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Yule's Coefficients of Association (Q and Y)

Source: R/yule.R
yule.Rd

Computes Yule's Q or Y for a 2x2 contingency table, optionally with asymptotic confidence interval based on the log odds ratio.

Usage

yuleQ(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  correct = FALSE,
  ...
)

yuleY(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  correct = FALSE,
  ...
)

Arguments

x

a 2x2 contingency table (matrix or table). If y is supplied, x and y are cross-tabulated via table().

y

optional second variable for cross-tabulation

conf.level

confidence level of the interval. If set to NA (the default), only the point estimate is returned.

sides

character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See details in ConfidenceIntervals().

correct

logical; if TRUE, applies the Haldane–Anscombe correction by adding 0.5 to all cells

...

further arguments passed to table()

Value

if conf.level = NA, a numeric scalar. Otherwise a named numeric vector with elements:

est

point estimate of Yule's Q or Y

lci

lower confidence interval bound

uci

upper confidence interval bound

Details

For a 2x2 table with cell counts \(a, b, c, d\):

Odds ratio: $$OR = \frac{ad}{bc}$$

Yule's Q: $$Q = \frac{OR - 1}{OR + 1} = \tanh\left(\frac{1}{2}\log(OR)\right)$$

Yule's Y: $$Y = \frac{\sqrt{OR} - 1}{\sqrt{OR} + 1} = \tanh\left(\frac{1}{4}\log(OR)\right)$$

Both coefficients are computed from the tanh form, which stays well defined when a zero cell drives the odds ratio to 0 or Inf (the coefficient is then -1 or 1).

Confidence intervals are obtained from the asymptotic normal approximation: $$\log(OR) \pm z \cdot \sqrt{1/a + 1/b + 1/c + 1/d}$$ and then transformed to the selected coefficient. With a zero cell the standard error is infinite and the interval degenerates to \([-1, 1]\); use correct = TRUE to obtain a finite interval.

For a one-sided interval the open side is reported at the range limit (-1 resp. 1), not at \(\pm\infty\).

References

Yule, G. U. (1912). On the methods of measuring association between two attributes.

See also

Other assoc.nominal: contCoef(), cramerV(), gkTau(), lambda(), mutInf(), phi(), tschuprowT(), uncertCoef()

Examples

m <- matrix(c(12, 5, 3, 20), nrow = 2)
yuleQ(m)                        # 0.8823529
#> [1] 0.8823529
yuleY(m, conf.level = 0.95)     # 0.6
#>       est       lci       uci 
#> 0.6000000 0.2849591 0.7980515 

# a zero cell yields the limiting value 1 (and not NaN)
yuleQ(matrix(c(12, 5, 0, 20), nrow = 2), conf.level = NA)
#> [1] 1

# ... a finite interval requires the Haldane-Anscombe correction
yuleQ(matrix(c(12, 5, 0, 20), nrow = 2), correct = TRUE)
#> [1] 0.9787645