Obtains the power and sample size for one-way repeated measures analysis of variance. Each subject takes all treatments in the longitudinal study.
getDesignRepeatedANOVA(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
corr = 0,
rounding = TRUE,
alpha = 0.05
)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.
The type II error.
The total sample size.
The number of treatment groups.
The treatment group means.
The total standard deviation.
The correlation among the repeated measures.
Whether to round up sample size. Defaults to 1 for sample size rounding.
The two-sided significance level. Defaults to 0.05.
Kaifeng Lu, kaifenglu@gmail.com
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 $$\lambda = \beta^T V_{\hat{\beta}}^{-1} \beta = \frac{n \sum_{j=1}^{k} (\mu_j - \bar{\mu})^2}{\sigma^2(1-\rho)}.$$
(design1 <- getDesignRepeatedANOVA(
beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
stDev = 5, corr = 0.2, alpha = 0.05))
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