Easy API Reference for BioequivalencePower

The key functions of the package are power, samplesize, power_analysis, adjusted_α, adjusted_α_with_samplesize, confint, and pvalue. Here are the doc strings for the easy-API methods of these functions.

Power

BioequivalencePower.power — Method
power(::Type{BS}, ::Type{D} = EDDefaultDesign, 
      ::Type{BW} = EmptyBayesianWrapper, ::Type{PP} = EmptyPriorParameters;
      kwargs...
      ) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign, BW <: BayesianWrapper, PP <:PriorParameters}

This is the easy-API power method. Its interface is designed to be as easy and simple to use as possible. However, since it encompasses all of the power functionality of the package, the docstring is long.

There are two positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

power(RatioTOST, ED2x2x3ReplicateCrossover, ...)

Another two optional positional arguments are the prior type BW and its parameters PP. Use these only if a you wish to compute expected power in the Bayesian context. For example: 

power(RatioTOST, ED2x2x3ReplicateCrossover, CVSEPrior, DegreesOfFreedomPriorParameters, ...)

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the number of subjects, n, and the variability parameter, CV (can also be SE in certain cases). The order of the keywords does not matter. Many other keywords are also possible (see below). Here is a minimal working example:

power(RatioTOST, ED2x2x3ReplicateCrossover, n = 20, CV = 0.32)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here, the number of subjects, n = 20 is automatically set in an HypothesisTestMetaParams object, and the coefficient of variation CV = 0.32 is set in a RatioTOST{ED2x2x3ReplicateCrossover} object. These objects are hidden from the user. The easy-API sees which arguments are supplied and constructs the objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: bioequivalence power study type. One of:

    • Standard (average) bioequivalence: RatioTOST or DifferenceTOST
    • Reference scaling and/or expanding limits: ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST
    • Tests for NTID (Narrow Therapeutic Index Drug): NarrowTherapeuticIndexDrugRatioTOST
    • Tests in the context of fiducial inference: FiducialRatioTOST
    • Tests with two end points: TwoDifferenceTOSTs or TwoRatioTOSTs
    • Dose proportionality: DoseProportionalityStudy
    • Non-inferiority tests: NonInferiorityRatioOST or NonInferiorityDifferenceOST

  • 2nd argument: design.

    • For the study types RatioTOST, DifferenceTOST, TwoDifferenceTOSTs, TwoRatioTOSTs, NonInferiorityRatioOST or NonInferiorityDifferenceOST, use one of: ED2Parallel, ED2x2Crossover, ED3x3Crossover, ED3x6x3Crossover, ED4x4Crossover,ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, ED2x4x4ReplicateCrossover, ED2x3x3PartialReplicate, ED2x4x2Balaam,ED2x2x2RepeatedCrossover, or EDPaired.
    • For the study types ExpandingLimitsRatioTOST or ReferenceScaledRatioTOST use one of: ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.
    • For the study type NarrowTherapeuticIndexDrugRatioTOST use one of ED2x2x3ReplicateCrossover or ED2x2x4ReplicateCrossover.
    • For the study type FiducialRatioTOST use one of ED2Parallel or ED2x2Crossover.
    • For the study type DoseProportionalityStudy use one of EDGeneralCrossover, EDGeneralParallel, or EDGeneral.

  • 3rd argument (optional) prior type:

    • This argument can only be used with study types RatioTOST, DifferenceTOST, NonInferiorityRatioOST, or NonInferiorityDifferenceOST and should be one of CVSEPrior, θ0Prior, or BothCVSEandθ0Prior, to indicate a prior on the variability parameter (CV or SE), on θ₀, or on both, respectively.
    • If this argument is set, you must set the 4th argument.

  • 4th argument (optional) prior distribution type:

    • This argument is relevant if and only if the 3rd argument (prior type) is set.
    • If the prior type is CVSEPrior use DegreesOfFreedomPriorParameters or CategoricalPriorParameters.
    • If the prior type is θ0Prior use StandardErrorPriorParameters or CategoricalPriorParameters.
    • If the prior type is BothCVSEandθ0Prior use DegreesOfFreedomAndStandardErrorPriorParameters or TwoWayCategoricalPriorParameters.

The keyword arguments following the initial arguments are always set in the form KEY = VALUE, where for example KEY can be n and value can be 20 for indicating the number of subjects. Here is a full list, starting with mandatory arguments and following with optional arguments.

Mandatory Arguments

  • n: Can be an integer denoting the total number of subjects or an array of the number of subjects per sequence group, e.g. n = [12, 8]. In the typical case where the total number of subjects is given, the subjects are distributed as uniformly as possible among sequences of the design using the distribute_groups function.
  • Variability arguments: The typical variability argument is CV, but in certain cases SE is used instead. In other cases secondary variability arguments are required or optional. Here is a complete list of variability arguments. Note that variability arguments are always positive and in certain cases they are given as a vector.
    • RatioTOST, DoseProportionalityStudy, and NonInferiorityRatioOST: Specify CV as a single value.
    • ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, and NarrowTherapeuticIndexDrugRatioTOST: Specify CV as a single value, or if wishing to consider heteroscedasticity, specify as a vector of length 2, with the first entry for the test formulation and the second entry for the reference formulation.
    • DifferenceTOST, and NonInferiorityDifferenceOST: Specify SE as a single value.
    • FiducialRatioTOST: Specify both SE and SEb, each as a single value.
    • TwoRatioTOSTs: Specify CV as a vector of length 2 matching the two end points of the study.
    • TwoDifferenceTOSTs: Specify SE as a vector of length 2 matching the two end points of the study.
    • DoseProportionalityStudy: Specify CV as a single value. Optionally, specify CVb as a single value (default value of CVb if not specified is twice CV.)
  • For DoseProportionalityStudy you must also specify the vector of doses via doses. This is a vector of integer values, e.g. doses = [1, 2, 8].
  • In case of a prior distribution (3rd and 4th arguments used) there are other mandatory arguments. See specific description below.

