Power Analysis for One-, Two-, Three-Way ANCOVA Using Means, Standard Deviations, and (Optionally) Contrasts (F test)
ancova.shieh.RdCalculates power or sample size for one-, two-, three-way ANCOVA. For factorial designs, use the argument factor.levels but note that unique combination of levels (cells in this case) should follow a specific order for the test of interaction. The order of marginal means and standard deviations is printed as a warning message.
Note that R has a partial matching feature which allows you to specify shortened versions of arguments, such as mu or mu.vec instead of mu.vector, or such as k or k.cov instead of k.covariates.
Formulas are validated using examples and tables in Shieh (2020).
Usage
power.f.ancova.shieh(mu.vector, sd.vector,
n.vector = NULL, p.vector = NULL,
factor.levels = length(mu.vector),
r.squared = 0, k.covariates = 1,
contrast.matrix = NULL,
power = NULL, alpha = 0.05,
ceiling = TRUE, verbose = TRUE,
pretty = FALSE)Arguments
- mu.vector
vector; adjusted means (or estimated marginal means) for each level of a factor.
- sd.vector
vector; unadjusted standard deviations for each level of a factor. If a pooled standard deviation is provided, repeat its value to match the number of group means. A warning will be issued if group standard deviations differ substantially beyond what is expected due to sampling error.
- n.vector
vector; sample sizes for each level of a factor.
- p.vector
vector; proportion of total sample size in each level of a factor. These proportions should sum to one.
- factor.levels
integer; number of levels or groups in each factor. For example, for two factors each having two levels or groups use e.g. c(2, 2), for three factors each having two levels or groups use e.g. c(2, 2, 2).
- r.squared
explanatory power of covariates (R-squared) in the ANCOVA model.
- k.covariates
integer; number of covariates in the ANCOVA model.
- contrast.matrix
vector or matrix; contrasts should not be confused with the model (design) matrix. Rows of contrast matrix indicate independent vector of contrasts summing to zero. The default contrast matrix is constructed using deviation coding scheme (a.k.a. effect coding). Columns in the contrast matrix indicate number of levels or groups (or cells in factorial designs).
- 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 \(\alpha\).
- ceiling
logical;
TRUEby default. IfFALSEsample size in each cell is NOT rounded up.- verbose
logical;
TRUEby default. IfFALSEno output is printed on the console.- pretty
logical; whether the output should show Unicode characters (if encoding allows for it).
FALSEby default.
Value
- parms
list of parameters used in calculation.
- test
type of the statistical test (F-Test)
- eta.squared
(partial) eta-squared.
- f
Cohen's f statistic.
- df1
numerator degrees of freedom.
- df2
denominator degrees of freedom.
- ncp
non-centrality parameter for the alternative.
- null.ncp
non-centrality parameter for the null.
- power
statistical power \((1-\beta)\).
- n.total
total sample size.
