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Consider independent binomial counts $X_i\sim \mathrm{Bin}(n_i,p)$ under the common-probability null hypothesis, with fixed positive row sizes $n_i$. Let $N={\sum }_i{n}_i$, $K={\sum }_i{X}_i$ and assume $0<K<N$. Conditional on $K$, the probability mass function is the displayed multivariate hypergeometric distribution: in the joint likelihood, the factor $p^K(1-p{)}^{N-K}$ is common to all admissible tables and cancels. The Vandermonde identity gives the denominator, so the conditional distribution is free of the unknown nuisance parameter $p$.

For a table $x$, its Pearson chi-squared statistic isAn exact conditional p-value sums the displayed probabilities over all tables with the same margins and $Q(x)\ge Q(x_{\mathrm{obs}})$. The ordering is by the Pearson chi-squared statistic, rather than by the probability of an individual table; different exact-test orderings can give different answers. Enumeration is practical when $K$ and the number of rows are small. Monte Carlo tests can approximate the same tail by sampling tables with fixed margins.^[ref]

An exact arithmetic comparison is easy too. Let $M$ be the least common multiple of the row sizes. Compare the integersince every table has the same positive denominator $MK(N-K)$. This includes boundary ties without a floating-point tolerance. Conditional p-values are discrete and can be conservative; “exact” concerns their null calibration, not equality of rejection probability with every chosen significance level.

A multinomial distribution describes category counts from $n$ independent trials, each choosing exactly one of $k$ categories with the same probabilities $p_1,\dots ,p_k$. Require $n\ge 0$ to be an integer, $p_i\ge 0$, and ${\sum }_i{p}_i=1$. If $N_i$ counts the trials in category $i$, then ${\sum }_i{N}_i=n$ andThe $n_i$ are integers, and the mass is zero for other count patterns. A category with $p_i=0$ has count zero almost surely, with $0^0=1$ in this formula. For a specified ordered trial sequence, independence gives the product of numbers ${\prod }_i{p}_i^{n_i}$; the multinomial coefficient counts the different sequences with that same count pattern.

For four trials with category probabilities $(1/2,3/10,1/5)$, the count pattern $(2,1,1)$ can occur in $4!/(2!1!1!)=12$ orders, each of probability $(1/2{)}^2(3/10)(1/5)=0.015$. Thus its total probability is $12(0.015)=0.18$. This is a constructed independent-trial example, distinct from drawing four objects without replacement from a ten-object population of composition $(5,3,2)$, where the multivariate hypergeometric distribution gives $2/7$.

The diagram shows all fifteen possible count patterns. Coordinates $(N_1,N_2)$ determine $N_3=4-N_1-N_2$; the triangle is a constraint on integer counts, not a continuous probability density. The mode is $(N_1,N_2,N_3)=(2,1,1)$, where the probability mass function is largest, but the expected values $(2,1.2,0.8)$ are not a possible count pattern.

Each single count has a binomial distribution, because recording category $i$ versus all other categories reduces every trial to a Bernoulli trial. Consequently,These statements about the joint count model are summarized in Aditya Guntuboyina's lecture notes.^[ref]

Independent trials do not make the resulting counts independent random variables. Within one trial the indicator random variables for two distinct categories cannot both equal one, so their product is zero and their covariance is $-p_i{p}_j$. Summing over independent trials gives $-n{p}_i{p}_j$. In the four-trial example, the variances are $(1,0.84,0.64)$ and the pairwise covariances are $-0.6,-0.4,-0.24$. Knowing that all four trials were in the first category fixes the other two counts at zero.

The attraction is that one formula describes any fixed-total list of category counts while preserving the constraint that they add up to $n$. It underlies multinomial likelihood calculations and uniform balls-in-bins allocation. The multinomial aggregation rule lets us merge categories without leaving this family of probability distributions. Allowing independent category counts with an unfixed total instead leads to a different model, related through Poisson-multinomial conditioning.