Argument for "test/reference" ratio or difference

A key input to the power computation is the ratio or difference between the test and reference formulation. That is, power is computed based on such a postulated point. It is typically denoted θ₀ with variable name peθ₀, except for the context of dose proportionality studies where it is denoted β₀ with variable name peβ₀. All of the study types set default for this argument, and you can naturally override that default.

Ratio formulation defaults:

  • peθ₀ for RatioTOST, FiducialRatioTOST, NonInferiorityRatioOST: Default is at 0.95.
  • peθ₀ for ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST: Default is at 0.9.
  • peθ₀ for NarrowTherapeuticIndexDrugRatioTOST: Default is at 0.975 if highly_variable = false (default), and at 0.95 otherwise (the HVNTID case).
  • peθ₀ for TwoRatioTOSTs: Default is at the vector [0.95, 0.95].

Difference formulation defaults:

  • peθ₀ for DifferenceTOST: Default is at 0.05.
  • peθ₀ for NonInferiorityDifferenceOST: Default is at -0.05.
  • peθ₀ for TwoDifferenceTOSTs: Default is at the vector [0.05, 0.05].

Dose proportionality default:

  • peβ₀ for DoseProportionalityStudy is set at 1 + log(0.95)/log(maximum(doses) / minimum(doses)).

Argument for limits

Most of the study types have set limits values θ₁ and θ₂ with θ₁ < θ₂. The non-inferiority types has the single limit θmargin, and the two simultaneous end points studies have θ₁ and θ₂, each as a vector of length 2. Default values are set for all of these, and you can set your own values to override the defaults. Note that in most cases θ₂ is defined in terms of θ₁. Hence you may set θ₁ and as a result, θ₂ will be adjusted accordingly, or you may set only θ₂, or set both.

  • For RatioTOST, ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, NarrowTherapeuticIndexDrugRatioTOST, FiducialRatioTOST,and DoseProportionalityStudy we have that θ₁ and θ₂ are each scalar values with defaults θ₁ = 0.8 and θ₂ = 1/θ₁ (this is 1.25).
  • For DifferenceTOST we have that θ₁ and θ₂ are each scalar values with defaults θ₁ = -0.2 and θ₂ = -θ₁ (this is 0.2).
  • For TwoRatioTOSTs and TwoDifferenceTOSTs we have that θ₁ and θ₂ are each vectors of length 2 with default individual entries corresponding to RatioTOST and DifferenceTOST. For example for TwoRatioTOSTs we have θ₁ = [0.8, 0.8] and θ₂ = 1 ./ θ₁.
  • For NonInferiorityRatioOST the default value is θmargin = 0.8.
  • For NonInferiorityDifferenceOST the default value is θmargin = -0.2.

Optional Arguments for hypothesis test meta parameters (including assumptions)

  • α: By default, α = 0.05 for most studies with the exception of FiducialRatioTOST, NonInferiorityRatioOST, and NonInferiorityDifferenceOST where it is set at 0.025 (note that this adjusted default is only applicable via the easy-API and not via the complete-API).
  • n_g: This is the number of groups in the study (it is the number of location sites and should not to be confused with the number of sequences in a design). By default n_g = 1. For standard (average) bioequivalence formulated as a ratio, namely RatioTOST, we may use other values of n_g if carrying out subject level simulations (subject_level = true). See more details with subject_level below.
  • robust: Set this to true for a determination of degrees of freedom using a robust formulation. Default is false.
  • test_group_effect: Set this to true for the specific case RatioTOST with n_g set to greater than one and subeject_level = true. In this case a model for testing group effect is first executed. Default is false.

Special arguments for reference scaling and/or expanding limits

  • reg_symbol: This is the symbol for the RegulatoryConstants object set in either ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST studies. For the former the default is :fda and for the latter it is :ema. You may set other regulatory constants objects if you wish.

Special arguments for narrow therapeutic drug index products

  • highly_variable: This boolean value with default false determines if the study is an NTDI or HVNTDI study. The case highly_variable = true yields different behavior for the hypothesis test.
  • regulatory_constant: Default is -log(0.9)/0.10 (which is about 1.053605), used only in highly_variable = false.
  • ratio_threshold: Default is 2.5.

Special arguments for two simultaneous end points

  • ρ: When using TwoRatioTOSTs or TwoDifferenceTOSTs, this is the assumed correlation coefficient between the two endpoints. Default is 0.0 and it can be set to any value such that -1 ≤ ρ ≤ 1.

Special arguments for dose proportionality study

  • design_matrix: Use this only with DoseProportionalityStudy and EDGeneral specified in the 1st and 2nd arguments respectively. In this case design_matrix should be set to an EDGeneral design object which encapsulates a design_matrix that can be specified.

Arguments for simulation and computation

Note that the arguments num_sims, seed, and subject_level are specific to a case where Monte Carlo power calculation is used. In study types where exact power calculation is used by default and Monte Carlo is also supported, the default is overridden and the calculation is changed to Monte Carlo if one of these arguments is set. In other cases, where Monte Carlo is already the default, setting these parameters just updates the nature of the Monte Carlo.

The study types with default as exact calculation and an option for Monte Carlo are: RatioTOST, DifferenceTOST.

The study types with only numerical calculation are: DoseProportionalityStudy (note that this is an approximate calculation), NonInferiorityRatioOST, and NonInferiorityDifferenceOST.

