rbe(df; dvar::Symbol,
    subject::Symbol,
    formulation::Symbol,
    period::Symbol,
    sequence::Symbol,
    g_tol::Float64 = 1e-8, x_tol::Float64 = 0.0, f_tol::Float64 = 0.0,
    iterations::Int = 100,
    store_trace = false, extended_trace = false, show_trace = false,
    memopt = true,
    init = [],
    postopt = false, maxopttry = 100)

Mixed model fitting function for replicate bioequivalence without data preparation (apply categorical! for each factor and sort! to dataframe).

Mixed model in matrix form:

with covariance matrix for each subject:

Arguments

• df - DataFrame with data;

Keywords:

• dvar - dependent variable;
• subject - subject factor;
• formulation - formulation factor;
• period - period factor;
• sequence - sequence factor;
• g_tol = 1e-8
• x_tol = 0.0
• f_tol = 0.0
• iterations = 100 - maximum iteration for optimization
• store_trace = false
• extended_trace = false
• show_trace = false
• memopt = true - memory optimization (function cache)
• init = [] - initial variance paremeters
• postopt = false - post optimization

This function can convert non-categorical to categorical and sort dataframe.

It can takes more time, but can help to avoid some errors like: "ERROR: type ContinuousTerm has no field contrasts".

coef(rbe::RBE)

Return model coefficients.

confint(obj::RBE; level::Real=0.95, expci::Bool = false, inv::Bool = false, df = :sat)

Compute confidence intervals for coefficients, with confidence level level (by default 95%).

Arguments

• expci = true: return exponented CI.

• inv = true: return -estimate ± t(alpha, df)*SE

• df = :sat: use Satterthwaite DF approximation.

• df = :df3 or df = :cont: DF (contain) = N - rank(ZX).

dof(rbe::RBE)

Return the number of degrees of freedom for the coefficients of the model.

StatsBase.stderror(rbe::RBE)

Return the standard errors for the coefficients of the model.

where

coefnum(rbe::RBE)

Return number of coefficients (length β).

contrast(rbe::RBE, L::Matrix; name = "Contrast", memopt = true)::ContrastTable

Return contrast table for L matrix. Table include:

• F
• NumDF
• DF
• P|f|

where

DF for one-dimetion case:

where $A = 2H^{-1}$

where $g = \triangledown _{\theta}(LC_{\theta}L')$

DF for multi-dimention case see Schaalje et al 2002.

p value calculated with:

pval    = ccdf(FDist(numdf, df), F)

design(rbe::RBE)::Design

Return design information object.

estimate(rbe::RBE, L::Matrix; df = :sat, name = "Estimate", memopt = true, alpha = 0.05)

Return estimate table for L 1xp matrix. Table include:

• estimate value
• SE
• DF
• t
• P|t|
• CI Upper
• CI Lower

For df = :sat:

where

where $A = 2H^{-1}$

where $g = \triangledown _{\theta}(LC_{\theta}L')$

For df = :cont (contain):

CI estimate is:

Example of L matrix if length of fixed effect vector is 6, estimate for 4-th value:

L = [0 0 0 1 0 0]

fixed(rbe::RBE)

Return fixed effect table (β).

optstat(rbe::RBE)

Return optimization status.

• true - converged
• false - not converged

    randrbeds(;n=24, sequence=[1,1],
        design = ["T" "R" "T" "R"; "R" "T" "R" "T"],
        inter=[0.5, 0.4, 0.9], intra=[0.1, 0.2],
        intercept = 0, seqcoef = [0.0, 0.0], periodcoef = [0.0, 0.0, 0.0, 0.0],
        formcoef = [0.0, 0.0],
        dropobs::Int = 0, seed::Int = 0)

Random dataset generation for bioequivalence.

Keywords

• $n=24$ - number of subjects;
• $sequence = [1,1]$ - distribution in sequences [1,1] means 1:1, [1,3] - 1:4 etc.;
• $design = ["T" "R" "T" "R"; "R" "T" "R" "T"]$ - desin matrix, each line is a sequence, each column - periods, cell - formulation id;
• $inter=[0.5, 0.4, 0.9]$ - inter-subject variance vector for G matrix (length 3): [σ₁, σ₂, ρ], where σ₁, σ₂ - formulation inter-subject variance, ρ - covariance coefficient;
• $intra=[0.1, 0.2]$ - intra-subject variance vector for R matrix (length 2): [σ₁, σ₂], where σ₁, σ₂ - formulation intra-subject variance for each formulation;
• $intercept = 0$ - intercept effect value;
• $seqcoef = [0.0, 0.0]$ - coefficients of sequences, additive (length(sequence) == length(seqcoef) == size(design, 1));
• $periodcoef = [0.0, 0.0, 0.0, 0.0]$ - coefficients of periods, additive (length(periodcoef) == size(design, 2));
• $formcoef = [0.0, 0.0]$ - coefficients of formulations, additive ;
• $dropobs = 0$ number of randomly dropped observations;
• $seed = 0$ - seed for random number generator (0 - random seed).

Multivariate normal disribution:

Where V:

Another way to use:

    randrbeds(n::Int, sequence::Vector,
        design::Matrix,
        θinter::Vector, θintra::Vector,
        intercept::Real, seqcoef::Vector, periodcoef::Vector, formcoef::Vector,
        dropsubj::Float64, dropobs::Int, seed::Int)

Using with RandRBEDS object:

randrbeds(task::RandRBEDS)

randrbetask(;n=24, sequence=[1,1],
        design = ["T" "R" "T" "R"; "R" "T" "R" "T"],
        inter=[0.5, 0.4, 0.9], intra=[0.1, 0.2],
        intercept = 0, seqcoef = [0.0, 0.0], periodcoef = [0.0, 0.0, 0.0, 0.0], formcoef = [0.0, 0.0],
        dropsubj = 0, dropobs::Int = 0, seed = 0)::RandRBEDS

Make task for random dataset generation.

reml2(rbe::RBE, θ::Vector{T}) where T <: AbstractFloat

Returm -2logREML for rbe model with θ variance vector.

reml2(rbe::RBE)

Returm -2logREML for rbe model

-2 REML function for ForwardDiff

simulation(task::RandRBEDS; io = stdout, verbose = false, num = 100,
    l = log(0.8), u = log(1.25), seed = 0)

Count successful BE outcomes.

Parameters

• task - RandRBEDS object

Keywords

• $io = stdout$ - text output
• $verbose = false$ - print messages to io
• $num = 100$ - number of simulations
• $l = log(0.8)$ - lower bound
• $u = log(1.25)$ - upper bound
• $seed = 0$ - seed for random number generator (0 - random seed)

theta(rbe::RBE)

Return theta (θ) vector (vector of variation parameters from optimization procedure).

typeiii(rbe::RBE)

Return TYPE III table.

(see contrast)