Download Statistical Methods for Interval Censored Complex Survey Data

Statistical Methods for Interval
Censored Complex Survey Data
C. M. Suchindran
Department of Biostatistics
Carolina Population Center
1
Motivating Example
Outcome of Interest
Time ( Age at onset ) to first transition to obesity
Data
Add Health Study ( Wave I –IV) ( Complex Survey
Design)
Recorded Data of age at first transition to obesity:
Obesity Status at each wave
2
Motivating Example
Add Health Study
Age
Data Format
Wave I
II
III
IV
(12-20)
(13-21)
(18-26)
(24-32)
Case 1 Age at Wave I 19 Obese at Wave I
Case 2 Age at Wave I 16 Not obese at Wave I;
Obese at Wave II (age 17)
Case 3 Age at Wave II 15 Not obese at Wave II;
Obese at Wave III ( age 20)
Case 4 Age at Wave III 18 Not obese at Wave III;
Obese at Wave IV (age 24)
Case 5 Age at Wave IV 28 Not obese at Wave IV
Case
Ltime
Rtime
1
.
19
2
16
17
3
15
20
4
18
24
5
28
.
Need to take into account sampling weights ( design)
Subject Information in overlapping intervals
3
Special Cases
1. Status determination at one time point only (Current
Status Data)
Obesity status ascertained at Wave IV only
Time to obesity is smaller or larger than age at Wave IV
2. Observations in non-overlapping Intervals
All subjects were observed in fixed intervals
Baseline, one year, two years , etc
In this talk we will examine interval censored data from an arbitrary
number of observation times with overlap
4
Available Software
SAS
Macro %EMIMC ; %ICSTEST
PROC LIFETEST, LIFEREG, MCMC , NLMIXED
STATA
Stset, streg INTCENS , STPM ( Parametric models), GLAMM
Ml for svy ( Tools for programmers of new survey commands)
R package
Icens Intcox smoothSurv BayesSurv
( Needs to manipulate the procedures to include survey design)
MPlus
5
Conventional Survival Analysis
Use of right censored data (Know the exact time of event or censoring
time)
1. Descriptive Analysis
Kaplan-Meier Estimate of Survival ( Survival rate estimated at times
of occurrence of events) ( Non-parametric estimation)
2. Hazards Modeling
log=
λi (t )
3.
log λ0i (t ) + β + β x + ... + β x
*
0
*
1 i1
*
p ip
Accelerated Failure time models
log Ti =β 0 + β1 xi1 + ...... + β p xip + σε i
When the distribution is Weibull
λ0 (t ) =
α log t β
*
j
βj
=
−
σ
1
α =
σ
Choice of log logistic results in proportional odds model
6
Descriptive Analysis
Estimation of Survival Function
Reference: Ying So, Gordon Johnston and Se Hee Kim. (2010).
“Analyzing Interval Censored Survival Data,” SAS Global Forum
Paper 257-2010.
Outcome variable of interest: Time to event (denoted as T)
Descriptive Measures:
1.
2.
3.
Survival Function S(t) = P (T >t) Probability (Time to event exceeds t)
Cumulative distribution function F(t)= P(T ≤ t ) (Cum Failure by time t)
F(t) =1-S(t)
Probability of event occurrence in the interval ( L, R) = P(L <T<R)
S(L)-S(R) or F (R)-F(L)
7
Nonparametric Estimation of Survival Function
Notations:
T ~ Time to event
Survival Function S (t) = P( T>t); Probability that event time exceeds t
Cumulative distribution function F (t) = 1-S(t) = Prob (event occurs before t)
Probability of event occurring in a interval (t1 t2 ) = S (t1 ) − S (t2 )
( Li , Ri ]
Interval in which event is known to occur
Li Left end point (may be zero or .)
Ri
Right end point (may be · or
usually observed ( L , R ]
i
∞
)
i
8
Example Data 1; Estimation of Turnbull Intervals
I = {( q1 , p1 ],( q2 , p2 ], ...( qm , pm ]}
id ltime rtime scw
1
.
7
5
2
.
8
10
3
6
10
15
Set of all left and right censored in
4
7
16
15
such a way that qj is a left end
5
7
14
25
point, pj is the right end point and
6
17
.
33
there is no other left or end point
7
37
44
33
between qj and pj
8
45
.