The study types with only Monte Carlo are: ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, NarrowTherapeuticIndexDrugRatioTOST, FiducialRatioTOST, TwoRatioTOSTs, and TwoDifferenceTOSTs.

When using Monte Carlo, these are the possible arguments.

  • num_sims: This is the number of simulation runs to execute. The default is 1e5.
  • seed: This is the simulation seed to use for reproducibility. The default value is an arbitrary 1984 value, yet can be set to any integer value. Setting seed = nothing, implies no random seed is set.
  • details: Set this to true to output additional simulation and/or computation output. Default is false.
  • subject_level: This argument is by default false and this implies simulations are based on randomly generating statistics. If set to true then it means that each simulation run randomly generates sample data and uses this random data to estimate the power. In general, subject_level = true is slower, yet in certain cases it yields better accuracy. You may set subject_level = true only for the following cases:
    • RatioTOST, yet only with one of the designs ED2x2Crossover, ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.
    • ReferenceScaledRatioTOST with any of the supported designs of the study.
    • ExpandingLimitsRatioTOST with any of the supported designs of the study.

Arguments for Bayesian computation

When setting the 3rd and 4th parameters for Bayesian computation, you must also supply prior parameters. See the Priors on parameters in power studies section of the docs for more details.

Return value

The return value is always an HypothesisTestMetaParams object. The achieved_power field specifies the power.

Sample size

BioequivalencePower.samplesize — Method
samplesize(::Type{BS}, ::Type{D} = EDDefaultDesign, 
           ::Type{BW} = EmptyBayesianWrapper, ::Type{PP} = EmptyPriorParameters;
           kwargs...
           ) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign, BW <: BayesianWrapper, PP <:PriorParameters}

This is the easy-API samplesize method. Its interface is designed to be as easy and simple to use as possible. However, since it encompasses all of the sample size functionality of the package, the docstring is long.

There are two mandatory positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

samplesize(RatioTOST, ED2x2x3ReplicateCrossover, ...)

Another two optional positional arguments are the prior type BW and its parameters PP. Use these only if a you wish to compute sample size using expected power in the Bayesian context. For example: 

samplesize(RatioTOST, ED2x2x3ReplicateCrossover, CVSEPrior, DegreesOfFreedomPriorParameters, ...)

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the variability parameter, CV (can also be SE in certain cases). The order of the keywords does not matter. Many other keywords are also possible (see below). Here is a minimal working example:

samplesize(RatioTOST, ED2x2x3ReplicateCrossover, CV = 0.32)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here, behind the scenes, CV = 0.32 is set in a RatioTOST{ED2x2x3ReplicateCrossover} object which is passed to a samplesize method for such objects. These details are hidden from the user. The easy-API sees which arguments are supplied and constructs the objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: bioequivalence power study type. One of:

    • Standard (average) bioequivalence: RatioTOST or DifferenceTOST
    • Reference scaling and/or expanding limits: ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST
    • Tests for NTID (Narrow Therapeutic Index Drug): NarrowTherapeuticIndexDrugRatioTOST
    • Tests in the context of fiducial inference: FiducialRatioTOST
    • Tests with two end points: TwoDifferenceTOSTs or TwoRatioTOSTs
    • Dose proportionality: DoseProportionalityStudy
    • Non-inferiority tests: NonInferiorityRatioOST or NonInferiorityDifferenceOST

  • 2nd argument: design.

    • For the study types RatioTOST, DifferenceTOST, TwoDifferenceTOSTs, TwoRatioTOSTs, NonInferiorityRatioOST or NonInferiorityDifferenceOST, use one of: ED2Parallel, ED2x2Crossover, ED3x3Crossover, ED3x6x3Crossover, ED4x4Crossover,ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, ED2x4x4ReplicateCrossover, ED2x3x3PartialReplicate, ED2x4x2Balaam,ED2x2x2RepeatedCrossover, or EDPaired.
    • For the study types ExpandingLimitsRatioTOST or ReferenceScaledRatioTOST use one of: ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.
    • For the study type NarrowTherapeuticIndexDrugRatioTOST use one of ED2x2x3ReplicateCrossover or ED2x2x4ReplicateCrossover.
    • For the study type FiducialRatioTOST use one of ED2Parallel or ED2x2Crossover.
    • For the study type DoseProportionalityStudy use one of EDGeneralCrossover, EDGeneralParallel, or EDGeneral.

  • 3rd argument (optional) prior type:

    • This argument can only be used with study types RatioTOST, DifferenceTOST, NonInferiorityRatioOST, or NonInferiorityDifferenceOST and should be one of CVSEPrior, θ0Prior, or BothCVSEandθ0Prior, to indicate a prior on the variability parameter (CV or SE), on θ₀, or on both, respectively.
    • If this argument is set, you must set the 4th argument.

  • 4th argument (optional) prior distribution type:

    • This argument is relevant if and only if the 3rd argument (prior type) is set.
    • If the prior type is CVSEPrior use DegreesOfFreedomPriorParameters or CategoricalPriorParameters.
    • If the prior type is θ0Prior use StandardErrorPriorParameters or CategoricalPriorParameters.
    • If the prior type is BothCVSEandθ0Prior use DegreesOfFreedomAndStandardErrorPriorParameters or TwoWayCategoricalPriorParameters.

The keyword arguments following the initial arguments are always set in the form KEY = VALUE, where for example KEY can be CV and value can be 0.32. Here is a full list, starting with mandatory arguments and following with optional arguments.

