lecture7: survival analysis - university of southampton · lecture7: survival analysis...
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Lecture7: Survival Analysis
Lecture7: Survival Analysis
Antonello Maruotti
Lecturer in Medical Statistics, S3RI and School of MathematicsUniversity of Southampton
Southampton, 6 February 2013
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Lecture7: Survival Analysis
Outline
Introduction
Basic definitions
The hazard
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Lecture7: Survival Analysis
Introduction
A couple of questions and...
I What makes survival data so special that their analysis needsa special treatment, even as long as a one-term course?
I Why isn’t it simply covered as a sub-topic in, let’s say,regression analysis?
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Lecture7: Survival Analysis
Introduction
...a clarification
I Survival data subsume more than only times from birth todeath for some individuals.
I Analysis of duration data, that is the time from a well-definedstarting point until the event of interest occurs.
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Lecture7: Survival Analysis
Introduction
Examples
I how long patients survived after diagnosis or treatment
I the length of unemployment spells
I how long a marriage lasts
I how long PhD students need to finish writing their theses
I and more...
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Lecture7: Survival Analysis
Introduction
Features
I Survival data result from a dynamic process and we want tocapture these dynamics in the analysis properly.
I The observation scheme for duration data can be rathercomplex, leading to data that are somehow cut.
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Lecture7: Survival Analysis
Basic definitions
The basic functions
In the following we will assume that time is running continuously,and we therefore will describe duration by a continuous randomvariable, denoted by T .
I T ≥ 0
I f (t) ⇒ density function
I F (t) ⇒ cumulative density function (cdf)
I S(t) ⇒ survival function
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Lecture7: Survival Analysis
Basic definitions
Recall that...
I The density function f (t) describes how the total probabilityof 1 is distributed over the domain of T .
I The function f (t) itself is not a probability and can takevalues bigger than 1. But still one can derive basic propertiesfrom looking at the density.
I For regions where the density has large values the area underthe curve over an interval of given length will be larger ascompared to an interval of same length where the density islower.
I Regions over which the density is high are regions where weexpect to observe more data points than in regions with lowdensities.
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Lecture7: Survival Analysis
Basic definitions
Recall that...
I The cdf F (t) is defined as F (t) := P(T ≤ t) which can becomputed from the density as
F (t) =
∫ t
0f (s)ds
.
I A cdf is an increasing function, even strictly increasing if thedensity f (t) > 0 everywhere.
I F (0) = 0 and limt→∞ F (t) = 1.
I There is a one-to-one link between f (t) and F (t) asF ′(t) = f (t). Knowing one of the functions means, at least inprinciple, knowing the other (you may have to take thederivative or perhaps solve an ugly integral).
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Lecture7: Survival Analysis
Basic definitions
Recall that...
I Instead of looking at the cdf, which gives the probability ofsurviving at most t time units, one prefers to look at survivalbeyond a given point in time. This is described by the survivalfunction S(t):
S(t) = P(T > t) = 1− P(T ≤ t) = 1− F (t)
I Consequently, S(t) starts at 1 for t = 0 and then declines to 0for t → ∞.
I It should be obvious that knowing any one of f (t), F (t) andS(t) allows to derive the other two functions.
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Lecture7: Survival Analysis
Basic definitions
To summarize
Pr(a ≤ T ≤ b)
All the three functions introduced so far allowed to describe, in oneway or another, how the survival times are distributed over thepotential range.
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Lecture7: Survival Analysis
The hazard
The dynamic process
I Density, cdf and survival function look at the marginaldistribution
I Conditioning on the survival experience so far, we have
Pr(t < T ≤ t +∆t | T > t)
I Defining the Hazard Rate
h(t) = lim∆t→0
Pr(t < T ≤ t +∆t | T > t)
∆t
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Lecture7: Survival Analysis
The hazard
The hazard in more details
The basic information in the hazard is, first of all, its qualitativebehavior.
