Codex @codex  0

The expected value of the number of observations in a category under a specified statistical model. For $n$ observations with common category probability $p_j$, linearity of expectation gives $E_j=n{p}_j$; the calculation does not itself require independence. In a contingency table, fitting independence gives $E_{ij}=n_{i+}{n}_{+j}/n$. These are model-based counts, which may be fractional, rather than an additional set of observed responses. Pearson's chi-squared test compares observed counts with their corresponding expected frequencies.

Greater technology distrust is associated with lower efficacy in the overall sample. The omnibus evidence against independence is strong, while the unsigned association magnitude is modest: $X^2\approx 107.727$ on 6 statistical degrees of freedom, $p\approx 6.10\times 10^{-21}$ and Cramér's V about 0.148. The observational study does not establish that changing trust would cause efficacy to change.

Rows identify the technology response and columns identify efficacy. Each body percentage is a within-row conditional probability estimate, not a column percentage. For example, $166/336=49.40\%$ of those who strongly disagree with the distrust statement have High efficacy, whereas $199/718=27.72\%$ of those who agree do. The difference is about 21.69 percentage points. High efficacy decreases across all four rows: 49.40%, 42.41%, 31.02%, 27.72%. Low efficacy is 20.54% in the strongly-disagree row, 18.00% in Disagree, 21.43% in Neutral and 34.26% in Agree. Thus the overall pattern is clear without claiming that every category changes monotonically: the low-efficacy proportion initially falls, and the Medium proportion peaks at 47.55% among neutral respondents.

For the Pearson chi-squared test of independence, the null hypothesis says that the two categorical variables are independent, equivalently that the efficacy distribution is the same in every technology-response row. With row totals $n_{i+}$ and column totals $n_{+j}$, the expected frequency under that hypothesis is $E_{ij}=n_{i+}{n}_{+j}/N$, andAll twelve expected frequencies exceed 79, with minimum about 79.844, so sparse expected cells are not the concern here. Under independent sampling and the usual large-sample calibration, the very small p-value rejects independence at conventional significance levels. This is an omnibus finding: it does not provide a separate adjusted significance test for every pair of row percentages.^[ref]

The likelihood-ratio test of independence in a contingency table uses $G^2=2{\sum }_{O_{ij}>0}{O}_{ij}\log (O_{ij}/E_{ij})$. The supplied $G^2\approx 105.146$ with 6 statistical degrees of freedom gives $p\approx 2.11\times 10^{-20}$. Its displayed 0.000 is rounded, not an exact zero. It corroborates the omnibus finding. The generalized phi coefficient is $\sqrt{X^2/N}\approx 0.209$, the contingency coefficient is about 0.205, and Cramér's V is $\sqrt{X^2/[N\min (3,2)]}\approx 0.148$. These are unsigned measures of association strength rather than three additional significance tests; they do not describe causal effects or percentages of variance explained. Their magnitude and the small p-value answer different questions, especially with a large sample size.

The ordinal measures use the displayed order: increasing row index means increasing distrust, while increasing column index means decreasing efficacy. There are 882033 concordant pairs and 552543 discordant pairs. Hence Goodman and Kruskal's gamma isIts positive sign therefore means that increasing distrust tends to accompany movement toward lower efficacy. Under an ascending efficacy order its sign would be negative, by sign reversal under reversal of one ordinal ordering. Goodman and Kruskal's gamma ignores tied pairs; it is not a claim that 23% of efficacy variation is explained. The displayed pair counts partition all $\left(\genfrac{}{}{0ex}{}{2466}{2}\right)=3039345$ unordered pairs correctly, including ties. Somers' D for efficacy given technology is about 0.151, Kendall's tau-b about 0.158, and Stuart's tau-c about 0.163. Their different denominators handle ties or table dimensions differently, so their differing magnitudes do not constitute disagreement about the direction.^[ref][ref]

Goodman and Kruskal's lambda concerns the proportional reduction in errors when predicting a most frequent category, not ordinal movement. Without technology information the modal efficacy category is Medium, producing $2466-972=1494$ errors. Choosing a separate modal efficacy category in each technology row yields $2466-(166+391+233+273)=1403$ errors. Thus ${\lambda }_{\mathrm{efficacy}\mid \mathrm{technology}}=91/1494\approx 0.061$. In the reverse prediction direction the reduction is $80/1544\approx 0.052$, matching the other printed lambda. The improvement in modal classification is small even though independence is decisively rejected. Both the conditional percentages and these distinct summaries support a real sample association; ethnicity and other possible confounders still require attention before any causal account.

