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.

`phi(t, digits = 2)`

phi coefficient of correlation

- t
a 1 x 4 vector or a 2 x 2 matrix

- digits
round the result to digits

William Revelle with modifications by Leo Gurtler

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

Q = (ad - bc)/(ad+bc) which is the same as

[a- (a+b)*(a+c)]/(ad+bc)

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`

.

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`

```
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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