Survival Analysis: Overview of Parametric, Nonparametric and Semiparametric approaches

Survival Analysis: Overview of Parametric, Nonparametric and Semiparametric approaches
SAS Global Forum 2010
Statistics and Data Analysis
Paper 252-2010
Survival Analysis: Overview of Parametric, Nonparametric and Semiparametric approaches
and New Developments
Joseph C. Gardiner, Division of Biostatistics, Department of Epidemiology,
Michigan State University, East Lansing, MI 48824
ABSTRACT
Time to event data arise in several fields including biostatistics, demography, economics, engineering and
sociology. The terms duration analysis, event-history analysis, failure-time analysis, reliability analysis, and transition
analysis refer essentially to the same group of techniques although the emphases in certain modeling aspects
could differ across disciplines. SAS® procedures LIFETEST, LIFEREG, PHREG, RELIABILITY, and
QLIM have different capabilities for analyzing duration data. Methods include Kaplan-Meier estimation,
accelerated life-testing models, and the ubiquitous Cox model. Recent developments in SAS extend their
reach to include analyses of multiple failure times, recurrent events, frailty models, Markov models and use of
Bayesian methods. We present an overview of these methods with examples illustrating their application in
the appropriate context.
INTRODUCTION
Survival Analysis is a collection of methods for the analysis of data that involve the time to occurrence of some
event, and more generally, to multiple durations between occurrences of different events or a repeatable
(recurrent) event. From their extensive use over decades in studies of survival times in clinical and health related
studies and failures times in industrial engineering (e.g., reliability studies), these methods have evolved to
special applications in several other fields, including demography (e.g., analyses of time intervals between
successive child births), sociology (e.g., studies of recidivism, duration of marriages), and labor economics
(e.g., analysis of spells of unemployment, duration of strikes). Books and monographs continue to be
published in this area that attest to its rich methodology and versatility. See references for a partial list.
The typical context in biostatistics is a data gathering process that records an event time T measured from a
specified time origin in a sample of patients. However, when follow up ends the event may not have occurred
in some patients resulting in right censored event times. What we know is that T exceeds U, where U is the follow
up time. The survival times of these patients are censored, and U is called the censoring time. Censoring will also
occur if say a patient dies from causes unrelated to the endpoint under study, or withdraws from study for
reasons not related to the endpoint. Such patients are lost to follow up. When there is a competing risk for the
endpoint of death, it is important to ascertain whether death is due to the cause under study. Other forms of
censoring are possible depending on the type of study. For example, if the true event time T is not observed
but is known to be less than or equal to V, we have a case of left censoring. If all that is known about T is that it
is somewhere between two times U and V (U<V), we say it is interval censored.
Generally, one records a number of covariates z (e.g., age, gender, comorbidity, treatment assignment etc.)
whose influence on the distribution of T is of interest. Due to the longitudinal feature of the data gathering
process some covariates are time-invariant while others could be time-varying. The latter may arise from
intermediate events that influence the distribution of T. Multi-state models provide a means of analyzing data
with multiple event times.
Despite our best intention in recording all covariates relevant to a specific analysis, we might encounter
heterogeneity in patient samples that cannot be explained by the observed covariates alone. Unobserved
heterogeneity is likely in observational studies. Frailty models and finite-mixture models can be very informative in
this regard.
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Statistics and Data Analysis
For time-fixed covariates z, the survival distribution S( t | z ) = P [T > t | z ] = exp( − H ( t | z )) is expressed in
t
terms of the cumulative hazard H ( t | z ) = ∫ h ( u | z )du where h ( t | z ) denotes the hazard function. (The
0
relationship between S and H is more subtle when the distribution T is not continuous). We may interpret
h ( t | z )∆t as a conditional probability because h ( t | z )∆t ≈ P [T < t + ∆t |T ≥ t , z ] . For this reason h ( t | z ) is
often referred to as the instantaneous risk of the event happening at time t. Other useful summary quantities in
survival analysis are (suppressing dependence on z):
=
µ E=
(T )
Mean survival time,
∫
∞
0
S( t )dt
(min(T , L))
Mean survival restricted =
to time L, µ L E=
∫
L
0
S( t )dt
t p inf{t > 0 : S( t ) ≤ 1 − p } , 0<p<1
Percentiles of survival distribution, =
∞
Mean residual life at time t, r ( t )= E(T − t |T > t )= {S( t )}−1 ∫ S( u )du
t
(
t
)
Just as H determines S, the relationship between
r and S is S( t ) r (0){r ( t )}−1 exp − ∫ {r ( u )}−1 du . Because
=
0
survival data are often quite skewed with long right tails, the restricted mean survival or the median survival
time are generally preferred as summary statistics.
The objectives in a survival analysis may include estimation of one or more of these statistics at specified
covariate profiles and quantifying the influence of z (e.g., treatments, demographics) on survival. These goals
can be achieved through modeling how z impacts T directly or indirectly through for example, the hazard
h ( t | z ) . However, an initial analysis would typically employ nonparametric methods to estimate the survival
function and summary statistics, and a comparison across several groups or sub-populations.
I.
NONPARAMETRIC ANALYSIS
Procedure LIFETEST is the mainstay of nonparametric survival analysis. For right censored data it computes
the Kaplan-Meier (product limit) estimator of the survival distribution S, its quartiles and the restricted mean
µ L . It provides tests of comparison of the survival distribution across two or more populations including
adjustment of the p-value for multiple comparisons if warranted, and tests of trend for ordered alternatives.
Using ODS graphics with the PLOTS= option can produce exquisite graphs for estimates for S, its
derivatives, and pointwise confidence intervals or a confidence band for S.
ILLUSTRATIVE EXAMPLE 1
McGilchrist & Aisbett (1991) describe a study in 38 kidney dialysis patients where the time in days to
infection at the catheter insertion point was recorded. Each patient (i) has two times. After the first insertion
of the catheter, the time to infection Ti 1 if observed is recorded. If the catheter is removed for any reason
other than infection, Ti 1 is considered right censored at the removal time U i 1 . If infection occurs, the
catheter is removed, the patient is treated and cleared of the infection and then after some time, a second
catheter is inserted. The second time to infection Ti 2 , measured from the time of second insertion, is either
observed or right censored if the catheter removed at time U i 2 for any reason other than infection.
The sample data are {( Xi 1 , δ i 1 , Xi 2 , δ i 2 , z i ) : 1 ≤ i ≤ n } where X=
min( Tij , U ij ), δ=
[ Tij ≤ U ij ] j=1,2 and δ ij
ij
ij
denotes the event indicator. Censoring is assumed independent of infection times. The data set KIDNEY has
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two records per patient with variables TIME, FAIL and covariates AGE (=average age between the
insertions of the catheter), and GENDER. PATIENT and INSERT identify patient and the two infection
times. Nonparametric methods use the accumulating count of events up to time t,
N j=
(t )
a.