34
9
46
.
8
10 46
.
12
Possible Event times
0 6 7 8 10 14 16 17 37 44 45 46 (INFINITY)
L L L R
R R
R L
L R L
L
R
L
L
R
Possible probability estimation intervals (6, 7) (7. 8) (37. 44) ( 46. infinity)
9
Contribution of Each Individual to Turnbull Intervals
ID
Event times
(6, 7]
(7, 8]
(37, 44)
(46, infinity)
1
(0, 7)
1
0
0
0
2
(0, 8)
1
1
0
0
3
(6, 10)
1
1
0
0
4
(7, 16)
0
1
0
0
5
(7, 14)
0
1
0
0
6
(17, .)
0
0
1
1
7
(37, 44)
0
0
1
0
8
(45, .)
0
0
0
1
9
(46, .)
0
0
0
1
10
(46, .)
0
0
0
1
10
EM Algorithm (Iterative Procedure) for Nonparametric Estimation
Step 1 : Start with an initial probability distribution (P)
(Uniform in all Turnbull Intervals) [.25 .25 .25 .25]
Step 2 : For Each ID calculate the probability (E Step)

ˆP (t ) = P(t j ) I {t j ∈ ( Li , Ri )}
i j
∑ Pˆ (tk )
tk ∈( Li , Ri )
For ID 2
Pˆ (t j ) = [0 .25/(.25+.25)
.25/(.25+.25)
0]
Step 3 : M Step Update P̂
for each Turnbull Interval
n
1
Pˆ (t j ) = ∑ Pˆi (t j )
n i =1
Iterate until convergence
11
Proc IML;
Q =(1 1 1 1);
N=10;
a={1 0 0 0,
1 1 0 0,
1 1 0 0,
0 1 0 0,
0 1 0 0,
0 0 1 1,
0 0 1 0,
0 0 0 1,
0 0 0 1,
0 0 0 1};
p = {0.2 0.3 0.15 0.35};
x=J(4,1,1);
EPS =.0001;
y =J(1,10,1);
do until (ABS(Q-p) <EPS);
Q =p;
c=a#p;
rowsum = c*x;
astar=c/rowsum;
p =y*astar/10;
END;
print p;
Lower Upper Probability Surv
6
7
37
46
7
8
44
+
0.1667
0.3333
0.1250
0.3750
.8333
.5000
.3750
12
SAS EMICM MACRO
%inc "H:/cpcnew2/xmacro.txt";
%inc "H:/cpcnew2/emicm.txt";
%EMICM(data = example,
left =ltime,
right = rtime,
group =group,
options =plot,
title ="NPMLE",
title2="example ",
timelabel="time to event"
);
Standard
Cumulative Survival
Error
Lower Upper Probability Probability Probability Survival
6
7
0.1667
0.1667
0.8333
0.1362
7
8
0.3333
0.5000
0.5000
0.1581
37
44
0.1250
0.6250
0.3750
0.1575
46
.
0.3750
1.0000
0.0000
0.0000
13
LIFEREG PROCEDURE IN SAS
ods graphics on;
proc lifereg;
model ( ltime rtime) = /d = weibull ;
probplot
pupper =10
printprobs
maxitem =(1000, 25)
ppout;
inset;
run;
14
The LIFEREG Procedure
Cumulative Probability Estimates
Pointwise 95%
Confidence
Lower
Lifetime
.
7
8
44
Upper Cumulative
Lifetime Probability
6
7
37
46
0.0000
0.1667
0.5000
0.6250
Limits
Lower Upper
0.0000
0.0249
0.2245
0.3032
0.0000
0.6106
0.7755
0.8645
Standard
Error
0.0000
0.1459
0.1581
0.1606
15
Introducing Sampling Weights in IML Macro
Proc IML;
step1 = J(10,4, 1);
weight = {5 ,10, 15, 15, 25, 33, 33, 34, 8, 12};
pweight = step1#weight;
N= sum(weight);
Q = {1 1 1 1};
anw={1 0 0 0,
1 1 0 0,
1 1 0 0,
0 1 0 0,
0 1 0 0,
0 0 1 1,
0 0 1 0,
0 0 0 1,
0 0 0 1,
0 0 0 1};
a= anw#pweight;
p = {0.25 0.25 0.25 0.25};
x=J(4,1,1);
EPS =.0001;
y =J(1,10,1);
do until (ABS(Q-p) <EPS);
Q =p;
c=a#p;
rowsum = c*x;
astar=c/rowsum;
p =weight`*astar/N;
END;
print pweight;
Interval
Probability
6-7
0.0409
7-8
0.3274
37-44
0.2396
46+
0.3920
16
Sampling Weights and SAS EMICM MACRO
id ltime rtime group weight
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
5
5
5
5
5
10
10
10
10
10
10
10
10
10
10
Standard
Cumulative Survival
Error
Lower Upper Probability Probability Probability Survival
6
7
37
46
7
8
44
.
