| getDesignRepeatedANOVA {lrstat} | R Documentation |
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, 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))