Mandatory Arguments

  • Variability arguments: The typical variability argument is CV, but in certain cases SE is used instead. In other cases secondary variability arguments are required or optional. Here is a complete list of variability arguments. Note that variability arguments are always positive and in certain cases they are given as a vector.
    • RatioTOST, DoseProportionalityStudy, and NonInferiorityRatioOST: Specify CV as a single value.
    • ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, and NarrowTherapeuticIndexDrugRatioTOST: Specify CV as a single value, or if wishing to consider heteroscedasticity, specify as a vector of length 2, with the first entry for the test formulation and the second entry for the reference formulation.
    • DifferenceTOST, and NonInferiorityDifferenceOST: Specify SE as a single value.
    • FiducialRatioTOST: Specify both SE and SEb, each as a single value.
    • TwoRatioTOSTs: Specify CV as a vector of length 2 matching the two end points of the study.
    • TwoDifferenceTOSTs: Specify SE as a vector of length 2 matching the two end points of the study.
    • DoseProportionalityStudy: Specify CV as a single value. Optionally, specify CVb as a single value (default value of CVb if not specified is twice CV.)
  • For DoseProportionalityStudy you must also specify the vector of doses via doses. This is a vector of integer values, e.g. doses = [1, 2, 8].
  • In case of a prior distribution (3rd and 4th arguments used) there are other mandatory arguments. See specific description below.

Argument for "test/reference" ratio or difference

A key input to the sample size computation is the ratio or difference between the test and reference formulation. That is, sample size is computed based on such a postulated point. It is typically denoted θ₀ with variable name peθ₀, except for the context of dose proportionality studies where it is denoted β₀ with variable name peβ₀. All of the study types set default for this argument, and you can naturally override that default.

Ratio formulation defaults:

  • peθ₀ for RatioTOST, FiducialRatioTOST, NonInferiorityRatioOST: Default is at 0.95.
  • peθ₀ for ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST: Default is at 0.9.
  • peθ₀ for NarrowTherapeuticIndexDrugRatioTOST: Default is at 0.975 if highly_variable = false (default), and at 0.95 otherwise (the HVNTID case).
  • peθ₀ for TwoRatioTOSTs: Default is at the vector [0.95, 0.95].

Difference formulation defaults:

  • peθ₀ for DifferenceTOST: Default is at 0.05.
  • peθ₀ for NonInferiorityDifferenceOST: Default is at -0.05.
  • peθ₀ for TwoDifferenceTOSTs: Default is at the vector [0.05, 0.05].

Dose proportionality default:

  • peβ₀ for DoseProportionalityStudy is set at 1 + log(0.95)/log(maximum(doses) / minimum(doses)).

Argument for limits

Most of the study types have set limits values θ₁ and θ₂ with θ₁ < θ₂. The non-inferiority types has the single limit θmargin, and the two simultaneous end points studies have θ₁ and θ₂, each as a vector of length 2. Default values are set for all of these, and you can set your own values to override the defaults. Note that in most cases θ₂ is defined in terms of θ₁. Hence you may set θ₁ and as a result, θ₂ will be adjusted accordingly, or you may set only θ₂, or set both.

  • For RatioTOST, ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, NarrowTherapeuticIndexDrugRatioTOST, FiducialRatioTOST,and DoseProportionalityStudy we have that θ₁ and θ₂ are each scalar values with defaults θ₁ = 0.8 and θ₂ = 1/θ₁ (this is 1.25).
  • For DifferenceTOST we have that θ₁ and θ₂ are each scalar values with defaults θ₁ = -0.2 and θ₂ = -θ₁ (this is 0.2).
  • For TwoRatioTOSTs and TwoDifferenceTOSTs we have that θ₁ and θ₂ are each vectors of length 2 with default individual entries corresponding to RatioTOST and DifferenceTOST. For example for TwoRatioTOSTs we have θ₁ = [0.8, 0.8] and θ₂ = 1 ./ θ₁.
  • For NonInferiorityRatioOST the default value is θmargin = 0.8.
  • For NonInferiorityDifferenceOST the default value is θmargin = -0.2.

Optional Arguments for hypothesis test meta parameters (including assumptions)

  • target_power: This is the target power for which sample size should be determined. Default value is 0.8. The function will search for n that has target_power ≤ achieved_power.
  • α: By default, α = 0.05 for most studies with the exception of FiducialRatioTOST, NonInferiorityRatioOST, and NonInferiorityDifferenceOST where it is set at 0.025 (note that this adjusted default is only applicable via the easy-API and not via the complete-API).
  • n_g: This is the number of groups in the study (it is the number of location sites and should not to be confused with the number of sequences in a design). By default n_g = 1. For standard (average) bioequivalence formulated as a ratio, namely RatioTOST, we may use other values of n_g if carrying out subject level simulations (subject_level = true). See more details with subject_level below.
  • robust: Set this to true for a determination of degrees of freedom using a robust formulation. Default is false.
  • test_group_effect: Set this to true for the specific case RatioTOST with n_g set to greater than one and subeject_level = true. In this case a model for testing group effect is first executed. Default is false.

Special arguments for reference scaling and/or expanding limits

  • reg_symbol: This is the symbol for the RegulatoryConstants object set in either ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST studies. For the former the default is :fda and for the latter it is :ema. You may set other regulatory constants objects if you wish.

Special arguments for narrow therapeutic drug index products

  • highly_variable: This boolean value with default false determines if the study is an NTDI or HVNTDI study. The case highly_variable = true yields different behavior for the hypothesis test.
  • regulatory_constant: Default is -log(0.9)/0.10 (which is about 1.053605), used only in highly_variable = false.
  • ratio_threshold: Default is 2.5.

Special arguments for two simultaneous end points

  • ρ: When using TwoRatioTOSTs or TwoDifferenceTOSTs, this is the assumed correlation coefficient between the two endpoints. Default is 0.0 and it can be set to any value such that -1 ≤ ρ ≤ 1.

