Power and Sample Size for Repeated-Measures ANOVA — getDesignRepeatedANOVA • lrstat

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.

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.

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

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


(design1 <- getDesignRepeatedANOVA(
  beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
  stDev = 5, corr = 0.2, alpha = 0.05))
#>   alpha     power  n ngroups stDev corr effectsize
#> 1  0.05 0.9027338 83       4     5  0.2      0.175