Hierarchical Multiple Regression Sample Size Formulas

Hierarchical Multiple Regression Sample Size Formulas

Below you will find descriptions and details for the 9 formulas that are used to compute sample sizes for hierarchical multiple regression studies.

Beta function:

Cumulative distribution function (CDF) for the F-distribution:

Variable definitions:
d₁ and d₂ = the degrees of freedom
I = regularized lower incomplete beta function

Cumulative distribution function (CDF) for the noncentral F-distribution:

Variable definitions:
d₁ and d₂ = the numerator and denominator degrees of freedom
λ = noncentrality parameter
F = Fisher F-value
I = regularized lower incomplete beta function

Cumulative distribution function (CDF) for the normal distribution:

Variable definitions:
µ = mean
σ = standard deviation
erf = error function

Effect size for hierarchical multiple regression studies:

Variable definitions:
R²_A = variance accounted for by a set of one or more independent variables A
R²_AB = combined variance accounted for by A and another set of one or more independent variables B

Error function:

Lower incomplete beta function:

Noncentrality parameter for the noncentral F-distribution:

Variable definitions:
f² = effect size
n = sample size

Regularized lower incomplete beta function:

Variable definitions:
the numerator = lower incomplete beta function
the denominator = beta function