References
Shieh, G. (2020). Power analysis and sample size planning in ANCOVA designs. Psychometrika, 85(1), 101-120. doi:10.1007/s11336-019-09692-3
Examples
###################################################################
########################## main effect ##########################
###################################################################
# power for one-way ANCOVA (two levels or groups)
power.f.ancova.shieh(mu.vector = c(0.20, 0), # marginal means
sd.vector = c(1, 1), # unadjusted standard deviations
n.vector = c(150, 150), # sample sizes
r.squared = 0.50, # proportion of variance explained by covariates
k.covariates = 1, # number of covariates
alpha = 0.05)
#> +--------------------------------------------------+
#> | POWER CALCULATION |
#> +--------------------------------------------------+
#>
#> One-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 300
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.316
#> Statistical Power = 0.684 <<
#>
# sample size for one-way ANCOVA (two levels or groups)
power.f.ancova.shieh(mu.vector = c(0.20, 0), # marginal means
sd.vector = c(1, 1), # unadjusted standard deviations
p.vector = c(0.50, 0.50), # allocation, should sum to 1
r.squared = 0.50,
k.covariates = 1,
alpha = 0.05,
power = 0.80)
#> +--------------------------------------------------+
#> | SAMPLE SIZE CALCULATION |
#> +--------------------------------------------------+
#>
#> One-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 396 <<
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.199
#> Statistical Power = 0.801
#>
###################################################################
####################### interaction effect ######################
###################################################################
# sample size for two-way ANCOVA (2 x 2)
power.f.ancova.shieh(mu.vector = c(0.20, 0.25, 0.15, 0.05), # marginal means
sd.vector = c(1, 1, 1, 1), # unadjusted standard deviations
p.vector = c(0.25, 0.25, 0.25, 0.25), # allocation, should sum to 1
factor.levels = c(2, 2), # 2 by 2 factorial design
r.squared = 0.50,
k.covariates = 1,
alpha = 0.05,
power = 0.80)
#> Elements of `mu.vector`, `sd.vector`, `n.vector` or `p.vector` should follow this specific order:
#> A1:B1 A1:B2 A2:B1 A2:B2
#> +--------------------------------------------------+
#> | SAMPLE SIZE CALCULATION |
#> +--------------------------------------------------+
#>
#> Two-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 2796 <<
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.200
#> Statistical Power = 0.8
#>
# Elements of `mu.vector`, `sd.vector`, `n.vector` or `p.vector` should follow this specific order:
# A1:B1 A1:B2 A2:B1 A2:B2
###################################################################
####################### planned contrasts #######################
###################################################################
#########################
## dummy coding scheme ##
#########################
contrast.object <- factorial.contrasts(factor.levels = 3, # one factor w/ 3 levels
coding = "treatment") # use dummy coding scheme
#> A1 A2 A3
#> A1 1 0 -1
#> A2 0 1 -1
# get contrast matrix from the contrast object
contrast.matrix <- contrast.object$contrast.matrix
# calculate sample size given design characteristics
ancova.design <- power.f.ancova.shieh(mu.vector = c(0.15, 0.30, 0.20), # marginal means
sd.vector = c(1, 1, 1), # unadjusted standard deviations
p.vector = c(1/3, 1/3, 1/3), # allocation, should sum to 1
contrast.matrix = contrast.matrix,
r.squared = 0.50,
k.covariates = 1,
alpha = 0.05,
power = 0.80)
#> +--------------------------------------------------+
#> | SAMPLE SIZE CALCULATION |
#> +--------------------------------------------------+
#>
#> One-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 1245 <<
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.199
#> Statistical Power = 0.801
#>
# power of planned contrasts, adjusted for alpha level
power.t.contrasts(ancova.design, adjust.alpha = "fdr")
#> +--------------------------------------------------+
#> | POWER CALCULATION |
#> +--------------------------------------------------+
#>
#> Multiple Contrast Analyses (T-Tests)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : psi = 0
#> H1 (Alt. Claim) : psi != 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> contr comparison psi d ncp n.total power
#> 1 A1 <=> A3 -0.05 -0.071 -1.018 1245 0.111
#> 2 A2 <=> A3 0.10 0.141 2.036 1245 0.418
#>
###########################
## Helmert coding scheme ##
###########################
contrast.object <- factorial.contrasts(factor.levels = 3, # one factor w/ 4 levels
