powerMediation (version 0.3.4)

powerInteract2by2: Power Calculation for Interaction Effect in 2x2 Two-Way ANOVA Given Effect Sizes

Description

Power calculation for interaction effect in 2x2 two-way ANOVA given effect sizes.

Usage

powerInteract2by2(n, tauBetaSigma, alpha = 0.05, nTests = 1, verbose = FALSE)

Arguments

n

integer. Number of subjects per group.

tauBetaSigma

Effect sizes ${\left(\tau \beta \right)}_{ij}/\sigma ,i=1,\dots ,a,j=1,\dots ,b$, where $a=b=2$ and $\sigma$ is the standard deviation of random error. Rows are for factor 1 and columns are for factor 2. Note that $\sum _{i=1}^a{\left(\tau \beta \right)}_{ij}=\sum _{j=1}^b{\left(\tau \beta \right)}_{ij}=0$. We can get ${\left(\tau \beta \right)}_{11}=\theta$, ${\left(\tau \beta \right)}_{12}=-\theta$, ${\left(\tau \beta \right)}_{21}=-\theta$, ${\left(\tau \beta \right)}_{22}=\theta$. So tauBetaSigma=$\theta /\sigma$

alpha

family-wise type I error rate.

nTests

integer. For high-throughput omics study, we perform two-way ANOVA for each of 'nTests' probes. We use Bonferroni correction to control for family-wise type I error rate. That is, for each probe, type I error rate would be alpha/nTests.

verbose

logical. Indicating if intermediate results should be printed out.

Value

A list with 5 elements:

power

the power of the two-way ANOVA test

df1

the first degree of freedom of the F test statistic (df1=(a-1)(b-1))

df2

the second degree of freedom of the F test statistic (df1=a*b(n-1))

F0

the rejection region boundary

ncp

the non-centrality parameter

Details

We assume the following model: $$y_{ijk}=\mu +{\tau }_i+{\beta }_j+{\left(\tau \beta \right)}_{ij}+{ϵ}_{ijk},$$ where $i=1,\dots ,a,j=1,\dots ,b,k=1,\dots ,n$, $\sum _{i=1}^a{\tau }_i=0$, $\sum _{j=1}^b{\beta }_j=0$, $\sum _{i=1}^a{\left(\tau \beta \right)}_{ij}=0$, $\sum _{j=1}^b{\left(\tau \beta \right)}_{ij}=0$, and ${ϵ}_{ijk}\stackrel{i.i.d}{\sim }N(0,{\sigma }^2)$.

The group means are $${\mu }_{ij}=\mu +{\tau }_i+{\beta }_j+{\left(\tau \beta \right)}_{ij},i=1\dots ,a,j=1,\dots ,b.$$ Note that $\mu =\sum _{i=1}^a\sum _{j=1}^b{\mu }_{ij}/(ab)$, ${\tau }_i=\sum _{j=1}^b{\mu }_{ij}/b-\mu$, and ${\beta }_j=\sum _{i=1}^a{\mu }_{ij}/a-\mu$.

The null hypothesis $H_0$: all ${\left(\tau \beta \right)}_{ij},i=1,\dots ,a,j=1,\dots ,b$ are equal to zero. The alternative hypothesis $H_a$: at least one ${\left(\tau \beta \right)}_{ij}$ is different from zero.

The F test statistic is $$F=M{S}_{AB}/M{S}_E\stackrel{H_a}{\sim }{F}_{(a-1)(b-1),ab(n-1),ncp},$$ where ncp is the non-centrality parameter of the F test statistic: $$ncp=n\sum _{i=1}^a\sum _{j=1}^b{\left[\frac{{\left(\tau \beta \right)}_{ij}}{\sigma }\right]}^2.$$

For the scenario $a=b=2$, we have ${\left(\tau \beta \right)}_{11}=\theta$, ${\left(\tau \beta \right)}_{12}=-\theta$, ${\left(\tau \beta \right)}_{21}=-\theta$, ${\left(\tau \beta \right)}_{22}=\theta$. Hence, the non-centrality parameter can be simplified to $$ncp=4n{\left(\frac{\theta }{\sigma }\right)}^2.$$

The power for testing the null hypothesis $H_0$ versus the alternative hypothesis $H_a$ is $$power=Pr(F>F_0|H_a),$$ where the rejection region boundary $F_0$ satisfies: $$Pr(F>F_0|H_0)=\alpha /nTests.$$

References

Chow SC, Shao J, and Wang H. Sample size calculations in clinical research. 2nd edition. Chapman & Hall/CRC. 2008

Montgomery DC. Design and Analysis of Experiments. 8th edition. John Wiley & Sons. Inc.

Examples

# NOT RUN {
n = 25
tauBetaSigma = 0.3

# power = 0.8437275
res2 = powerInteract2by2(n = n, tauBetaSigma = tauBetaSigma, 
    alpha = 0.05, nTests = 1, verbose = TRUE)

# }