Standard error of Yule's Q and Yule's Y

[Dirk Enzmann]

Is there someone who has access to a more recent edition of Fisher (1934) to look up Fisher's formula for the standard error of Yule's Q and Yule's Y?

The standard errors given by Yule (1912, p. 593) are \[SE(Q) = 0.5 \cdot (1 - Q^2) \cdot \sqrt{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}}\] (with a, b, c, and d the cell frequencies of a 2 by 2 table) and \[SE(\omega) = 0.25 \cdot (1 - \omega^2) \cdot \sqrt{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}}\] where omega meanwhile is called Yule's Y.

I wonder whether the formula in Fisher (later than 1934, p. 302?) is the same because it seems that Yule's (1912) formula is based on the delta method avant la lettre.

References:

• Fisher, R. A. (1934 ff.) Statistical Methods for Research Workers. Edinburgh (5th ed.): Oliver and Boyd. [it should be a more recent edition!]
• Yule, G. U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75(6), 579–642. https://doi.org/10.2307/2340126

[Chen Samulsion]

Hi Enzmann, could you provide a snapshot of page 302 in Fisher's book of 5th edtion? I suspect that in later edtions he had deleted relevant content of Yule's Q and Yule's Y.

[Dirk Enzmann]

Here page 302 from Fisher (1934):

I hope my post was not misleading: I remember that some time ago I saw (somewhere, I can't find it again) a formula for the standard error of Yule's Q (or Y) and the reference was "Fisher: Statistical Methods for Resarch Workers (p. 302)". Because it is not in the 1934 book, I believe that it can be found in a more recent edition.

[Chen Samulsion]

In the twelfth edition of Fisher's book (published in 1954), he mentioned Yule only for 8 times, including 4 times in reference and index. And in the reference, he did not mention Yule's article of 1912.
The snapshot in #3 was in page 327 in the 12th ed. And its page 302 seems not what you are asking for either. I attach p302, p318 and p327 here.

[Dirk Enzmann]

Thank you for your efforts! Well, it seems that the source I saw was mistaken. It is unlikely that it has been removed from earlier editions. The only possibility I can think of is he 14th (revised and enlarged edition) published 1970 or the latest 1990 publication under the title "Statistical Methods, Experimental Design, and Statistical Reference" (Oxford University Press, edited by Bennett) that actually contains a re-issue of 14th edition.

[Dirk Enzmann]

Add on:

If you want to calculate Yule's Q (denoted as Q to give credit to the French -- now Belgian -- statistician Quételet, still famous in criminology) or Yule's Y (originally denoted as \(\omega\) in his original publication, see Yule, 1912) it is helpful to know that Q is identical to Goodman and Kruskal's (1972) gamma, so that Yule's Y can be obtained from gamma via
\[ Y=\frac{1-\sqrt{1-Q^2}}{Q} \]

Example:

program define yule_qy
  
   syntax varlist(min=2 max=2 numeric)
  
   tempname Q Y term se_Q se_Y

   tabulate `varlist', gamma matcell(T)
   scalar `Q' = r(gamma)
   scalar `Y' = (1 - sqrt(1 - r(gamma)^2))/r(gamma)
   scalar `term' = sqrt(1/T[1,1] + 1/T[1,2] + 1/T[2,1] + 1/T[1,1])
   scalar `se_Q' = 0.5*(1 - `Q'^2)*`term'
   scalar `se_Y' = 0.25*(1 - `Y'^2)*`term'
  
   di _n "Yule's Q = " `Q' ", SE(Q) = " `se_Q' ", p = " 2*(1-normal(abs(`Q'/`se_Q')))
   di    "Yule's Y = " `Y' ", SE(Y) = " `se_Y' ", p = " 2*(1-normal(abs(`Y'/`se_Y')))
   di as txt "Note: SEs are not aymptotic as in Goodman & Kruskal (1972)"
end

clear
input y x freq
  0 0 5
  0 1 7
  1 0 3
  1 1 2
end
qui expand freq

yule_qy y x

results in:

. yule_qy y x

           |           x
         y |         0          1 |     Total
-----------+----------------------+----------
         0 |         5          7 |        12
         1 |         3          2 |         5
-----------+----------------------+----------
     Total |         8          9 |        17

                    gamma =  -0.3548  ASE = 0.474

Yule's Q = -.35483871, SE(Q) = .40909594, p = .38573775
Yule's Y = -.18338605, SE(Y) = .22614267, p = .41740544
Note: SEs are not aymptotic as in Goodman & Kruskal (1972)

References:

• Goodman, L. A. & Kruskal, W. H. (1972). Measures of association for cross classifications IV: Simplification of asymptotic variances. Journal of the American Statistical Association, 67, 415–421.
• Yule, G. U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75, 579–642.

[Bruce Weaver]

Hello Dirk Enzmann. This online calculator gives a different SE for Y than you show in #6. It reports:

• SE(Y) via delta method: 0.262013

I believe problem is that in your program, you need to change this:

   scalar `term' = sqrt(1/T[1,1] + 1/T[1,2] + 1/T[2,1] + 1/T[1,1])

to this:

   scalar `term' = sqrt(1/T[1,1] + 1/T[1,2] + 1/T[2,1] + 1/T[2,2])

With that change, your program returns this output:

. yule_qy y x

           |           x
         y |         0          1 |     Total
-----------+----------------------+----------
         0 |         5          7 |        12 
         1 |         3          2 |         5 
-----------+----------------------+----------
     Total |         8          9 |        17 

                    gamma =  -0.3548  ASE = 0.474

Yule's Q = -.35483871, SE(Q) = .47398515, p = .45408125
Yule's Y = -.18338605, SE(Y) = .26201254, p = .48398149
Note: SEs are not aymptotic as in Goodman & Kruskal (1972)

And of course, SE(Q) changes too, as do the two p-values.

Cheers,
Bruce

[Dirk Enzmann]

Bruce Weaver : Of course, your are right. Thanks for spotting this!

And: Now the SE of Yule's Q corresponds to Goodman and Kruskal's asymptotic SE, i.e. the SEs are asymptotic.