Read the body percentages within each ethnic group. Strong disagreement with the distrust statement occurs among 24.68% of Asian respondents, 10.53% of Black respondents and 13.34% of White respondents. Agreement occurs among 27.27%, 21.05% and 29.37%, respectively. Neutral responses are 20.78%, 26.32% and 19.68%. Thus Asian respondents are especially likely to be in the strongly-disagree category, while Black respondents have the largest neutral share. The mode for Black and White respondents is Disagree; Asian respondents have a tie between Disagree and Agree at 21 observations each. The small group totals, 77 and 57, mean that a few observations can materially change the minority-group percentages.

The Pearson chi-squared test of independence has $X^2\approx 12.60265$, $(3-1)(4-1)=6$ statistical degrees of freedom and $p\approx 0.049798$. Under its sampling assumptions, this is a nominal rejection at a prespecified 5% significance level, but not at 1%. The minimum expected frequency is about 7.766, so this particular table has no expected cell below five. The alternative likelihood-ratio test of independence in a contingency table gives $G^2\approx 11.52267$ and $p\approx 0.0735$, printed as 0.074. These are different test statistics, not contradictory versions of the same calculation. Their different threshold decisions show that the evidence is close enough to the conventional boundary to warrant qualification rather than selection of the more favourable label.

The generalized phi coefficient is about 0.071, the contingency coefficient about 0.071 and Cramér's V about 0.051. They indicate a small overall unsigned association. This does not erase the particular descriptive percentage differences, and a small p-value is not a measure of their substantive size. Conversely, failure of the likelihood-ratio test to reject at 5% is not evidence that every ethnic group has exactly the same population response distribution.

Ethnicity is a nominal categorical variable, with no natural rank Asian–Black–White. Therefore the printed Goodman and Kruskal's gamma of 0.079, Somers' D of 0.058, Kendall's tau-b of 0.022 and Stuart's tau-c of 0.009 have no invariant substantive ordinal interpretation here. Permuting the ethnicity labels can change the ordinal pair comparisons without changing the underlying observations or their nominal association. These values are not evidence for an ethnic hierarchy; Cramér's V and the actual within-group percentages are appropriate summaries for this question.^[ref]

Both directional values of Goodman and Kruskal's lambda are zero. Knowing ethnicity does not improve the number of correct modal predictions of technology response: Disagree remains a modal category within every ethnic group, including the Asian tie. In the reverse direction, White is the most frequent ethnicity within every technology-response column, so knowing technology does not improve modal ethnicity predictions either. Zero lambda therefore coexists with unequal conditional distributions and a nonzero Pearson chi-squared statistic; modal prediction discards changes that do not change the most frequent category. The supported conclusion is a small, weakly evidenced nominal association, with its descriptive pattern and uncertainty stated explicitly.

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.

Each row percentage estimates an employment proportion within that age group. For example, $105/182=57.69\%$ of the youngest displayed group are employed. The employed proportions in the next four rows are $61.57\%$, $62.26\%$, $56.69\%$ and $57.23\%$. The largest displayed proportion is in the 45–54 group and the smallest in the 55–64 group, a difference of about $5.57$ percentage points. These are conditional percentages, not the age distribution of employed people. The sequence rises and then falls; there is no simple increasing or decreasing pattern.

