Application of Survival Data Analysis- Introduction and Discussion (存活数据分析及应用- 简介和讨论), will give an overview of survival data analysis, including parametric and non-parametric approaches and proportional hazard model, providing a real life example of survival data-based field return analysis. Several common issues in survival data analysis will also be discussed.
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4. Outline
Introduction
Measurements of ARR and reliability
Survival data – a glance
Special Features in survival data
Overview of Statistical Methods
Parametric approach
Distribution based approach
Semi-parametric approach – Cox PH model
Accelerated Failure Time Model
Frailty Model
Non-parametric approach
Kaplan–Meier curve
Log-Rank Test
Examples
Discussion
Summary
5. Measurements of Field Failure and
Reliability
ARR (Annual Return Rate) – based on field returns in one year.
How to define one year in field? - Shipments can go out at different times, so one year
in the field may mean different starting date in calendar. One year from the first
shipment, one year for every shipment, or one year of continuous operation for every
unit included in the shipments considered?
Many different ARR calculations by applying different adjustments
Linear extrapolation
Prediction based on survival curve
Reliability Prediction
MTTF – MTTF is estimated based on reliability tests. For example, MTTF of a hard
disk drive can be millions of hours. However, the reliability may only cover thousands
of hours (in field). How accurate is the estimation?
Multiple distributions for failure time
Multiple failure modes may govern failures at different life time.
Example – bathtub hazard curve
6. ARR Calculation
Annual Returns
• Shipment 1 Actual
Estimated
• Shipment 2
• Shipment 3 Estimated
• Shipment 4 Estimated
• Shipment 5 Estimated
• Shipment 6 Estimated
How to estimate or predict survival at a future time point?
7. Survival Data – A Glance
What is survival data?
Data measuring the time to event Number alive
and under
Events: death, failure, received, a complication, etc. Year since observation at
Incomplete data in terms of event time entry into the beginning Number dyning Numbercensor 1- Mortality Survival
study of interval during interval ed or withdraw Mortality rate Rate function
[0,1) 146 27 3 0.18 0.82 0.82
An example [1,2) 116 18 10 0.16 0.84 0.69
[2,3) 88 21 10 0.24 0.76 0.52
[3,4) 57 9 3 0.16 0.84 0.44
[4,5) 45 1 3 0.02 0.98 0.43
Year since Number alive and Number dying Number [5,6) 41 2 11 0.05 0.95 0.41
entry into under observation during intervalcensored or [6,7) 28 3 5 0.11 0.89 0.37
study withdrawn [7,8) 20 1 8 0.05 0.95 0.35
at the beginning of [8,9) 11 2 1 0.18 0.82 0.28
interval [9,10) 8 2 6 0.25 0.75 0.21
[0,1) 146 27 3
[1,2) 116 18 10 1
[2,3) 88 21 10
0.8
[3,4) 57 9 3 Survival Function
[4,5) 45 1 3 0.6
[5,6) 41 2 11
[6,7) 28 3 5 0.4
[7,8) 20 1 8
0.2
[8,9) 11 2 1
[9,10) 8 2 6 0
0 1 2 3 4 5 6 7 8 9 10
Data cited from a clinical trial on myocardial infarction
(MI) (Svetlana, S., 2002) Year after enter into study
8. Special Features of Survival Data?
Time-to-event - The primary interest of the survival analysis is
time to event.
Time to event can be modeled by a distribution function
Random variable
The „time to event‟ for every unit is available as time goes infinity (or approaching
to a limit)
The time to event is usually not normally distributed
Censored - with incomplete information about the „time to
event‟.
General issues in survival data analysis
The non-normality aspect of the survival data violates the normality
assumption of most commonly used statistical model such as regression
or ANOVA, etc.
Incompleteness may cause issues such as:
Estimation bias.
Difficulty in validating the assumption
9. Censoring
A censored observation is defined as an
observation with incomplete information
about the „time-to-event‟
Different types of censoring, such as
right censoring, left censoring, and interval
censoring, etc.
Right censoring --- The information about
time to event is incomplete because the
subject did not have an event during the
time when the subject was studied.
