Test For Association/Correlation Between Paired Binary Variables

This function calculates a p-value for the signifinace of a phi correlation coefficient (or mean square contingency coefficient, \(\phi\) or \(r\phi\)), the measure of association between two binary variables, and calculates a confidence interval.

Usage

phi_coef.test(
  x,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

x

a square matrix containing the observations of two binary variables as a two-by-two table of counts.

alternative

a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.

conf.level

numeric between 0 and 1, the confidence level required; default 0.95.

Value

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

statistic

the value of the test statistic.

parameter

the number of (paired) observations.

p.value

the p-value of the test.

conf.int

confidence interval of the \(\phi\) correlation coefficient (95% or other specified level).

estimate

the \(\phi\) correlation coefficient.

null.value

the value of the association measure under the null hypothesis, always 0.

alternative

a character string describing the alternative hypothesis.

method

the character string "Phi correlation coefficient with confidence interval".

data.name

a character string giving the name of the data.

Details

The phi coefficient is calculated using phi_coef. For derivation of the standard error and confidence interval, see Bishop et al. (2003), and Bonett (2021). See also ci.phi(), the confidence interval for a phi correlation in the reference manual for package statpsych.

References

Yule, G.U. (1912). On the Methods of Measuring Association Between Two Attributes. J Royal Stat Soc. 75 (6): 579–652. doi:10.2307/2340126 .

Bishop, Y.M.M., Fienberg, S.E., Holland, P.W. (1975). Discrete Multivariate Analysis. MIT Press. (See Ch.11.) ISBN 978-0-387-72805-6.

Bonett, Douglas G. (2021). Statistical Methods for Psychologists, Volume 3: Introduction to Introduction to Categorical Data Analysis. University of California, Santa Cruz. (See 3.4 Measures of Association for 2 × 2 Tables.)

See also

matrix, mcnemar.test

Other correl_coef: cor_coef.test(), phi_coef()

Examples

## Example from Wikipedia
twobytwo <- matrix(c(6, 1, 2, 3), nrow = 2, dimnames = rep(list(c("Cat", "Dog")), 2) |>
              setNames(c("Actual", "Predicted")))
addmargins(twobytwo)
#>       Predicted
#> Actual Cat Dog Sum
#>    Cat   6   2   8
#>    Dog   1   3   4
#>    Sum   7   5  12

phi_coef.test(twobytwo)
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = 1.6528, n = 12, p-value = 0.09837
#> alternative hypothesis: true phi is not equal to 0
#> 95 percent confidence interval:
#>  -0.08885493  1.04503781
#> sample estimates:
#>       phi 
#> 0.4780914 
#> 
phi_coef.test(twobytwo, alternative = "less")
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = 1.6528, n = 12, p-value = 0.9508
#> alternative hypothesis: true phi is less than 0
#> 95 percent confidence interval:
#>  -1.0000000  0.9538878
#> sample estimates:
#>       phi 
#> 0.4780914 
#> 
phi_coef.test(twobytwo, alternative = "greater")
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = 1.6528, n = 12, p-value = 0.04919
#> alternative hypothesis: true phi is greater than 0
#> 95 percent confidence interval:
#>  0.002295051 1.000000000
#> sample estimates:
#>       phi 
#> 0.4780914 
#> 

## Example from Statology
twobytwo <- matrix(c(4, 8, 9, 4), nrow = 2, dimnames =
              list(Gender = c("Male", "Female"), Party = c("Dem", "Rep")))
addmargins(twobytwo)
#>         Party
#> Gender   Dem Rep Sum
#>   Male     4   9  13
#>   Female   8   4  12
#>   Sum     12  13  25

phi_coef.test(twobytwo)
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = -1.919, n = 25, p-value = 0.05499
#> alternative hypothesis: true phi is not equal to 0
#> 95 percent confidence interval:
#>  -0.725613555  0.007664837
#> sample estimates:
#>        phi 
#> -0.3589744 
#> 

## Setting confidence level to 1 - p-value gives upper bound of confidence interval close to zero
pval <- phi_coef.test(twobytwo)$p.value
phi_coef.test(twobytwo, conf.level = 1 - pval)
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = -1.919, n = 25, p-value = 0.05499
#> alternative hypothesis: true phi is not equal to 0
#> 94.50144 percent confidence interval:
#>  -7.179487e-01 -1.110223e-16
#> sample estimates:
#>        phi 
#> -0.3589744 
#> 
## Similarly, with one-tailed tests setting confidence level to 1 - p-value/2 conserves the upper
## or lower CI bound with alternative = "less" or alternative = "greater" respectively
phi_coef.test(twobytwo, alternative = "less", conf.level = 1 - pval/2)
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = -1.919, n = 25, p-value = 0.02749
#> alternative hypothesis: true phi is less than 0
#> 97.25072 percent confidence interval:
#>  -1.000000e+00 -1.110223e-16
#> sample estimates:
#>        phi 
#> -0.3589744 
#> 
phi_coef.test(twobytwo, alternative = "greater", conf.level = 1 - pval/2)
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = -1.919, n = 25, p-value = 0.9725
#> alternative hypothesis: true phi is greater than 0
#> 97.25072 percent confidence interval:
#>  -0.7179487  1.0000000
#> sample estimates:
#>        phi 
#> -0.3589744 
#> 

## Example from statpsych::ci.phi(), which should return: -
##       Estimate         SE         LL        UL
## [1,] 0.1229976 0.05746271 0.01037273 0.2356224

twobytwo <- matrix(c(229, 28, 96, 24), nrow = 2, dimnames = rep(list(c("Zero", "One")), 2))
addmargins(twobytwo)
#>      Zero One Sum
#> Zero  229  96 325
#> One    28  24  52
#> Sum   257 120 377

phi_coef.test(twobytwo)
#> 
#> 	Phi correlation coefficient with confidence interval
#> 
#> data:  twobytwo
#> z = 2.1405, n = 377, p-value = 0.03232
#> alternative hypothesis: true phi is not equal to 0
#> 95 percent confidence interval:
#>  0.01037273 0.23562243
#> sample estimates:
#>       phi 
#> 0.1229976 
#> 
## Check standard error as expected
with(phi_coef.test(twobytwo), c(stderr = estimate/statistic))
#> stderr.phi 
#> 0.05746271 

rm(twobytwo, pval)