getDesignRepeatedANOVA: Power and Sample Size for Repeated-Measures ANOVA

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Power and Sample Size for Repeated-Measures ANOVA

Description

Obtains the power and sample size for one-way repeated measures analysis of variance. Each subject takes all treatments in the longitudinal study.

Usage

getDesignRepeatedANOVA(
  beta = NA_real_,
  n = NA_real_,
  ngroups = 2,
  means = NA_real_,
  stDev = 1,
  corr = 0,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

beta	The type II error.
n	The total sample size.
ngroups	The number of treatment groups.
means	The treatment group means.
stDev	The total standard deviation.
corr	The correlation among the repeated measures.
rounding	Whether to round up sample size. Defaults to 1 for sample size rounding.
alpha	The two-sided significance level. Defaults to 0.05.

Details

Let y_{ij} denote the measurement under treatment condition j (j=1,\ldots,k) for subject i (i=1,\ldots,n). Then

y_{ij} = \alpha + \beta_j + b_i + e_{ij},

where b_i denotes the subject random effect, b_i \sim N(0, \sigma_b^2), and e_{ij} \sim N(0, \sigma_e^2) denotes the within-subject residual. If we set \beta_k = 0, then \alpha is the mean of the last treatment (control), and \beta_j is the difference in means between the jth treatment and the control for j=1,\ldots,k-1.

The repeated measures have a compound symmetry covariance structure. Let \sigma^2 = \sigma_b^2 + \sigma_e^2, and \rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma_e^2}. Then Var(y_i) = \sigma^2 \{(1-\rho) I_k + \rho 1_k 1_k^T\}. Let X_i denote the design matrix for subject i. Let \theta = (\alpha, \beta_1, \ldots, \beta_{k-1})^T. It follows that

Var(\hat{\theta}) = \left(\sum_{i=1}^{n} X_i^T V_i^{-1} X_i\right)^{-1}.

It can be shown that

Var(\hat{\beta}) = \frac{\sigma^2 (1-\rho)}{n} (I_{k-1} + 1_{k-1} 1_{k-1}^T).

It follows that \hat{\beta}^T \hat{V}_{\hat{\beta}}^{-1} \hat{\beta} \sim F_{k-1,(n-1)(k-1), \lambda}, where the noncentrality parameter for the F distribution is

\lambda = \beta^T V_{\hat{\beta}}^{-1} \beta = \frac{n \sum_{j=1}^{k} (\mu_j - \bar{\mu})^2}{\sigma^2(1-\rho)}.

Value

An S3 class designRepeatedANOVA object with the following components:

• power: The power to reject the null hypothesis that there is no difference among the treatment groups.

• alpha: The two-sided significance level.

• n: The number of subjects.

• ngroups: The number of treatment groups.

• means: The treatment group means.

• stDev: The total standard deviation.

• corr: The correlation among the repeated measures.

• effectsize: The effect size.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, [e-mail]

Examples

(design1 <- getDesignRepeatedANOVA(
  beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
  stDev = 5, corr = 0.2, alpha = 0.05))

lrstat documentation built on Oct. 18, 2024, 9:06 a.m.