The association remains positive within each income group, although its estimated size is smaller in the higher-income groups. The two extreme trust categories give the following conditional vaccination percentages and differences:

In the lowest-income group, the intervening percentages are 42.86%, 44.12% and 82.54%, so the increase is monotone. Its Pearson chi-squared statistic is approximately 62.520 on four statistical degrees of freedom, with the printed asymptotic $p\approx 8.56\times 10^{-13}$. Cramér's V is approximately 0.441, Goodman and Kruskal's gamma is approximately 0.626 and the column-dependent Somers' D is approximately 0.259. The column-dependent Goodman and Kruskal's lambda is approximately 0.158: trust improves the modal prediction more here than in the pooled table.

In the middle-income group, the intervening vaccination percentages are 61.54%, 81.82% and 92.04%. The printed Pearson chi-squared statistic is approximately 24.033 on four statistical degrees of freedom, with asymptotic $p\approx 7.87\times 10^{-5}$. Cramér's V is approximately 0.298, Goodman and Kruskal's gamma approximately 0.454 and column-dependent Somers' D approximately 0.100. Column-dependent Goodman and Kruskal's lambda is zero: vaccinated respondents are the majority in every row, so the modal prediction never changes. A zero lambda does not mean independence.

In the highest-income group, the intermediate vaccination percentages are 71.43%, 76.92% and 94.55%; the final category falls slightly to 91.14%. Thus the observed pattern is broadly positive rather than strictly monotone. The printed Pearson chi-squared statistic is approximately 8.199 on four statistical degrees of freedom, with asymptotic $p\approx 0.0846$. It does not reject independence at 5%. The corresponding Cramér's V is approximately 0.229, Goodman and Kruskal's gamma approximately 0.281 and column-dependent Somers' D approximately 0.058. The column-dependent Goodman and Kruskal's lambda is again zero.

Small counts limit the asymptotic tests, especially in the middle- and highest-income tables. Their minimum expected counts are approximately 0.644 and 0.306, with respectively 30% and 40% of cells below five. The lowest-income table has one expected count below five, with minimum approximately 3.315. These diagnostics concern expected counts under independence, not just the observed row sizes.

The supplied counts are sufficient to make an exact conditional Pearson test for a binary contingency table, without needing individual records or assuming a chi-squared distribution approximation. Conditioning on the row totals and total unvaccinated count removes the common unknown binomial probability. For each admissible table, sum its multivariate hypergeometric distribution probability when its Pearson chi-squared statistic is at least the observed one. Complete enumeration gives:

These are new exact conditional calculations, rather than the paper's printed asymptotic p-values. They confirm evidence of association in the first two groups and nonrejection at 5% in the third, while showing the numerical effect of sparse cells. The tail is ordered by the Pearson chi-squared statistic; an exact test with a different ordering need not have the same p-value. Fixed-margin Monte Carlo tests approximate this same Pearson-ordered tail when enumeration is impractical.^[ref]

The original count-based figure below includes pointwise approximate 95% Wilson intervals. The very wide intervals for sparsely populated trust categories make the uncertainty visible; they are not simultaneous intervals or a formal test of differences between income groups.

In the elaboration model, the positive direction is replicated after conditioning on income: income does not fully explain away the observed trust–vaccination relationship. The decreasing estimated contrasts and ordered coefficients suggest possible specification by income, or effect modification on an explicitly chosen scale. They do not establish it. Different p-values can arise from different sample sizes and standard errors; significant and nonsignificant results need not differ significantly. A formal statistical interaction test or a confidence interval for differences between comparable contrasts would address that question.^[ref]

Nor do these conditional tables prove that income is a causal confounder or causal mediator. Those labels require assumptions about causal order, relevant confounding and how variables were measured. This is an observational study of recorded vaccination status, not a randomized change in trust or a test of vaccine effectiveness.

Every seven-element subset of the hundred individually identifiable tiles is equally likely. Choose three of the thirty-four vowels and four of the sixty-four consonants; the favourable draw has no blank tiles. ThusThis is a multivariate hypergeometric distribution count, including the implicit factor $\left(\genfrac{}{}{0ex}{}{2}{0}\right)=1$ for blanks. Dividing by $\left(\genfrac{}{}{0ex}{}{98}{7}\right)$ instead would incorrectly condition on having drawn no blanks.