10. Overview of Statistical Methods
Objectives:
Characterize and estimate the distribution of the failure time;
Compare failure times among different groups, e.g. generations of products (old vs.
new), treatment vs. control, etc.
Assess the relationship of covariates to time-to-event, e.g. which factors
significantly affect the distribution of time-to-event?
Approaches:
To estimate the survival (hazard) function:
parametric approach: specify a parametric model, i.e. a specific distribution
(exponential, Weibull, etc.)
empirical approach: use nonparametric or semi-parametric estimation (more
popular in biomedical sciences), such as Kaplan–Meier estimator
To compare two survival functions:
Log-rank test
To model the relationship between failure time and covariates:
Cox proportional hazard model
Accelerate failure-time model
Frailty model
11. Parametric Survival Model
Parametric Survival Model
Assumption on underlying distribution
Hazard function, h(t), and survival function, S(t), is completely
specified
Continuous process
Prediction possible
Main Assumption
The survival time t is assumed to follow a distribution with density
function f (t). Specifying one of the three functions f(t), S(t), or h(t)
means to specify the other two functions.
S (t ) P (T t ) f (u )du
t
d
S (t ) t
f (t ) h(u )du
h(t ) dt S (t ) exp
S (t ) S (t ) 0
13. Weibull Model
Assumption:
Time to event, t, follows Weibull ( , ) with probability
function:
f (t ) t 1 exp(t ), where , 0
The hazard function is given by:
h(t ) t 1
The survival function
S (t ) exp(t ) S (t ) exp( t ) log( S (t )) t
log( log( S (t ))) log( ) log(t )
Exponential Distribution – nice properties
Flexible
Graphical evaluation
14. Likelihood and Censored Survival Data
Likelihood estimate (right censored data):
The likelihood function of parameter(s) :
n
L( , t ) f (ti , ) i [ S (ti , )]1 i
i 1
MLE ˆ of :
( ; t )
U ( ; t ) 0 where ( ; t ) is the log likelihood function
ˆ
~ N ( ,V ) where V J 1 and J denotes Fisher informatio n matrix
Hypothesis Tests
Score test
Likelihood ratio test
15. Semi-Parametric Model
Cox PH Model - a very popular model in Biostatistics
Distribution of time-to-event unknown but proportional hazard ratio is assumed.
Baseline hazard is not needed in the estimation of hazard ratio
Semi-parametric - The baseline hazard can take any form, the covariates enter
the model linearly
Proportional hazard assumption
h(t | X ) h0 (t ) exp( X )
h(t | X 1 ) h0 (t ) exp( X 1 )
exp(( X 1 X 0 ) )
h(t | X 0 ) h0 (t ) exp( X 0 )
Parameter estimation – based on partial likelihood function
k
exp( X [ j ] )
L
j 1 lR exp( X l )
j
where X [ j ] denotes the covariate vector for the observation which actually experience d
the event at t j ; R j denotes the risk set at time t j ; k denotes dictinct event time s.
16. Cox PH Model
Effect of treatment vs. control (X=1 vs. X=0)
ˆ
HR exp( )
ˆ is
exp( ) the relative odds of observations from the treatment group,
relative to observations from the control group. An intuitive way of
understanding the influence of covariates on the hazard
Weibull model and proportional hazard
If the shape parameter does not change but the scale parameter is influenced by
the covariates, Weibull model implies the assumption of proportional hazard
holds.
Let exp( X ) in the Weibull Model, we have
h(t | X 1 ) exp( X 1 )t 1
1
exp(( X 1 X 0 ) )
h(t | X 2) exp( X 0 )t
17. Accelerate Failure Time Model
Accelerated failure time model (AFT)
A parametric model that describes covariate effects in terms of
survival time instead of relative hazard as Cox PH model. A
distribution has a scale parameter.
Log-logistic distribution
Other distributions, such as Weibull distribution Gamma distribution, etc.
Assumption:
The influence of a covariate is to multiply the predicted time to event (not
hazard) by some constant. Therefore, it can be expressed as a linear
model for the logarithm of the survival time.
