Association/Correlation Test from Summary Statistics

Test for association between paired samples using only the correlation coefficient and sample size. Supports Pearson's product moment correlation, Kendall's \(\tau\) (tau), or Spearman's \(\rho\) (rho). This is the updated version of the TOSTr function.

Usage

corsum_test(
  r,
  n,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  method = c("pearson", "kendall", "spearman"),
  alpha = 0.05,
  null = 0
)

Arguments

r

correlation coefficient (the estimated value)

n

sample size (number of pairs)

alternative

a character string specifying the alternative hypothesis:

"two.sided": correlation is not equal to null (default)
"greater": correlation is greater than null
"less": correlation is less than null
"equivalence": correlation is within the equivalence bounds (TOST)
"minimal.effect": correlation is outside the equivalence bounds (TOST)

You can specify just the initial letter.

method

a character string indicating which correlation coefficient is to be used for the test. One of "pearson", "kendall", or "spearman", can be abbreviated.

alpha

alpha level (default = 0.05)

null

a number or vector indicating the null hypothesis value(s):

For standard tests: a single value (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

Value

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

statistic: z-score with name "z".
p.value: the p-value of the test.
parameter: the sample size with name "N".
conf.int: a confidence interval for the correlation appropriate to the specified alternative hypothesis.
estimate: the estimated correlation coefficient, with name "cor", "tau", or "rho" corresponding to the method employed.
stderr: the standard error of the test statistic.
null.value: the value(s) of the correlation coefficient under the null hypothesis.
alternative: character string indicating the alternative hypothesis.
method: a character string indicating how the correlation was measured.
data.name: a character string giving the names of the data.
call: the matched call.

Details

This function uses Fisher's z transformation for the correlations, but uses Fieller's correction of the standard error for Kendall's \(\tau\) or Spearman's \(\rho\).

Unlike z_cor_test, which requires raw data, this function only needs the correlation value and sample size. This is particularly useful when:

You only have access to summary statistics (correlation coefficient and sample size)
You want to reanalyze published results within an equivalence testing framework

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

For standard tests (two.sided, less, greater), the function tests whether the correlation differs from the null value (typically 0).
For equivalence testing ("equivalence"), it determines whether the correlation falls within the specified bounds, which can be set asymmetrically.
For minimal effect testing ("minimal.effect"), it determines whether the correlation 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

References

Goertzen, J. R., & Cribbie, R. A. (2010). Detecting a lack of association: An equivalence testing approach. British Journal of Mathematical and Statistical Psychology, 63(3), 527-537. https://doi.org/10.1348/000711009X475853, formula page 531.

Examples

# Example 1: Standard significance test for Pearson correlation
corsum_test(r = 0.45, n = 30, method = "pearson", alternative = "two.sided")
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  x and y
#> z = 2.5186, N = 30, p-value = 0.01178
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  0.1070928 0.6972330
#> sample estimates:
#>  cor 
#> 0.45 
#> 

# Example 2: Equivalence test for Spearman correlation
# Testing if correlation is equivalent to zero within ±0.3
corsum_test(r = 0.15, n = 40, method = "spearman",
            alternative = "equivalence", null = 0.3)
#> 
#> 	Spearman's rank correlation rho
#> 
#> data:  x and y
#> z = -0.93572, N = 40, p-value = 0.1747
#> alternative hypothesis: equivalence
#> null values:
#>  rho  rho 
#>  0.3 -0.3 
#> 90 percent confidence interval:
#>  -0.1265834  0.4049426
#> sample estimates:
#>  rho 
#> 0.15 
#> 

# Example 3: Minimal effect test for Kendall's tau
# Testing if correlation is meaningfully different from ±0.25
corsum_test(r = 0.42, n = 50, method = "kendall",
            alternative = "minimal.effect", null = 0.25)
#> 
#> 	Kendall's rank correlation tau
#> 
#> data:  x and y
#> z = 1.9727, N = 50, p-value = 0.02426
#> alternative hypothesis: minimal.effect
#> null values:
#>   tau   tau 
#>  0.25 -0.25 
#> 90 percent confidence interval:
#>  0.2797138 0.5427266
#> sample estimates:
#>  tau 
#> 0.42 
#> 

# Example 4: One-sided test with non-zero null
# Testing if correlation is greater than 0.3
corsum_test(r = 0.45, n = 35, method = "pearson",
            alternative = "greater", null = 0.3)
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  x and y
#> z = 0.99097, N = 35, p-value = 0.1608
#> alternative hypothesis: true correlation is greater than 0.3
#> 90 percent confidence interval:
#>  0.1915334 0.6501001
#> sample estimates:
#>  cor 
#> 0.45 
#> 

# Example 5: Using asymmetric bounds for equivalence testing
corsum_test(r = 0.1, n = 60, method = "pearson",
            alternative = "equivalence", null = c(-0.2, 0.3))
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  x and y
#> z = -1.5793, N = 60, p-value = 0.05713
#> alternative hypothesis: equivalence
#> null values:
#> correlation correlation 
#>        -0.2         0.3 
#> 90 percent confidence interval:
#>  -0.1169926  0.3078798
#> sample estimates:
#> cor 
#> 0.1 
#>