∑
n
t, Y j ( t )
[ X ij ≤ t ,=
δ ij 1] and the number at risk at time=
i =1
∑
n
[ X ij ≥ t ] .
i =1
ESTIMATION OF SURVIVAL CURVES
The following syntax will produce the product-limit estimates of infection time by insertion for females and
males. Use of formats when applicable makes the output display more readable.
proc format;
value gender 0='male' 1='female';
value insert 1='first' 2='second';
run;
ods graphics on;
proc lifetest data=kidney
plots=survival(nocensor cb=hw cl strata=panel atrisk=0 to 600 by 50);
strata insert gender;
time time*fail(0);
format gender gender. insert insert.;
run;
ods graphics off;
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The NOCENSOR option suppresses the display of censored times (with the symbol + ); the CB=HW option
displays the simultaneous Hall-Wellner 95% confidence band for the survival curves; CL displays the lower
and upper limits of the pointwise 95% confidence interval, and at the foot of each plot the ATRISK option
shows the number of patients at risk of infection at specified times. STRATA=panel requests display of the
four individual plots in a 2×2 panel, instead of the default overlay. For the first catheter insertion it appears
that females have a longer infection-free duration than males. Stratified by INSERT the default logrank test
for comparing the time to infection distributions for males and females is requested by
strata insert/group=gender test=logrank;
The test is significant (p=.0009). However, when comparisons are made separately at each catheter insertion
this significance is seen in the figure below for the first insertion (left panel) and not the second insertion
(right panel) (p=.2343). Pointwise 95% confidence intervals are also displayed.
b. CUMULATIVE HAZARD AND KERNEL-SMOOTHED HAZARD
The option method=pl nelson in the LIFTEST statement adds to the default table of Kaplan-Meier
estimates the Nelson-Aalen estimate of the cumulative hazard for each stratum specified in the strata
t
statement. With strata defined by INSERT, the estimate is Hˆ ( t ) = {Y ( u )}−1 dN ( u ) , j=1, 2. It could be
j
∫
0
j
j
used to obtain an estimator of the survival curve Sˆ j=
( t ) exp( −Hˆ j ( t )) . In addition, the PLOTS option in the
syntax below produces an estimate of the kernel-smoothed hazard function using the Epanechnikov kernel
and bandwidth of 35 days. Optimal selection of a bandwidth by minimizing the mean integrated squared
error was not feasible with this small data set. The risk of infection appears to first increase and then taper
off, but another increase is seen for the second catheter insertion. The sharp rise towards the end is
somewhat typical in this context. However, another choice of kernel or bandwidth might depict a different
pattern. Larger bandwidth produces smoother curves.
ods graphics on;
proc lifetest data=kidney method=pl nelson
plots(only)=hazard(kernel=e bw=35);
strata insert;
time time*fail(0);
format insert insert.;
run;
ods graphics off;
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II.
Statistics and Data Analysis
PARAMETRIC MODELS-ACCELERATED FAILURE TIME MODEL
Procedures LIFEREG and RELIABILITY can be used for inference from survival data that have a
combination of left, right and interval censored observations. The accelerated failure time (AFT) model is
specified by logT= µ + σε with location and scale parameters µ, σ, respectively. Covariate effects are
modeled by µ = z′β1 , and additionally if plausible heteroscedasticity by log σ = z′β 2 By specifying a
distribution for the random variable ε, independent of z, one induces a distribution on T. Estimation of
parameters ( β1 , β 2 ) is via maximum likelihood. Survival distributions within the AFT class are the
exponential, Weibull, lognormal and loglogistic. All distributions have the functional form S( t ) = S0 (( t / α )γ )
where σ= γ −1 , µ= log α , α > 0, γ > 0 , and S0 is a known survival distribution
SAS also allows the generalized gamma (GG) distribution which has an additional shape parameter. Here ε
has the one-parameter log-gamma distribution with shape parameter k>0, i.e., S0 is the gamma survival
distribution with shape parameter k. A re-parameterization suggested by Prentice (1974), recasts the GG in
T z′β1 + σ 0 Z where k = δ −2 , σ 0 = σδ and the distribution of Z is defined for all δ≠0. In
the AFT form log=
the limit as δ→0, Z converges to the standard normal. SAS calls δ the shape and σ 0 the scale of the GG.
Defined in this way, GG returns three special cases: with δ=0 the log normal; with δ=1 the Weibull; with
δ=1 and σ 0 =1 the exponential. Testing of these restrictions within the parent GG is valid under maximum
likelihood (ML) via for example, likelihood ratio and Lagrangian multiplier (score) tests.
a. FITTING PARAMETRIC MODELS
Initially we assume the within-patient times ( Ti 1 , Ti 2 ) are independent, making our sample comprise of 76
individual catheter insertions. The dist=gamma option requests fitting the GG to the model with covariates
age and gender (Table 1, column 1). Wald tests produced by default indicate that age is not significant
(p=.57), but gender is strongly significant (p<.0001). The positive estimated β-coefficient for female gender
shows that female dialysis patients have a longer infection-free time compared to male patients.
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proc lifereg data=kidney;
class gender;
model time*fail(0)=age gender/dist=gamma;
format gender gender.;
run;
Table 1: Summary of results of fitting parametric AFT models to infection times
Maximum likelihood estimate (standard error)
Parameter
Intercept
AGE
GG
Lognormal
Weibull
Exponential
Loglogistic
3.4188(0.5322)
3.4490(0.4939)
4.2916(0.5505)
4.4025(0.4971)
3.4052(0.4636)
–0.0054(0.0097) –0.0054(0.0097) –0.0042(0.0103) –0.0046(0.0094) –0.0073(0.0093)
GENDERfemale
1.3830(0.3283)
1.3269(0.3261)
0.9655(0.3248)
0.8853(0.2871)
1.5526(0.3265)
Scale
1.1863(0.1085)
1.1847(0.1077)
1.1031(0.1035)
1(fixed)
0.6793(0.0720)
Shape
–0.0473(0.3164)
0(fixed)
1(fixed)
1(fixed)
na
–2 log L
197.032
197.053
206.197
207.348
198.532
BIC
218.686
214.375
223.520
220.340
215.855
na
.887
<.0001
Shape <.001
Scale .316
na
LM test
p-value
The log-normal, Weibull and exponential models can be fitted directly by changing the dist= option (e.g.,
dist=lnormal) or by restricting the GG model’s shape and scale parameters. Then Lagrangian multiplier (LM)
1-degree of freedom chi-square tests are produced.
