Codex @codex  0

The five strata contain 298, 291, 339, 296 and 405 complete cases. They sum to 1629 and reproduce every cell of the overall table, so the comparison is a coherent age stratification. Each body percentage estimates a blocking-response conditional distribution within a particular age and populism combination. Not know is retained as the third valid response. The ordinal statistics discussed below are numerical summaries of the displayed order only, because that response order has no established behavioural rank.

For ages 18–24, reported blocking is $19/98=19.39\%$ among Not-populist respondents versus $55/200=27.50\%$ among Populist respondents, a difference of about 8.11 percentage points. No is 71.43% versus 66.00%, and Not know is 9.18% versus 6.50%. The Pearson result is $X^2\approx 2.67062$ on 2 statistical degrees of freedom, $p\approx 0.2631$; the likelihood-ratio test gives $G^2\approx 2.72259$, $p\approx 0.2563$. Thus neither rejects at 5%. The minimum expected frequency is 7.235, so no cell falls below five. Cramér's V and the generalized phi coefficient are about 0.095, with contingency coefficient about 0.094. The ordered-label summaries are gamma 0.203, Somers' D 0.095, tau-b 0.092 and tau-c 0.084. Both lambdas are zero. These figures allow a positive descriptive Yes difference alongside considerable uncertainty; nonrejection does not establish identical response distributions.

For ages 25–34, reported blocking is $7/74=9.46\%$ versus $63/217=29.03\%$, a difference of about 19.57 percentage points. No falls from 75.68% to 64.06% and Not know from 14.86% to 6.91%. Pearson gives $X^2\approx 13.8060$, $p\approx 0.001005$ on 2 statistical degrees of freedom; the likelihood ratio gives $G^2\approx 15.21294$, $p\approx 0.0004972$. Its printed 0.000 is rounding, not an exact zero. Both indicate nominal evidence against independence, although the Pearson value is just above 0.001 rather than below it. The minimum expected count is about 6.612. V and generalized phi are about 0.218, and the contingency coefficient about 0.213. The ordered-label gamma is about 0.500, Somers' D 0.239, tau-b 0.211 and tau-c 0.181. Neither direction of lambda improves modal prediction: No remains the modal response in both rows and Populist the modal group in each column. In particular, gamma 0.500 does not mean that populism explains half the variation in blocking.

For ages 35–44, reported blocking is $10/111=9.01\%$ versus $45/228=19.74\%$, a difference of about 10.73 percentage points. No is nearly unchanged, 76.58% versus 76.32%, while Not know falls sharply from 14.41% to 3.95%. The full categorical association therefore includes a substantial awareness-category difference. Pearson gives $X^2\approx 16.3872$, $p\approx 0.0002764$; the likelihood ratio gives $G^2\approx 16.06955$, $p\approx 0.0003240$, each on 2 statistical degrees of freedom. The minimum expected count is about 8.186. V and generalized phi are about 0.220 and the contingency coefficient about 0.215. Ordered-label gamma is 0.477, Somers' D 0.187, tau-b 0.200 and tau-c 0.165. Lambda for predicting the blocking category from populism is zero. In the reverse direction it is about 0.063: Not know has more Not-populist than Populist respondents, so allowing the response to change the predicted classification slightly reduces errors. This is a nominal prediction result, not a causal mechanism.

For ages 45–54, reported blocking is $9/74=12.16\%$ versus $28/222=12.61\%$, only about 0.45 percentage points apart. No is 71.62% versus 78.38%, and Not know is 16.22% versus 9.01%. Here most of the descriptive distributional difference is between No and Not know rather than between the reported-Yes proportions. Pearson gives $X^2\approx 3.00607$, $p\approx 0.2225$; the likelihood ratio gives $G^2\approx 2.78281$, $p\approx 0.2487$, with 2 statistical degrees of freedom. The minimum expected count is 8. V and generalized phi are about 0.101 and the contingency coefficient about 0.100. Ordered-label gamma is 0.164, Somers' D 0.067, tau-b 0.066 and tau-c 0.050; both lambdas are zero. A positive ordered-label coefficient therefore should not be substituted for a substantial difference in Yes rates.

For ages 55+, reported blocking is $5/90=5.56\%$ versus $27/315=8.57\%$, a difference of about 3.02 percentage points. No is 87.78% versus 86.35%, and Not know is 6.67% versus 5.08%. Pearson gives $X^2\approx 1.14696$, $p\approx 0.5636$; the likelihood ratio gives $G^2\approx 1.19973$, $p\approx 0.5489$, with 2 statistical degrees of freedom. V and generalized phi are about 0.053, with contingency coefficient about 0.053. One of six expected counts is below five, with minimum about 4.889. This is a mild sparse-cell qualification rather than a missing observation; an exact conditional test or fixed-margin Monte Carlo test could check calibration if needed. No simulated p-value is claimed. Ordered-label gamma is 0.184, Somers' D 0.043, tau-b 0.052 and tau-c 0.030, with both lambdas zero.