Special arguments for dose proportionality study

  • design_matrix: Use this only with DoseProportionalityStudy and EDGeneral specified in the 1st and 2nd arguments respectively. In this case design_matrix should be set to an EDGeneral design object which encapsulates a design_matrix that can be specified.

Arguments for simulation and computation

Note that the arguments num_sims, seed, and subject_level are specific to a case where Monte Carlo sample size calculation is used (this means the power calculation used by samplesize uses Monte Carlo). In study types where exact power calculation is used by default and Monte Carlo is also supported, the default is overridden and the calculation is changed to Monte Carlo if one of these arguments is set. In other cases, where Monte Carlo is already the default, setting these parameters just updates the nature of the Monte Carlo.

The study types with default as exact calculation and an option for Monte Carlo are: RatioTOST, DifferenceTOST.

The study types with only numerical calculation are: DoseProportionalityStudy (note that this is an approximate calculation), NonInferiorityRatioOST, and NonInferiorityDifferenceOST.

The study types with only Monte Carlo are: ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, NarrowTherapeuticIndexDrugRatioTOST, FiducialRatioTOST, TwoRatioTOSTs, and TwoDifferenceTOSTs.

When using Monte Carlo, these are the possible arguments.

  • num_sims: This is the number of simulation runs to execute for each individual power calculation. The default is 1e5.
  • seed: This is the simulation seed to use for reproducibility. The default value is an arbitrary 1984 value, yet can be set to any integer value. Setting seed = nothing, implies no random seed is set. Warning: sample size search without a fixed seed is not recommended..
  • details: Set this to true to output additional simulation and/or computation output. Default is false.
  • subject_level: This argument is by default false and this implies simulations are based on randomly generating statistics. If set to true then it means that each simulation run randomly generates sample data and uses this random data to estimate the power. In general, subject_level = true is slower, yet in certain cases it yields better accuracy. You may set subject_level = true only for the following cases:
    • RatioTOST, yet only with one of the designs ED2x2Crossover, ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.
    • ReferenceScaledRatioTOST with any of the supported designs of the study.
    • ExpandingLimitsRatioTOST with any of the supported designs of the study.

Arguments for Bayesian computation

When setting the 3rd and 4th parameters for Bayesian computation, you must also supply prior parameters. See the Priors on parameters in power studies section of the docs for more details.

Counting iterations

An additional argument is count_iter with default value false. If set to true the return value is a tuple where the second argument is the number of power computations computed until finding the sample size.

Return value

The return value is an HypothesisTestMetaParams object. In this object, n is the determined sample size and the achieved_power field specifies the power obtained with such a sample size. If count_iter = true, the return value is a tuple with the first argument being an HypothesisTestMetaParams object, and the second argument being the number of iterations.

Power Analysis

BioequivalencePower.power_analysis — Method
power_analysis(::Type{BS}, ::Type{D} = EDDefaultDesign; 
               kwargs...
               ) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign}

This is the easy-API power_analysis method. Its interface is designed to be as easy and simple to use as possible. However, since it encompasses all of the power_analysis functionality of the package, the docstring is long.

There are two mandatory positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

power_analysis(RatioTOST, ED2x2x3ReplicateCrossover, ...)

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the variability parameter, CV. The order of the keywords does not matter. Many other keywords are also possible (see below). Here is a minimal working example:

power_analysis(RatioTOST, ED2x2x3ReplicateCrossover, CV = 0.32)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here the coefficient of variation CV = 0.32 is set in a RatioTOST{ED2x2x3ReplicateCrossover} object. This object is hidden from the user. The easy-API sees which arguments are supplied and constructs objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: bioequivalence power study type (note that not only some of the BioequivalencePowerStudy sub-types are supported). One of:

    • Standard (average) bioequivalence: RatioTOST
    • Reference scaling and/or expanding limits: ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST
    • Tests for NTID (Narrow Therapeutic Index Drug): NarrowTherapeuticIndexDrugRatioTOST

  • 2nd argument: design.

    • For the study type RatioTOST use one of: ED2Parallel, ED2x2Crossover, ED3x3Crossover, ED3x6x3Crossover, ED4x4Crossover,ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, ED2x4x4ReplicateCrossover, ED2x3x3PartialReplicate, ED2x4x2Balaam,ED2x2x2RepeatedCrossover, or EDPaired.
    • For the study types ExpandingLimitsRatioTOST or ReferenceScaledRatioTOST use one of: ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.
    • For the study type NarrowTherapeuticIndexDrugRatioTOST use one of ED2x2x3ReplicateCrossover or ED2x2x4ReplicateCrossover.

The keyword arguments following the initial two arguments are always set in the form KEY = VALUE, where for example KEY can be CV and value can be 0.32. Here is a full list, starting with mandatory arguments and following with optional arguments.

Mandatory Arguments

  • Variability argument: This is CV. It should always be a single positive value. Note that vector CV for heteroscedasticity is not supported as part of power_analysis.

Argument for "test/reference" ratio

A key input to the power analysis is the ratio between the test and reference formulation. That is, power is computed based on such a postulated point, and that point is varied. It is denoted θ₀ with variable name peθ₀. All of the study types set default for this argument, and you can naturally override that default.

  • peθ₀ for RatioTOST: Default is at 0.95.
  • peθ₀ for ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST: Default is at 0.9.
  • peθ₀ for NarrowTherapeuticIndexDrugRatioTOST: Default is at 0.975 if highly_variable = false (default), and at 0.95 otherwise (the HVNTID case).

Argument for limits

These are set limits values θ₁ and θ₂ with θ₁ < θ₂. Default values are set for all of these with defaults θ₁ = 0.8 and θ₂ = 1/θ₁ (this is 1.25). You can set your own values to override the defaults. Note that in most cases θ₂ is defined in terms of θ₁. Hence you may set θ₁ and as a result, θ₂ will be adjusted accordingly, or you may set only θ₂, or set both.