For lognormal: H 0 : δ = 0. Use dist=gamma noshape1 shape1=0;
For Weibull: H 0 : δ = 1. Use dist=gamma noshape1 shape1=1;
For exponential: H 0=
: δ 1,=
σ 0 1. Use dist=gamma noshape1 shape1=1 noscale scale=1;
For GG vs exponential, SAS does not produce a joint test 2 df LM test. However, in all situations except for
comparing GG to loglogistic one can perform a likelihood ratio test (LRT). The LM test and LRT are
asymptotically equivalent. In this example, compared to the GG model the simpler lognormal model is
acceptable. It also has the lowest BIC.
b. ESTIMATION OF PERCENTILES
T z′β + σε for a specified covariate profile z the 100(1−p)-th percentile t p of the
In the AFT model log=
t p exp( z′β + σ w p ) where w p is the corresponding percentile of ε . Although
=
event time T is obtained from
statistics computed via the output statement from the fitted model in LIFEREG may be used for this
purpose, an easier approach is to use PROC RELIABILITY. Suppose we want estimates and 95%
confidence intervals for the 25th, 50th and 75th percentiles for females age=44 years and males age= 49 years.
These are approximately the median ages in the data set. We add these two profiles to the data set kidney:
data covar;
input gender age @@;
datalines;
1 44 0 49
;
run;
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data kidney2;
set covar(in=one) kidney;
if one then control=1;
else control=0;
run;
The same lognormal model statistics of Table 1 column 2 are obtained using the syntax below. The
OBSTATS options produce the desired estimates. The ODS statements are added to permit some editing of
the output data set ModObstats. The control= option reduces observation-wise calculations to the six
records in ModObstats for the control variable value=1 only. Results are shown in Table 2.
ods select ModObstats;
proc reliability data=kidney2;
class gender;
distribution lognormal;
model time*fail(0)=age gender/obstats(quantiles=.25 .50 .75 control=control);
format gender gender.;
run;
Table 2: Estimates of percentiles in lognormal model
Age Gender
95% Lower 95% Upper
CL
CL
p
Estimate
44 female
0.25
44.2
30.9
63.3
44 female
0.50
98.3
69.6
138.9
44 female
0.75
218.7
148.3
322.5
49 male
0.25
10.9
6.2
19.2
49 male
0.50
24.2
13.9
42.0
49 male
0.75
53.7
30.2
95.4
The LOGSCALE statement in RELIABILITY permits modeling heteroscedasticity in the scale parameter σ .
For example logscale age gender; models log σ =
β 20 + β 21 Age + β 22 [ Gender =
female ] resulting in a
6-parameter model. It turns out that ( β 21 , β 22 ) are not significant indicating that our simpler model is
adequate. Generally, when specifying the covariates for (µ,σ) one should consider exclusion restrictions where
at least one covariate present in µ = z1′ β1 is excluded in log σ = z′2 β 2 , and vice versa. This could ensure
stability in ML estimates and estimated standard errors. Exclusions restrictions are informed by the subject
matter rather than statistical considerations.
c. JOINT MODELING OF INFECTION TIMES
Consider a joint model for the infection times ( Ti 1 , Ti 2 ) allowing correlation between them. Create one record
for both infection times, transform to the log scale y ij∗ = log Tij and create a variable UBij (upperbound) as
UBij = log X ij if δ ij = 0 (censored);=
UBij log( X ij + c ) if δ ij = 1 (infection time) where c is an arbitrary
y ij* z′ij β + u ij with ( u i 1 , u i 2 ) ~Normal (0, ρ , σ 1 , σ 2 ) , but the observed
positive constant. Our model is =
analysis variable is:
UBij if y ij* ≥ UBij
.
y ij =  *
*
 y i 1 if y ij < UBij
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The following syntax sets up the data set BIVAR with one record per patient.
proc sort data=kidney; by patient; run;
data bivar;
merge kidney(keep=insert patient time fail age gender where=(insert=1))
kidney(keep=insert patient time fail where=(insert=2)
rename=(time=time2 fail=fail2));
by patient;
drop insert;
ub=log(time+10); ub2=log(time2+10); /*c=10*/
lgtime=log(time); lgtime2=log(time2);
if fail=0 then ub=lgtime;
if fail2=0 then ub2=lgtime2;
run;
We use the same covariates (AGE, GENDER) in the two-equation model although generally covariate
specification should consider exclusion restrictions. PROC QLIM estimates the joint model by maximum
likelihood (Table 3).
proc qlim data=bivar;
class gender;
format gender gender.;
endogenous lgtime~censored(UB=UB);
endogenous lgtime2~censored(UB=UB2);
model lgtime=age gender;
model lgtime2=age gender;
run;
Table 3. Parameter estimates for joint model for infection times
Parameter
DF
Standard
Error
Estimate (Hessian)
Approx
t Value Pr > |t|
Standard
Error
(QML)
lgtime.Intercept
1
3.411765
0.702542
4.86
<.0001
0.532027
lgtime.age
1 –0.013135
0.013567
–0.97
0.3330
0.011349
lgtime.gender
female
1
1.743168
0.453233
3.85
0.0001
0.434777
lgtime.gender
male
0
0
.
.
.
.
_Sigma.lgtime
1
1.205189
0.150094
8.03
<.0001
0.117485
lgtime2.Intercept
1
3.481873
0.662375
5.26
<.0001
0.649470
lgtime2.age
1
0.004995
0.013432
0.37
0.7100
0.013230
lgtime2.gender
female
1
0.866471
0.457612
1.89
0.0583
0.506512
lgtime2.gender
male
0
0
.
.
.
.
_Sigma.lgtime2
1
1.102781
0.148801
7.41
<.0001
0.131952
_Rho
1
0.144550
0.181132
0.80
0.4249
0.209544
The Wald test for no correlation H 0 : ρ = 0 is not significant. The gender effect is strong for the first
insertion but weak for the second supporting what we had seen in our nonparametric analysis. Standard
errors are obtained from the Hessian matrix based on the second derivative of the log-likelihood. Optionally,
with covest=qml added to the PROC QLIM statement, quasi-maximum likelihood (QML) standard errors
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are obtained as a ‘sandwich’ of the Hessian and outer product (OP) matrices. QML standard errors are shown
in the last column of Table 3. To obtain standard errors from OP only use covest=op. Although
asymptotically equivalent, in finite samples the results from the three methods could differ.
d. FRAILTY MODEL
Another approach to incorporating correlation between the infection times is through a shared frailty ν i , a
random effect, via log Tij = z′ij β + ν i + σε ij . Under the assumption that ( Ti 1 , Ti 2 ) are conditionally independent
given (ν i , zi 1 , zi 2 ), ML estimation of the marginal model based on the data {( X ij , δ ij , zij ) :=
j 1, 2,1 ≤ i ≤ n} can
be carried out under assumed parametric distributions on (ν i , ε ij ). Currently there is no direct SAS procedure
to carry out the computations. However, informed by LIFEREG for suitable starting values of the model’s
parameters, PROC NLMIXED can be used to optimize the marginal likelihood. Post estimation provides
empirical Bayes (EB) estimates of ν i , ie, E(ν i |data ). The plot below shows the EB estimates and 95%
confidence limits for the 38 patients in the sample for the Weibull model with ν i ~ N ( −½σν2 , σν2 ). The frailty
effect is strong with a LRT p-value<.01, but the effect appears to be influenced by patient #21.