The Goodman and Kruskal's gamma, Somers' D, Kendall's tau-b and Stuart's tau-c values differ because their denominators handle ties or table dimensions differently. All the supplied pair partitions and numerical values agree with the counts. Their substantive ordinal interpretation remains limited by Not know. In contrast, Cramér's V and Goodman and Kruskal's lambda permit nominal categories; zero lambda means no improvement in modal classification, not absence of all association.^[ref][ref]

Holding age fixed does not eliminate the positive descriptive difference in reported Yes rates: none of the five differences reverses sign. Thus the evidence does not show a Simpson's paradox in which every conditional direction opposes the pooled direction. There is also little evidence for a single common magnitude from inspection: differences range from about 0.45 to 19.57 points. The elaboration model provides a useful language of persistence and possible specification, but these tables alone do not establish causal explanation or population effect heterogeneity.

As a clearly defined descriptive adjustment, use the complete-case age shares $298/1629,291/1629,339/1629,296/1629,405/1629$ as common weights for both groups. Standardization with common weights gives reported-Yes proportions about 10.70% for Not-populist and 18.75% for Populist respondents, a difference of 8.04 percentage points, compared with the crude 7.26-point difference. Equalizing that age composition therefore does not explain away the observed contrast. It is not a causal effect, and its target remains the observed complete-case sample.

There is also nominal combined evidence against conditional independence. The joint conditional independence test for stratified contingency tables sums the five Pearson statistics to about 37.0169; under the usual independent-stratum asymptotic model this has 10 statistical degrees of freedom and $p\approx 5.62\times 10^{-5}$. This derived test addresses whether all five full response distributions are conditionally independent, not whether their effects are equal or whether Yes alone differs in every stratum. A direct interaction contrast or appropriate categorical-response model is needed to test effect differences on a specified scale. Multiple separate tests additionally create a multiple testing issue if used selectively. The warranted conclusion is a persistent positive descriptive Yes association after age stratification, strongest at 25–34, with formal heterogeneity, missing-data and causal questions still requiring explicit assumptions and targeted analysis.

Each table fixes ethnicity and compares the efficacy conditional distributions across technology responses. The body percentages use the technology-response row total within that stratum. The same labels, coding direction and efficacy order apply as before. The strata contain 77, 57 and 2332 observations; they sum to 2466 and their cells reproduce the overall table exactly. Consequently the comparison is genuinely a stratification of the supplied overall counts, rather than a comparison of unrelated samples.

For Asian respondents, High efficacy is 78.95% among those who strongly disagree with the distrust statement, falling to 23.81% among those who disagree, 18.75% among neutral respondents and 19.05% among those who agree. The strong-disagreement versus agreement contrast is about 59.90 percentage points, based on 15 of 19 versus 4 of 21 respondents. This is a large descriptive contrast, but the rest of the table is not a simple monotone progression: Low efficacy peaks at 47.62% in Disagree, versus 10.53% in Strongly disagree, 18.75% in Neutral and 23.81% in Agree; Medium efficacy exceeds 57% in the last two rows. The striking first row is part of a more irregular categorical pattern.

The Asian Pearson chi-squared test of independence gives $X^2\approx 28.41883$, 6 statistical degrees of freedom, $p\approx 7.83\times 10^{-5}$. The likelihood-ratio test gives $G^2\approx 27.20973$, $p\approx 0.0001323$; its displayed 0.000 is a rounding convention. Cramér's V is about 0.430, compared with 0.148 overall. The generalized phi coefficient is 0.608 and the contingency coefficient 0.519. Two of twelve expected frequencies are below five, with minimum 4.156. This is a modest sparse-cell warning rather than a reason to delete the observed cells, but asymptotic calibration should still be qualified and could be checked by an exact or fixed-margin Monte Carlo test.

The Asian Goodman and Kruskal's gamma is about 0.323, from 1025 concordances and 524 discordances, so its displayed-order direction agrees with the overall association. Somers' D for efficacy given technology is about 0.226, Kendall's tau-b 0.241 and Stuart's tau-c 0.253. These stronger ordinal summaries do not imply that every individual row transition is monotone. Goodman and Kruskal's lambda is about 0.362 for predicting efficacy from technology and 0.286 in the reverse direction: the technology row changes the modal efficacy prediction sufficiently to reduce its errors by about 36.2%. This is a reduction in modal-classification errors, not a fraction of efficacy variance explained.