0.0409
0.3275
0.2396
0.3920
0.0409
0.3684
0.6080
1.0000
0.9591
0.6316
0.3920
0.0000
0.0163
0.0350
0.0396
0.0000
Replicate Observations
17
Estimation of Survival Function with PROC LIFEREG
Incorporating Sampling Weights
The LIFEREG Procedure
ods graphics on;
proc lifereg data = example ;
model ( ltime rtime) = /d = weibull ;
weight scw;
probplot
pupper =10
printprobs
maxitem =(1000, 25)
ppout;
inset;
run;
quit;
ods graphics off;
Cumulative Probability Estimates
Pointwise 95%
Confidence
Lower
Upper Cumulative
Limits
Lifetime Lifetime Probability Lower
Upper
.
7
8
44
6
7
37
46
0.0000
0.0409
0.3684
0.6080
0.0000 0.0000
0.0173 0.0936
0.3028 0.4392
0.5287 0.6819
Standard
Error
0.0000
0.0177
0.0350
0.0394
18
id
ltime
rtime
group
1
45
.
1
2
25
37
1
3
37
0
1
4
6
10
1
5
46
.
1
6
0
5
1
7
0
7
1
8
26
40
1
9
18
.
1
10
46
.
1
47
8
12
0
48
0
5
0
49
30
34
0
50
0
22
0
51
5
8
0
52
13
.
0
DATA SET 2
19
Standard_
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
group
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
Lower
4
5
8
11
12
16
18
19
21
22
23
24
30
31
33
34
35
44
48
4
6
7
11
15
17
24
25
33
34
Upper
5
8
9
12
13
17
19
20
22
23
24
25
31
32
34
35
36
48
60
5
7
8
12
16
18
25
26
34
35
Probability
0.0433
0.0433
0.0000
0.0692
0.0000
0.1454
0.1411
0.1157
0.0000
0.0000
0.0000
0.0999
0.0709
0.0000
0.0000
0.0000
0.1608
0.0552
0.0552
0.0526
0.0343
0.1066
0.0673
0.0000
0.0000
0.0900
0.0000
0.0794
0.0000
Cumulative_
Probability
0.0433
0.0866
0.0866
0.1558
0.1558
0.3012
0.4423
0.5580
0.5580
0.5580
0.5580
0.6579
0.7288
0.7288
0.7288
0.7288
0.8896
0.9448
1.0000
0.0526
0.0870
0.1935
0.2609
0.2609
0.2609
0.3509
0.3509
0.4303
0.4303
Survival_
Probability
0.9567
0.9134
0.9134
0.8442
0.8442
0.6988
0.5577
0.4420
0.4420
0.4420
0.4420
0.3421
0.2712
0.2712
0.2712
0.2712
0.1104
0.0552
0.0000
0.9474
0.9130
0.8065
0.7391
0.7391
0.7391
0.6491
0.6491
0.5697
0.5697
std Error_
Survival
0.0332
0.0415
0.0415
0.0564
0.0564
0.0776
0.0825
0.0763
0.0763
0.0763
0.0763
0.0756
0.0670
0.0670
0.0670
0.0670
0.0483
0.0358
0.0000
0.0368
0.0483
0.0608
0.0647
0.0647
0.0647
0.0715
0.0715
0.0742
0.0742
Unweighted
Analysis
SAS MACRO
EMICM
20
21
Generalized Log-rank Test I (Zhao & Sun, 2004)
Test Statistic and Covariance Matrix
0
1
group
U
8.5121
-8.5121
12.0650
-12.0650
Chi-Square
6.0055
cov(U)
-12.0650
12.0650
DF Pr > Chi-Square
1
0.0143
Generalized Log-Rank Test II (Sun, Zhao & Zhao, 2005)
xi(x)=xlog(x)
Test Statistic and Covariance Matrix
group
0
1
U
9.2978 13.8302
-9.2978 -13.8302
ChiSquare
6.2507
cov(U)
-13.8302
13.8302
DF Pr>ChiSquare
1
0.0124
22
Regression Analysis
Approach 1 (Marginal Models)
Treat cluster correlation as a nuisance parameter and solve weighted
likelihood. Obtain variance estimates using anyone of the following
method: Robust Variance (Huber method); Jackknife or
Bootstrap
Approach 2 (Random Effects models)
