compare_cor: Comparing Two Independent Correlation Coefficients

In Lakens/TOSTER: Two One-Sided Tests (TOST) Equivalence Testing

Comparing Two Independent Correlation Coefficients

Description

A function to compare correlations between independent studies. This function is intended to be used to compare the compatibility of original studies with replication studies (lower p-values indicating lower compatibility).

Usage

compare_cor(
  r1,
  df1,
  r2,
  df2,
  method = c("fisher", "kraatz"),
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  null = 0
)

Arguments

r1	Correlation from study 1.
df1	Degrees of freedom from study 1 (if a simple correlation the df is N-2).
r2	Correlation from study 2.
df2	Degrees of freedom from study 2 (if a simple correlation the df is N-2).
method	Method for determining differences: "fisher": uses Fisher's z transformation (default) "kraatz": uses the Kraatz method
alternative	A character string specifying the alternative hypothesis: "two.sided": difference is not equal to null (default) "greater": difference is greater than null "less": difference is less than null "equivalence": difference is within the equivalence bounds (TOST) "minimal.effect": difference is outside the equivalence bounds (TOST) You can specify just the initial letter.
null	A number or vector indicating the null hypothesis value(s): For standard tests: a single value representing the null difference (default = 0) For equivalence/minimal effect tests: either a single value (symmetric bounds ±value will be created) or a vector of two values representing the lower and upper bounds

Details

This function tests for differences between correlation coefficients from independent studies. It is particularly useful for:

• Comparing an original study with its replication

• Meta-analytic comparisons between studies

• Testing if correlations from different samples are equivalent

The function offers two methods for comparing correlations:

1. Fisher's z transformation (default): Transforms correlations to stabilize variance

2. Kraatz method: Uses a direct approach that may be more appropriate for larger correlations

The function supports both standard hypothesis testing and equivalence/minimal effect testing:

• For standard tests (two.sided, less, greater), the function tests whether the difference between correlations differs from the null value (typically 0).

• For equivalence testing ("equivalence"), it determines whether the difference falls within the specified bounds, which can be set asymmetrically.

• For minimal effect testing ("minimal.effect"), it determines whether the difference falls outside the specified bounds.

When performing equivalence or minimal effect testing:

• If a single value is provided for null, symmetric bounds ±value will be used

• If two values are provided for null, they will be used as the lower and upper bounds

Value

A list with class "htest" containing the following components:

• statistic: z-score with name "z"

• p.value: numeric scalar containing the p-value for the test under the null hypothesis

• estimate: difference in correlation coefficients between studies

• null.value: the specified hypothesized value(s) for the null hypothesis

• alternative: character string indicating the alternative hypothesis

• method: description of the method used for comparison

• data.name: "Summary Statistics" to denote summary statistics were utilized

• cor: list containing the correlation coefficients used in the comparison

• call: the matched call

References

Counsell, A., & Cribbie, R. A. (2015). Equivalence tests for comparing correlation and regression coefficients. The British journal of mathematical and statistical psychology, 68(2), 292-309. https://doi.org/10.1111/bmsp.12045

Anderson, S., & Hauck, W. W. (1983). A new procedure for testing equivalence in comparative bioavailability and other clinical trials. Communications in Statistics-Theory and Methods, 12(23), 2663-2692.

See Also

Other compare studies: boot_compare_cor(), boot_compare_smd(), compare_smd()

Examples

# Example 1: Comparing two correlations (standard test)
compare_cor(r1 = 0.45, df1 = 48, r2 = 0.25, df2 = 58,
            method = "fisher", alternative = "two.sided")

# Example 2: Testing for equivalence between correlations
# Testing if the difference between correlations is within ±0.15
compare_cor(r1 = 0.42, df1 = 38, r2 = 0.38, df2 = 42,
            method = "fisher", alternative = "equivalence", null = 0.15)

# Example 3: Testing for minimal effects using Kraatz method
# Testing if the difference between correlations is outside ±0.2
compare_cor(r1 = 0.53, df1 = 28, r2 = 0.22, df2 = 32,
            method = "kraatz", alternative = "minimal.effect", null = 0.2)

# Example 4: One-sided test (are correlations different in a specific direction?)
compare_cor(r1 = 0.65, df1 = 48, r2 = 0.45, df2 = 52,
            method = "fisher", alternative = "greater")

# Example 5: Using asymmetric bounds for equivalence testing
compare_cor(r1 = 0.35, df1 = 48, r2 = 0.25, df2 = 52,
            method = "fisher", alternative = "equivalence", null = c(-0.05, 0.2))

Lakens/TOSTER documentation built on June 9, 2025, 8:57 p.m.