Past exam of the digital social research: statistics core course of the University of Oxford / 2018 / 4 / Solution

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 $X^2\approx 15.8680$ on $(5-1)(2-1)=4$ statistical degrees of freedom, $p\approx 0.003201$. The likelihood-ratio test gives $G^2\approx 15.8572$ and $p\approx 0.003217$. 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 $(444-424)/(1692-424)\approx 0.0158$, matching the displayed 0.016. Direction must therefore be stated, rather than treating both lambda numbers as interchangeable. 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.