Power Analysis for One-, Two-, Three-Way ANCOVA Contrasts and Multiple Comparisons (T-Tests)
ancova.contrasts.RdCalculates power or sample size for one-, two-, three-Way ANCOVA contrasts and multiple comparisons. The pwrss.t.contrast() function allows a single contrast. For multiple contrast testing (multiple comparisons) use the pwrss.t.contrasts() function. The pwrss.t.contrasts() function also allows adjustment to alpha due to multiple testing. All arguments are the same as with the earlier function except that it accepts an object returned from the pwrss.f.ancova.shieh() function for convenience. Beware that, in this case, all other arguments are ignored except alpha and adjust.alpha.
The pwrss.t.contrasts() function returns a data frame with as many rows as number of contrasts and eight columns with names 'contrast', 'comparison', 'psi', 'd', 'ncp', 'df', 'n.total', and 'power'. 'psi' is the contrast estimate. 'd'
is the standardized contrast estimate. Remaining parameters are explained elsewhere.
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 (2017).
Usage
power.t.contrast(mu.vector,
sd.vector,
contrast.vector,
n.vector = NULL, p.vector = NULL,
r.squared = 0, k.covariates = 1,
power = NULL, alpha = 0.05,
tukey.kramer = FALSE,
ceiling = TRUE, verbose = TRUE, pretty = FALSE)
power.t.contrasts(x = NULL,
mu.vector = NULL,
sd.vector = NULL,
n.vector = NULL, p.vector = NULL,
r.squared = 0, k.covariates = 1,
contrast.matrix = NULL,
power = NULL, alpha = 0.05,
adjust.alpha = c("none", "tukey", "bonferroni",
"holm", "hochberg", "hommel",
"BH", "BY", "fdr"),
ceiling = TRUE, verbose = TRUE, pretty = FALSE)Arguments
- x
object; an object returned from
pwrss.f.ancova.shieh()function.- mu.vector
vector; adjusted means (or estimated marginal means) for each level of a factor. Ignored when 'x' is specified.
- sd.vector
vector; unadjusted standard deviations for each level of a factor. Ignored when 'x' is specified.
- n.vector
vector; sample sizes for each level of a factor. Ignored when 'x' is specified.
- p.vector
vector; proportion of total sample size in each level of a factor. These proportions should sum to one. Ignored when 'x' is specified.
- r.squared
explanatory power of covariates (R-squared) in the ANCOVA model. Ignored when 'x' is specified.
- k.covariates
Number of covariates in the ANCOVA model. Ignored when 'x' is specified.
- contrast.vector
vector; a single contrast in the form of a vector with as many elements as number of levels or groups (or cells in factorial designs). Ignored when 'x' is specified.
- 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). Ignored when 'x' is specified.
- power
statistical power, defined as the probability of correctly rejecting a false null hypothesis, denoted as \(1 - \beta\). Ignored when 'x' is specified.
- alpha
type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\). Note that this should be specified even if 'x' is specified. The 'alpha' value within the 'x' object pertains to the omnibus test, NOT the test of contrasts.
- tukey.kramer
logical;
TRUEby default. IfFALSEno adjustment will be made to control Type 1 error.- adjust.alpha
character; one of the methods in c("none", "tukey", "bonferroni", "holm", "hochberg", "hommel", "BH", "BY", "fdr") to control Type 1 error. See
?stats::p.adjust.- 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 (T-Test).
- psi
contrast-weighted mean difference.
- d
contrast-weighted standardized mean difference.
- df
degrees of freedom.
- t.alpha
critical values.
- 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. (2017). Power and sample size calculations for contrast analysis in ANCOVA. Multivariate Behavioral Research, 52(1), 1-11. doi:10.1080/00273171.2016.1219841
Examples
###################################################################
####################### 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
#>