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Lecture7: Survival Analysis
The hazard
Some useful identities
I h(t) = f (t)S(t) ⇒ f (t) = h(t)S(t)
I h(t) = [− log S(t)]′
I S(t) = exp{−∫ t0 h(s)ds
}I Define the cumulative hazard H(t)
H(t) =
∫ t
0h(s)ds ⇒ S(t) = exp{−H(t)}or log S(t) = −H(t)
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Lecture7: Survival Analysis
The hazard
By using the definition of conditional probabilities
Pr(t < T ≤ t +∆t | T > t) =Pr([t < T ≤ t +∆t] ∩ [T > t])
Pr(T > t)
=Pr(t < T ≤ t +∆t | T > t)
Pr(T > t)
It may be helpful to sketch this relation graphically
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Lecture7: Survival Analysis
The hazard
An example
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Survival Analysis: Non Parametric Estimation
Survival Analysis:Non Parametric Estimation
Antonello Maruotti
Lecturer in Medical Statistics, S3RI and School of MathematicsUniversity of Southampton
Southampton, 6 February 2013
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Survival Analysis: Non Parametric Estimation
Outline
General Concepts
Non Parametric Estimation (no censoring)
Non Parametric Estimation (including censoring)
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Survival Analysis: Non Parametric Estimation
General Concepts
Few remarks before starting
I Each subject has a beginning and an end anywhere along thetime line of the complete study.
I In many clinical trials, subjects may enter or begin the studyand reach end-point at vastly differing points.
I Each subject is characterized by
1. Survival time2. Status at the end of the survival time (event occurrence or
censored)3. The study group they are in.
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Survival Analysis: Non Parametric Estimation
General Concepts
Censoring
I The total survival time for that subject cannot be accuratelydetermined.
I The subject drops out, is lost to follow-up, or required data arenot available
I The study ends before the subject had the event of interestoccur, i.e., they survived at least until the end of the study,
I There is no knowledge of what happened thereafter.
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Survival Analysis: Non Parametric Estimation
General Concepts
Censoring
I Right censoring: the period of observation expires, or anindividual is removed from the study, before the event occurs.
I Left censoring: the initial time at risk is unknown.
I Interval censoring: both right and left censored
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (no censoring)
Estimation
I Random variable T with cdf F (t)
I S(t) = 1− F (t)
I With no censored observations:
S(t) = 1− F (t)
I To estimate F (t) at each time t:I data t1, . . . , tnI parameter of interest θ = F (t) = Pr(T ≤ t)
I θ = #obs.≤tn =
∑ni=1 I(0,ti )
(t)
n
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (no censoring)
Confidence intervals
I Confidence interval for F (t):
θ ∓ zα/2
√θ(1− θ)
n
I Confidence interval for S(t):
1− θ ∓ zα/2
√θ(1− θ)
n
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (no censoring)
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
Estimation
I To estimate the proportions θiI ni = # of individuals at risk at the beginning of the i-th
intervalI di = # of individuals experiencing the event
θi =ni − di
ni
I Kaplan Meier estimator
S(t) =∏i :ti≤t
ni − dini
I It reduces to 1− F (t) with no censored observations
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
Example
Subject Group Survival # surviving Event # surviving Cumulativetime at risk after event survival
in the interval rate
1 1 1 6 1 5 1× 56
2 1 2 5 1 4 1× 56× 4
53 1 3 4 1 3 1× 5
6× 4
5× 3
44 1 4 3 1 2 1× 5
6× 4
5× 3
4× 2
35 1 4.5 2 1 1 1× 5
6× 4
5× 3
4× 2
3× 1
26 1 5 0
7 2 0.5 6 1 5 1× 56
8 2 0.75 5 1 4 1× 56× 4
59 2 1 4 1 3 1× 5
6× 4
5× 3
410 2 1.5 0
11 2 2 2 1 1 1× 56× 4
5× 3
4× 1
212 2 3.5 1 1 0 1× 5
6× 4
5× 3
4× 1
2
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
Example
1
1
0.00
0.25
0.50
0.75
1.00
0 1 2 3 4 5analysis time
group = 1 group = 2
Surviving functions by group
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
Understanding KM analysis
I The lengths of the horizontal lines represent the survivalduration for that interval.
I The interval is terminated by the occurrence of the event ofinterest.
I The vertical distances between horizontal lines illustrate thechange in the cumulative probability.
I The KM curve is a step-wise estimator, not a smooth function.
I What about estimate of point survival?
I Which is the effect of censoring?
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
Comparison of KM estimates
I It is simple to visualize the difference between two survivalcurves.
I The difference must be quantified in order to assess statisticalsignificance.
I MethodsI log-rank test ⇒ Most sensitive to consistent differenceI Wilcoxon test ⇒ Most sensitive to early differencesI hazard ratio ⇒ gives relative event rate in the groups
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
Log-Rank test: Example
Time Group 1 Group 2 Group 1 Group 2 Group 1 Group 2Event Event At Risk At Risk Expected Expected
0,5 0 1 6 6 0,50 0,500,75 0 1 6 5 0,55 0,451 1 1 6 4 1,20 0,802 1 1 5 2 1,43 0,573 1 0 4 1 0,80 0,203,5 0 1 3 1 0,75 0,254 1 0 3 0 1,00 0,004,5 1 0 2 0 1,00 0,00
The logrank test statistic is constructed by computing the observedand expected number of events in one of the groups at each
observed event time and then adding these to obtain an overallsummary across all time points where there is an event.
