Detecting nonrandom associations between proportions by tests of remaining-space variables

Abstract

Descriptive and inferential applications of the “remaining-space” transformation Z_j = Y_j/(1 — Y_i),where the Ys are members of the same set of proportions, are briefly reviewed in connection with petrographic variation diagrams. The sign reversal of the Al ₂ O ₃-SiO ₂ correlations in the Taupo complex, first noted by Darroch and Ratcliff, is probably the rule rather than the exception in subalkaline basalt-rhyolite or gabbro-granite associations. Darroch argues that the Snow test of Y_j/(1 — Y_j)versus Y_i/(1 — Y_j)would be in general more conservative than combined tests of Y_j versus Y_i/(1 — Y_j)and Y_i versus Y_j/(1 — Y_i).At Taupo, however, the two procedures give identical results for all pairings of six major compositional variables. To first-order approximation, the use of zero as null value in the testing of correlations involving remaining-space variables is valid only if, as postulated both by Connor and Mosimann and by Darroch and Ratcliff, variances are proportional to means in the initial open array. It is argued here that the perfect rank correlation of means and variances thus postulated would survive closure. In Taupo, as in most Harker arrays, the contribution of alumina to the total variance is very small. In the Taupo suite, in addition, the contribution of combined alkalies to the total variance is anomalously large; the result is that at Taupo the rank correlation of means and variances is very weak.