coding = "helmert") # use helmert coding scheme
#> A1 A2 A3
#> A1 -0.500 0.500 0.000
#> A2 -0.167 -0.167 0.333
# get contrast matrix from the contrast object
contrast.matrix <- contrast.object$contrast.matrix
# calculate sample size given design characteristics
ancova.design <- power.f.ancova.shieh(mu.vector = c(0.15, 0.30, 0.20), # marginal means
sd.vector = c(1, 1, 1), # unadjusted standard deviations
p.vector = c(1/3, 1/3, 1/3), # allocation, should sum to 1
contrast.matrix = contrast.matrix,
r.squared = 0.50,
k.covariates = 1,
alpha = 0.05,
power = 0.80)
#> +--------------------------------------------------+
#> | SAMPLE SIZE CALCULATION |
#> +--------------------------------------------------+
#>
#> One-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 1245 <<
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.199
#> Statistical Power = 0.801
#>
# power of planned contrasts
power.t.contrasts(ancova.design)
#> +--------------------------------------------------+
#> | POWER CALCULATION |
#> +--------------------------------------------------+
#>
#> Multiple Contrast Analyses (T-Tests)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : psi = 0
#> H1 (Alt. Claim) : psi != 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> contr comparison psi d ncp n.total power
#> 1 A2 <=> A1 0.075 0.106 3.055 1245 0.863
#> 2 A3 <=> A1 A2 -0.008 -0.012 -0.588 1245 0.090
#>
##############################
## polynomial coding scheme ##
##############################
contrast.object <- factorial.contrasts(factor.levels = 3, # one factor w/ 4 levels
coding = "poly") # use polynomial coding scheme
#> A1 A2 A3
#> A.L -0.707 0.000 0.707
#> A.Q 0.408 -0.816 0.408
# get contrast matrix from the contrast object
contrast.matrix <- contrast.object$contrast.matrix
# calculate sample size given design characteristics
ancova.design <- power.f.ancova.shieh(mu.vector = c(0.15, 0.30, 0.20), # marginal means
sd.vector = c(1, 1, 1), # unadjusted standard deviations
p.vector = c(1/3, 1/3, 1/3), # allocation, should sum to 1
contrast.matrix = contrast.matrix,
r.squared = 0.50,
k.covariates = 1,
alpha = 0.05,
power = 0.80)
#> +--------------------------------------------------+
#> | SAMPLE SIZE CALCULATION |
#> +--------------------------------------------------+
#>
#> One-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 1245 <<
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.199
#> Statistical Power = 0.801
#>
# power of the planned contrasts
power.t.contrasts(ancova.design)
#> +--------------------------------------------------+
#> | POWER CALCULATION |
#> +--------------------------------------------------+
#>
#> Multiple Contrast Analyses (T-Tests)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : psi = 0
#> H1 (Alt. Claim) : psi != 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> contr comparison psi d ncp n.total power
#> 1 A3 <=> A1 0.035 0.050 1.018 1245 0.174
#> 2 A1 A3 <=> A2 -0.102 -0.144 -2.939 1245 0.836
#>
######################
## custom contrasts ##
######################
# custom contrasts
contrast.matrix <- rbind(
cbind(A1 = 1, A2 = -0.50, A3 = -0.50),
cbind(A1 = 0.50, A2 = 0.50, A3 = -1)
)
# labels are not required for custom contrasts,
# but they make it easier to understand power.t.contrasts() output
# calculate sample size given design characteristics
ancova.design <- power.f.ancova.shieh(mu.vector = c(0.15, 0.30, 0.20), # marginal means
sd.vector = c(1, 1, 1), # unadjusted standard deviations
p.vector = c(1/3, 1/3, 1/3), # allocation, should sum to 1
contrast.matrix = contrast.matrix,
r.squared = 0.50,
k.covariates = 1,
alpha = 0.05,
power = 0.80)
#> +--------------------------------------------------+
#> | SAMPLE SIZE CALCULATION |
#> +--------------------------------------------------+
#>
#> One-way Analysis of Covariance (F-Test)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : eta.squared = 0
#> H1 (Alt. Claim) : eta.squared > 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> Total Sample Size = 1245 <<
#> Type 1 Error (alpha) = 0.050
#> Type 2 Error (beta) = 0.199
#> Statistical Power = 0.801
#>
# power of the planned contrasts
power.t.contrasts(ancova.design)
#> +--------------------------------------------------+
#> | POWER CALCULATION |
#> +--------------------------------------------------+
#>
#> Multiple Contrast Analyses (T-Tests)
#>
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#> H0 (Null Claim) : psi = 0
#> H1 (Alt. Claim) : psi != 0
#>
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#> contr comparison psi d ncp n.total power
#> 1 A1 <=> A2 A3 -0.100 -0.141 -2.351 1245 0.652
#> 2 A1 A2 <=> A3 0.025 0.035 0.588 1245 0.090
#>