III.
SEMIPARAMETRIC MODEL-PROPORTIONAL HAZARDS MODEL
The workhorse of survival analysis for over three decades, the proportional hazards model (PHM) assumes
h( t |z ) = h0 ( t )exp( z′β ) where h0 is an unspecified baseline hazard function and the parameter β is unknown.
′
For time-invariant covariates, S ( t |z ) = S0 ( t )exp( z β ) where S0 is the survival function corresponding to h0 .
( t |z2 ) exp(( z1 − z2 )′β ) is constant in time.
Given two covariate profiles z1 , z2 the hazard ratio h( t |z1 ) / h=
The stratified PHM given by hk ( t |z ) = h0 k ( t )exp( z′β ) maintains the proportional hazards assumption in each
stratum k for a K-level stratification factor. For example, survival data from a multicenter clinical trial are
often analyzed with center as the stratifying variable.
In addition to analysis based on the traditional PHM, enhancements to PROC PHREG allow for several
additional data structures. These include time-dependent covariates, multiple failure times, recurrent events,
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and delayed entry or left censoring. Although many covariates of interest are assessed at t=0, for example,
age at entry, gender, race, comorbid conditions, baseline clinical measurements, we may have some covariates
measured during the period of follow-up making them time-dependent. Intermediate events that may occur
during follow-up could influence occurrence of the primary event of interest. Multiple events of different
types or recurrences of the same event are typical in longitudinal studies or in data structures that are
clustered (e.g., animals within the same litter). A unified approach to analysis of event history data has been
explicated (Anderson et al, 1993) based on the theory of multivariate counting processes.
Suppose there are K event types. Let N k ( t ) denote the number of type k events that have occurred by time t;
Yk ( t ) denotes the number of individuals at risk for the type k event just before t; and z(t) the covariate
history observed just prior to t. Conditional on the prior history (denoted by ℑt − ) the multiplicative intensity
model (MIM) is E( dN k ( t )| ℑt − ) =
Yk ( t )α k ( t |z( t ))dt where α k ( t |z( t )) = α k 0 ( t )exp( z′( t )β k ) . For a single event
type, the MIM reduces to the previously described PHM. To harness the power of PHREG to fit the MIM,
some preliminary data processing may be required to structure the event history and covariate data
appropriately to permit the correct evaluation of the at-risk sets.
a. FITTING THE PHM
Consider again the two times to infection since insertion of the catheter in 38 dialysis patients. The following
=
syntax fits the PHM to both infection
times, hk ( t |z ) h0 k ( t )exp( β1k AGE + β 2 k GENDER ) where
INSERT=k. Because ( β1k , β 2 k , k = 1, 2) may be correlated a robust covariance is requested by the
COVSANDWICH (AGGREGATE) option and all standard errors used in subsequent inference will use this
covariance. The class statement uses GLM coding. Results of maximum partial likelihood estimation are in
Table 4. We notice that the effect of gender is strong for the first infection time, with a lower infection rate
among female patients compared to male patients. The comparison for the second infection time is not
significant. These conclusions are in line with our previous nonparametric and parametric analyses.
proc phreg data=kidney covsandwich(aggregate);
id patient;
class gender insert/param=glm;
strata insert;
model time*fail(0)=age*insert gender*insert;
format gender gender. insert insert.;
run;
Table 4: Parameter estimates in PHM model for infection times
Parameter
DF
Parameter
Estimate
Standard
Error
StdErr
Ratio
ChiSquare
pvalue
age*insert
first
1
0.00964
0.01115
0.901
0.7467
0.3875
age*insert
second
1
–0.00332
0.01064
0.751
0.0971
0.7553
gender*insert
female
first
1
–1.38599
0.44621
1.062
9.6479
0.0019
gender*insert
female
second
1
–0.54276
0.60239
1.316
0.8118
0.3676
gender*insert
male
first
0
0
.
.
.
.
gender*insert
male
second
0
0
.
.
.
.
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Because we have used GLM coding in the class statement, all contrast statements shown below must be
provided in the comparative style: female vs male. For computing hazard ratios (HR) and 95% confidence
intervals use:
contrast 'HR female vs male at INSERT=1' gender*insert 1 0 -1/estimate=exp;
contrast 'HR female vs male at INSERT=2' gender*insert 0 1 0 -1/estimate=exp;
Estimate
Standard
Error
HR female vs male at INSERT=1
0.2501
0.1116
0.1043
0.5996
0.0019
HR female vs male at INSERT=2
0.5811
0.3501
0.1785
1.8925
0.3676
Contrast
95% Confidence
Limits
p-value
A forthcoming enhancement to the HAZARDRATIO statement would give the same results from a single
statement: hazardratio "Gender effect" gender/cl=wald; The label is optional.
Because the gender effect is dissimilar for the two catheter insertions, a test of equality of the gender effect is
unwarranted. However, for illustration this test of equality H 0 : β 21 = β 22 is obtained from
contrast "Same Gender Effect" gender*insert 1 -1 -1 1;
The resulting Wald test is barely significant (p=0.038).
ILLUSTRATIVE EXAMPLE 2
Data set BMT contains follow up data on 137 patients who underwent a bone marrow transplant for
treatment of acute leukemia (Klein & Moeschberger, 1997). These data have been analyzed extensively to
meet different objectives using different strategies. We focus here on two events death/relapse combined and
the event of platelet recovery when a patient’s platelets return to normal levels. It is an important indicator of
prognosis of survival. Initially following surgery all patients have depressed platelet count. Subsequently, in
120 patients recovery to normal levels was observed (PRI=1). TRETP is the recovery time. For the other 17
patients without platelet recovery (PRI=0), TRETP is set to missing. TFREEST denotes the time to
death/relapse which was observed in 83 patients (DFI=1), 67 of whom had platelet recovery. Event times
are in days from transplant. TFREEST is censored (DFI=0) if the event death/relapse has not occurred at
the end of follow-up.
b. FITTING A PHM WITH TIME-DEPENDENT COVARIATES
In the analysis of TFREEST we will create a multiple record file to handle the time-dependent status of
platelet recovery. Details of SAS code to create the long file BMT_LG is given in Gardiner, Luo & Lin
(2008). All patients begin at TSTART=0. For a patient who had platelet recovery we create two records one
of each time interval (0, TRETP] and (TRETP, TFREEST]. For the first record define TSTOP=TRETP,
PLSTATUS=0, STATUS=0 and STRATUM=‘01’. For the second record define TSTART=TRETP,
TSTOP=TFREEST, PLSTATUS=1, STATUS=DFI and STRATUM=‘12’.
