`## Loading required package: idem`

`## Loading required package: Rcpp`

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.

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
```

There are four major steps in conducting imputation and inference using **idem**. First, a class *IDEMDATA* object should be generated by the function *imData*. Second, the imputation models will be fit to the data observed from the completers by the function *imFitModel*. Third, imputation can be conducted by the function *imImpAll*. Lastly, treatment effect estimation and hypothesis testing can be performed by function *imInfer*.

In this step, the original dataset with specification parameters will be combined and checked. These parameters include variable names in the dataset, endpoint specification, duration of the study, etc.. If there is mis-specification, error messages will be generated. Otherwise, a class *IDEMDATA* object will be generated with certain data visulation functions implemented as its S3 methods.

```
rst.data <- imData(abc, trt="TRT", outcome=c("Y1","Y2"), y0=NULL,
endfml="Y3", bounds=c(10,20), duration=365,
err.terminate = FALSE);
print(rst.data);
```

```
## Model specification is invalid. Please check the following:
## No survival time specified
## Endpoint formula error: Error in eval(substitute(expr), data, enclos = parent.frame()) : object 'Y3' not found
## Upper bound is smaller than some observed outcomes
```

```
rst.data <- imData(abc, trt="TRT", surv="SURV", outcome=c("Y1","Y2"),
y0=NULL, endfml="Y2",
trt.label = c("UC+SBT", "SAT+SBT"),
cov=c("AGE"), duration=365, bounds=c(0,100));
```

The class *IDEMDATA* provides S3 plot and summary methods with multiple options for the visualization of the data.

`plot(rst.data, opt = "survivor");`

`summary(rst.data, opt = "misstable");`

```
## Y1 Y2 UC.SBT SAT.SBT
## 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
```

`plot(rst.data, opt = "missing", cols = c("blue", "gray"));`

`plot(rst.data, opt = "KM");`

To fit the imputation model to data observed from the completers, i.e.Â the subjects who were alive at the end of the study without missing data, the class *IDEMDATA* object needs to be passed to the function *imFitModel* as parameters. The result has class name **IDEMFIT**, which will be passed to imputation functions.

`rst.fit <- imFitModel(rst.data);`

The goodness of fit diagnostics plots can be generated by the S3 plot method implemented for class *IDEMFIT*:

`plot(rst.fit, mfrow=c(2,4));`

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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```

`plot(rst.mixing);`

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

The result from *imIMPALL* is class *IDEMFIT*. Density plots the imputed outcomes and the imputed functional endpoint can be generated by the S3 plot method associated with class *IDEMFIT*.

`plot(rst.imp, opt = "imputed", deltas = c(-0.25,0,0.25), xlim=c(0,100), endp=FALSE);`

`plot(rst.imp, opt = "imputed", deltas = c(-0.25,0,0.25), xlim=c(0,100), endp=TRUE);`

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, can be plotted by the S3 plot method of class *IDEMFIT*.

`plot(rst.imp, delta=0);`

The function *imInfer* implements bootstrap analysis for hypothesis testing, point estimation and confidence intervals of the treatment effects.

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

`rst.test <- imInfer(rst.imp, n.boot = 2);`

```
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 3e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 4e-06 seconds
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## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 3e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 6e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 5e-06 seconds
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## Chain 3: Gradient evaluation took 5e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 1.4e-05 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 1.6e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.16 seconds.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 5e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 4e-06 seconds
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## Chain 4: Gradient evaluation took 3e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 6e-06 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## ---- Bootstrap 2
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 5e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 1.4e-05 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 1.5e-05 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 6e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 1.4e-05 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 1.4e-05 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: Gradient evaluation took 5e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 6e-06 seconds
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## Chain 3: Gradient evaluation took 4e-06 seconds
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## Chain 1: Gradient evaluation took 1.4e-05 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 3: Gradient evaluation took 4e-06 seconds
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: Gradient evaluation took 7e-06 seconds
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```

`rst.test`

```
##
## The sensitivity parameters considered were
## [1] -0.25 0.00 0.25
##
## Treatment effect (theta) under different
## sensitivity parameters are:
##
## Delta0 Delta1 Theta SD Q2.5 Q97.5
## [1,] -0.25 -0.25 -0.16266301 0.0008897477 -0.16440688 -0.16091913
## [2,] 0.00 -0.25 -0.13040494 0.0102078330 -0.15041193 -0.11039796
## [3,] 0.25 -0.25 -0.08679936 0.0013103557 -0.08936761 -0.08423111
## [4,] -0.25 0.00 -0.23088538 0.0133462158 -0.25704348 -0.20472728
## [5,] 0.00 0.00 -0.19281629 0.0296205106 -0.25087142 -0.13476116
## [6,] 0.25 0.00 -0.13534660 0.0210142235 -0.17653372 -0.09415948
## [7,] -0.25 0.25 -0.28931595 0.0014074191 -0.29207444 -0.28655746
## [8,] 0.00 0.25 -0.26039808 0.0126991266 -0.28528791 -0.23550825
## [9,] 0.25 0.25 -0.20901396 0.0116961382 -0.23193797 -0.18608995
## PValue
## [1,] 0.000000e+00
## [2,] 2.261832e-37
## [3,] 0.000000e+00
## [4,] 4.729365e-67
## [5,] 7.537466e-11
## [6,] 1.189128e-10
## [7,] 0.000000e+00
## [8,] 1.934890e-93
## [9,] 2.007705e-71
##
## Treatment effect (quantiles) under different
## sensitivity parameters are:
##
## Delta TRT Q QuantY QuantSurv Q2.5 Q97.5 Q2.5_Surv
## 3 -0.25 0 0.5 NA 72 236.00000 236.00000 1
## 8 -0.25 1 0.5 20.21470 NA 28.28173 28.28173 0
## 13 0.00 0 0.5 NA 72 236.00000 236.00000 1
## 18 0.00 1 0.5 29.02872 NA 34.00000 34.00000 0
## 23 0.25 0 0.5 NA 72 236.00000 236.00000 1
## 28 0.25 1 0.5 34.00000 NA 39.25126 39.25126 0
## Q97.5_Surv
## 3 1
## 8 0
## 13 1
## 18 0
## 23 1
## 28 0
##
##
## The hypothesis testing and confidence intervals are
## based on 2 bootstrap samples. Please consider more
## bootstrap samples (e.g. >100) for the validity
## of the results.
```

A contour plot of p-values in the sensitivity analysis results can be generated by the S3 method of the result returned by **imInfer**:

`plot(rst.test, nlevels = 30, con.v=0.05, zlim=c(0, 0.05));`

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

`imShiny();`