# 567Unit17 ```Unit 17: Failure-Time
Regression Analysis
Ram&oacute;n V. Le&oacute;n
Notes largely based on
“Statistical Methods for Reliability Data”
by W.Q. Meeker and L. A. Escobar,
Wiley, 1998 and on their class notes.
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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
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Computer Program Execution Time
• Time to complete a computationally
• Information from the Unix uptime
command
• Predictions needed for scheduling
subsequent steps in a multi-step
computational process
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Scatter Plot of Computer Program
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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
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Failure-Time Regression Analysis
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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.
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Nickel-Based Super-Alloy Fatigue Data
(Nelson 1984, 1990)
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Scale Accelerated Failure Time (SAFT) Models
Illustration of
Acceleration and
Deceleration
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Scale Accelerated Failure Time (SAFT) Model
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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
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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 
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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)
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Some Properties of SAFT Models
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Weibull Probability Plot of Two Members from an SAFT
Model with a Baseline Lognormal Distribution
p
t p ( x)
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t p ( x0 )
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 log t p ( x)   ( x) 
since p   nor 




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1
since log t p ( x)    0  1 x   nor
( p)  1 x  log t p (0) 
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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 




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1
log tˆp  x    ˆ0  ˆ1 x   nor
 p  ˆ
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Likelihood for Lognormal Distribution Simple
Regression Model with Right Censored Data
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Estimated Parameter VarianceCovariance Matrix
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Standard Errors and Confidence
Intervals for Parameters
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SPLIDA Output
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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
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Standard Errors and Confidence Intervals
for Quantities at Specific Explanatory
Variable Conditions
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Regression Model with Right Censored Data
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JMP Loss Function
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Stanford Heart Transplant Data
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Stanford Heart Transplant Data
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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
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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).
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Definition of Residuals
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Plot of Standardized Residuals Versus Fitted Values for
the Log-Quadratic Weibull Regression Model Fit to the
Super Alloy Data on Log-Log Axes
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Probability Plot of the Standardized Residuals for
the Log-Quadratic Weibull Regression Model Fit to
the Super Alloy Data
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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
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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.
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Piccioto Data Showing Fraction
Failing at Each Length
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Picciotto Data
Multiple Weibull Probability Plot
with Weibull ML Estimates
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Picciotto Data
Multiple Lognormal Probability Plot
with Lognormal ML Estimates
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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
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Picciotto Data Multiple Lognormal Probability Plot with
Lognormal ML Log-Linear Regression Model Estimates
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 Model: log T   
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0
 1 log  Length    , 
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N(0,1) 
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Picciotto Data
Log-Linear Model Residual vs. Fitted Values Plot
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Picciotto Data
Multiple Lognormal Probability Plot with Lognormal ML
Square Root-Linear Regression Model Estimates
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 Model: log T  
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0
 1 log Length   Z , Z~N(0,1)
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
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Picciotto Data
Square Root – Linear Model Residuals vs.
Fitted Values Plot
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Examples of Power Transformations
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Box-Cox Transformation
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Picciotto Data
Relationship Sensitivity Analysis for the .1 Quantile at Length 30 mm.
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Picciotto Data
Relationship Sensitivity Analysis for the .1 Quantile at Length 10 mm.
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Picciotto Data
Profile Plot for Different Box-Cox
Parameters for the Length Relationship
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Picciotto Data
Relationship Sensitivity Analysis .1, .5, .9
Quantiles at 10 mm Length
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Picciotto Data
Relationship/Distribution Sensitivity Analysis .1
Quantile at 10 mm Length
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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)
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Scatter Plot the Effect of Voltage and Temperature
on Glass Capacitor Life (Zelen 1959)
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Weibull Probability Plots of Glass Capacitor Life Test Results at
Individual Temperature and Voltage Test Conditions
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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.
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Two-Variable Regression Models
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-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
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Independent Models at each
Experimental Condition
Test statistic:
2   231.6   244.24  25.28
2
2
.95,16


4
.95,12  5.226
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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
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Estimates of Weibull t.5 Plotted for each Combination of the
Glass Capacitor Test Conditions
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Superalloy JMP Loss Function
Let’s think
of some
interesting
tests?
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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)
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 S (t; x )  S (t , x0 ) ( x )
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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
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Weibull Probability Plot of Two Members from a
PH/SAFT Model with Baseline Weibull Distribution
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Contrasting SAFT and PH Models
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Interpreting PH Models as
Failure Time Transformation
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Proportional Hazard Model (Lognormal
Baseline) as a Time Transformation
Amount of acceleration
(deceleration) depends
on the position in time.
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General Failure Time Transformation Model
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General Failure Time Transformation Graph
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General Failure Time Transformation Model
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Hazard Failure Time Model
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Cox’s Semiparametric
Proportional Hazard rates Model
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Simple Example
h(t , x)  h0 (t )e  x
(x is a scalar in this example)
Consider the following data:
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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.
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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
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Risk Sets
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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
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Partial Likelihood Calculation
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Partial Likelihood Calculation
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A Game
Three players throw three balls in a paper basket. We have the
following probabilities:
Player
1
p1
2
p2
3
p3
Suppose exactly one ball landed in the basket, what is the
pi
p1  p2  p3
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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
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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
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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
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Comments on the Cox PH Model Likelihood
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Picciotto Cox PH Model Survival Estimates
(Linear Axes)
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Picciotto Cox PH Model Survival Estimates (on
Weibull Probability Scales)
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Cox Proportional Hazards Model in JMP
We analyze the failure stresses of a single carbon fiber as a function
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
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JMP Analysis
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Effect Likelihood Ratio Tests
Source
Nparm
Length
1
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DF L-R ChiSquare Prob&gt;ChiSq
1
140.962733
Stat 567: Unit 17 - Ram&oacute;n Le&oacute;n
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
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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
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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)
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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
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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&gt;Chisq
3
&lt;.0001
Effect Likelihood Ratio Tests
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Source
Nparm
Length
3
DF L-R ChiSquare Prob&gt;ChiSq
3
210.898899
Stat 567: Unit 17 - Ram&oacute;n Le&oacute;n
0.0000
108
Parameter Estimates
Term
Estimate
Std Error
Lower CL
Upper CL
-1.9230939 0.163213 -2.250564 -1.610452
Length
-0.1355
Length -0.3619084 0.1178567 -0.598202
Length 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
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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
0.0
1
2
3
4
5
6
Stress
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Risk Ratios
Term
Risk Ratio Lower CL
Length
Length
Length
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
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0.10534 0.199797
0.5498 0.873279
1.917115 3.109734
0.36
h(t , 20)  h0 (t )e
Upper CL
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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.
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The Proportional Hazard Failure Time Model
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Statistical Methods for
Semiparametric (Cox) PH Model
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Cox PH Model Likelihood (Probability of the Data)
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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
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