idem: Inference in Randomized Controlled Trials with Death and Missingness

Chenguang Wang

2017-06-30

## Loading required package: idem
## Loading required package: Rcpp
## Loading required package: rstan
## Loading required package: ggplot2
## Loading required package: StanHeaders
## rstan (Version 2.15.1, packaged: 2017-04-19 05:03:57 UTC, GitRev: 2e1f913d3ca3)
## For execution on a local, multicore CPU with excess RAM we recommend calling
## rstan_options(auto_write = TRUE)
## options(mc.cores = parallel::detectCores())

Introduction

In randomized studies involving severely ill patients, functional outcomes are often unobserved due to missed clinic visits, premature withdrawal or death. It is well known that if these unobserved functional outcomes are not handled properly, biased treatment comparisons can be produced.

R package idem implement a procedure for comparing treatments that is based on the composite endpoint of both the functional outcome and survival. The procedure considers missing data imputation with a sensitivity analysis strategy to handle the unobserved functional outcomes not due to death.

Data accepted by idem

In dataset accepted by idem, each row should represent a subject with treatment assignment, baseline coveraites, baseline outcome, post-randomization outcomes and survival time.

The idem package provides dataset abc from ABC trial as an example data set.

head(abc);
##     AGE TRT SURV Y1 Y2
## 1 59.63   1  999 NA NA
## 2 66.89   0  999 49 52
## 3 59.70   1    1 NA NA
## 4 81.41   0   72 NA NA
## 5 66.52   1  999 51 45
## 6 40.27   0   65 NA NA

Basic setting

Parameters of most functions in idem are organized and passed by a list. These parameters include variable names in the dataset, endpoint specification, duration of the study, etc..

The function chkPara checks the specification in the list of parameters for errors:

 err.lst.var <- list(trt="TRT", outcome=c("Y1","Y2"),
                 y0=NULL, endfml="Y2", bounds=c(10,20),
                 duration=365);
 cat(imChkPars(abc, err.lst.var));
## Model specification in not valid. Please check the following: 
## ----No survival time specified
## ----Endpoint does not involve outcome
## ----Upper bound is bigger than some observed outcomes

Data visualization

idem provides several functions for the visualization of the data.

Spaghetti plot for completers

lst.var <- list(trt="TRT", surv="SURV", outcome=c("Y1","Y2"), y0=NULL,
                endp=c("Y2"), unitTime="days",
                trt.label = c("UC+SBT", "SAT+SBT"),
                cov=c("AGE"), endfml="Y2", duration=365, bounds=c(0,100));

imPlotCompleters(abc, lst.var);

Missing pattern frequency table

imMisTable(abc, lst.var);
##                 Y1         Y2         Control    Intervention
## Deaths on study ""         ""         "58 (62%)" "38 (41%)"  
## S=1             "Observed" "Observed" "18 (19%)" "32 (34%)"  
## S=2             "Observed" "Missing"  "8 (9%)"   "8 (9%)"    
## S=3             "Missing"  "Observed" "1 (1%)"   "0 (0%)"    
## S=4             "Missing"  "Missing"  "9 (10%)"  "15 (16%)"  
## Total           ""         ""         "94"       "93"

Missing pattern heatmap

imPlotMisPattern(abc, lst.var);

Kaplan-Meier curves

imPlotSurv(abc, lst.var);

Missing data imputation

Model fitting

The first step in missing data imputation is to fit the imputation model to data observed from the completers, i.e. the subjects who were alive at the end of the study without missing data. The result has class name IDEM.FIT, which will be passed to imputation functions.

rst.fit <- imFitModel(abc, lst.var);

Check convergence

The MCMC sampling is primarily done by rstan. It is suggested that the convergence of the MCMC chains should be checked. This can be done by the imImpSingle function which imputes missing data for an individual subject under the benchmark assumption.

rst.mixing <- imImpSingle(abc[1,], rst.fit, chains = 4, iter = 2000, warmup = 1000);
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rstan::traceplot(rst.mixing$rst.stan, "YMIS");

Imputation

The following code shows how to use imImpAll to get the imputed complete datasets under benchmark assmption delta=0 and for sensitivity analysis. We use 300 iterations to reduce the computation time.

rst.imp <- imImpAll(abc, rst.fit, deltas=c(-0.25,0,0.25),
                    normal=TRUE, chains = 4, iter = 300, warmup = 100);

The following results show the completed datasets:

tail(rst.imp$complete, n=10);
##          ID DELTA IMP   AGE TRT SURV Y1       Y2 ORGY1 ORGY2     ENDP
## 187.69  187  0.00   1 66.12   1  999 26 23.40310    26    NA 23.40310
## 187.12  187  0.00   2 66.12   1  999 26 28.00054    26    NA 28.00054
## 187.514 187  0.00   3 66.12   1  999 26 30.90857    26    NA 30.90857
## 187.156 187  0.00   4 66.12   1  999 26 31.66782    26    NA 31.66782
## 187.235 187  0.00   5 66.12   1  999 26 32.66910    26    NA 32.66910
## 187.412 187  0.25   1 66.12   1  999 26 37.89598    26    NA 37.89598
## 187.184 187  0.25   2 66.12   1  999 26 30.55835    26    NA 30.55835
## 187.317 187  0.25   3 66.12   1  999 26 27.68812    26    NA 27.68812
## 187.232 187  0.25   4 66.12   1  999 26 34.78296    26    NA 34.78296
## 187.446 187  0.25   5 66.12   1  999 26 32.07419    26    NA 32.07419

