R: From a two by two table, find the Yule coefficients of...

Yule {psych} | R Documentation

## From a two by two table, find the Yule coefficients of association, convert to phi, or tetrachoric, recreate table the table to create the Yule coefficient.

### Description

One of the many measures of association is the Yule coefficient. Given a two x two table of counts

| a | b | R1
| c | d | R2
| C1 | C2 | n

Yule Q is (ad - bc)/(ad+bc).
Conceptually, this is the number of pairs in agreement (ad) - the number in disagreement (bc) over the total number of paired observations. Warren (2008) has shown that Yule's Q is one of the “coefficients that have zero value under statistical independence, maximum value unity, and minimum value minus unity independent of the marginal distributions" (p 787).
ad/bc is the odds ratio and Q = (OR-1)/(OR+1)
Yule's coefficient of colligation is Y = (sqrt(OR) - 1)/(sqrt(OR)+1) Yule.inv finds the cell entries for a particular Q and the marginals (a+b,c+d,a+c, b+d). This is useful for converting old tables of correlations into more conventional `phi` or tetrachoric correlations `tetrachoric`
Yule2phi and Yule2tetra convert the Yule Q with set marginals to the correponding phi or tetrachoric correlation.

Bonett and Price show that the Q and Y coefficients are both part of a general family of coefficients raising the OR to a power (c). If c=1, then this is Yule's Q. If .5, then Yule's Y, if c = .75, then this is Digby's H. They propose that c = .5 - (.5 * min(cell probabilty)^2 is a more general coefficient. YuleBonett implements this for the 2 x 2 case, YuleCor for the data matrix case.

### Usage

    YuleBonett(x,c=1,bonett=FALSE,alpha=.05) #find the generalized Yule cofficients
    YuleCor(x,c=1,bonett=FALSE,alpha=.05) #do this for a matrix
    Yule(x,Y=FALSE)  #find Yule given a two by two table of frequencies
     #find the frequencies that produce a Yule Q given the Q and marginals
    Yule.inv(Q,m,n=NULL)
    #find the phi coefficient that matches the Yule Q given the marginals
    Yule2phi(Q,m,n=NULL)
    Yule2tetra(Q,m,n=NULL,correct=TRUE)

       #Find the tetrachoric correlation given the Yule Q and the marginals
    #(deprecated) Find the tetrachoric correlation given the Yule Q and the marginals
    Yule2poly(Q,m,n=NULL,correct=TRUE)

### Arguments

`x` | A vector of four elements or a two by two matrix, or, in the case of YuleBonett or YuleCor, this can also be a data matrix
`c` | 1 returns Yule Q, .5, Yule's Y, .75 Digby's H
`bonett` | If FALSE, then find Q, Y, or H, if TRUE, then find the generalized Bonett cofficient
`alpha` | The two tailed probability for confidence intervals
`Y` | Y=TRUE return Yule's Y coefficient of colligation
`Q` | Either a single Yule coefficient or a matrix of Yule coefficients
`m` | The vector c(R1,C2) or a two x two matrix of marginals or a four element vector of marginals. The preferred form is c(R1,C1)
`n` | The number of subjects (if the marginals are given as frequencies
`correct` | When finding a tetrachoric correlation, should small cell sizes be corrected for continuity. See `{link{tetrachoric}` for a discussion.

### Details

Yule developed two measures of association for two by two tables. Both are functions of the odds ratio

### Value

`Q` | The Yule Q coefficient
`R` | A two by two matrix of counts
`result` | If given matrix input, then a matrix of phis or tetrachorics
`rho` | From YuleBonett and YuleCor
`ci` | The upper and lower confidence intervals in matrix form (From YuleBonett and YuleCor).

### Note

Yule.inv is currently done by using the optimize function, but presumably could be redone by solving a quadratic equation.

### Author(s)

William Revelle

### References

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

Bonett, D.G. and Price, R.M, (2007) Statistical Inference for Generalized Yule Coefficients in 2 x 2 Contingency Tables. Sociological Methods and Research, 35, 429-446.

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

### See Also

See Also as `phi`, `tetrachoric`, `Yule2poly.matrix`, `Yule2phi.matrix`

### Examples

    Nach <- matrix(c(40,10,20,50),ncol=2,byrow=TRUE)
    Yule(Nach)
    Yule.inv(.81818,c(50,60),n=120)
    Yule2phi(.81818,c(50,60),n=120)
    Yule2tetra(.81818,c(50,60),n=120)
    phi(Nach)  #much less
    #or express as percents and do not specify n
    Nach <- matrix(c(40,10,20,50),ncol=2,byrow=TRUE)
    Nach/120
    Yule(Nach)
    Yule.inv(.81818,c(.41667,.5))
    Yule2phi(.81818,c(.41667,.5))
    Yule2tetra(.81818,c(.41667,.5))
    phi(Nach)  #much less
    YuleCor(ability[,1:4],,TRUE)
    YuleBonett(Nach,1)  #Yule Q
    YuleBonett(Nach,.5)  #Yule Y
    YuleBonett(Nach,.75)  #Digby H
    YuleBonett(Nach,,TRUE)  #Yule* is a generalized Yule

[Package psych version 1.7.8 Index]