psych (version 2.3.3)

# phi: Find the phi coefficient of correlation between two dichotomous variables

## Description

Given a 1 x 4 vector or a 2 x 2 matrix of frequencies, find the phi coefficient of correlation. Typical use is in the case of predicting a dichotomous criterion from a dichotomous predictor.

## Usage

`phi(t, digits = 2)`

## Value

phi coefficient of correlation

## Arguments

t

a 1 x 4 vector or a 2 x 2 matrix

digits

round the result to digits

## Author

William Revelle with modifications by Leo Gurtler

## Details

In many prediction situations, a dichotomous predictor (accept/reject) is validated against a dichotomous criterion (success/failure). Although a polychoric correlation estimates the underlying Pearson correlation as if the predictor and criteria were continuous and bivariate normal variables, and the tetrachoric correlation if both x and y are assumed to dichotomized normal distributions, the phi coefficient is the Pearson applied to a matrix of 0's and 1s.

The phi coefficient was first reported by Yule (1912), but should not be confused with the `Yule` Q coefficient.

For a very useful discussion of various measures of association given a 2 x 2 table, and why one should probably prefer the `Yule` Q coefficient, see Warren (2008).

Given a two x two table of counts

 a b a+b (R1) c d c+d (R2) a+c(C1) b+d (C2) a+b+c+d (N)

convert all counts to fractions of the total and then
Phi = [a- (a+b)*(a+c)]/sqrt((a+b)(c+d)(a+c)(b+d) ) =
(a - R1 * C1)/sqrt(R1 * R2 * C1 * C2)

This is in contrast to the Yule coefficient, Q, where

Since the phi coefficient is just a Pearson correlation applied to dichotomous data, to find a matrix of phis from a data set involves just finding the correlations using cor or `lowerCor` or `corr.test`.

## References

Warrens, Matthijs (2008), On Association Coefficients for 2x2 Tables and Properties That Do Not Depend on the Marginal Distributions. Psychometrika, 73, 777-789.

Yule, G.U. (1912). On the methods of measuring the association between two attributes. Journal of the Royal Statistical Society, 75, 579-652.

`phi2tetra`, `AUC`, `Yule`, `Yule.inv` `Yule2phi`, `comorbidity`, `tetrachoric` and `polychoric`

## Examples

Run this code
``````phi(c(30,20,20,30))
phi(c(40,10,10,40))
x <- matrix(c(40,5,20,20),ncol=2)
phi(x)

``````

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