For a patient who did not have platelet recovery we create a single record: TSTOP=TFREEST,
PLSTATUS=0, STATUS=DFI and STRATUM=‘02’. PLSTATUS defines the platelet recovery status just
prior to TSTOP and STATUS indicates whether or not death/relapse occurred at TSTOP. All time-invariant
covariates are retained on each record. For this illustration we consider disease group (DGROUP) only. The
variable STRATUM is created for convenience. It can be used to verify counts of events and censored values.
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SAS Global Forum 2010
Statistics and Data Analysis
proc format;
value dgroup 1='ALL' 2='AML low risk' 3='AML high risk';
value plstatus 0='before' 1='after';
run;
Consider estimation of the PHM h( t |z( t )) = h0 ( t )exp( z′( t )β ) for the risk of death/relapse. With dummy
variables AML H and AML L for AML-high risk and AML-low risk the linear predictor z′( t )β is defined as:
β1 AML H + β 2 AML L + β 3 PLSTATUS ( t ) + β 4 AML H × PLSTATUS ( t ) + β5 AML L × PLSTATUS ( t ) which
is β1 AML H + β 2 AML L for t <TRETP, and ( β1 + β 4 ) AML H + ( β 2 + β 5 ) AML L + β 3 for t ≥TRETP.
Estimation of the β-parameters is carried out by maximum partial likelihood estimation: the counting process
style input must be used in the model statement to create the appropriate risk sets at each death/relapse time.
proc phreg data=bmt_lg;
class plstatus(ref='before') dgroup(ref='ALL')/param=ref;
model (tstart, tstop)*status(0)=dgroup|plstatus/rl;
hazardratio dgroup/diff=ref cl=wald;
format dgroup dgroup. plstatus plstatus.;
run;
Table 5: Parameter estimates from PHM for time to death/relapse
Parameter
DF
Parameter Standard
Estimate
Error
ChiSquare
Pr > ChiSq
dgroup
AML high risk
1
0.83069
0.69324
1.4358
0.2308
dgroup
AML low risk
1
1.05334
0.71832
2.1503
0.1425
plstatus
after
1
–0.45715
0.62896
0.5283
0.4673
plstatus*dgroup
after
AML high risk
1
–0.52620
0.75215
0.4894
0.4842
plstatus*dgroup
after
AML low risk
1
–1.85157
0.78769
5.5254
0.0187
The HAZARDRATIO statement is needed to produce the estimates of hazard ratios and 95% confidence
intervals (Table 6). They are not produced by default because the dgroup|plstatus specification in the model
is viewed as containing an interaction of dgroup with plstatus. The option DIFF=ref requests hazard ratios
for the two AML disease groups with ALL as referent, and CL=wald gives the 95% confidence limits.
With the aforementioned parameterization, in the first row of Table 6 the point estimate is obtained from
Table 5: exp ( βˆ1 + βˆ4 ) =exp(0.83069−0.52620)=1.356. Compared to the ALL patient group, the AML low
risk group has improved prognosis for survival after platelet recovery (p=0.012). The same results can be
obtained from contrast statements including p-values.
Table 6: Hazard Ratios for Disease Group
Point
95% Wald
Estimate Confidence Limits
Description
dgroup AML high risk vs ALL At plstatus=after
1.356
0.765
2.403
dgroup AML low risk vs ALL At plstatus=after
0.450
0.241
0.840
dgroup AML high risk vs ALL At plstatus=before
2.295
0.590
8.930
dgroup AML low risk vs ALL At plstatus=before
2.867
0.701 11.719
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c.
Statistics and Data Analysis
PLOTTING SURVIVAL CURVES
The PLOTS=survival option in the PROC PHREG statement produce graphs of estimated survival curves at
specified covariate profiles. Consider the six profiles defined by DGROUP and PLSTATUS output to the
data set COVAR.
proc sort data=bmt_lg out=covar(keep=dgroup plstatus) nodupkey;
by dgroup plstatus;
format dgroup dgroup. plstatus plstatus.;
run;
The BASELINE statement and its options produce the Nelson-Aalen estimator of the survival function at a
t
fixed profile z using: Sˆ( t |=
z ) exp − H ( t , βˆ )exp( z′ βˆ ) where H ( t , βˆ ) = {S ( 0 ) ( u , βˆ )}−1 dN ( u ),
0
0
(
0
0
)
0
∫
0
S ( t , βˆ ) = ∑ i =1 Yi ( t )exp( z′( t )βˆ ) and N ( t ) is the counting process for death/relapse events in the sample.
(0)
n
Survival curves derived from a PHM with time-dependent covariates should be interpreted with caution. For
(
)
t
example, the relationship S ( t |z(=
t )) exp − ∫ h0 ( u )exp( z′( u )β )du holds under the assumption of strict
0
exogeneity of the accumulating covariate process t→ z( t ). By strict exogeneity we mean that t→ z( t ) evolves
( t )] P[ z( t + ∆t )|z( t )] . See Lancaster (1990) for further discussion.
as P[ z( t + ∆t )|T ≥ t + ∆t , z=
The following syntax will produce the plots shown next.
ods graphics on/width=4in height=4in;
proc phreg data=bmt_lg plots(overlay=group)=survival;
class plstatus(ref='before') dgroup(ref='ALL')/param=ref;
model (tstart, tstop)*status(0)=dgroup|plstatus;
baseline covariates=covar out=surv survival=survival/method=ch group=dgroup
rowid=plstatus;
format dgroup dgroup. plstatus plstatus.;
run;
ods graphics off;
Six curves are displayed in three panels. The GROUP= option collates the curves by disease group, and
ROWID appropriately labels the curves. Further modification of the plots (e.g., changing colors, title, line
type, axes and legends) would need more manipulation of the output graphics file through PROC
TEMPLATE, or some editing of the plot with the ODS graphics editor. See Statistical Graphics using ODS in
SAS/STAT® User’s Guide. An alternative is to use the SURV output dataset to plot the curves by PROC
GPLOT.
d. MULTI-STATE MODELS
Although not discussed in detail here, the same technique of expanding the data set appropriately could be
used in other settings including analyses of recurrent events, multiple failure times and competing risks
models. For example, if platelet recovery is viewed as an intermediate event along with the terminal event
death/relapse, another expansion of the data set BMT_LG would place all 137 patients at risk of each event,
together with 120 records for the post-recovery transition to the terminal event. The data set will have 394
records. This is a three-state model with transitions 0→1 (platelet recovery), 0→2 (death/relapse without
platelet recovery), and 1→2. The multiplicative intensity model α hj ( t |z( t )) = α hj 0 ( t )exp( z′( t )β hj ) with
stratum-specific covariates and hj denoting the h→j transition can be analyzed using PHREG. See Gardiner et
al, (2008).
