= Solution
\b[Age and observed populism classification are associated, but the overall association is small and not a strictly monotone age trend. Populist classification is most common in the 55+ group, about 77.12%, versus 66.77% at ages 18–24; it also falls back to 66.38% at ages 35–44.]
This table has 1692 observations, all respondents with an observed populism classification, because age has no missing values. The within-age Populist percentages are 66.77%, 73.18%, 66.38%, 74.03% and 77.12%, in ascending age-group order. Their Not-populist complements are 33.23%, 26.82%, 33.62%, 25.97% and 22.88%. Comparing the oldest and youngest Populist percentages gives about 10.35 <percentage points>, but the intermediate reversal prevents describing a uniform rise at every age transition. The row totals 310, 302, 348, 308 and 424 are the classified cases; their differences from the age totals in Figure 1 sum to the 92 missing classifications.
The <Pearson chi-squared test of independence> gives $X^2\approx15.8680$ on $(5-1)(2-1)=4$ <statistical degrees of freedom>, $p\approx0.003201$. The <likelihood-ratio test> gives $G^2\approx15.8572$ and $p\approx0.003217$. Both reject <independence> at 5% and 1% under the usual sampling assumptions. The minimum <expected frequency> is about 85.317, so the large-sample approximation is not threatened by sparse expected counts here. Neither test is specifically a test of a linear or monotone age trend.
The <generalized phi coefficient> and <Cramér's V> are about 0.097, and the <contingency coefficient> about 0.096: small unsigned association magnitudes. If the binary classification is coded Not populist before Populist and age groups are ordered younger to older, the ordinal output has a meaningful conventional direction. The 258482 concordant and 205602 discordant pairs give <Goodman and Kruskal's gamma> about 0.114; <Somers' D> for classification given age is about 0.046, <Kendall's tau-b> 0.065 and <Stuart's tau-c> 0.074. These small positive summaries describe an overall older–Populist tendency while allowing the middle-age reversal. They use different tie or table-size adjustments and are not percentages of respondents caused to change classification by ageing.
<Goodman and Kruskal's lambda> for predicting classification from age is zero because Populist remains the modal classification in every age row. In the reverse direction, predicting age group from classification improves modal predictions slightly: without classification the most frequent age group has 424 cases, whereas choosing a separate modal age group for each classification yields $117+327=444$ correct predictions. The error reduction is $(444-424)/(1692-424)\approx0.0158$, matching the displayed 0.016. Direction must therefore be stated, rather than treating both lambda numbers as interchangeable. \b[Age is a relevant control variable for the later comparison, but these cross-sectional associations do not establish an age-related causal mechanism or remove other confounders.]
Back to article page