Plot denisity of the imputed data

Function imPlotImputed plots the densities of the imputed outcomes and the imputed functional endpoint.

imPlotImputed(rst.imp, deltas = c(-0.25,0,0.25), xlim=c(0,100), endp=FALSE);

imPlotImputed(rst.imp, deltas=c(-0.25,0,0.25), xlim=c(0,100), endp=TRUE);

Composite endpoint analysis

Plot the cumulative distribution of the compositve endpoint

Function imPlotComposite generates the treatment-specific cumulative distribution functions of the composite endpoint, where the values of the composite endpoint are labeled according to the survival time and functional endpoint among survivors.

imPlotComposite(rst.imp, delta=0);
## Loading required package: DBI
## Loading required package: tcltk

Estimation of \(\theta\) and quantiles

Given a complete dataset with imputed outcomes, idem estimates treatment effect and quantiles of the composite endpoint using function imEstimate:

rst.est <- imEstimate(rst.imp, quantiles=c(0.25,0.5,0.75));
print(rst.est$theta);
##   Delta0 Delta1       Theta
## 1  -0.25  -0.25 -0.17359872
## 2   0.00  -0.25 -0.12656143
## 3   0.25  -0.25 -0.09055136
## 4  -0.25   0.00 -0.23816060
## 5   0.00   0.00 -0.18371082
## 6   0.25   0.00 -0.13676504
## 7  -0.25   0.25 -0.29874171
## 8   0.00   0.25 -0.25989476
## 9   0.25   0.25 -0.21496225
print(rst.est$quantiles);
##    Delta TRT    Q   QuantY QuantSurv
## 3  -0.25   0 0.25       NA        14
## 8  -0.25   0 0.50       NA        72
## 13 -0.25   0 0.75 34.20458        NA
## 18 -0.25   1 0.25       NA        61
## 23 -0.25   1 0.50 20.88373        NA
## 28 -0.25   1 0.75 39.30426        NA
## 33  0.00   0 0.25       NA        14
## 38  0.00   0 0.50       NA        72
## 43  0.00   0 0.75 38.57076        NA
## 48  0.00   1 0.25       NA        61
## 53  0.00   1 0.50 30.00000        NA
## 58  0.00   1 0.75 43.00000        NA
## 63  0.25   0 0.25       NA        14
## 68  0.25   0 0.50       NA        72
## 73  0.25   0 0.75 40.30951        NA
## 78  0.25   1 0.25       NA        61
## 83  0.25   1 0.50 34.00000        NA
## 88  0.25   1 0.75 48.32770        NA

Bootstrap analysis

idem evaluates the uncertainty of the estimated \(\theta\) and quantiles by bootstrap analysis.

For illustration, we run 2 bootstrap samples by the following code:

rst.boot <- imBs(rst.imp,
                 n.boot = 2,
                 n.cores = 1,
                 quantiles = c(0.25,0.5,0.75));
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
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##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Adjust your expectations accordingly!
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## WARNING: There aren't enough warmup iterations to fit the
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.16 seconds.
## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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##          three stages of adaptation as currently configured.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
## Adjust your expectations accordingly!
## 
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##          three stages of adaptation as currently configured.
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## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
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## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
## 
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##          three stages of adaptation as currently configured.
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## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
## 
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Adjust your expectations accordingly!
## 
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## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Adjust your expectations accordingly!
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## WARNING: There aren't enough warmup iterations to fit the
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
## 
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
## 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## Adjust your expectations accordingly!
## 
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
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## 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
## Adjust your expectations accordingly!
## 
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##          three stages of adaptation as currently configured.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## WARNING: There aren't enough warmup iterations to fit the
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
## Adjust your expectations accordingly!
## 
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## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
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## WARNING: There aren't enough warmup iterations to fit the
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Gradient evaluation took 5e-06 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
## 
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##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Gradient evaluation took 1.5e-05 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
## Adjust your expectations accordingly!
## 
## 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## 
## Gradient evaluation took 4e-06 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## 
## Gradient evaluation took 4e-06 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
##          Reducing each adaptation stage to 15%/75%/10% of
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Gradient evaluation took 1.4e-05 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
##          Reducing each adaptation stage to 15%/75%/10% of
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Gradient evaluation took 3e-06 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Gradient evaluation took 1.3e-05 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
## Adjust your expectations accordingly!
## 
## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## 
## Gradient evaluation took 1.3e-05 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
## Adjust your expectations accordingly!
## 
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## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## 
## WARNING: There aren't enough warmup iterations to fit the
##          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
## Adjust your expectations accordingly!
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## WARNING: There aren't enough warmup iterations to fit the
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
## 
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## Adjust your expectations accordingly!
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## Adjust your expectations accordingly!
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Hypothesis testing

