# SampleSize

Sample size estimation.

## Introduction

### Hypothesis

#### Equality

Two-sided hypothesis:

H₀: A - B = 0 Hₐ: A - B ≠ 0

100*(1 - α)% two-sided confidence interval used to demonstrate it.

#### Equivalence | Bioequivalence

Two-sided hypothesis or two one-sided hypothesis (TOST):

H₀: |A − B| ≥ δ Hₐ: |A − B| < δ

or

H₀: A - B ≤ δˡ || A - B ≥ δᵘ Hₐ: δˡ < A - B < δᵘ

100*(1 - 2*α)% two-sided confidence interval used to demonstrate it.

See Walker&Nowacki, 2011, Understanding Equivalence and Noninferiority Testing.

Chow at al. Sample Size Calculations in Clinical Research 3-rd ed. for sample size calculation used alpha level for testing corresponding 100\(1 - α)% two-sided confidence interval*

For bioequivalence clinical trial 90% (100\(1 - 2*α)%)* two-sided confidence interval is used (α = 0.05), but for therapeutic equivalence 95% two-sided confidence interval should be used (in this case use α = 0.025).

#### Non-inferiority | Superiority

One-sided hypothesis:

H₀: A − B ≤ δ Hₐ: A − B > δ

100*(1 - α)% one-sided confidence interval used to demonstrate it.

In clinical research α = 0.025 or less should be used, or 100*(1 - α)% two-sided confidence interval should be used.

### ctsamplen

ClinicalTrialUtilities.ctsamplenFunction
ctsamplen(;param::Symbol, type::Symbol, group::Symbol = :notdef,
alpha::Real = 0.05, beta::Real = 0.2, diff::Real = 0, sd::Real = 0,
a::Real = 0, b::Real = 0,
k::Real = 1, logscale::Bool = true)::TaskResult

Clinical trial sample size estimation.

param (Parameter type):

• :mean - Means;
• :prop - Proportions;
• :or - Odd Ratio;

type (Hypothesis type):

• :ea - Equality (two-sided hypothesis, 100*(1 - α)% two-sided CI);
• :ei - Equivalence (two-sided hypothesis, 100*(1 - 2*α)% two-sided CI);
• :ns - Non-Inferiority / Superiority (one-sided hypothesis, one side of 100*(1 -2*α)% two-sided CI);
• :mcnm - McNemar's Equality test;

group (group num):

• :one - One sample;
• :two - Two sample, result is for one group, second group size = n * k;

alpha - Alpha (o < α < 1) (default = 0.05);

beta - Beta (o < β < 1) (default = 0.2); power = 1 - β;

diff - difference/equivalence margin/non-inferiority/superiority margin;

sd - Standard deviation (σ, for Means only);

a - Group A (μ₀/p₀) - Test group;

b - Group B (μ₁/p₁) - Reference group or reference value;

k - Na/Nb (after sample size estimation second group size: Na=k*Nb, only for two sample design) (default=1);

logscale - diff is log transformed for OR:

• true - diff is already in log-scale, no transformation required (default);
• false - diff is not in log-scale, will be transformed;

### besamplen

ClinicalTrialUtilities.besamplenFunction
besamplen(;alpha::Real=0.05, beta::Real=0.2,
theta0::Real=0.95, theta1::Real=0.8, theta2::Real=1.25,
cv::Real=0.0, sd::Real=0.0, design::Symbol=:d2x2,
method::Symbol=:owenq, logscale::Bool=true)::TaskResult

Bioequivalence sample size estimation.

alpha - Alpha (o < α < 1) (default = 0.05);

beta - Beta (o < β < 1) (default=0.2); power = 1 - β;

theta0 - T/R Ratio (default=0.95);

theta1 - Lower Level (default=0.8);

theta2 - Upper level (default=1.25);

cv - coefficient of variation;

logscale - theta1, theta2, theta0: if true - make log transformation (default true);

design - trial design;

• :parallel
• :d2x2 (default)
• :d2x2x4
• :d2x4x4
• :d2x3x3
• :d2x4x2
• :d3x3
• :d3x6x3

method - calculating method: Owen's Q Function | NonCentral T | Shifted;

• :owenq (default)
• :nct
• :shifted