## Summary

Analysis models define statistical methods (e.g., significance tests or descriptive statistics) that are applied to the study data in a clinical trial.

## Initialization

An analysis model can be initialized using the following command:

It is highly recommended to use this command to initialize an analysis model as it will simplify the process of specifying components of the data model, including the MultAdj, MultAdjProc, MultAdjStrategy, Test, Statistic objects.

## Components of an analysis model

After an AnalysisModel object has been initialized, components of the analysis model can be specified by adding objects to the model using the ‘+’ operator as shown below.

### Test object

#### Description

This object specifies a significance test that will be applied to one or more samples defined in a data model. A Test object is defined by the following four arguments:

• id defines the test’s unique ID (label).

• method defines the significance test.

• samples defines the IDs of the samples (defined in the data model) that the significance test is applied to.

• par defines the parameter(s) of the statistical test.

Several commonly used significance tests are already implemented in the Mediana package. In addition, the user can easily define custom significance tests (see User-defined functions). The built-in tests are listed below along with the required parameters that need to be included in the par argument:

• TTest: perform the two-sample t-test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• TTestNI: perform the non-inferiority two-sample t-test between the two samples defined in the samples argument. Required parameter: margin (positive non-inferiority margin). Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• WilcoxTest: perform the Wilcoxon-Mann-Whitney test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• PropTest: perform the two-sample test for proportions between the two samples defined in the samples argument. Optional parameter: yates (Yates’ continuity correction flag that is set to TRUE or FALSE) and larger (Larger value is expected in the second sample (TRUE or FALSE)).
• PropTestNI: perform the non-inferiority two-sample test for proportions between the two samples defined in the samples argument. Required parameter: margin (positive non-inferiority margin). Optional parameter: yates (Yates’ continuity correction flag that is set to TRUE or FALSE) and larger (Larger value is expected in the second sample (TRUE or FALSE)).
• FisherTest: perform the Fisher exact test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• GLMPoissonTest: perform the Poisson regression test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• GLMNegBinomTest: perform the Negative-binomial regression test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• LogrankTest: perform the Log-rank test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).
• OrdinalLogisticRegTest: perform an Ordinal logistic regression test between the two samples defined in the samples argument. Optional parameter: larger (Larger value is expected in the second sample (TRUE or FALSE)).

It needs to be noted that the significance tests listed above are implemented as one-sided tests and thus the sample order in the samples argument is important. In particular, the Mediana package assumes by default that a numerically larger value of the endpoint is expected in Sample 2 compared to Sample 1. Suppose, for example, that a higher treatment response indicates a beneficial effect (e.g., higher improvement rate). In this case Sample 1 should include control patients whereas Sample 2 should include patients allocated to the experimental treatment arm. The sample order needs to be reversed if a beneficial treatment effect is associated with a lower value of the endpoint (e.g., lower blood pressure), or alternatively (from version 1.0.6), the optional parameters larger must be set to FALSE to indicate that a larger value is expected on the first Sample.

Several Test objects can be added to an AnalysisModelobject.

For more information about the Test object, see the package’s documentation Test on the CRAN web site.

#### Example

Examples of Test objects:

Carry out the two-sample t-test:

# Placebo and Treatment samples were defined in the data model
Test(id = "Placebo vs treatment",
samples = samples("Placebo", "Treatment"),
method = "TTest")


Carry out the two-sample t-test with larger values expected in the first sample (from v1.0.6):

# Placebo and Treatment samples were defined in the data model
Test(id = "Placebo vs treatment",
samples = samples("Treatment", "Placebo"),
method = "TTest",
par = parameters(larger = FALSE))


#### Example

Examples of Test objects:

Carry out the two-sample t-test:

Carry out the two-sample t-test with larger values expected in the first sample (from v1.0.6):

Carry out the two-sample t-test for non-inferiority:

Carry out the two-sample t-test with pooled samples:

### Statistic object

#### Description

This object specifies a descriptive statistic that will be computed based on one or more samples defined in a data model. A Statistic object is defined by four arguments:

• id defines the descriptive statistic’s unique ID (label).

• method defines the type of statistic/method for computing the statistic.

• samples defines the samples (pre-defined in the data model) to be used for computing the statistic.

• par defines the parameter(s) of the statistic.

Several methods for computing descriptive statistics are already implemented in the Mediana package and the user can also define custom functions for computing descriptive statistics (see User-defined functions). These methods are shown below along with the required parameters that need to be defined in the par argument:

• MedianStat: compute the median of the sample defined in the samples argument.

• MeanStat: compute the mean of the sample defined in the samples argument.

• SdStat: compute the standard deviation of the sample defined in the samples argument.

• MinStat: compute the minimum value in the sample defined in the samples argument.

• MaxStat: compute the maximum value in the sample defined in the samples argument.

• DiffMeanStat: compute the difference of means between the two samples defined in the samples argument. Two samples must be defined.

• EffectSizeContStat: compute the effect size for a continuous endpoint. Two samples must be defined.

• RatioEffectSizeContStat: compute the ratio of two effect sizes for a continuous endpoint. Four samples must be defined.

• PropStat: compute the proportion of the sample defined in the samples argument.