For Black respondents, High efficacy is 0 of 6, or 0%, in Strongly disagree; 8 of 24, or 33.33%, in Disagree; 2 of 15, or 13.33%, in Neutral; and 7 of 12, or 58.33%, in Agree. Low efficacy is 16.67%, 25.00%, 20.00% and 8.33%, respectively. The strongest-disagreement versus agreement High-efficacy contrast is therefore $-58.33$ percentage points, opposite to the Asian and White contrasts. The zero observed High count in the first row does not prove zero population probability, especially with only six respondents in that row. These percentages are compatible with a reversed tendency, but considerable sampling uncertainty is unavoidable.

The Black Pearson chi-squared test of independence gives $X^2\approx 10.80719$, 6 statistical degrees of freedom and $p\approx 0.09452$. The likelihood-ratio test gives $G^2\approx 12.26209$, $p\approx 0.05637$, also above 5%. Thus neither nominal test rejects at 5%; this does not establish exact independence. Eight of twelve expected frequencies are below five, with minimum about 1.158, making the usual large-sample calibration particularly questionable. An exact conditional analysis or Monte Carlo test would be more defensible for a calibrated inferential claim; no uncomputed simulated p-value is asserted.^[ref]

The Black Cramér's V is about 0.308, the generalized phi coefficient 0.435 and the contingency coefficient 0.399. An unsigned magnitude of this size can coexist with a nonsignificant test in a small sample. There are 254 concordances and 457 discordances, giving Goodman and Kruskal's gamma about $-0.286$, Somers' D about $-0.179$, Kendall's tau-b about $-0.191$ and Stuart's tau-c about $-0.187$. Their negative signs agree with the reversed displayed-order direction. Goodman and Kruskal's lambda is about 0.107 for efficacy given technology and zero for technology given efficacy. In the latter direction, Disagree remains a modal technology category within every efficacy column, so modal prediction gains nothing despite the changed conditional percentages.

For White respondents, High efficacy decreases from 48.55% in Strongly disagree through 43.10% and 32.03% to 27.45% in Agree. The extreme-row contrast is about 21.11 percentage points, from 151 of 311 versus 188 of 685. Low efficacy is 21.22%, 17.10%, 21.57% and 35.04%, respectively. This closely resembles the overall distribution, including the fact that the Low category is not monotone at every step. The White Pearson chi-squared test of independence gives $X^2\approx 107.70835$, 6 statistical degrees of freedom and $p\approx 6.15\times 10^{-21}$; the likelihood-ratio test gives $G^2\approx 105.54201$, $p\approx 1.75\times 10^{-20}$. All expected frequencies exceed 74, so sparse cells are not the problem for these nominal tests.

The White Cramér's V is about 0.152, the generalized phi coefficient 0.215 and the contingency coefficient 0.210. Goodman and Kruskal's gamma is about 0.236, Somers' D 0.155, Kendall's tau-b 0.162 and Stuart's tau-c 0.167: close to the overall ordinal summaries and in the same direction. The two values of Goodman and Kruskal's lambda are about 0.061 for efficacy given technology and 0.062 for technology given efficacy. They again indicate only a small improvement in modal prediction. The large sample size helps explain why the White test has a much smaller p-value than the smaller strata, without implying that its association magnitude is the largest.

The original figure below plots selected empirical conditional probabilities from all the cells. It gives the opposite small-stratum pattern and the dominant White pattern a common scale; it is not a reproduction of a source figure, an estimated continuous trend or a display of uncertainty intervals.

The elaboration model asks how the original association changes when a third variable is held fixed. If the association disappeared throughout the strata, an explanation by the control variable might be plausible, subject to a credible causal ordering. Here it persists in the White group, which supplies 94.57% of the sample, and is stronger on several descriptive measures in the Asian group. Therefore ethnicity does not simply explain away the relationship. The White result is a replication of the broad pattern; the reversed Black direction and the Asian shape suggest descriptive specification or possible effect modification. The output supplies no temporal or causal design that would distinguish an actual causal mechanism from other confounding, measurement differences or sampling variation. It also does not demonstrate a Simpson's paradox in which the association reverses in every stratum: that is not what the displayed tables show.

The predominance of White respondents explains why the pooled pattern resembles theirs. By the law of total probability, each technology row's efficacy distribution is a mixture over ethnicity, using the ethnic proportions within that technology row. Those weights need not equal the overall ethnic proportions. As an optional descriptive comparison, standardization with common weights using the overall ethnic shares gives High-efficacy proportions of about 48.38% in Strongly disagree and 27.90% in Agree, a 20.48-percentage point contrast, compared with the crude 21.69-point contrast. This explicitly defined adjustment leaves the broad contrast intact; it is not a causal estimate and does not turn the three different stratum patterns into one common effect.