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Statistics and Data Analysis
Disease–free survival plots
dgroup=1: ALL
dgroup=2: AML-low risk
dgroup=3: AML-high risk
plstatus: Platelet recovery status (timedependent).
Within each disease group the survival plots are
for two “what if” scenarios: (i) for a patient
without platelet recovery (lower curve), (ii) for
another patient with platelet recovery (upper
curve). All curves are estimated at the same grid
of event times (76 distinct times for 83 events).
Plots are computed directly from the formula
Sˆ( t |=
z0 ) exp − H 0 ( t , βˆ )exp( z′0 βˆ )
(
for a fixed profile z0 .
14
)
SAS Global Forum 2010
IV.
Statistics and Data Analysis
BAYESIAN ANALYSES
Frequentist analyses are based on the distribution of the data y that leads to a likelihood function L(θ; y) with
parameters θ considered as fixed constants. It is the distribution of y that provides a basis for statistical
inference on the unknown θ. We cannot make probabilistic statements about θ. Rather, the distributional
properties of its estimator θˆ = θˆ( y ) such as consistency, asymptotic normality are used to make inferences
about θ. For example, the classical 100(1−α)% confidence interval for a 1-dimensional parameter θ with
ˆ UCL(θ)
ˆ and P[LCL(θ)<θ<UCL(θ)]=1
ˆ
ˆ
− α for all values of θ is a probability
confidence limits LCL(θ),
statement about the confidence limits and not the parameter θ.
The Bayesian paradigm on the other hand places a distribution on θ, the prior distribution π(θ) which
expresses our degree of belief in θ, and when combined with L(θ; y) gives the posterior distribution π(θ|y) of
θ given y. Because Bayes’ theorem gives π(θ|y) ∝ L(θ; y) π(θ) the term Bayesian analysis is applied to
inferences drawn from the posterior distribution. For instance, the aforementioned frequentist confidence
b
interval for θ can be replaced by a probability statement P[ a < θ < b ]=∫ π ( u | y )du based on the posterior
a
distribution. If this probability is (1−α) we call (a, b) a 100(1−α)% credible interval for θ.
A closed-form expression for π(θ|y) can only be derived in a relatively few cases. Therefore, a general
approach is to simulate π(θ|y) by drawing samples {θ( b ) : 1 ≤ b ≤ B} and use them for inference. For example,
the posterior mean is calculated as θ = B −1 ∑ b =1 θ( b ) and an equal-tail 95% credible interval for a one
B
dimensional θ is the interval between the 2.5-th and 97.5-th percentiles of the sample. The theory underlying
the simulation approach is the Markov Chain Monte Carlo (MCMC) method that constructs a Markov chain
whose stationary distribution is the posterior distribution. The process of drawing samples from the posterior
distribution is based on Metropolis-Hastings algorithms or its variants (e.g., Gibbs sampler). The MCMC
procedure designed to analyze Bayesian models fuels the capability of LIFEREG and PHREG to provide a
Bayes solution to several survival models.
The BAYES statement in both LIFEREG and PHREG invokes the Bayes engine. For most analyses none of
the myriad of options in the BAYES statement needs to be explicitly specified. However, a diligent
investigation of the results should be undertaken to ascertain convergence of the underlying Markov Chain to
its stationary distribution and whether the samples from the posterior exhibit dependencies. Several useful
diagnostics and plots are produced by default if ODS Graphics is enabled with the PLOTS request. Finally,
the posterior sample can be saved in a data set with the OUTPOST option for additional analyses. For
quantities of interest such as the hazard ratio and percentiles of the survival curve that can be expressed as a
function g(θ), the posterior sample { g ( θ( b ) ) : 1 ≤ b ≤ B} is used to describe summary statistics for g(θ).
The following are standard MCMC options in the BAYES statement (default in parenthesis).
SEED= sets the random number generator for simulating the Markov chain samples (time of day).
NBI= # burn-in iterations discarded before the samples are saved (2000).
NMC=# iterations after burn-in (10000)
THIN=k retains one in every k samples after burn-in (k=1)
The initial values θ( 0 ) = (θ1( 0 ) , θ 2( 0 ) , , θ K( 0 ) ) are arbitrary (can be set by INITIAL=). One iteration of the Gibbs
sampler produces θ(1) = (θ1(1) , θ 2(1) , , θ K(1) ) based on component-by-component random draws from
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Statistics and Data Analysis
(0)
(0)
(1)
(1)
(0)
(0)
conditional distributions: θ(1)
1 drawn from π (θ1 |θ 2 , , θ K , y ) , θ 2 drawn from π (θ 2 |θ1 , θ 3  , θ K , y ) ,…,
(1)
(1)
(1)
(b )
θ(1)
: 1 ≤ b ≤ B} .
K drawn from π (θ K |θ1 , θ 2  , θ K −1 , y ) . After B iterations this leads to the chain {θ
a.
BAYESIAN ANALYSIS WITH LIFEREG
Consider the model log=
Ti z′i β + σε i for the time to death/relapse (TFREEST) in bone marrow transplant
patients with disease group (DGROUP) as covariate. For a Bayes analysis, a prior distribution on
β = ( β 0 , β1 , β 2 ) is specified through COEFFPRIOR and for σ through SCALEPRIOR. The following
syntax fits a Weibull model with a normal prior β ~ N (0,106 I3 ), and gamma prior σ ~ G(10 −4 ,10 −4 ) . Several
options need not be explicitly stated as they are the defaults. After a burn-in of 4000, one-half of the 20000
samples is retained. The data set BMT is the single record per patient file (137 patients). The order=freq
option is used to preserve the previously used parameterization with the ALL group as referent.
ods graphics off;
proc lifereg data=bmt order=freq;
class dgroup;
format dgroup dgroup.;
model tfreest*dfi(0)=dgroup/dist=weibull;
bayes seed=3538623 outpost=post_w nbi=4000 nmc=20000 thin=2
coeffprior=normal(var=1E6)scaleprior=gamma(shape=1E-4, iscale=1E-4);
run;
ods graphics close;
Trace, autocorrelation and density plots are produced for each of the four parameters θ = ( β 0 , β1 , β 2 , σ ). It is
imperative that these (and other diagnostics) be examined before any conclusions are drawn from the
simulated posterior samples {θ( b ) : 1 ≤ b ≤ B} . The results shown on the next page are almost perfect. The trace
plot show excellent mixing, the autocorrelation decreases to near zero, and the density is bell-shaped. The
trace plots are centered near their respective posterior mean and traverse the posterior space with small
fluctuations. For the intercept β 0 which corresponds to the ALL group, the trace plot is centered near the
posterior mean of 7.0. Samples in both tails are covered. These results exhibit convergence of the Markov
chain to its stationary distribution. The Geweke test (not shown) produced by default, compares the posterior
mean from the early part (first 10%) of the Markov chain to posterior mean from the latter part (last 50%).