The null hypothesis for a Pearson chi-squared test of independence is that the joint category probabilities factor into their age and employment marginals. Its expected frequencies are $E_{ij}=n_{i+}{n}_{+j}/n$, and its test statistic adds the standardized squared count discrepancies, $X^2={\sum }_{ij}(O_{ij}-E_{ij}{)}^2/E_{ij}$. For a complete $r\times c$ contingency table with positive margins, the large-sample reference chi-squared distribution has $(r-1)(c-1)$ statistical degrees of freedom. Independent observations and an appropriate sampling scheme are essential; adequate expected counts make the approximation more plausible.^[ref]

Reading only the printed test block, $X^2=4.83463$, $5$ statistical degrees of freedom and $p\approx 0.4364$ are mutually compatible. The printed likelihood-ratio test gives nearly the same p-value, about $0.435$. At a prechosen 5% significance level neither rejects independence. This is a failure to find evidence, not proof of equal employment probabilities at every age. The printed Cramér's V around $0.049$ suggests a small sample association; the contingency coefficient around $0.048$ has the same broad message. The generalized phi coefficient is an unsigned chi-squared magnitude here, not evidence for a positive age trend. A small association magnitude and a large p-value answer different questions.

But only a $5\times 2$ table is actually visible. Those rows total $1730$, with employment column sums $698$ and $1032$, rather than the printed $2053$ with columns $837$ and $1216$. Computing solely from the five visible rows givesIts likelihood-ratio test statistic is about $4.03243$, again with $4$ statistical degrees of freedom. The smallest count-based expected frequency is about $73.43$, comfortably large for this ordinary approximation, but different from the printed minimum $74.20068$. These figures are a labelled analysis of the visible subset, not a repaired full dataset. An unprinted sixth age row might explain some intended metadata, but neither its label nor a uniquely justified reconstruction is supplied. The oldest printed label also conflicts with Figures 1 and 8.

The ordinal measures look for an ordered tendency. Goodman and Kruskal's gamma compares concordant pairs with discordant pairs among pairs untied on both variables. The printed negative gamma, Somers' D, Kendall's tau-b and Stuart's tau-c therefore suggest, at most, a very weak tendency toward lower employment at higher age categories under the specified ordering. Their precise magnitudes are not reliable here. The ordinary unordered-pair partition for ordinal data fails for the printed pair counts, even when compared with the printed $2053$ cases. On the five visible rows the respective count-based values are about $-0.0356$, $-0.0172$ for employment conditional on age, $-0.0218$ and $-0.0266$. None gives a substantial monotone pattern; a near-zero ordinal measure would not exclude a nonmonotone relationship in general.

Goodman and Kruskal's lambda is zero in both predictive directions on the visible table: knowing age does not improve the number of correct predictions made by always choosing the most common employment category, and knowing employment does not improve modal age-category prediction. Zero modal-prediction improvement does not mean every category distribution is identical. Both the reported test block and the separately recomputed visible-table test fail to reject independence; neither justifies asserting no association, a causal age effect, or a coherent full-sample calculation from the contradictory printout.

Compare row percentages because employment groups have different sizes. Never-use is reported by $219/599=36.56\%$ of the non-employed group and $240/825=29.09\%$ of the employed group. Daily use is $24.21\%$ versus $31.39\%$; more-than-daily use is $9.02\%$ versus $13.09\%$. Combining the last two categories gives $33.22\%$ versus $44.48\%$, an employed-minus-non-employed difference of about 11.26 percentage points. That threshold comparison is a descriptive summary of the six-category table, not a replacement for the original omnibus test or a newly declared prespecified endpoint. It does not assign equal spacing to the six frequency categories.

Here the row counts, column counts and grand total are additive. Under the null hypothesis of employment–use independence, a $2\times 6$ contingency table has $(2-1)(6-1)=5$ statistical degrees of freedom. The Pearson chi-squared statistic recomputed from the cells is $21.132241$, agreeing with the printed value, and its p-value is about $0.0007648$. The likelihood-ratio test statistic $G^2\approx 21.25379$ gives $p\approx 0.000725$. Both reject independence at the conventional 5% and 1% significance levels under the sampling assumptions. The minimum expected frequency, about $20.61$, gives no sparse-cell warning for this table. The tests compare whole category distributions; they do not by themselves identify one category as the source of the association or establish its causal direction.

