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A Connection between Correlation and Contingency

Published online by Cambridge University Press:  24 October 2008

H. O. Hirschfeld
H. O. Hirschfeld
Affiliation:
Fitzwilliam House
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Extract

Let us consider a discontinuous bivariate distribution. That is, let us consider N × Q non-negative values pνq (ν = 1, 2, …, N; q = 1, 2, …, Q), being the theoretical probabilities of the νth value of a variate Xμ (μ = 1, 2, …, N) concurring with the qth value of a second variate Ys (s = 1, 2, …, Q).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 4 , October 1935 , pp. 520 - 524
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* However, at most [min (N, Q) − 1] and at least one possibility is of practical use.

* If λ1 ≠ 0 and λ2 ≠ 0, then in addition to equation (4) a corresponding equation (4′), obtained by interchanging in (4) x ν − m x and y q − m ν, N and Q, and Greek and Latin indices, is found from (2) by eliminating the x ν − m x instead of the y q − m ν.

† It will be obvious by an argument parallel to the following one, that equations (1 b) and (4′) [see footnote* above] would be sufficient conditions for (1) and (2) as well. Thus we may assume from now on that N ≥ Q.

‡ Thus, if ρ2 = 0, then all y q = 0 = m ν. But we shall show later that we can always find at least one solution of (4) and (1 a) with ρ2 > 0 except in the case of absolute non-correlation, i.e. if p νq = P ντq. In this case, however, the problem is trivial.

* See p. 521, note †. If N > Q, then at least N − Q of the (ρ(i))2 must vanish. For if (ρi)2 > 0, then by (ii) λ2 = 1, λ1 = (ρ(i))2 > 0, but there are at most Q (linearly independent) characteristic sets of (4′) (see p. 521, note *). Furthermore it may occur that for some i = 2, …, N two coordinates and are equal. However, this property is of no statistical relevance, since it depends on the exact values of the p νq. The same applies if two characteristic-values are equal.

* Provided that the variance (7) is 1, which is true for the

† Usually Pearson's Mean Square Contingency is only defined for N = Q. But if N > Q (say) there are at least two possibilities of defining a measure having its properties: one is given by (18), another would have Q − 1 instead of N − 1 (as the corresponding argument for the y q would show).