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.
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.
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)
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 |
Yule developed two measures of association for two by two tables. Both are functions of the odds ratio
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). |
Yule.inv is currently done by using the optimize function, but presumably could be redone by solving a quadratic equation.
William Revelle
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 as phi, tetrachoric, Yule2poly.matrix, Yule2phi.matrix
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(psychTools::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
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