Power Analysis for Dependent Correlations (Steiger's Z-Test)

Calculates power or sample size (only one can be NULL at a time) to test difference between paired correlations (Pearson) using Fisher's Z-transformation.

Validated via PASS and G*Power.

power.z.twocors.steiger(rho12, rho13, rho23,
                        rho14 = NULL, rho24 = NULL, rho34 = NULL,
                        n = NULL, power = NULL, alpha = 0.05,
                        alternative = c("two.sided", "one.sided"),
                        pooled = TRUE, common.index = FALSE,
                        ceiling = TRUE, verbose = TRUE, pretty = FALSE)

Arguments

rho12

correlation between variable V1 and V2 (one common index and no common index). Check examples below.

rho13

correlation between variable V1 and V3 (one common index and no common index). Check examples below.

rho23

correlation between variable V2 and V3 (one common index and no common index). Check examples below.

rho14

correlation between variable V1 and V4 (no common index only). Check examples below.

rho24

correlation between variable V2 and V4 (no common index only). Check examples below.

rho34

correlation between variable V3 and V4 (no common index only). Check examples below.

n

integer; 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 \(\alpha\).

alternative

character; the direction or type of the hypothesis test: "two.sided" or "one.sided".

pooled

logical; whether standard error should be pooled. TRUE by default.

common.index

logical; whether calculations pertain to one common index. TRUE means calculations involve correlations with a common index (where both correlations share one variable). FALSE (default) means calculations pertain to correlations with no common index (where all relevant correlations must be explicitly specified). Check examples below.

ceiling

logical; if TRUE rounds up sample size.

verbose

logical; 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 (Z-Test)

mean

mean of the alternative distribution.

sd

standard deviation of the alternative distribution.

null.mean

mean of the null distribution.

null.sd

standard deviation of the null distribution.

z.alpha

critical value(s).

power

statistical power \((1-\beta)\).

n

sample size for the first and second groups, in the form of c(n1, n2).

References

Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87(2), 245-251. doi:10.1037/0033-2909.87.2.245

Examples

# example data for one common index
# compare cor(V1, V2) to cor(V1, V3)

# subject    V1       V2      V3
# <int>    <dbl>    <dbl>    <dbl>
#   1       1.2      2.3      0.8
#   2      -0.0      1.1      0.7
#   3       1.9     -0.4     -2.3
#   4       0.7      1.3      0.4
#   5       2.1     -0.1      0.8
#   ...     ...      ...      ...
#   1000   -0.5      2.7     -1.7

# V1: socio-economic status (common)
# V2: pretest
# V3: post-test

power.z.twocors.steiger(rho12 = 0.35, rho13 = 0.45, rho23 = 0.05,
                        n = 1000, power = NULL, alpha = 0.05,
                        alternative = "two.sided",
                        common.index = TRUE)
#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Dependent Correlations
#> 
#>   Common Index    : TRUE
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : rho12 - rho13 = 0
#>   H1 (Alt. Claim) : rho12 - rho13 != 0
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 1000
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.253
#>   Statistical Power    = 0.747  <<
#> 


# example data for no common index
# compare cor(V1, V2) to cor(V3, V4)

# subject    V1       V2       V3       V4
# <int>    <dbl>    <dbl>    <dbl>    <dbl>
#   1       1.2      2.3      0.8      1.2
#   2      -0.0      1.1      0.7      0.9
#   3       1.9     -0.4     -2.3     -0.1
#   4       0.7      1.3      0.4     -0.3
#   5       2.1     -0.1      0.8      2.7
#   ...     ...      ...      ...      ...
#   1000   -0.5      2.7     -1.7      0.8

# V1: pretest reading
# V2: pretest math
# V3: post-test reading
# V4: post-test math

power.z.twocors.steiger(rho12 = 0.45, rho13 = 0.45, rho23 = 0.50,
                        rho14 = 0.50, rho24 = 0.80, rho34 = 0.55,
                        n = 1000, power = NULL, alpha = 0.05,
                        alternative = "two.sided",
                        common.index = FALSE)
#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Dependent Correlations
#> 
#>   Common Index    : FALSE
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : rho12 - rho34 = 0
#>   H1 (Alt. Claim) : rho12 - rho34 != 0
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 1000
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.062
#>   Statistical Power    = 0.938  <<
#>