Power Analysis for Mixed-Effects Analysis of Variance (F-Test)

Calculates power or sample size for mixed-effects ANOVA design with two factors (between and within). When there is only one group observed over time, this design is often referred to as repeated-measures ANOVA.

Formulas are validated using G*Power and tables in PASS documentation.

NOTE: The pwrss.f.rmanova() function is deprecated and will no longer be supported, but it will remain available as a wrapper for power.f.mixed.anova() during the transition period.

Usage

power.f.mixed.anova(eta.squared, null.eta.squared = 0,
                    factor.levels = c(2, 2),
                    factor.type = c("between", "within"),
                    rho.within = 0.50, epsilon = 1,
                    n.total = NULL, power = NULL, alpha = 0.05,
                    effect = c("between", "within", "interaction"),
                    ceiling = TRUE, verbose = TRUE, pretty = FALSE)

Arguments

eta.squared: (partial) eta-squared for the alternative.
null.eta.squared: (partial) eta-squared for the null.
rho.within: Correlation between repeated measures. For example, for pretest/post-test designs, this is the correlation between pretest and post-test scores regardless of group membership. The default is 0.50. If eta.squared is already adjusted for this correlation specify 'rho.within = NA'.
factor.levels: vector; integer; length of two representing the number of levels for groups and measures. For example, in randomized controlled trials with two arms (treatment and control) where pre-test, post-test, and follow-up test are administered, this would be represented as c(2, 3).
factor.type: vector; character; length of two indicating the order of between-subject and within-subject factors. By default, the first value represents the between-subject factor and the second value represents the within-subject factor. This argument is rarely needed, except when unsure which element in 'factor.levels' represent between-subject or within-subject factors. Therefore, specify the 'factor.levels' accordingly.
epsilon: non-sphericity correction factor, default is 1 (means no violation of sphericity). Lower bound for this argument is epsilon = 1 / (factor.levels[2] - 1).
n.total: integer; total sample size.
power: statistical power, defined as the probability of correctly rejecting a false null hypothesis, denoted as \(1 - \beta\).
alpha: type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\).
effect: character; the effect of interest: "between", "within", or "interaction".
ceiling: logical; TRUE by default. If FALSE sample size in each group is NOT rounded up.
verbose: logical; TRUE by default. If FALSE no output is printed on the console.
pretty: logical; whether the output should show Unicode characters (if encoding allows for it). FALSE by default.

Value

parms: list of parameters used in calculation.
test: type of the statistical test (F-Test).
df1: numerator degrees of freedom.
df2: denominator degrees of freedom.
ncp: non-centrality parameter under alternative.
null.ncp: non-centrality parameter under null.
power: statistical power \((1-\beta)\).
n.total: total sample size.

References

Bulus, M., & Polat, C. (2023). pwrss R paketi ile istatistiksel guc analizi [Statistical power analysis with pwrss R package]. Ahi Evran Universitesi Kirsehir Egitim Fakultesi Dergisi, 24(3), 2207-2328. doi:10.29299/kefad.1209913

Examples


######################################################
# pretest-post-test design with treatment group only  #
######################################################

# a researcher is expecting a difference of Cohen's d = 0.30
# between post-test and pretest score translating into
# Eta-squared = 0.022

# adjust effect size for correlation with 'rho.within'
power.f.mixed.anova(eta.squared = 0.022,
                    factor.levels = c(1, 2), # 1 between 2 within
                    rho.within = 0.50,
                    effect = "within",
                    alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Repeated Measures Analysis of Variance (F-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : eta.squared = 0 
#>   H1 (Alt. Claim) : eta.squared > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 90  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.196
#>   Statistical Power      = 0.804
#> 

# if effect size is already adjusted for correlation
# use 'rho.within = NA'
power.f.mixed.anova(eta.squared = 0.08255,
                    factor.levels = c(1, 2), # 1 between 2 within
                    rho.within = NA,
                    effect = "within",
                    alpha = 0.05, power = 0.80)
#> Warning: Assuming that 'eta.squared' and 'null.eta.squared' are already adjusted for within-subject correlation.
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Repeated Measures Analysis of Variance (F-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : eta.squared = 0 
#>   H1 (Alt. Claim) : eta.squared > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 90  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.196
#>   Statistical Power      = 0.804
#> 

##########################################################
# post-test only design with treatment and control groups #
##########################################################

# a researcher is expecting a difference of Cohen's d = 0.50
# on the post-test score between treatment and control groups
# translating into Eta-squared = 0.059
power.f.mixed.anova(eta.squared = 0.059,
                    factor.levels = c(2, 1),  # 2 between 1 within
                    effect = "between",
                    alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Analysis of Variance (F-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : eta.squared = 0 
#>   H1 (Alt. Claim) : eta.squared > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 128  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.197
#>   Statistical Power      = 0.803
#> 


#############################################################
# pretest-post-test design with treatment and control groups #
#############################################################

# a researcher is expecting a difference of Cohen's d = 0.40
# on the post-test score between treatment and control groups
# after controlling for the pretest translating into
# partial Eta-squared = 0.038
power.f.mixed.anova(eta.squared = 0.038,
                    factor.levels = c(2, 2),  # 2 between 2 within
                    rho.within = 0.50,
                    effect = "between",
                    alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Mixed-Effects Analysis of Variance (F-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : eta.squared = 0 
#>   H1 (Alt. Claim) : eta.squared > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 152  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.197
#>   Statistical Power      = 0.803
#> 

# a researcher is expecting an interaction effect
# (between groups and time) of Eta-squared = 0.01
power.f.mixed.anova(eta.squared = 0.01,
                    factor.levels = c(2, 2),  # 2 between 2 within
                    rho.within = 0.50,
                    effect = "interaction",
                    alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Mixed-Effects Analysis of Variance (F-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : eta.squared = 0 
#>   H1 (Alt. Claim) : eta.squared > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 198  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.196
#>   Statistical Power      = 0.804
#> 

# a researcher is expecting an interaction effect
# (between groups and time) of Eta-squared = 0.01
power.f.mixed.anova(eta.squared = 0.01,
                    factor.levels = c(2, 2),  # 2 between 2 within
                    rho.within = 0.50,
                    effect = "within",
                    alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Mixed-Effects Analysis of Variance (F-Test)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : eta.squared = 0 
#>   H1 (Alt. Claim) : eta.squared > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 198  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.196
#>   Statistical Power      = 0.804
#>