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.

Usage

power.z.twocors.steiger(
  rho12 = NULL,
  rho13 = NULL,
  rho23 = NULL,
  rho14 = NULL,
  rho24 = NULL,
  rho34 = NULL,
  req.sign = "+",
  n = NULL,
  power = NULL,
  alpha = 0.05,
  alternative = c("two.sided", "one.sided"),
  pooled = TRUE,
  common.index = FALSE,
  ceil.n = TRUE,
  verbose = 1,
  utf = 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.
req.sign: whether estimated rho is smaller or larger than the other (when minimum detectable rho is of interest). Sign comparison is between rho12 and rho13 with common index and between rho12 and rho34 with no common index. Note that sign comparison is relative to the known correlation.
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.
ceil.n: logical; if TRUE rounds up sample size.
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 (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. https://doi.org/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)        : rho12 - rho13  = 0
#>   H1 (Alternative) : rho12 - rho13 != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target Effect (rho12) = 0.350 (vs. rho13 = 0.450)
#>   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)        : rho12 - rho34  = 0
#>   H1 (Alternative) : rho12 - rho34 != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target Effect (rho12) = 0.450 (vs. rho34 = 0.550)
#>   Sample Size           = 1000
#>   Type 1 Error (alpha)  = 0.050
#>   Type 2 Error (beta)   = 0.062
#>   Statistical Power     = 0.938  <<
#>