χ2 = 3.07; p − value = 0.0798
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Survival Analysis: Non Parametric Estimation
Non Parametric Estimation (including censoring)
What to avoid
I Compare mean survival ⇒ Censoring makes this meaningless
I Overinterpret the tail of a survival curve ⇒ There aregenerally few subjects in tails
I Compare proportions surviving at a fixed time ⇒ Fine fordescription, not for hypothesis testing
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Cox Proportional Hazards Regression for Survival Data
Cox Proportional Hazards Regressionfor Survival Data
Antonello Maruotti
Lecturer in Medical Statistics, S3RI and School of MathematicsUniversity of Southampton
Southampton, 6 February 2013
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Cox Proportional Hazards Regression for Survival Data
Outline
Some simple distributions
The Cox PH model
Model diagnostics
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Cox Proportional Hazards Regression for Survival Data
Some simple distributions
Survival distributions
I Survival analysis focuses on the distribution of survival times.
I Although there are well known methods for estimatingunconditional survival distributions, most interesting survivalmodeling examines the relationship between survival and oneor more predictors.
I In principle, every distribution on R+ can serve to characterizesurvival data.
I Constant hazardI Gompertz distributionI Weibull distribution
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Cox Proportional Hazards Regression for Survival Data
Some simple distributions
Survival distributions
Modeling of survival data usually employs the hazard function
h(t) = lim∆t→0
Pr(t < T ≤ t +∆t | T > t)
∆t
I Constant hazard: h(t) = λ ⇒ S(t) = e−λt
I Gompertz: h(t) = aebt , a > 0, b > 0 ⇒ S(t) = eab[1−ebt ]
I Weibull: h(t) = λata−1 ⇒ S(t) = e−λta
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Cox Proportional Hazards Regression for Survival Data
The Cox PH model
Regression-like model
A parametric model based on the exponential distribution may bewritten as
log hi (t) = β0 + β1xi1 + · · ·+ βpxip
log-baseline hazard
The constant β0 represents a kind of log-baseline hazard
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Cox Proportional Hazards Regression for Survival Data
The Cox PH model
The Cox model
The Cox model leaves the baseline hazard functionβ0(t) = log h0(t) unspecified
log hi (t) = β0(t) + β1xi1 + · · ·+ βpxip
The model is semiparametric, because while the baseline hazardcan take any form, the covariates enter the model linearly.
I The baseline hazard does not depend on covariates, but onlyon time
I The covariates are time-constant
I Proportional hazard assumption follows
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Cox Proportional Hazards Regression for Survival Data
The Cox PH model
The hazard ratio
For two observations i and j , the hazard ratio
hi (t)
hj(t)=
h0(t) exp(β1xi1 + · · ·+ βpxip)
h0(t) exp(β1xj1 + · · ·+ βpxjp)
=exp(β1xi1 + · · ·+ βpxip)
exp(β1xj1 + · · ·+ βpxjp)
= exp
(p∑
l=1
βl(xil − xjl)
)
is independent of time t. Consequently, the Cox model is aproportional hazards model.
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Cox Proportional Hazards Regression for Survival Data
The Cox PH model
The hazard ratio: an example
I Only one covariate: TreatmentI xi = 1 ⇒ PlaceboI xj = 0 ⇒ Treatment
I Hazard ratio is then exp(β1)
I We expect that hazard is larger in the placebo group, i.e. thehazard ratio is expected grater than 1.
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Cox Proportional Hazards Regression for Survival Data
The Cox PH model
Time-constant covariates
I Not changing over time (e.g. gender)
I Values are set at time t = 0
I Variables unlikely to change are often consideredtime-constant
I Other variables are sometimes treated as time independent
I Time-dependent covariates are allowed, but PH assumptionsis not satisfied (an extended Cox model is needed)
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Cox Proportional Hazards Regression for Survival Data
The Cox PH model
Advantages
I Robustness
I Because of the model form, the estimated hazards are alwaysnon-negative
I We can estimate fixed effects and compute the hazard ratioeven though the baseline hazard is left unspecified
Can we use a logistic model?