Finally, a significant test in one group and a nonsignificant test in another is not itself a test that the groups' associations differ. A formal statistical interaction requires a directly specified heterogeneity test or model on a chosen effect scale. The omnibus tests also address whole categorical distributions, whereas gamma and the High-efficacy contrast describe particular ordered features. Multiple stratum tests create a multiple testing issue if used to select a discovery. The defensible result is an association that survives stratification in the dominant group, with important descriptive heterogeneity that merits a direct, sparse-cell-aware investigation rather than an assertion that ethnicity causes the difference.

For each income band, compare the response distribution within the two gender groups. The table below combines agree and strongly agree only to make one comparable contrast visible. The Pearson chi-squared statistic and its p-value still use all five original response categories, with four statistical degrees of freedom in every band. Income labels are thousands of USD per year; finite upper bounds are excluded.

In the lowest-income band, combined agreement differs by only about $4.39$ percentage points. The full distributions still differ descriptively: strong disagreement is $13.64\%$ in the male group and $27.40\%$ in the female group. The omnibus Pearson chi-squared test does not reject independence at 5%, but that decision does not prove equal distributions.

In the 10k–20k band, combined agreement is almost the same. Nevertheless, strong agreement is higher in the female group, $18.66\%$ versus $11.54\%$, while the less intense agreement category is higher in the male group, $24.36\%$ versus $16.42\%$. Collapsing the categories hides that distinction. The full-distribution p-value is approximately $0.2435$.

In the 20k–30k band, combined agreement is higher in the male group by about $12.96$ percentage points. Strong agreement alone is $27.17\%$ versus $18.30\%$. The observed difference is therefore larger than in the two lowest bands, despite its full-table p-value remaining above 5%.

In the 30k–40k band, combined agreement differs by about $17.40$ percentage points, with fewer male neutral responses, $15.69\%$ versus $26.14\%$. The full-table p-value of $0.03851$ is nominally significant at 5%. In the 40k–50k band, the combined agreement gap is a similar $16.18$ percentage points, but the full-table p-value is $0.1534$. These labels are not a direct comparison between the two effects: significant and nonsignificant results need not differ significantly.

In the highest-income band, strong agreement is $41.84\%$ in the male group and $22.15\%$ in the female group. Combined agreement differs by about $20.95$ percentage points. This band has both the largest sample size and the largest Cramér's V of the six tables, with a very small p-value. Its particularly strong evidence reflects both precision and an observed difference in the distribution.

All six minimum expected frequencies exceed five: approximately $7.15$, $9.57$, $12.02$, $15.04$, $7.93$ and $36.03$. Thus sparse expected counts do not provide a reason to reject the usual chi-squared distribution approximation here. If the six hypothesis tests form one exploratory family, a Bonferroni correction would use $0.05/6\approx 0.00833$ for each test; only the highest-income result crosses that threshold. This qualifies the unadjusted 30k–40k result without turning its observed contrast into zero.

In the elaboration model, the larger agreement gaps in the middle and upper bands provide a descriptive pattern of specification: the conditional contrast varies with income. This is not a Simpson's paradox reversal, because the combined agreement difference has the same sign in every band. Nor is the larger high-income effect size established simply by comparing its p-value with the other p-values. A formal claim of effect modification requires a stated scale and an appropriate comparison of the effects.

The pooled comparison also mixes different income compositions. The highest-income band contains $631/1042\approx 60.56\%$ of the male group but $474/1093\approx 43.37\%$ of the female group. Average each group's six agreement proportions over the same pooled income weights using standardization with common weights. This givesThe crude difference was $19.56$ percentage points, so common weighting attenuates it by about $3.30$ percentage points but leaves a substantial descriptive contrast. With average-group rather than pooled-person weights, the symmetric decomposition of a stratified mean difference instead gives an exact split into approximately $16.29$ percentage points from conditional contrasts and $3.27$ percentage points from composition. These are different explicitly defined descriptive summaries, not estimates of a causally mediated share.

An antecedent income variable could be considered in a confounding explanation, whereas an income variable on an assumed causal pathway would enter an interpretation through causal mediation. The conditional tables do not distinguish these causal structures. They also do not identify whether any adjustment introduces collider bias. The supported conclusion is income-dependent descriptive association plus a measurable composition difference, with causal interpretation left conditional on additional assumptions.