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Let $C,D$ count concordant pairs and discordant pairs, and let $T_X,T_Y$ count pairs tied only on the indicated variable. Thenwhen both factors in the denominator are positive. Unlike Goodman and Kruskal's gamma, this coefficient accounts for ties. Its absolute value is at most one: form vectors of the comparison signs for all unordered pairs. Their inner product is $C-D$ and their squared norms are the two denominator factors, in reverse variable order. The Cauchy-Schwarz inequality supplies the bound. A variable constant throughout gives a zero denominator, not a meaningful zero association.^[ref]

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.

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.

For an $r\times c$ contingency table of two ordinal categorical variables, write $m=\min (r,c)>1$, $n$ for the sample size and $C,D$ for concordant pairs and discordant pairs. The coefficient isIt provides an ordinal association measure whose normalization accounts for the table dimensions. It generally differs from Kendall's tau-b and Goodman and Kruskal's gamma on a table with tied pairs. An association sign presupposes the chosen category ordering and is not a causal direction. The eponym is associated with Alan Stuart.^[ref]