Optional Arguments for hypothesis test meta parameters (including assumptions)

  • target_power: This is the target power for which sample size should be determined. Default value is 0.8. When initially determining the sample size, power_analysis will search for n that has target_power ≤ achieved_power.
  • α: By default, α = 0.05.
  • robust: Set this to true for a determination of degrees of freedom using a robust formulation. Default is false.

Special arguments for reference scaling and/or expanding limits

  • reg_symbol: This is the symbol for the RegulatoryConstants object set in either ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST studies. For the former the default is :fda and for the latter it is :ema. You may set other regulatory constants objects if you wish.

Special arguments for narrow therapeutic drug index products

  • highly_variable: This boolean value with default false determines if the study is an NTDI or HVNTDI study. The case highly_variable = true yields different behavior for the hypothesis test.
  • regulatory_constant: Default is -log(0.9)/0.10 (which is about 1.053605), used only in highly_variable = false.
  • ratio_threshold: Default is 2.5.

Arguments for simulation and computation

Note that the arguments num_sims, seed, and subject_level are specific to a case where Monte Carlo power calculation is used. In study types where exact power calculation is used by default and Monte Carlo is also supported, the default is overridden and the calculation is changed to Monte Carlo if one of these arguments is set. In other cases, where Monte Carlo is already the default, setting these parameters just updates the nature of the Monte Carlo.

In the context of power_analysis, the study type with default as exact calculation and an option for Monte Carlo is: RatioTOST.

Similarly, the study types with only Monte Carlo are: ReferenceScaledRatioTOST, ExpandingLimitsRatioTOST, and NarrowTherapeuticIndexDrugRatioTOST.

When using Monte Carlo, these are the possible arguments.

  • num_sims: This is the number of simulation runs to execute for each individual power calculation. The default is 1e5.
  • seed: This is the simulation seed to use for reproducibility. The default value is an arbitrary 1984 value, yet can be set to any integer value. Setting seed = nothing, implies no random seed is set. Warning: sample size search or power analysis without a fixed seed is not recommended..
  • details: Set this to true to output additional simulation and/or computation output. Default is false.
  • subject_level: This argument is by default false and this implies simulations are based on randomly generating statistics. If set to true then it means that each simulation run randomly generates sample data and uses this random data to estimate the power. In general, subject_level = true is slower, yet in certain cases it yields better accuracy. You may set subject_level = true only for the following cases:
    • ReferenceScaledRatioTOST with any of the supported designs of the study.
    • ExpandingLimitsRatioTOST with any of the supported designs of the study.

Return value

The return value is a tuple that represents the power analysis information.

Adjusted α

BioequivalencePower.adjusted_α — Method
adjusted_α(::Type{BS}, ::Type{D} = EDDefaultDesign;
           kwargs...
           ) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign}

This is the easy-API adjusted_α method. Its interface is designed to be as easy and simple to use as possible. However, since it encompasses all of the adjusted α functionality of the package, the docstring is long.

There are two mandatory positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

adjusted_α(ReferenceScaledRatioTOST, ED2x2x4ReplicateCrossover, ...)

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the variability parameter, CV. The order of the keywords does not matter. Many other keywords are also possible (see below). Here is a minimal working example:

adjusted_α(ReferenceScaledRatioTOST, ED2x2x4ReplicateCrossover, CV = 0.27)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here, the coefficient of variation CV = 0.27 is set in a ReferenceScaledRatioTOST{ED2x2x4ReplicateCrossover} object. This object is hidden from the user. The easy-API sees which arguments are supplied and constructs objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: bioequivalence power study type (note that not only Reference scaling and/or expanding limits BioequivalencePowerStudy sub-types are supported). Hence choose one of: ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST.

  • 2nd argument: design. Use one of: ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.

The keyword arguments following the initial arguments are always set in the form KEY = VALUE, where for example KEY can be CV and value can be 0.27. Here is a full list, starting with mandatory argument and following with optional arguments.

Mandatory Arguments

  • Variability argument: This is CV or if wishing to consider heteroscedasticity, specify as a vector of length 2, with the first entry for the test formulation and the second entry for the reference formulation.

Argument for "test/reference" ratio

A key input is the ratio between the test and reference formulation. That is, power is computed based on such a postulated point. It is denoted θ₀ with variable name peθ₀. Default is at 0.9 and you can naturally override that default.

Argument for limits

These are set limits values θ₁ and θ₂ with θ₁ < θ₂. Default values are set for all of these with defaults θ₁ = 0.8 and θ₂ = 1/θ₁ (this is 1.25). You can set your own values to override the defaults. Note that in most cases θ₂ is defined in terms of θ₁. Hence you may set θ₁ and as a result, θ₂ will be adjusted accordingly, or you may set only θ₂, or set both.

Type I error and α

  • α: By default, α = 0.05. This is the desired type I error.
  • pre_specified_α: Default is nothing. You may set this value and then the function will compute sample size using this pre specified value.

Further optional Arguments for hypothesis test meta parameters (including assumptions)

  • n: Can be an integer denoting the total number of subjects or an array of the number of subjects per sequence group, e.g. n = [12, 8]. In the typical case where the total number of subjects is given, the subjects are distributed as uniformly as possible among sequences of the design using the distribute_groups function. If this argument is not given, adjusted_α calls samplesize to obtain this value.

Special arguments for reference scaling and/or expanding limits

  • reg_symbol: This is the symbol for the RegulatoryConstants object set in either ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST studies. For the former the default is :fda and for the latter it is :ema. You may set other regulatory constants objects if you wish.