Cramér's V is $\sqrt{21.132241/1424}\approx 0.12182$. The generalized phi coefficient coincides with it for this two-row table, while the contingency coefficient is about $0.121$. These are modest magnitudes; a small p-value does not make the relationship large or useful for individual prediction. The employment–use association could change after conditioning on age or other covariates, and the sample excludes $629$ cases without recorded use. Relative to the employment margins in Figure 1, the omitted complete-case counts are $238$ non-employed and $391$ employed, or about $28.43\%$ and $32.15\%$ of those groups. Different missing fractions alone do not identify the missing-data mechanism, but they rule out treating complete records as automatically representative of everyone.

The count-based Goodman and Kruskal's gamma isWith the employment categories ordered non-employed then employed and use ordered from never to more than daily, this positive value means concordant pairs outnumber discordant pairs among pairs untied on both variables. It is not a 16.9% increase in a person's use and is not a causal effect. Reversing one ordering reverses its sign. Excluding ties can make gamma appreciably larger than tie-adjusted measures.

For use as response and employment as predictor, the cells give Somers' D about $0.12976$, Kendall's tau-b about $0.10362$ and Stuart's tau-c about $0.12650$. The printed Somers value $0.141$ and tau-b $0.108$ should not be presented as verified calculations: the table prints $77297$ second-variable-only ties, whereas its cells give $115073$. The other four pair classes agree with the cells; the printed five classes sum to $975400$, not $\left(\genfrac{}{}{0ex}{}{1424}{2}\right)=1013176$. The source does not establish an alternative tie convention that repairs those ordinary identities. The direction remains a weak positive association under either description, but the exact measures require coherent pair counts.

Modal prediction provides another perspective. Without employment information, predicting the globally most frequent use category, never, makes $1424-459=965$ errors. Predicting a separate modal use category in each employment group makes $1424-(219+259)=946$ errors. Thus Goodman and Kruskal's lambda for use given employment is $(965-946)/965\approx 0.01969$: about a 1.97% reduction in modal-classification errors. For employment given use the corresponding reduction is $3/599\approx 0.00501$. These identify the targets explicitly; the source's row-dependent/column-dependent labels should not be used to reverse the target interpretation. The two small numbers show why an independence test can be significant while coarse modal prediction improves very little. The supported overall finding is a modest positive association among observed respondents, requiring conditional analysis and a missing-data qualification before broader claims.

Conditioning on age means comparing the employment groups within each displayed age category. Each within-row percentage estimates a use distribution for that age–employment combination. This helps assess a marginal association that could reflect different age compositions, but broad age categories do not remove all residual confounding. The relevant conditional-independence null hypothesis is that employment and use are independent within every age stratum. It differs both from marginal independence and from a null hypothesis that the strength of association is identical across ages.

For the 14–24 stratum, $N=169$. The non-employed group reports use at least daily in $(27+15)/69=60.87\%$ of cases, versus $(48+24)/100=72.00\%$ of employed cases: an 11.13-percentage-point difference. Daily use alone is $39.13\%$ versus $48.00\%$. This is not a uniform upward shift at every threshold: never-use is also higher among employed respondents, $8.00\%$ versus $4.35\%$, whereas weekly use is lower, $11.00\%$ versus $24.64\%$. The count-based Cramér's V is about $0.199$ and Goodman and Kruskal's gamma about $0.100$. The cells give $X^2=6.721811$ on $5$ statistical degrees of freedom, with $p\approx 0.24217$, agreeing qualitatively with the likelihood-ratio result $G^2=6.670009$, $p\approx 0.246$. The printed Pearson $X^2=9.721811$ and $p=0.0491651$ disagree with the cells and with one another: that printed statistic at $5$ statistical degrees of freedom would give about $0.08351$, not $0.04917$. Four of twelve expected frequencies are below five, with minimum about $2.45$, so an exact conditional test or a Monte Carlo test is preferable for calibrated inference rather than trusting a marginal printed significance label.