• DiffPropStat: compute the difference of the proportions between the two samples defined in the samples argument. Two samples must be defined.

• EffectSizePropStat: compute the effect size for a binary endpoint. Two samples must be defined.

• RatioEffectSizePropStat: compute the ratio of two effect sizes for a binary endpoint. Four samples must be defined.

• HazardRatioStat: compute the hazard ratio of the two samples defined in the samples argument. Two samples must be defined. By default the Log-Rank method is used. Optional argument: method taking as value Log-Rank or Cox.

• EffectSizeEventStat: compute the effect size for a survival endpoint (log of the HR. Two samples must be defined. Two samples must be defined. By default the Log-Rank method is used. Optional argument: method taking as value Log-Rank or Cox.

• RatioEffectSizeEventStat: compute the ratio of two effect sizes for a survival endpoint based on the Log-Rank method. Four samples must be defined. By default the Log-Rank method is used. Optional argument: method taking as value Log-Rank or Cox.

• EventCountStat: compute the number of events observed in the sample(s) defined in the samples argument.

• PatientCountStat: compute the number of patients observed in the sample(s) defined in the samples argument

Several Statistic objects can be added to an AnalysisModel object.

For more information about the Statistic object, see the R documentation Statistic.

#### Example

Examples of Statistic objects:

Compute the mean of a single sample:

### MultAdjProc object

#### Description

This object specifies a multiplicity adjustment procedure that will be applied to the significance tests in order to protect the overall Type I error rate. A MultAdjProc object is defined by three arguments:

• proc defines a multiplicity adjustment procedure.

• par defines the parameter(s) of the multiplicity adjustment procedure (optional).

• tests defines the specific tests (defined in the analysis model) to which the multiplicity adjustment procedure will be applied.

If no tests are defined, the multiplicity adjustment procedure will be applied to all tests defined in the AnalysisModel object.

Several commonly used multiplicity adjustment procedures are included in the Mediana package. In addition, the user can easily define custom multiplicity adjustments. The built-in multiplicity adjustments are defined below along with the required parameters that need to be included in the par argument:

• BonferroniAdj: Bonferroni procedure. Optional parameter: weight (vector of hypothesis weights).

• HolmAdj: Holm procedure. Optional parameter: weight (vector of hypothesis weights).

• HochbergAdj: Hochberg procedure. Optional parameter: weight (vector of hypothesis weights).

• HommelAdj: Hommel procedure. Optional parameter: weight (vector of hypothesis weights).

• FixedSeqAdj: Fixed-sequence procedure.

• ChainAdj: Family of chain procedures. Required parameters: weight (vector of hypothesis weights) and transition (matrix of transition parameters).

• FallbackAdj: Fallback procedure. Required parameters: weight (vector of hypothesis weights).

• NormalParamAdj: Parametric multiple testing procedure derived from a multivariate normal distribution. Required parameter: corr (correlation matrix of the multivariate normal distribution). Optional parameter: weight (vector of hypothesis weights).

• ParallelGatekeepingAdj: Family of parallel gatekeeping procedures. Required parameters: family (vectors of hypotheses included in each family), proc (vector of procedure names applied to each family), gamma (vector of truncation parameters).

• MultipleSequenceGatekeepingAdj: Family of multiple-sequence gatekeeping procedures. Required parameters: family (vectors of hypotheses included in each family), proc (vector of procedure names applied to each family), gamma (vector of truncation parameters).

• MixtureGatekeepingAdj: Family of mixture-based gatekeeping procedures. Required parameters: family (vectors of hypotheses included in each family), proc (vector of procedure names applied to each family), gamma (vector of truncation parameters), serial (matrix of indicators), parallel (matrix of indicators).

Several MultAdjProc objects can be added to an AnalysisModelobject using the ‘+’ operator or by grouping them into a MultAdj object.

For more information about the MultAdjProc object, see the package’s documentation MultAdjProc.

#### Example

Examples of MultAdjProc objects:

Apply a multiplicity adjustment based on the chain procedure:

This procedure implementation is facilicated by the use of the FixedSeqAdj method intoduced in version 1.0.4.

Apply a multiple-sequence gatekeeping procedure:

### MultAdjStrategy object

#### Description

This object specifies a multiplicity adjustment strategy that can include several multiplicity adjustment procedures. A multiplicity adjustment strategy may be defined when the same Clinical Scenario Evaluation approach is applied to several clinical trials.

A MultAdjStrategy object serves as a wrapper for several MultAdjProc objects.

For more information about the MultAdjStrategy object, see the package’s documentation MultAdjStrategy.

#### Example

Example of a MultAdjStrategy object:

Perform complex multiplicity adjustments based on gatekeeping procedures in two clinical trials with three endpoints:

### MultAdj object

#### Description

This object can be used to combine several MultAdjProc or MultAdjStrategy objects and add them as a single object to an AnalysisModel object . This object is provided mainly for convenience and its use is optional. Alternatively, MultAdjProc or MultAdjStrategy objects can be added to an AnalysisModel object incrementally using the ‘+’ operator.

For more information about the MultAdj object, see the package’s documentation MultAdj.

#### Example

Example of a MultAdj object:

Perform Clinical Scenario Evaluation to compare three candidate multiplicity adjustment procedures: