Skip to contents

Equivalence Test for the Pearson Correlation via Two One-Sided Tests (TOST)

Source: R/equivalence_r.R
equivalence_r.Rd

Performs a two one-sided tests procedure for equivalence of a Pearson correlation \(\rho\) to zero against user-specified equivalence bounds \([-\rho_L, \rho_U]\) (Counsell & Cribbie, 2015; Goertzen & Cribbie, 2010). Uses the Fisher's \(Z\) transformation throughout, a large-sample approximation whose accuracy under bivariate normality improves quickly with n. Equivalence is declared when the 100(1 - 2\(\alpha\))% Fisher's \(Z\) CI on \(\rho\) lies entirely inside the equivalence region.

Usage

equivalence_r(
  r = NULL,
  n = NULL,
  x = NULL,
  y = NULL,
  rho_lower = NULL,
  rho_upper = NULL,
  alpha_level = 0.05
)

Arguments

r, n

Observed sample correlation r and sample size. Alternatively supply x and y to compute r from raw data.

x, y

Numeric vectors of paired observations. If supplied, r and n are computed from the data and the r/n arguments are ignored.

rho_lower, rho_upper

Equivalence bounds on the correlation scale, both positive. The equivalence region is \([-\rho_L, +\rho_U]\). If only rho_upper is supplied, the bounds are symmetric.

alpha_level

One-sided significance level. Default 0.05.

Value

A data.frame with rows for the observed r, the two one-sided test statistics on the Fisher's \(Z\) scale, their p-values, the joint TOST p-value, the 100(1 - 2\(\alpha\))% CI on \(\rho\), the equivalence bounds, a binary equivalence flag, and the sample size (n).

Details

Fisher's \(Z\) transformation. $$Z = \tfrac{1}{2} \log\left(\frac{1 + r}{1 - r}\right), \quad \mathrm{Var}(Z) = \frac{1}{n - 3}.$$ The TOST is run on the Fisher's \(Z\) scale:

  • Lower test: \((Z - Z_{-\rho_L}) \sqrt{n - 3}\) compared against the upper \(\alpha\) of \(N(0, 1)\).

  • Upper test: \((Z - Z_{\rho_U}) \sqrt{n - 3}\) compared against the lower \(\alpha\) of \(N(0, 1)\).

The CI bounds are back-transformed from the \(Z\) scale via \(r = \tanh(Z)\) so they remain in \([-1, 1]\).

Choosing \(\rho_L\) and \(\rho_U\). Common choices in psychology are \(0.1\), or domain-specific meaningfulness thresholds (e.g., \(0.2\) for cognitive task correlations). The bounds must be set before data collection.

References

Counsell, A., & Cribbie, R. A. (2015). Equivalence tests for comparing correlation and regression coefficients. British Journal of Mathematical and Statistical Psychology, 68(2), 292–309. doi:10.1111/bmsp.12045

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. doi:10.1348/000711009X475853

Lakens, D. (2017). Equivalence tests: A practical primer for t tests, correlations, and meta-analyses. Social Psychological and Personality Science, 8(4), 355–362. doi:10.1177/1948550617697177

See also

equivalence_smd, ci_r

Other hypothesis tests: adjusted_means(), ancova(), anova_within(), ci_dunnett(), ci_scheffe(), ci_tukey_kramer(), compare_cov_structures(), contrast_test(), correlations_test(), equivalence_smd(), factorial_anova(), manova_split_plot(), mauchly_test(), mixed_anova(), obrien_test(), pairwise_within(), randomization_test(), randomization_test_paired(), regions_of_significance(), simple_effects_AB(), summary_t_test(), welch_t()

Other equivalence testing: equivalence_c(), equivalence_smd(), plot_equivalence(), power_density_equivalence_md(), power_equivalence_c(), power_equivalence_md(), power_equivalence_md_plot(), ss_power_equivalence_c()

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Equivalence test that |rho| < 0.10 with n = 200 and r = 0.05:
equivalence_r(r = 0.05, n = 200, rho_upper = 0.10)
#>  term         value 
#>  r            0.05  
#>  z_lower_test 2.11  
#>  z_upper_test -0.706
#>  p_lower      0.0174
#>  p_upper      0.2401
#>  p_tost       0.2401
#>  lower_limit  -0.067
#>  upper_limit  0.166 
#>  rho_lower    -0.1  
#>  rho_upper    0.1   
#>  equivalent   0     
#>  n            200   
#> 
#> Confidence level: 90%

# 2. From raw data:
set.seed(113)
x <- rnorm(150); y <- 0.04 * x + rnorm(150)
equivalence_r(x = x, y = y, rho_upper = 0.15)
#>  term         value  
#>  r            -0.0329
#>  z_lower_test 1.43   
#>  z_upper_test -2.23  
#>  p_lower      0.0759 
#>  p_upper      0.0128 
#>  p_tost       0.0759 
#>  lower_limit  -0.167 
#>  upper_limit  0.102  
#>  rho_lower    -0.15  
#>  rho_upper    0.15   
#>  equivalent   0      
#>  n            150    
#> 
#> Confidence level: 90%