• UpSet bar ≠ heatmap cell. The UpSet bar for "A + B" counts papers whose exact topic set is {A, B} and nothing else (e.g. social power + identity ≈ 15 papers). The heatmap cell counts every paper with at least both tags, regardless of what else they carry (e.g. ≈ 131 papers). The heatmap value equals the sum of every UpSet bar whose dot pattern contains both topics.
• Raw count vs phi. Raw count is volume; phi is association above chance. Two common labels (like social power and identity) will co-occur a lot in absolute terms even if they're only modestly associated — their phi may be only ~0.20–0.30 even when their co-occurrence count is the corpus maximum. Rare labels that nonetheless travel together (like emotion + place-based study) can be invisible in raw count but jump out in phi.
• Cluster ordering surfaces block structure. Switching to cluster (hierarchical on 1−φ) reorders rows and columns so positively-associated labels land adjacent. Expect roughly three blocks on the diagonal: a social-political cluster (social power, identity, governance, mobility), a place/space-form cluster (place-based study, built environment, scale/location), and a nature-society cluster (human-environment, natural environment, climate and natural hazards, perhaps agriculture/food). The long-tail labels (methods, technology, energy, weather, recreation) typically sit at the edges and their clustering position is noisy because of small support.
• What strong negative phi (red) means. Red cells flag tag pairs that papers avoid combining — distinct analytical commitments. Expect mild red between method-of-paper-explanation labels (methods, technology) and affective labels (emotion), and between purely-physical labels (weather, climate and natural hazards) and identity/power labels.
• Sample-size caveat. weather (n=2), recreation (n=13), energy (n=12), natural environment (n=14) have low support, so their phi values are noisy and their clustered position can shift on minor data changes. Treat their positions as suggestive at best.

Pairs like social power and identity are the most common co-occurring tags, but that's not particularly meaningful — they're also the two most common individual tags. We instead want to examine the correlation between tags: how often they appear together relative to how often they appear separately. The phi coefficient does exactly this. Its focus is how much more likely it is that either both topics X and Y appear on a paper, or neither does, than that one appears without the other.

φ  =  n₁₁·n₀₀ − n₁₀·n₀₁ √(n_1··n_0··n_·0·n_·1)

The four counts come from a 2×2 contingency table for any pair of topics X and Y across the corpus:

Concretely, for X = social power and Y = identity on our 346 papers: n₁₁ ≈ papers with both, n₁₀ ≈ social-power-only, n₀₁ ≈ identity-only, n₀₀ ≈ neither. The numerator (n₁₁·n₀₀ − n₁₀·n₀₁) is positive when "both-or-neither" dominates "one-without-the-other", and negative in the reverse case. The denominator is a geometric mean of the marginal totals; dividing by it normalizes φ into [−1, 1] regardless of corpus size or how lopsided the individual labels are.

Properties worth knowing:

• φ is symmetric: φ(X, Y) = φ(Y, X). The heatmap is therefore symmetric across the diagonal.
• φ = 0 means topics co-occur exactly at chance frequency, given their marginals.
• For two binary variables, φ equals the Pearson correlation coefficient; both have the same meaning in this setting.
• In practice |φ| rarely exceeds ~0.5 even for strong associations in document-tagging data, because each tag has a wide range of papers it could appear on — the diverging colour scale on the heatmap amplifies the |φ| × 2.5 mapping so that mid-range associations are still visible.
• φ is sensitive to the marginal frequencies. A pair with low individual support that always co-occurs can show a very high φ on a small sample (the weather/climate-and-natural-hazards pair would be a candidate here). The sample-size caveat above applies.

Is |φ| = 0.3 surprising? Significance and bootstrap

A useful intuition for our 346-paper corpus: under the null hypothesis that two tags are independent, the standard error of φ is roughly 1/√n ≈ 0.054. So an observed φ = 0.3 sits about 5–6 standard errors away from zero — extremely unlikely under independence.

More precisely: the chi-square test statistic for a 2×2 contingency table is χ² = n · φ², with 1 degree of freedom. So:

• |φ| ≥ 0.10 → χ² ≥ 3.46, p ≈ 0.06 (marginal)
• |φ| ≥ 0.15 → χ² ≥ 7.79, p < 0.01
• |φ| ≥ 0.20 → χ² ≥ 13.8, p < 0.001
• |φ| ≥ 0.30 → χ² ≥ 31.1, p < 10⁻⁷

These thresholds are asymptotic and assume both tags have reasonable marginal support. They break down when one tag is rare. Click any heatmap cell to see its 95% bootstrap CI and its chi-square p-value — bootstrap (resampling papers with replacement, 1000 iterations) is the more honest measure when marginal support is small, because it captures the actual sampling variability of φ rather than assuming the asymptotic Normal approximation.