Arguments for simulation

Note that the arguments num_sims, seed, and subject_level are for Monte Carlo power calculation

When using Monte Carlo, these are the possible arguments.

  • num_sims: This is the number of simulation runs to execute. The default is 1e5.
  • seed: This is the simulation seed to use for reproducibility. The default value is an arbitrary 1984 value, yet can be set to any integer value. Setting seed = nothing, implies no random seed is set. Warning: sample size search or power analysis without a fixed seed is not recommended..
  • details: Set this to true to output additional simulation and/or computation output. Default is false.
  • subject_level: This argument is by default false and this implies simulations are based on randomly generating statistics. If set to true then it means that each simulation run randomly generates sample data and uses this random data to estimate the power. In general, subject_level = true is slower, yet in certain cases it yields better accuracy.

Return value

In case no adjustment is made, the return value is a named tuple similar to this:

(before_adjustment = (α = 0.05, type_I_error = 0.03331, achieved_power = 0.82244, n = [13, 13]), 
after_adjustment = :no_adjustment_made)

In case an adjustment is made, the return value is a named tuple similar to this:

(before_adjustment = (α = 0.05, type_I_error = 0.08269, achieved_power = 0.80225, n = [15, 15]), 
after_adjustment = (α = 0.028163596987724294, type_I_error = 0.05, achieved_power = 0.71536, n = [15, 15]))
BioequivalencePower.adjusted_α_with_samplesize — Method
adjusted_α_with_samplesize(::Type{BS}, ::Type{D} = EDDefaultDesign;
                           kwargs...
                           ) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign}

This is the easy-API adjusted_α_with_samplesize method.

There are two mandatory positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

adjusted_α_with_samplesize(ExpandingLimitsRatioTOST, ED2x3x3PartialReplicate, ...)

At the moment the only supported study type for BS is ExpandingLimitsRatioTOST.

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the variability parameter, CV. The order of the keywords does not matter. Many other keywords are also possible (see below). Here is a minimal working example:

adjusted_α_with_samplesize(ExpandingLimitsRatioTOST, ED2x3x3PartialReplicate, CV = 0.35)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here, the coefficient of variation CV = 0.35 is set in a ExpandingLimitsRatioTOST{ED2x3x3PartialReplicate} object. This object is hidden from the user. The easy-API sees which arguments are supplied and constructs objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: Must be ExpandingLimitsRatioTOST.

  • 2nd argument: design. Use one of: ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, or ED2x3x3PartialReplicate.

The keyword arguments following the initial arguments are always set in the form KEY = VALUE, where for example KEY can be CV and value can be 0.35. Here is a full list, starting with mandatory argument and following with optional arguments.

Mandatory Arguments

  • Variability argument: This is CV or if wishing to consider heteroscedasticity, specify as a vector of length 2, with the first entry for the test formulation and the second entry for the reference formulation.

Argument for "test/reference" ratio

A key input is the ratio between the test and reference formulation. That is, power is computed based on such a postulated point. It is denoted θ₀ with variable name peθ₀. Default is at 0.9 and you can naturally override that default.

Argument for limits

These are set limits values θ₁ and θ₂ with θ₁ < θ₂. Default values are set for all of these with defaults θ₁ = 0.8 and θ₂ = 1/θ₁ (this is 1.25). You can set your own values to override the defaults. Note that in most cases θ₂ is defined in terms of θ₁. Hence you may set θ₁ and as a result, θ₂ will be adjusted accordingly, or you may set only θ₂, or set both.

Type I error and α

  • α: By default, α = 0.05. This is the desired type I error.

Further optional Arguments for hypothesis test meta parameters (including assumptions)

  • n: Can be an integer denoting the total number of subjects or an array of the number of subjects per sequence group, e.g. n = [12, 8]. In the typical case where the total number of subjects is given, the subjects are distributed as uniformly as possible among sequences of the design using the distribute_groups function. If this argument is not given, adjusted_α_with_samplesize calls samplesize to obtain this value.

Special arguments for expanding limits

  • reg_symbol: This is the symbol for the RegulatoryConstants object set in either ReferenceScaledRatioTOST or ExpandingLimitsRatioTOST studies. For the former the default is :fda and for the latter it is :ema. You may set other regulatory constants objects if you wish.

Arguments for simulation

Note that the arguments num_sims, seed, and subject_level are for Monte Carlo power calculation

When using Monte Carlo, these are the possible arguments.

  • num_sims: This is the number of simulation runs to execute. The default is 1e5.
  • seed: This is the simulation seed to use for reproducibility. The default value is an arbitrary 1984 value, yet can be set to any integer value. Setting seed = nothing, implies no random seed is set. Warning: sample size search or power analysis without a fixed seed is not recommended..
  • details: Set this to true to output additional simulation and/or computation output. Default is false.
  • subject_level: This argument is by default false and this implies simulations are based on randomly generating statistics. If set to true then it means that each simulation run randomly generates sample data and uses this random data to estimate the power. In general, subject_level = true is slower, yet in certain cases it yields better accuracy.

Other arguments

The argument max_iter with a default value of 10 determines an upper bound on the number of adjusted_α function calls.

Return value

The same as adjusted_α yet with a corrected sample size if needed.

Confidence Interval

StatsAPI.confint — Method
confint(::Type{BS}, ::Type{D} = EDDefaultDesign;
        kwargs...
        ) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign}

This is the easy-API confint method.

There are two mandatory positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

confint(RatioTOST, ED2x2x3ReplicateCrossover, ...)