Model:
S (t | X 1 ) S (t | X 2 ) where is the accelerati factor
on
log(t ) X
Weibull distribution and AFT
1 1
Assume : exp( X ), we have : log( t ) X
1/
18. Frailty Model
Model Assumption:
h j (t | X i , j ) h0 (t ) j exp( X j )
It is assumed that the frailty factor j follow a distribution (such as Gamma
and inverse Gaussian) with mean of 1 and an unknown variance that can be
specified by a parameter.
Frailty model is usually used to a population that are likely to have a
mixture of hazards (with heterogeneity). Some subjects are more
failure-prone so that more „frail‟.
A random effect model - to count for unmeasured or unobserved
„frailties‟.
Weibull Model:
For Weibul l Model, with a simple gamma frailty assumption , ~ g (1 / , ), we have :
h(t ) (t ) 1 S (t ) , where S (t ) 1 t
1 /
19. Non-Parametric Approach
Kaplan-Meier survival curve
The approach was published in 1958 by Edward L. Kaplan and Paul Meier in
their paper, “Non-parametric estimation from incomplete observations”. J. Am.
Stat. Assoc. 53:457-481. Kaplan and Meier were interested in the lifetime of
vacuum tubes and the duration of cancer, respectively.
Also called product limit method, since
d
S (t ) 1 i
ˆ
ti t ni
where d i is number of events at time ti and ni is the number of subjects at risk
just prior to time t i .
Confidence interval: Kalbfleisch and Prentice (2002) suggested using:
ˆ ˆ
V log( log( S (t )))
1
ˆ
(log( S (t )) 2
n (n
di
di )
ˆ
to get a confidence for log( log( S (t ))).
ti t i i
ˆ
The confidence interval for s (t ) can be derived accordingly.
20. Non-Parametric Approach
Log-Rank test is used to test the equality of two survival
functions. For comparing two survival curves, we have:
Z
j
(o1 j e1 j )
j
v1 j
Z 2 ~ 1
2
v1 j is estimated based on a hypergeometric distribution.
21. Example 1
Example 1: Field survival data can be used to
further evaluate product quality and may indicate
possible quality related issues. The hazard
function for hard disk drive field returns (or Weibull fit
failures) shows a significant peak at early life
time.
Lognormal fit
Commonly used parametric distribution models
such as Weibull, Lognormal, or Logistic model
fit such a hazard function poorly. Therefore,
Kaplan-Meier and Log-Rank test are used to Logistic fit
describe survival functions and evaluate the
effects of two interested factors on drive‟s field
survivals, respectively.
22. Example 1
In addition, field survival data is observational. Propensity score matching is
applied to balance out possible effect from other factors (covariates). Both
before and after matching results are presented here.
Chi-
Test Chi-Square DF ProbChiSq Test Square DF ProbChiSq
Matched Sample
Log-Rank 138.5724 1 <.0001 Log-Rank 1.2565 1 0.2613
Original Data
Description HazardRatio WaldLower WaldUpper Description HazardRatio WaldLower WaldUpper
GROUP1 vs. GROUP2 2.287 1.971 2.653 GROUP1 vs. 1.151 0.643 2.060
GROUP2
23. Example 2
This is an example to demonstrate Cox PH model
application. The time to event is the disease free
time for a Acute Myelocytic Leukemia (AML) patient
after a special treatment. It is interested to evaluate
if the disease free time after the treatment may vary
by gender and by age.
