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

Source: R/anova.mixed.R
power.f.mixed.anova.Rd

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 the PASS documentation.

Usage

power.f.mixed.anova(
  eta.squared,
  null.eta.squared = 0,
  factor.levels = c(2, 2),
  factor.type = c("between", "within"),
  rho.within = 0.5,
  epsilon = 1,
  n.total = NULL,
  power = NULL,
  alpha = 0.05,
  effect = c("between", "within", "interaction"),
  ceiling = TRUE,
  verbose = 1,
  utf = FALSE
)

Arguments

eta.squared

(partial) eta-squared for the alternative.

null.eta.squared

(partial) eta-squared for the null.

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.

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.

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

1 by default (returns test, hypotheses, and results), if 2 a more detailed output is given (plus key parameters and definitions), if 0 no output is printed on the console.

utf

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.

Details

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

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. https://doi.org/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",
                    power = 0.80, alpha = 0.05)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Repeated Measures Analysis of Variance (F-Test)
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : eta.squared = 0
#>   H1 (Alternative) : 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",
                    power = 0.80, alpha = 0.05)
#> 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)        : eta.squared = 0
#>   H1 (Alternative) : 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",
                    power = 0.80, alpha = 0.05)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Analysis of Variance (F-Test)
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : eta.squared = 0
#>   H1 (Alternative) : 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",
                    power = 0.80, alpha = 0.05)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Mixed-Effects Analysis of Variance (F-Test)
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : eta.squared = 0
#>   H1 (Alternative) : 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",
                    power = 0.80, alpha = 0.05)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Mixed-Effects Analysis of Variance (F-Test)
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : eta.squared = 0
#>   H1 (Alternative) : 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",
                    power = 0.80, alpha = 0.05)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Mixed-Effects Analysis of Variance (F-Test)
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : eta.squared = 0
#>   H1 (Alternative) : eta.squared > 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Total Sample Size    = 198  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.196
#>   Statistical Power    = 0.804
#>