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the number of subjects, n, the variability parameter, CV (can also be SE in certain cases), and the point estimate peθ₀ (note that the point estimate has a default value when we use the complete-API but with the easy-API we purposefully do not have a default value). The order of the keywords does not matter. Several other keywords are also possible (see below). Here is a minimal working example:

confint(RatioTOST, ED2x2x3ReplicateCrossover, n = 20, CV = 0.32, peθ₀ = 0.9)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here, the number of subjects, n = 20 is automatically set in an HypothesisTestMetaParams object, and the coefficient of variation CV = 0.32 is set in a RatioTOST{ED2x2x3ReplicateCrossover} object, similarly for peθ₀. These objects are hidden from the user. The easy-API sees which arguments are supplied and constructs the objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: bioequivalence power study type. One of:

    • Standard (average) bioequivalence: RatioTOST or DifferenceTOST
    • Tests in the context of fiducial inference: FiducialRatioTOST

  • 2nd argument: design.

    • For the study types RatioTOST or DifferenceTOST, use one of: ED2Parallel, ED2x2Crossover, ED3x3Crossover, ED3x6x3Crossover, ED4x4Crossover,ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, ED2x4x4ReplicateCrossover, ED2x3x3PartialReplicate, ED2x4x2Balaam,ED2x2x2RepeatedCrossover, or EDPaired.
    • For the study type FiducialRatioTOST use one of ED2Parallel or ED2x2Crossover.

The keyword arguments following the initial arguments are always set in the form KEY = VALUE, where for example KEY can be n and value can be 20 for indicating the number of subjects. Here is a full list, starting with mandatory arguments and following with optional arguments.

Mandatory Arguments

  • n: Can be an integer denoting the total number of subjects or an array of the number of subjects per sequence group, e.g. n = [12, 8]. In the typical case where the total number of subjects is given, the subjects are distributed as uniformly as possible among sequences of the design using the distribute_groups function.
  • Variability arguments: The typical variability argument is CV, but in certain cases SE is used instead. In other cases secondary variability arguments are required. Note that variability arguments are always positive.
    • RatioTOST: Specify CV as a single value.
    • DifferenceTOST: Specify SE as a single value.
    • FiducialRatioTOST: Specify both SE and SEb, each as a single value.
  • peθ₀: This is the observed point estimate of the "test/reference" ratio or difference.

Optional Arguments for hypothesis test meta parameters (including assumptions)

  • α: By default, α = 0.05 unless it using FiducialRatioTOST, where it is set at 0.025 (note that this adjusted default is only applicable via the easy-API and not via the complete-API).
  • robust: Set this to true for a determination of degrees of freedom using a robust formulation. Default is false.

Return value

The return value is a tuple that has the lower and upper limits of the confidence interval.

p-value

HypothesisTests.pvalue — Method
pvalue(::Type{BS}, ::Type{D} = EDDefaultDesign;
       kwargs...) where {BS <: BioequivalencePowerStudy, D <: ExperimentalDesign}

This is the easy-API pvalue method.

There are two mandatory positional arguments, first the study type BS and then the design type D. Type names are used for these arguments. For example:

pvalue(RatioTOST, ED2x2x3ReplicateCrossover, ...)

After these positional arguments, the remaining arguments are indicated in the method signature via kwargs (key word arguments). With these keywords we always need to set the number of subjects, n, the variability parameter, CV or SE, and the point estimate peθ₀ (note that the point estimate has a default value when we use the complete-API but with the easy-API we purposefully do not have a default value). The order of the keywords does not matter. Several other keywords are also possible (see below). Here is a minimal working example:

pvalue(RatioTOST, ED2x2x3ReplicateCrossover, n = 20, CV = 0.32, peθ₀ = 0.9)

The essence of the easy-API is that the it uses the complete-API under the hood, yet presents a simple interface. For example here, the number of subjects, n = 20 is automatically set in an HypothesisTestMetaParams object, and the coefficient of variation CV = 0.32 is set in a RatioTOST{ED2x2x3ReplicateCrossover} object, similarly for peθ₀. These objects are hidden from the user. The easy-API sees which arguments are supplied and constructs the objects based on these arguments before invoking the complete-API internally.

Here is a full list of arguments:

  • 1st argument: bioequivalence power study type and only standard (average) bioequivalence is supported. Hence the argument is one of RatioTOST or DifferenceTOST.

  • 2nd argument: design. One of: ED2Parallel, ED2x2Crossover, ED3x3Crossover, ED3x6x3Crossover, ED4x4Crossover,ED2x2x3ReplicateCrossover, ED2x2x4ReplicateCrossover, ED2x4x4ReplicateCrossover, ED2x3x3PartialReplicate, ED2x4x2Balaam,ED2x2x2RepeatedCrossover, or EDPaired.

The keyword arguments following the initial arguments are always set in the form KEY = VALUE, where for example KEY can be n and value can be 20 for indicating the number of subjects. Here is a full list, starting with mandatory arguments and following with optional arguments.

Mandatory Arguments

  • n: Can be an integer denoting the total number of subjects or an array of the number of subjects per sequence group, e.g. n = [12, 8]. In the typical case where the total number of subjects is given, the subjects are distributed as uniformly as possible among sequences of the design using the distribute_groups function.
  • Variability arguments: The typical variability argument is CV, but in certain cases SE is used instead. Note that variability arguments are always positive.
    • RatioTOST: Specify CV as a single value.
    • DifferenceTOST: Specify SE as a single value.
  • peθ₀: This is the observed point estimate of the "test/reference" ratio or difference.

Optional Arguments for hypothesis test meta parameters (including assumptions)

  • robust: Set this to true for a determination of degrees of freedom using a robust formulation. Default is false.

Additional arguments

  • both_p_values: Set this to true if you wish for the return value to be both the p-values (from each of the one sided tests that make up a TOST). Default is 'false'.

Return value

The return value is a p-value, unless both_p_values = true in which case the return value is a tuple of two p-values.