As we have seen, the raw PMI score is not invariant to the distribution of the marginals. This can be seen in Table 1, which concerns the association between vaccination and death from smallpox; the original proportions in panel a are based on the Sheffield data in Table I of Yule (1912). In panel b, the marginals of the table with regard to vaccination have been adjusted to 50/50 (as may have happened if these data had been collected in a randomized, controlled trial), and in panel c, we have the canonical table where both marginals are 50/50.3 Notice that the PMI is highest for the original (unbalanced) table and decreases as the marginals are balanced. Conversely, the MI is lowest in the the original (unbalanced) table and increases as the marginals are balanced. Of course, Yule's $Y$ is constant throughout, by construction.

Table 1:

$2×2$ Contingency Tables for the Association between Vaccination and Death from Smallpox.

(a) Original Table(b) Vaccination Rate 50%(c) Canonical Table
PMI $=$ 2.300, MI $=$ 0.108PMI $=$ 0.866, MI $=$ 0.205PMI $=$ 0.705, MI $=$ 0.310
RecoverDieMarginalsRecoverDieMarginalsRecoverDieMarginals
Vaccinated 0.840 0.043 0.883 0.476 0.024 0.500 0.408 0.092 0.500
Unvaccinated 0.059 0.058 0.117 0.252 0.248 0.500 0.092 0.408 0.500
Marginals 0.899 0.101  0.728 0.272  0.500 0.500
(a) Original Table(b) Vaccination Rate 50%(c) Canonical Table
PMI $=$ 2.300, MI $=$ 0.108PMI $=$ 0.866, MI $=$ 0.205PMI $=$ 0.705, MI $=$ 0.310
RecoverDieMarginalsRecoverDieMarginalsRecoverDieMarginals
Vaccinated 0.840 0.043 0.883 0.476 0.024 0.500 0.408 0.092 0.500
Unvaccinated 0.059 0.058 0.117 0.252 0.248 0.500 0.092 0.408 0.500
Marginals 0.899 0.101  0.728 0.272  0.500 0.500

Note: Panel a is the original table based on the data in Yule (1912), panel b adjusts the marginals for vaccinated/unvaccinated to be 50/50, and panel c is the canonical table where the marginals are both 50/50. In all three tables, Yule's $Y=0.630$.

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