Allow for explicit estimation of within and between cluster variance
Random effects Models are used. Variance estimation uses Robust
“sandwich” estimators, bootstrap and Jackknife procedures
(For multilevel model sandwich estimators are difficult to compute;
Jackknife may be unreliable; bootstrap
Approach 3 (Bayesian Models)
23
Partial List of Example Data ( 94 observations , 8 clusters )
id
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
ltime
45
25
37
6
46
26
18
46
46
24
46
27
36
7
rtime
group
37
10
5
7
40
34
16
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
village
1
1
1
1
1
2
2
2
2
3
3
3
3
3
3
3
ctype
2
4
2
4
2
3
3
4
2
2
2
2
2
4
2
4
w1
2268
684
5809
436
3949
1666
1775
1962
4519
2196
1635
4895
5072
5303
1440
3388
w2
88
88
88
88
88
76
76
76
76
64
64
64
64
64
64
64
w
199584
60192
511192
38368
347512
126616
134900
149112
343444
140544
104640
313280
324608
339392
92160
216832
scw
weight2
0.86
0.54
0.26
0.16
2.21
1.38
0.17
0.1
1.5
0.94
0.67
0.55
0.72
0.58
0.79
0.64
1.82
1.48
0.64
0.53
0.48
0.4
1.43
1.18
1.48
1.23
1.55
1.28
0.42
0.35
0.99
0.82
SCW (scaled weight Method I; weight2 ( Scaled weight method II)
24
Weibull Regression ( Marginal Model )
Accelerated Failure Time Model (PROC LIFEREG)
β 0 + β1 (Group )i + σε
log(Tij ) =
where ε is distributed as extreme value distribution.
Hazards Model
log λ (=
t ) α log t + β 0* + β1* (Group )
Note
βj
1
β =
− ,α =
−1
σ
σ
*
j
25
Jackknife
(with no stratification)
A single replicate is created by removing from the sample,
all units associated with a given PSU and inflating the
original weights by a factor to keep the sum of weights
the same.
The standard deviation of the estimates derived after
removing one PSU at a time is an estimate of the
variance (Typically average deviation square from the
parameter estimate based on all data)
26
PROC LIFEREG
proc lifereg data = onelevel;
model ( ltime rtime) = group /d = weibull ;
weight w;
run;
27
The LIFEREG Procedure
Model Information
Data Set
Dependent Variable
Dependent Variable
Weight Variable
Number of Observations
Noncensored Values
Right Censored Values
Left Censored Values
Interval Censored Values
Name of Distribution
Log Likelihood
WORK.ONELEVEL
Log(ltime)
Log(rtime)
w
94
0
37
6
51
Weibull
-135.0600573
Analysis of Maximum Likelihood Parameter Estimates
95% Confidence
Standard
Parameter
DF Estimate
Error
Limits
Intercept
1
3.4272
0.2063
3.0228 3.8316
group
1
0.5785
0.2453
0.0976 1.0594
Scale
1
0.7306
0.1028
0.5544 0.9627
Weibull Shape 1 1.3688
0.1927
1.0387 1.8036
Jackknife standard Error of Group parameter estimate = 0.1914
ChiSquare
Pr > ChiSq
275.96
5.56
<.0001
0.0184
28
Random Effects Models
L. Grilli and M. Pratesi (2004), “Weighted Estimation in Multilevel
Ordinal and Binary Models in the Presence of Informative Sampling
Designs,” Survey Methodology, 30(1): 93-103.