Hypothesis testing results and the confidence intervals of and quantiles of the composite endpoint are obtained by summarizing the results from the bootstrap analysis and the analysis on the original dataset:

rst.final <- imTest(rst.boot);
print(rst.final);
## $theta
##   Delta0 Delta1       Theta          SD        PValue
## 1  -0.25  -0.25 -0.17359872 0.014608040  1.437197e-32
## 2   0.00  -0.25 -0.12656143 0.002410407  0.000000e+00
## 3   0.25  -0.25 -0.09055136 0.009140136  3.881098e-23
## 4  -0.25   0.00 -0.23816060 0.024767341  6.849876e-22
## 5   0.00   0.00 -0.18371082 0.006745905 2.646842e-163
## 6   0.25   0.00 -0.13676504 0.009722516  6.072579e-45
## 7  -0.25   0.25 -0.29874171 0.024282024  8.727312e-35
## 8   0.00   0.25 -0.25989476 0.009431326 3.694202e-167
## 9   0.25   0.25 -0.21496225 0.004804638  0.000000e+00
## 
## $quantiles
##    Delta TRT    Q   QuantY QuantSurv   BSQ2.5  BSQ97.5 BSQ2.5_Surv
## 3  -0.25   0 0.25       NA        14 13.00000 13.00000           1
## 8  -0.25   0 0.50       NA        72 91.00000 91.00000           1
## 13 -0.25   0 0.75 34.20458        NA 37.00000 37.00000           0
## 18 -0.25   1 0.25       NA        61 69.00000 69.00000           1
## 23 -0.25   1 0.50 20.88373        NA 25.75686 25.75686           0
## 28 -0.25   1 0.75 39.30426        NA 41.00000 41.00000           0
## 33  0.00   0 0.25       NA        14 13.00000 13.00000           1
## 38  0.00   0 0.50       NA        72 91.00000 91.00000           1
## 43  0.00   0 0.75 38.57076        NA 39.67093 39.67093           0
## 48  0.00   1 0.25       NA        61 69.00000 69.00000           1
## 53  0.00   1 0.50 30.00000        NA 36.00000 36.00000           0
## 58  0.00   1 0.75 43.00000        NA 45.24981 45.24981           0
## 63  0.25   0 0.25       NA        14 13.00000 13.00000           1
## 68  0.25   0 0.50       NA        72 91.00000 91.00000           1
## 73  0.25   0 0.75 40.30951        NA 42.91333 42.91333           0
## 78  0.25   1 0.25       NA        61 69.00000 69.00000           1
## 83  0.25   1 0.50 34.00000        NA 40.00000 40.00000           0
## 88  0.25   1 0.75 48.32770        NA 48.51478 48.51478           0
##    BSQ97.5_Surv
## 3             1
## 8             1
## 13            0
## 18            1
## 23            0
## 28            0
## 33            1
## 38            1
## 43            0
## 48            1
## 53            0
## 58            0
## 63            1
## 68            1
## 73            0
## 78            1
## 83            0
## 88            0
## 
## $survivor
##   Delta0 Delta1       Diff         SD        PValue
## 1  -0.25  -0.25  -3.870369 3.00788014  1.981839e-01
## 2  -0.25   0.00   8.092589 1.35737646  2.492702e-09
## 3  -0.25   0.25  24.828136 3.21537497  1.154632e-14
## 4   0.00  -0.25 -13.247132 4.41963005  2.723471e-03
## 5   0.00   0.00  -1.284175 0.05437345 2.536524e-123
## 6   0.00   0.25  15.451373 1.80362507  0.000000e+00
## 7   0.25  -0.25 -21.213851 5.20882597  4.647640e-05
## 8   0.25   0.00  -9.250894 0.84356937  5.545399e-28
## 9   0.25   0.25   7.484654 1.01442915  1.605382e-13
## 
## $lst.var
## $lst.var$trt
## [1] "TRT"
## 
## $lst.var$surv
## [1] "SURV"
## 
## $lst.var$outcome
## [1] "Y1" "Y2"
## 
## $lst.var$y0
## NULL
## 
## $lst.var$endp
## [1] "Y2"
## 
## $lst.var$unitTime
## [1] "days"
## 
## $lst.var$trt.label
## [1] "UC+SBT"  "SAT+SBT"
## 
## $lst.var$cov
## [1] "AGE"
## 
## $lst.var$endfml
## [1] "Y2"
## 
## $lst.var$duration
## [1] 365
## 
## $lst.var$bounds
## [1]   0 100
## 
## 
## attr(,"class")
## [1] "IDEM.TEST"

A contour plot of p-values in the sensitivity analysis results can be generated by plotContour:

imPlotContour(rst.final, nlevels = 30, con.v=0.05, zlim=c(0, 0.05));

Graphical user interface (GUI)

The idem package provides a web-based GUI for composite endpoint analysis. The GUI can be accessed by

imShiny();