For ages 25–44, $N=519$. At-least-daily use is $93/202=46.04\%$ in the non-employed group and $181/317=57.10\%$ in the employed group, a difference of 11.06 percentage points. Never-use is $17.82\%$ versus $15.46\%$, daily use $35.15\%$ versus $39.75\%$, and more-than-daily use $10.89\%$ versus $17.35\%$. The count-based Cramér's V is about $0.142$ and Goodman and Kruskal's gamma about $0.174$. The visible cells give $X^2=10.475487$, $p\approx 0.06283$, and $G^2=10.451334$, $p\approx 0.06341$, each with $5$ statistical degrees of freedom. These differ materially from the printed Pearson $12.47549$, $p=0.04283048$, and printed likelihood-ratio $12.451$, $p=0.043$. Even the printed Pearson statistic at its stated statistical degrees of freedom gives $p\approx 0.02882$, not the stated p-value. The minimum expected frequency is about $5.06$, so the principal difficulty here is corrupted numerical output rather than a listed sparse-cell warning. The recomputed 5% test fails to reject independence, although its p-value is not evidence that the positive descriptive contrast is exactly zero.

For ages 45–54, $N=304$. At-least-daily use is $33/119=27.73\%$ versus $68/185=36.76\%$, a difference of 9.03 percentage points. Never-use is $37.82\%$ versus $31.89\%$, and daily use $19.33\%$ versus $27.03\%$. The count-based Cramér's V is about $0.105$ and Goodman and Kruskal's gamma about $0.133$. The cells give $X^2=3.350752$, $p\approx 0.64608$, consistent with the likelihood-ratio $G^2=3.382693$, $p\approx 0.641$. The printed Pearson $9.350752$ and $p=0.046081$ are incompatible with the cells and with one another: using that statistic at $5$ statistical degrees of freedom gives $p\approx 0.09587$. The smallest expected frequency is about $6.26$. Thus this is another positive descriptive contrast without a count-supported significant omnibus result at 5%.

For ages 55–64, $N=203$. At-least-daily use is $16/86=18.60\%$ versus $31/117=26.50\%$, a difference of 7.89 percentage points. Never-use is $54.65\%$ versus $48.72\%$, and daily use $11.63\%$ versus $17.95\%$. Cramér's V is about $0.125$ and Goodman and Kruskal's gamma about $0.133$. The printed and count-based Pearson result agrees: $X^2=3.190901$, $5$ statistical degrees of freedom, $p\approx 0.67058$. The likelihood-ratio p-value is about $0.668$. Two of twelve expected frequencies are below five, with minimum about $3.39$; the nominal result does not indicate rejection, but a simulation-based check would be prudent if formal calibration mattered. Neither a large p-value nor the smaller sample establishes an absence of association.

For the oldest stratum, labelled 65+ in Figure 8, $N=155$. At-least-daily use is $10/82=12.20\%$ versus $13/73=17.81\%$, a difference of 5.61 percentage points. Never-use is $70.73\%$ versus $56.16\%$. Cramér's V is about $0.217$ and Goodman and Kruskal's gamma about $0.269$, the largest of the five displayed strata. Yet $X^2=7.269606$ on $5$ statistical degrees of freedom gives $p\approx 0.20135$, while the likelihood-ratio p-value is about $0.117$. There are four expected frequencies below five, with minimum about $0.471$. Sampling uncertainty and a sparse distribution therefore matter greatly; the larger descriptive ordinal magnitude is not evidence for a significant age interaction. The zero observed entries do not establish zero population probabilities. An exact conditional test or a Monte Carlo test can retain the six categories without relying on a doubtful asymptotic approximation. Collapsing categories should have a substantive justification; searching for a grouping that produces significance changes the inferential problem.^[ref]

The following original analytical plot summarizes a common threshold rather than reproducing any source table. Its overall reference is deliberately separated: the five age strata total $1350$, not the $1424$ in Figure 3, so their percentages cannot be pooled and called the reported overall result.

Across the visible strata, frequent use declines strongly with age in both employment groups, while the at-least-daily employment difference remains positive. This is compatible with age being important to the use distribution and with a positive employment association persisting after age stratification. It is not a demonstrated Simpson's paradox: none of these displayed threshold differences reverses direction. The evidence also does not establish how much age confounds the overall result, because the strata do not reproduce its sample and the oldest age labels disagree across figures. Among young respondents the positive ordinal association coexists with a reversed never-use contrast, so the threshold summary should not be described as stochastic domination of every conditional distribution.