Obs Group gender age Time Status
1 AML-Low Risk M 24 3395 0
2 AML-Low Risk F 26 3471 0
3 AML-Low Risk F 26 3618 0
4 AML-Low Risk M 27 3286 0
5 AML-Mediate Risk F 29 3034 0
6 AML-Mediate Risk F 31 3676 0
7 AML-Low Risk M 31 2547 0
8 AML-Low Risk M 32 3183 0
9 AML-High Risk F 32 4123 0
10 AML-Low Risk M 33 2569 0
11 AML-Low Risk M 33 2900 0
12 AML-Low Risk F 33 2805 1
13 AML-Low Risk M 34 3691 0
14 AML-Low Risk F 34 3179 0
15 AML-Low Risk F 34 2246 0
16 AML-High Risk F 34 3328 0 Test of Equality over Strata
17 AML-High Risk F 35 2640 0 Test Chi-Square DF Pr >Chi-Square
18 AML-Low Risk M 39 1760 1
… … … … … Log-Rank 26.9998 5 <.0001
273 AML-High Risk M 74 16 1
Part of the data used in this example is from an
example published by SAS
24. Example 2
• SAS codes
proc phreg data=Example2;
class gender group;
model Time*Status(0)=age group gender
/selection=stepwise;
run;
Analysis of Maximum Likelihood Estimates
Parameter DF Paramete Standard Chi-Square Pr > ChiS Hazard
r Error q Ratio
Estimate
age 1 0.15180 0.01229 152.5961 <.0001 1.164
Group AML-High 1 0.46243 0.19063 5.8844 0.0153 1.588
Risk
Group AML-Low 1 -0.18436 0.20569 0.8034 0.3701 0.832
Risk
Summary of Stepwise Selection
Step Effect DF Number Score Wald Pr > ChiSq
In Chi-Square Chi-Square
Entered Removed
1 age 1 1 169.3010 <.0001
2 Group 2 2 13.1022 0.0014 Test of Equality over Strata
Test Chi-Square DF Pr >Chi-Square
The modeling result suggests that the effect of gender on
Log-Rank 17.1657 2 <.0002
survival function after the transplant is not statistically significant,
but the effects of age and severity group are significant.
25. Discussion – Parametric Models
Nice properties
Efficient data reduction – a function with a few parameters completely
describes a survival pattern.
Enable Standardized comparison – evaluation and comparison based on
statistics such as MTTF
Prediction into future possible
Possible issues
Assumptions
Non-informative censoring
Parametric distribution
Exponential family – flexible enough?
One vs. multiple distributions – three Weibull distributions for describing a bathtub hazard?
How confident we are about future survival path?
Estimation
Distribution – usually non symmetric
Sample size and time period covered by observations
Censoring
26. Discussion – Cox PH Model
Nice properties:
Parametric distribution assumption is not needed.
Easy to evaluate or test the hypotheses about the effect of a covariate on survival
Very popular in clinical trail analysis and outcome studies
Possible issues:
Proportional hazard – a strong assumption
When violated, stratified or extended Cox models may be used.
Tests of the assumption
log(-log(S(t))) plot
Including interactions with time in the model
Scaled Schoenfeld residuals plot
Estimation
Censored observation – not informative
Similar issues as seen in a multivariate regression model
27. Discussion – Non-Parametric Approach
Nice properties:
Distribution free
Graphical and intuitive
Describe well observed survival
Possible issues
Not continuous
Estimates can be biased when improperly stratified– For example,
survival function estimates on the tail can be poor.
Smoothing is usually needed when estimating hazard function
Not informative in terms of future survival function
In cases with cross survival or hazard curves, Log-Rank test is not
appropriate.
28. Discussion – Estimation Improvement
Bayesian based survival analysis approaches
Introducing prior knowledge to improve parameter
estimation
Application of multiple imputation to survival
analysis
May reduce the effect of censored observations.
The availability of large historical observations may be
informative to the imputation.
29. Summary
Survival analysis – has found its applications in many fields. It can be powerful in
providing insightful information to evaluate a product reliability, monitoring field
quality, assisting in making warranty policy, and validating new drug efficacy, etc.
Parametric distribution based approach would be the most popular survival
analysis approach in reliability engineering while Cox PH model and non-
parametric approach are usually favored in biostatistical survival analysis.
Each approach comes with its own assumptions and is designed to meet a
specified purpose. Validation of these assumptions should always be conducted to
ensure the appropriate applications of an approach.
Censored data – a major characteristic for survival data that contributes to the
uniqueness of survival data analysis and possible issues in model estimation. It
should always be kept in mind when designing related experiments and analyzing
survival data.