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Cox Proportional Hazards Regression for Survival Data
Model diagnostics
Checking proportional hazards
I Test and graphical diagnostic for PH may be based on scaledSchoenfeld residuals
I Influential observations
I Nonlinearity
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Survival Analysis - Stata
Survival Analysis - Stata
Antonello Maruotti
Lecturer in Medical Statistics, S3RI and School of MathematicsUniversity of Southampton
Southampton, 6 February 2013
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Survival Analysis - Stata
Outline
Introduction
Coding
Kaplan-Meier
PH Cox model
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Survival Analysis - Stata
Introduction
Aim
Illustrate how to use Stata to
I prepare survival data for analysis
I estimate hazard and survival functions
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Survival Analysis - Stata
Introduction
Data manipulationA manipulation of the data is needed to facilitate summary andanalysis.
help st
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Survival Analysis - Stata
Introduction
Assumptions
I Continuous time survival data
I Single failure data, i.e. one record per unit
I No complications such as truncation and/or missing values
I Data do not need to be weighted
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Survival Analysis - Stata
Introduction
Data structure
Data have a very simple structure
I One row per unit (e.g. subject)
I The survival time and the censoring status must be includedas variables (1= failure, 0 = otherwise)
I Covariates (explanatory variables) could be included
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Survival Analysis - Stata
Introduction
Data description
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Survival Analysis - Stata
Coding
stset
stset declares the data in memory to be st dataI Main
I Time variable ⇒ survival timeI Failure variable ⇒ censoring status
I OptionsI Origin time expression sets when a subject becomes at riskI Enter time expressions specifies when a subject first comes
under observationI Exit time expression specifies the latest time under which the
subject is both under observation and at risk.
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Survival Analysis - Stata
Coding
stset in practice
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Survival Analysis - Stata
Coding
stset in practice
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Survival Analysis - Stata
Coding
stset: example
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Survival Analysis - Stata
Coding
Using stsetNew variables in the data, why? Which is your meaning? Shouldyou use them?
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Survival Analysis - Stata
Coding
Using stset
I st is a binary variable indicating cases included (1) orexcluded (0) from the analysis
I d is a censoring indicator
I t is the survival time
I t0 is the time at which units are observed to be at risk
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Survival Analysis - Stata
Coding
Using stset
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Survival Analysis - Stata
Coding
Summary statisticsYou must stset your data before using
I stdescribe produces a summary of the st data
I stsum summarizes survival-time data
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Survival Analysis - Stata
Kaplan-Meier
Kaplan-Meier
I Simple single-spell type
I Right censoring
I No left censoring (truncation)
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Survival Analysis - Stata
Kaplan-Meier
sts
Survival times are treated as observations on a continuous variable
I sts list
I sts graph
I sts test
I sts generate
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Survival Analysis - Stata
Kaplan-Meier
sts list
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Survival Analysis - Stata
Kaplan-Meier
sts list: example
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Survival Analysis - Stata
Kaplan-Meier
sts graph
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Survival Analysis - Stata
Kaplan-Meier
sts graph: example
0.00
0.25
0.50
0.75
1.00
0 200 400 600 800 1000analysis time
Kaplan−Meier survival estimate
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Survival Analysis - Stata
Kaplan-Meier
sts graph: example
0.00
1.00
2.00
3.00
0 200 400 600 800 1000analysis time
Nelson−Aalen cumulative hazard estimate
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Survival Analysis - Stata
Kaplan-Meier
sts graph: example
.001
.002
.003
.004
.005
0 200 400 600 800analysis time
Smoothed hazard estimate
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Survival Analysis - Stata
Kaplan-Meier
sts graph: stratification
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Survival Analysis - Stata
Kaplan-Meier
sts graph: stratification
11
1
3
2
1
21
22
112
1
1
22
1
3
11
1
2
1
11
2
11 3
211
22 3
11 1
111
1
0.00
0.25
0.50
0.75
1.00
0 200 400 600 800 1000analysis time
sex = 1 sex = 2
Kaplan−Meier survival estimates
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Survival Analysis - Stata
Kaplan-Meier
sts test
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Survival Analysis - Stata
Kaplan-Meier
sts test
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Survival Analysis - Stata
PH Cox model
stcox
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Survival Analysis - Stata
PH Cox model
stcox: options for model checking
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Survival Analysis - Stata
PH Cox model
stcox: example
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Survival Analysis - Stata
PH Cox model
stphplot: model checking
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Survival Analysis - Stata
PH Cox model
stphplot: model checking
−2
02
4−
ln[−
ln(S
urvi
val P
roba
bilit
y)]
2 3 4 5 6 7ln(analysis time)
sex = 1 sex = 2
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Survival Analysis - Stata
PH Cox model
estat phtest: model checking
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Survival Analysis - Stata
PH Cox model
estat phtest: model checking
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