There are no differences for each of the parameters.
Table 7 reports the simple statistics, percentiles, credible intervals, and high probability density (HPD)
intervals for each of the parameters based on the posterior sample of 10000. Because the priors used are noninformative, the mean, standard deviation and credible interval should be fairly close to the corresponding
maximum likelihood estimates (estimate, standard error, 95% CI).
Table 7. Posterior Statistics for parameters from Bayes analysis of the Weibull model
Percentiles
Parameter
N
Standard
Mean Deviation
Posterior Intervals
25%
50%
75%
Equal-Tail
Interval
HPD Interval
ALL
10000
7.0373
0.3506
6.7969
7.0253
7.2601
6.3857
7.7702
6.3528
7.7304
AML low risk
10000
1.1346
0.4947
0.8006
1.1330
1.4620
0.1684
2.1239
0.1479
2.0888
AML high risk
10000 –0.4810
0.4517 –0.7835 –0.4756 –0.1777 –1.3781
0.3898 –1.3442
0.4197
Scale
10000
0.1621
2.0383
2.0048
1.6934
1.5804
16
1.6826
1.7967
1.4026
1.3750
SAS Global Forum 2010
b.
Statistics and Data Analysis
ESTIMATION OF PERCENTILES
=
For the Weibull, the p-th percentile
is t p exp( z′β + σ w p ) where w p =log( − log(1 − p )) . A Bayes estimate is
constructed from the posterior samples t (pb ) = exp( z′β ( b ) + σ ( b )w p ), b = 1, , B . Results are shown for the
median in Table 8, right hand side panel with corresponding nonparametric and MLE estimates for
comparison. The results are obtained by processing the OUTPOST=post_w data set. Similar, but not
necessarily identical results can be derived using PROC MCMC using its MONITOR option.
Table 8: Estimate of Median disease-free survival (in days)
MLE (Weibull)
Nonparametric
Median
ALL
95% Confidence
Interval
Median
Bayes (Weibull)
95% Confidence
Interval
Posterior
Mean
Equal-Tail
Credible Interval
418
192
…
590.58
307.42
1134.53
650.88
317.45
1256.91
AML low risk
2204
641
…
1810.64
951.49
3445.55
2023.39
1010.18
3898.97
AML high risk
183
113
390
376.73
214.80
660.72
395.38
211.05
682.01
Likewise, disease-free survival at t days can be estimated from S ( t |z, θ ) = exp( − exp( − y )) b= 1,…,B
where =
y ( b ) (log t − z′β ( b ) ) / σ ( b ) . For an example see SAS/STAT®: The MCMC Procedure.
(b )
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(b )
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Statistics and Data Analysis
c. BAYESIAN ANALYSIS IN PHREG
There are two approaches to a Bayes analysis of the PHM h( t |z( t )) = h0 ( t )exp( z′( t )β ) .The first is based on
the partial likelihood L(β; y) combined with a prior π(β) which produces the posterior π(β|y). The baseline
hazard h0 ( t ) is left unspecified. Inference is made from samples {β( b ) : 1 ≤ b ≤ B} drawn from π(β|y). Thus the
partial likelihood is treated as a likelihood function just as in the previous analysis. The second approach
discussed later parameterizes h0 ( t ) by a finite-dimensional parameter λ, h0 ( t ) = h0 ( t , λ ) producing a full
likelihood L(θ; y), where θ=( β,λ).
Returning to our analysis of the time to death/relapse among bone marrow transplant patients we consider
the PHM with disease group DGROUP, the time-dependent indicator PLSTATUS(t) of return of platelets to
normal levels, and their interaction. This is a 5 parameter model. The extended file BMT_LG is used. The
BAYES statement invokes the analysis. Options are the same as in LIFEREG. We only need to specify a
prior for β which is taken here as β ~ N (0,106 I5 ).
ods graphics on;
proc phreg data=bmt_LG;
class plstatus(ref='before') dgroup(ref='ALL')/param=ref;
model (tstart, tstop)*status(0)=dgroup|plstatus/ ties=breslow;
format dgroup dgroup. plstatus plstatus.;
bayes seed=4112010 outpost=postsample nbi=4000 nmc=20000 thin=2
coeffprior=normal(var=1E6);
hazardratio dgroup/diff=ref cl=wald;
run;
ods graphics off;
Before drawing inferences from the posterior sample, we should examine the trace, autocorrelation and
density plots for each parameter to be content that the underlying chain has converged. The plots for the two
parameters involving the AML low risk group shown below suggest that the mixing in the chain is acceptable,
although we notice long correlation times. Plots for the 3 other parameters (not shown) are very similar.
The HAZARDRATIO statement delivers the Bayes solution corresponding to the previous classical ML
analysis in Table 6. These results (Table 9) can also be derived from the OUTPOST=postsample data set. We
can also use postsample to assess the posterior probability that the HR for AML low risk vs ALL after
platelet recovery is <1. The probability is over 99%.
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Statistics and Data Analysis
Table 9: Hazard Ratios for Disease Group (10000 samples)
Quantiles
Mean
Std
Dev 25%
50% 75%
AML high risk vs ALL At plstatus=after
1.411
0.423 1.111
1.351 1.647
0.768
2.392 0.691
2.241
AML low risk vs ALL At plstatus=after
0.470
0.153 0.361
0.449 0.556
0.234
0.827 0.208
0.773
AML high risk vs ALL At plstatus=before
3.711
3.827 1.625
2.674 4.420
0.649
13.191 0.294 10.046
AML low risk vs ALL At plstatus=before
4.631
4.863 1.961
3.264 5.542
0.757
17.112 0.275 12.778
Description
95% EqualTail Interval
95% HPD
Interval
d. SURVIVAL CURVES (FROM BAYES ANALYSIS)
Disease-free survival at t days for a specified covariate profile z0 is estimated from the posterior sample
(b )
) exp( − H 0 ( t , β ( b ) )exp( z′0 β ( b ) )) b= 1,…,B. This approach is similar to the classical estimates
S ( t | z0 , β =
Sˆ( t |=
z ) exp − H ( t , βˆ )exp( z′ βˆ ) where β̂ denotes is the maximum partial likelihood estimator.
0
(
0
0
)
We use the same COVAR data set with six profiles defined by disease groups and platelet recovery status.