Maximization of the weighted log-likelihood of the model using SAS
Procedure NLMIXED
(Provides unbiased parameter estimates. Need re-sampling methods
such as bootstrapping or Jackknife to get proper variance
estimates)
The procedure requires sampling weights at each level
(For Add Health data see Chantala, Blanchet and Suchindran ( 2006),
“Software to compute sampling weights for Multilevel Analysis,” CPC
Website)
29
Multilevel model
Choice of weights (In a two level set up)
ω
Level 2 (cluster level) weights to adjust for differential probabilities of
cluster selection (Level 2 weights denoted as
j)
Level 1 weights: weight for individual I in cluster j denoted as
The final weight is
ωi| j
ω j * ωi| j
30
Scaling of Weights
For multilevel modeling scaling of weights is recommended:
Scaling of level 1 weights (Pfeffermann et al 1998)
Method I
ωi| j
ω= =
ωi| j
*
i| j
ωi| j
nj
∑ω
i =1
* nj
i| j
Note that the sum of weights in cluster j is the number of
subjects in cluster j ( n j )
Method I is recommended when the outcome variable depends on the
selection of subjects at level 1
31
Scaling of Weights
Method II ( Pfefefrmann et al 1998)
nj
ωi*| j =
∑ω
i =1
nj
i| j
2
ω
∑ i| j
* ωi| j
i =1
Method II is recommended when the outcome depends on
sampling at both levels
(For Add Health Data):
Kim Chantala, Dan Blanchette and C. M. Suchindran,
“Software to compute sampling weights in Multilevel
Analysis,” CPC website .
32
Scaling of Weights
Method III (Korn and Graubard, 2003)
Level I weights = 1
And level 2 weights =
nj
ω *j = ∑ ω j * ωi| j
i =1
Use when level 1 weights and outcome are not correlated
References:
Pfeffermann, D. et al (1998), “Weighting for unequal selection
probabilities in multilevel models,” JRSS-B, 60: 23-40.
Korn, E. L. and Graubard (2003), “Estimating variance components by
using survey data,” JRSS-B, 65:175-190.
33
Weibull Regression (Random Effects) Model
Accelerated Failure Time Model (NLMIXED)
log(Tij ) =
β 0 + β1 (Group )i + σε i
where ε is distributed as extreme value distribution
Random Effects Model (NLMIXED)
Add a random effect term to the linear predictor.
We assume that the random variable is distributed as
Normal (0, θ)
34
PROC NLMIXED with both level weights
proc nlmixed data = combweight qpoints =10;
parms beta0 0 beta1 = 0 p = .92 theta =.01;
bounds p >0 , theta >0;
ebetaxb = exp(-(beta0+beta1*group+b));
lamda= exp(-beta0);
s_1 = exp(-(ltime*ebetaxb)**(1/p));
S_u = exp (-(rtime*ebetaxb)**(1/p));
f_t = (lamda*p)*(lamda*ltime)**(p-1)*ebetaxb**1/p;
if ( ltime ^=. and rtime ^= . and ltime =rtime)
then lik = f_t;
else if ( ltime^=. and rtime =.) then
lik = S_1;
else if ( ltime =. and rtime ^=.) then
lik = 1-s_u;
else lik = S_1 -S_u;
llik =scw*log(lik);
model ctype ~ general(llik);
random b ~normal ( 0, theta) subject =village;
replicate w2;
run;
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
9245.6
9253.6
9254.0
9269.1
Parameter Estimates
Standard
Parameter Estimate Error
beta0
3.4697 0.04048
beta1
0.5625 0.05186
p
0.6159 0.01394
theta
0.09658 0.01932
DF
357
357
357
357
t Value
85.72
10.85
44.18
5.00
Pr > |t|
<.0001
<.0001
<.0001
<.0001
35
Use of Cfactor Option to make level 2 weight integer
proc nlmixed data = combweight qpoints =10 cfactor=100 ;
parms beta0 0 beta1 = 0 p = .92 theta =.01;
bounds p >0 , theta >0;
ebetaxb = exp(-(beta0+beta1*group+b));
