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Each frequency table describes a marginal distribution, rather than a relationship among variables. The recorded two-category gender variable has 1,042 male and 1,093 female responses: 48.81% and 51.19%. It is a nominal categorical variable. Its ordering in the table supplies no quantitative ranking; a cumulative percentage over those arbitrarily ordered labels has no invariant substantive meaning.

Household income is an ordinal categorical variable with six bands. Their shares, from the lowest band upward, are $5.48\%$, $9.93\%$, $11.48\%$, $13.02\%$, $8.34\%$ and $51.76\%$. The highest band alone contains 1,105 respondents, slightly more than half the sample. The bands are not a numerical income measurement with equal steps: the top band is open-ended, so an exact sample mean income cannot be recovered from these category counts alone.

The agreement item is also an ordinal categorical variable. Combining the first two categories givesNeutral responses account for $447/2135\approx 20.94\%$; the two disagreement categories together account for $570/2135\approx 26.70\%$. These percentages describe the measured vaccination intention item, rather than actual vaccination status.

All three tables have 2,135 valid observations and no missing values for these variables, so their percentages and valid percentages coincide. This is different from the seven missing group-indicator values in question 1. The income and agreement columns sum to $100.01\%$ because the displayed percentages were rounded separately. That rounding is compatible with counts summing exactly to 2,135.

The tables alone cannot show whether the gender composition differs by income, or whether vaccination intention differs between groups. Those are questions about a joint contingency table and its conditional probabilities, not about separate marginal distributions.

Source paper: released examination archive. The cover identifies Michaelmas Term 2022. All four questions are addressed; statements and the original residual figure are omitted. The supplied tables concern trust and recorded vaccination status; they do not support the introduction's separate reference to gender or vaccine willingness. The inconsistent residual graphic and a qualification about the first test's degrees of freedom are discussed where they matter.

These are row percentages, so the comparisons estimate the conditional probability of recorded vaccination within each NHS trust category. The percentages increase across the ordered trust categories asThe extreme-category difference is approximately 49.31 percentage points, rather than a 49.31% relative increase. The ordinal categorical variable records an order; the categories do not establish equal numerical trust increments.

Under the null hypothesis of independence, the expected count in a cell is its row total times its column total divided by 748. The Pearson chi-squared statistic compares observed and expected counts throughThere are $(5-1)(2-1)=4$ statistical degrees of freedom, giving $p\approx 6.49\times 10^{-21}$. The minimum expected count is approximately 5.015, so the usual expected-count concern is much less acute here than in the income strata. The table gives strong evidence of a statistical association. The likelihood-ratio test of independence in a contingency table agrees: its statistic is approximately 81.361 on four statistical degrees of freedom, with a very small p-value. The displayed 0.000 is rounding, not a zero probability.

For magnitude, Cramér's V isIt describes the size of the departure from independence without using the category order or giving a direction. Calling it “moderate” is only a rough contextual description; the actual percentage differences are easier to interpret.

The ordered coefficients describe a positive direction. Goodman and Kruskal's gamma uses only concordant pairs and discordant pairs:It excludes tied pairs, which are frequent when the response has only two categories. It is therefore not a claim that 54.7% of respondents were vaccinated because of their trust. The directional Somers' D for vaccination as response includes pairs tied on that response in its denominator:The smaller value reflects the retained ties; it does not contradict the positive statistical association.

The small column-dependent Goodman and Kruskal's lambda has a useful prediction interpretation. Without trust information, always predicting the more frequent vaccinated category gives 121 mistakes. Predicting the majority category separately in each trust row gives $13+15+26+34+28=116$ mistakes. HenceTrust reduces this particular modal-prediction error by only about 4.13%, despite the strong evidence against independence. Most rows retain the same majority category. Evidence of association, its ordinal direction and improvement in a chosen prediction rule are different questions. None establishes that changing trust would change an individual's vaccination status.

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.

A person's reported intention or willingness to receive a vaccine under the conditions of a question. It is distinct from actual vaccination status. The response depends on the question, time and circumstances; an intention score does not directly measure later receipt or a medical effect.