The BASELINE statement requests a data set SURV_BAYES be formed to contain the output. With
SURVIVAL=_ALL_ we obtain at each event time the posterior mean, standard error, equal-tailed credible
interval limits, and HPD interval limits. For the purpose of plotting the survival curves we can use SGPLOT
or GPLOT. However, when fully operational, under ODS graphics the PLOTS option in the PHREG
statement together with additional options in the BASELINE statement would also yield the desired results.
proc phreg data=bmt_LG ;
class plstatus(ref='before') dgroup(ref='ALL')/param=ref;
model (tstart, tstop)*status(0)=dgroup|plstatus;
format dgroup dgroup. plstatus plstatus.;
bayes seed=5808208 outpost=postsample nbi=5000 nmc=25000 thin=2
coeffprior=normal(var=1E6);
baseline covariates=covar out=surv_bayes survival=_ALL_;
run;
The plots shown next are obtained by combining into one data set the survival estimates from the classical
analysis with the posterior means from the Bayes analysis. We use GPLOT (with two plot statements) to
exploit various options for axes, colors, legends etc.
The following syntax plots the Bayes estimates and pointwise HPD bands for the DGROUP=1 (ALL
patients). The plot is not shown.
ods graphics on;
proc sgplot data=surv_bayes(where=(dgroup=1));
band x=tStop lower=lowerHPDSurvival upper=upperHPDSurvival /
group=plstatus modelname="Survival" transparency=.8;
step x=tStop y=Survival / group=plstatus name="Survival";
title "DISEASE GROUP = ALL";
run;
ods graphics off;
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Statistics and Data Analysis
Disease Group ALL
Disease Group AML Low Risk
Disease Group AML High Risk
Disease/relapse-free estimates of survival
Plots for the classical analysis are obtained from the
z0 ) exp − H 0 ( t , βˆ )exp( z′0 βˆ ) where
estimates Sˆ( t |=
(
)
β̂ denotes the maximum partial likelihood
estimator.
Plots for the Bayes analysis are derived from the
posterior samples
(b )
S ( t | z0 , β =
) exp( − H 0 ( t , β ( b ) )exp( z′0 β ( b ) ))
1 ≤ b ≤ B which are obtained from {β( b ) : 1 ≤ b ≤ B} .
All calculations are made at a fixed profile z0 and at
the same grid of event times. The two sets of
estimates track each other, especially for after
platelet recovery.
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Statistics and Data Analysis
e. PIECEWISE CONSTANT HAZARD
The second approach to a Bayes analysis includes a parameterization of the baseline hazard h0 ( t ) in the PHM
a 0 < a1 < .... < a J −1 < a J =
∞ denote a partition of the time axis into Jas a piecewise constant function. Let 0 =
intervals [ a j −1 , a j ), j = 1,..., J . The piecewise constant hazard =
is h0 ( t , λ )
∑
J
j =1
λ j [ a j −1 ≤ t < a j ] , with
parameters λ = ( λ1 ,..., λ J ) , λ j > 0 for all j. An alternative parameterization uses log-hazards
=
α (α
=
log λ j . Because the PHM is parametric in θ = ( λ , β ) or θ = ( α , β ), the likelihood L(θ; y)
1 ,..., α J ), α j
can be constructed for the observed data y. Together with a specified prior π(θ) on θ we obtain the posterior
π(θ|y)∝ L(θ; y) π(θ). The basis for inference is the sample {θ( b ) : 1 ≤ b ≤ B} drawn from this distribution using
the Gibbs sampler. The MLE of θ obtained by maximizing L(θ; y) are produced which serve as the default
initial values for the sampler.
Simply adding PIECEWISE alone to the Bayes statement triggers the following: (i) log-hazard
parameterization (ii) J=8 intervals (iii) uniform prior π ( α j ) ∝ 1 for all j. This is the same as an improper prior
on λ j , that is, π ( λ j ) ∝ λ −j 1 . Interval cut-points are chosen by default to have approximately an equal
number of events in each interval. Of course, all of these can be changed by options. The total number of
events in the BMT data set is 83. By default 8 intervals are constructed to have about 10-11 events in each
interval. Increasing the number of intervals could produce unstable estimates of λ . Too few intervals could
lead to poor fit. To obtain a feasible solution for λ the intervals must have at least one event. After trial and
error, we use J=12. The following syntax specifies independent normal priors for ( α , β ) . Correlation times are
still large, but Geweke diagnostics are quite good. Results are in Table 10.
ods graphics on;
proc phreg data=bmt_LG;
class plstatus(ref='before') dgroup(ref='ALL')/param=ref;
model (tstart, tstop)*status(0)=dgroup|plstatus;
format dgroup dgroup. plstatus plstatus.;
bayes seed=4122010 outpost=postsample nbi=5000 nmc=30000 thin=2
coeffprior=normal(var=1E6)
piecewise=loghazard(Ninterval=12 prior=normal(var=1e6));
run;
ods graphics off;
Table 10: Piecewise constant hazard model
Maximum likelihood estimates
Estimate
Standard
Error
AML: high risk
0.8414
0.6925
–0.5158
2.1986
0.8947
0.7604
–0.5284
2.4939
AML: low risk
1.0552
0.7158
–0.3477
2.4582
1.1162
0.7902
–0.4504
2.6721
PLSTATUS: after
recovery
–0.4296
0.6301
–1.6646
0.8054
–0.3246
0.6861
–1.6085
1.0542
PLSTATUS × AML
high risk
–0.5548
0.7515
–2.0277
0.9181
–0.5987
0.8205
–2.3587
0.9024
PLSTATUS × AML
low risk
–1.8651
0.7852
–3.4040 –0.3261
–1.9215
0.8614
–3.7017 –0.2770
Parameter
95% Confidence
Limits
Bayes estimates
21
Posterior Standard 95% HPD Interval
Mean Deviation
SAS Global Forum 2010
Statistics and Data Analysis
Diagnostic plots are shown below for 2 of the 5 regression parameters. They could be compared with the
corresponding plots shown earlier for the Cox model on page 18.
A BASELINE statement is used to save the Bayes estimates of the survival curves and other optional
quantities. Depicted below are curves for the AML low risk and AML high risk groups, paralleling the
corresponding plots shown on page 20. The patterns are very similar, but with slightly more separation
between estimates from the Bayes and classical analyses.
Disease Group AML high Risk
Disease Group AML low risk
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SAS Global Forum 2010
Statistics and Data Analysis
DATA SETS
The two data sets KIDNEY and BMT used in this paper are widely circulated via the world-wide-web. We
used the original sources McGilchrist & Aisbett (1991) and Klein & Moeschberger (1997).
ACKNOWLEDGEMENTS
I wish to thank Ying So, Gordon Johnston and Jan Chvosta of the SAS Institute who have provided valuable
feedback on my numerous questions and comments over many years. It is a pleasure to have learnt from
them the many intricacies, nuances and versatility of SAS procedures. All errors and omissions in this paper
are solely mine. This research was supported in part by the Agency for Healthcare Research & Quality under
Grant 1R01 HS14206.
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SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS
Institute Inc. in the USA and other countries. ® indicates USA registration.
CONTACT INFORMATION
Joseph C. Gardiner
Division of Biostatistics, Department of Epidemiology
Michigan State University, East Lansing MI 48824
[email protected]
23
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