The Measurement of Association of Rows and Columns for an r × s Contingency Table

Abstract

The generalization of Edwards's argument for the measure of association of the rows and columns of 2 x 2 table, to that of an r x s table whose rows and columns are assumed unordered, shows, not surprisingly, that association ought to be measured by some function of the (r-l) (s-1) cross-ratios. Such a function is suggested by the introduction of a metric on certain equivalence classes. The properties of such metrics are examined, and in particular comparisons are made with Good's suggestion of the use of the algebraic rank of the contingency table, and with Lindley's significance test for association in the r x stable. GENERALIZATION OF EDWARD'S ARGUMENTFOR THE 2 x 2 TABLE EDWARDS (1963) has shown that any measure of association of rows and columns for a 2 x 2 contingency table should be a function of the cross-ratio (Pll P22)/(P12 P21)' where the table is represented by (Pij; I~i,j~2). This argument is generalized below to the r x s contingency table, and some suggestions for measures of association are given.Let Pr,s be the class of r x s contingency tables, that is, Pr,s={(Pij):Pij>O,I~i~r,l~j~s and~~Pij= I} It is presumed that no orderings of the rows and columns of tables of P, s are available.Let~be an equivalence relation such that for p,qEPr,s,p~q means that the "association" of rows and columns of P is the same as the "association" of rows and columns of q. A precise definition of "association" is deliberately omitted at this point, but following Edwards, two propositions about it are made.Proposition I If P, q E Pr,s are such thatPij qij -= -, l~i~r, l~j~s-l, Pi. qi.then p~q (employing the usual dot notation for summation over a suffix). 64ALTHAM -Column and Row Measurements for a Contingency Table [No.1, Proposition II If P, q E Pr,s are such that Pij _ qij ---, 1~i~r-l, 1~j~s, P.j q,j A then p--q. Let O,.!! be a relation such that for p,qEPr,s' po"!!q iff PijPrs qijqrs--= --, 1~i~r-l, 1~j~s-l. PisPrj qisqrjThis is easily seen to be an equivalence relation. The generalization of Edwards's result is the following. OR ATheorem. If p,q EP r.s and p--q, then p-q. Proof. Define . PijPrs qijqrs ()(.ij=--=--, l~i~r, l~j~s.PisPrj qisqrj Define 7Tij = 7To ()(.ijPisqrj, where 7To is the normalizing constant, i.e. such that 7TEP r,s' Then 7Tij/7Ti. = qij/qi., 1~i~r, 1~j~s-l, so that 7T1.,q and 7Tij/7T.j = Pij/P.j, 1~i~r-l, 1~j~s, so that 7T1.,p.Hence, by the transitivity of 1." p1.,q as required.Similar versions of this result for r x s x t tables are easily obtained, given suitable modification of Propositions I and II. There are essentially four different types of dependence of rows, columns and layers to be considered, and accordingly four different types of cross-ratio arise as being relevant for the description of these four types of association. An example follows. LetPr,s,!={(Pijk):Pijk>O, l~i~r, l~j~s, l~k~t, and :S:S:SPijk= I}.Let !!., be an equivalence relation such that for P, q E P; s t>P!!., q means that the second order interaction of the rows, columns and layers of P' is the same as that of q. The...

Some Models for the Analysis of Association in Multiway Cross-Classifications Having Ordered Categories

“…Note that the €Ii, exploit the ordinal character of the variables, and note that all other odds ratios can be computed from them. See Goodman (1969) and Altham (1970). There are certainly other ways that ordinality can be taken into account (see Plackett 1965;Altham 1970;Fienberg 1980), but the mathematical tractability of this procedure of forming the Oij appears to be a definite advantage.…”

“…See Goodman (1969) and Altham (1970). There are certainly other ways that ordinality can be taken into account (see Plackett 1965;Altham 1970;Fienberg 1980), but the mathematical tractability of this procedure of forming the Oij appears to be a definite advantage. Each €Iij describes the association present in a particular region of the Z x J table; the €Ii, thus measure the local association between the row and column variables.…”

Intergenerational Occupational Mobility in Great Britain and the United States Since 1850: Comment

“…Also focusing on odds-ratios, the Long-Ferrie study relies on Altham’s (1970) index as the main method of yielding findings. While it has not been widely used in previous research on social mobility, Altham’s index, denoted as d , is a sensible summary measure comparing two tables, involving comparisons of all possible odds-ratios.…”

Measures, Models, and Graphical Displays in the Analysis of Cross-Classified Data

“…With the correlation model, it is based on the /),ij and the marginals. There are some: advantages in using models in which the measurement of this relationship is based only on the log-odds-ratios [see, for example, Edwards (1963); Goodman (1965Goodman ( , 1969; Mosteller (1968); Altham (1970); Becker and Clogg (1989)]; there are also some advantages in using models that take the marginals into account. The marginalfree unweighted log-linear approach and corresponding unweighted association models are advantageous when the marginals are not fixed and cross-classification tables having different marginals are to be compared; the marginaldependent weighted log-linear approach and corresponding weighted association models are advantageous when, for example, the cross-classification Tables 6, 7, 10, and 11), and we shall discuss this matter further next.…”