ramón León
112
The Proportional Hazard Failure Time Model
11/14/2010
Stat 567: Unit 17 - Ramón León
113
Statistical Methods for
Semiparametric (Cox) PH Model
11/14/2010
Stat 567: Unit 17 - Ramón León
114
Cox PH Model Likelihood (Probability of the Data)
11/14/2010
Stat 567: Unit 17 - Ramón León
115
Other Topics in Failure-Time Regression
• Models with non-location-scale
distributions
• Nonlinear relationship regression model
• Fatigue-limit regression model
• Special topics in regression: random
effects models, nonparametric SAFT
regression models, parametric PH
regression models
11/14/2010
Stat 567: Unit 17 - Ramón León
116