lamda= exp(-beta0);
s_1 = exp(-(ltime*ebetaxb)**(1/p));
S_u = exp (-(rtime*ebetaxb)**(1/p));
f_t = (lamda*p)*(lamda*ltime)**(p-1)*ebetaxb**1/p;
if ( ltime ^=. and rtime ^= . and ltime =rtime)
then lik = f_t;
else if ( ltime^=. and rtime =.) then
lik = S_1;
else if ( ltime =. and rtime ^=.) then
lik = 1-s_u;
else lik = S_1 -S_u;
llik =scw*log(lik);
model ctype ~ general(llik);
random b ~normal ( 0, theta) subject =village;
replicate w2;
run;
36
Original Weights (No multiplier to make
integer weights)
With Cfactor statement (original
weights*100)
Standard
Parameter
beta0
beta1
p
theta
Estimate
3.4697
0.5625
0.6159
0.09659
Error
0.04048
0.05186
0.01394
0.01932
Parameter
beta0
beta1
p
theta
Estimate
3.4697
0.5625
0.6159
0.09659
Standard
Error
0.04048
0.05186
0.01394
0.01932
Jackknife Variance 0.1826
37
Alternate Level 1 Weights
( Method II)
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
7898.7
7906.7
7907.1
7922.2
Parameter Estimates
Parameter
beta0
beta1
p
theta
Standard
Estimate
Error
DF t Value
3.4851
0.03044 357 114.51
0.4589
0.04293
357 10.69
0.6109
0.01481
357 41.26
0.03480 0.01230 357
2.83
Pr > |t|
<.0001
<.0001
<.0001
0.0049
38
Markov Chain Monte Carlo (MCMC) Simulation
Procedure
Designed to fit Bayesian Models
Makes inference based on posterior distributions of the
parameters
MCMC procedure generates samples from the desired
posterior distributions and make inference
Priors for parameters (Vague priors for the parameters)
Gamma prior for positive density support (for variance)
39
SAS PROC MCMC
ods graphics on;
proc mcmc outpost =postout seed =1234 nbi=6000 nmc=60000 ntu =3000 missing=AC DIC
statistics = (summary interval);
ods select postSummaries Postintervals dic;
array var1[8];
parms p = .92 beta0 = 3.3582 beta1 = .5213 var1: 0;
parms s2g .1239;
prior beta: ~ normal (0, sd =10000);
prior var1: ~normal(0, var =s2g)
;
prior p ~gamma (shape =0.001, is =.001);
prior s2g: ~ general(-log(s2g));
ebetaxb = exp(-(beta0+beta1*group+var1[village]));
lamda= exp(-beta0);
s_1 = exp(-(ltime*ebetaxb)**(1/p));
S_u = exp (-(rtime*ebetaxb)**(1/p));
f_t = (lamda*p)*(lamda*ltime)**(p-1)*ebetaxb**1/p;
if ( ltime ^=. and rtime ^= . and ltime =rtime)
then lik = f_t;
else if ( ltime^=. and rtime =.) then
lik = S_1;
else if ( ltime =. and rtime ^=.) then
lik = 1-s_u;
else lik = S_1 -S_u;
llik = scw*log(lik);
40
The MCMC Procedure
Posterior Summaries
Parameter
N
Mean
Standard
Deviation
p
beta0
beta1
s2g
60000
60000
60000
60000
0.6096
3.3764
0.5476
0.0206
0.0793
0.1280
0.2235
0.0191
Percentiles
25%
50%
0.5555
3.2958
0.3868
0.00143
0.6041
3.3782
0.5303
0.0123
75%
0.6542
3.4646
0.6937
0.0377
Posterior Intervals
Parameter
Alpha
Equal-Tail
Interval
p
beta0
beta1
s2g
0.050
0.050
0.050
0.050
0.4745
3.1165
0.1769
6.374E-6
0.7893
3.6251
1.0318
0.0567
HPD
0.4730
3.1118
0.1737
2.009E-6
Interval
0.7859
3.6191
1.0022
0.0530
41
Summary of results
( Testing of Group Variable )
Method
Estimate
Std. Error
Log rank test
P value
0.0124
NLMIXED (
Unweighted )
0.5585
0.2088
0.0318
LIFEREG
( weighted level 1 No
random effect
0.5374
0.1783
0.0026
NLMIXED ( Both level
weights; Level I
Method I )
0.5625
0.0518
(P < .0001)
0.1826
Jackknife std error
NLMIXED ( Scaling
Method II )
0.4589
0.0429
Jackknife
PROC MCMC
(P<.0001)
?
0.5476
0.1280
Credible Interval
( (0.1769, 1.0318)
42
Questions?
43