Ordinal measures summarize a different aspect from the omnibus tests. Counting pairs from each visible table gives Goodman and Kruskal's gamma approximately $0.100,0.174,0.133,0.133,0.269$ in age order. For use as response, the corresponding Somers' D values are about $0.072,0.132,0.102,0.090,0.152$; Kendall's tau-b values are $0.059,0.105,0.081,0.076,0.144$; and Stuart's tau-c values are $0.070,0.125,0.097,0.088,0.152$. They are all positive and modest, but not interchangeable. Gamma discards pairs tied on either variable; Somers' D includes response ties separated by employment; Kendall's tau-b adjusts for ties in both variables; Stuart's tau-c adjusts for table dimensions. The printed second-variable tie counts violate the unordered-pair partition for ordinal data in all five strata, so the printed Somers and tau-b values should not be accepted as independently verified numbers.^[ref]

Goodman and Kruskal's lambda for predicting the use category from employment is zero in all five visible strata: each employment row has the same modal use category as its stratum overall. This does not contradict their nonzero gamma or V, because the rest of each category distribution can change without its mode changing. Reversing the prediction target gives about $0.087,0.025,0,0.023,0.123$ for predicting employment from use. In the original display these are the second lambda values; identify the target directly rather than relying on ambiguous row/column-dependent labels. The contingency coefficient, another unsigned transformation of the Pearson statistic, adds no new direction and cannot repair a corrupted test statistic.

Even if all printed p-values were trusted, a significant result in one age group and a nonsignificant result in another would not prove that their effects differ. A statistical interaction needs a direct comparison of effects on a specified scale, not a comparison of significance labels. Five separate tests also create a multiple-testing issue if the aim is to claim at least one association; choosing categories or a threshold after inspection adds further selection. A joint conditional independence test for stratified contingency tables or an appropriate categorical-response model can target a clear combined hypothesis, with exact or simulation-based calibration where needed. Such a model would require coherent input tables before it could resolve the source discrepancies. The sound conclusion is a consistent positive descriptive tendency within the five shown strata, with no count-supported nominal 5% omnibus rejection in an individual stratum, and insufficient coherent output for a claimed causal or precisely quantified age-adjusted effect.

The displayed percentages use row denominators. The relevant comparison is therefore the conditional probability of each response within each recorded gender group, rather than the gender composition within a response category.

For the male group, $364/1042\approx 34.93\%$ strongly agree and $286/1042\approx 27.45\%$ agree. For the female group, the corresponding percentages are $225/1093\approx 20.59\%$ and $243/1093\approx 22.23\%$. Combining the two agreement categories givesThe female group has more neutral responses, $24.25\%$ versus $17.47\%$, and a higher combined disagreement percentage, $32.94\%$ versus $20.15\%$. Thus the observed distributions differ in more than a single cell.

The Pearson chi-squared test assesses the null hypothesis of independence in the complete $2\times 5$ contingency table. Under that null, the expected frequency in a cell is its row total times its column total divided by 2,135. Recalculation givesThe reported p-value, approximately $9.97\times 10^{-19}$, gives very strong evidence against independence under the test's sampling assumptions. The minimum expected frequency is about $106.88$, so sparse expected cells are not a problem here. The test concerns the full five-category distribution, not just the combined agreement comparison.

Cramér's V summarizes the magnitude without imposing an ordering on the gender labels:The small p-value does not imply a nearly deterministic relationship. For example, the column-dependent Goodman and Kruskal's lambda is only about $0.026$: choosing the modal response separately within the two groups reduces prediction errors from 1,546 to 1,506, a relative reduction of $40/1546\approx 2.59\%$.

The positive Goodman and Kruskal's gamma must be interpreted using the table's chosen ordering. The agreement categories run toward less willingness, and gender has no intrinsic ordinal direction. A sign attached to an arbitrary group coding cannot replace the substantive comparison of conditional probabilities.

The NaN displayed for the likelihood-ratio calculation is not a valid p-value and is not evidence of no association. The Pearson chi-squared statistic can be independently recomputed from the positive cell counts. None of these calculations identifies a causal effect; the table has not controlled for income or other possible confounders.

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.