AIC Foundations

CATDAP applies Akaike's Information Criterion (AIC) to contingency-table models.

The two-way AIC

Given a response variable \(E\) and an explanatory variable \(F\), the AIC for the contingency-table model is:

\[ AIC(E; F) = -2 \sum_{i,j} n_{EF}(i,j) \ln \frac{n_{EF}(i,j)}{n_F(j)} + 2 (C_E - 1) C_F \]

where:

• \(n_{EF}(i, j)\) — cross-tab frequency for response category \(i\) and explanatory category \(j\)
• \(n_F(j)\) — marginal frequency of explanatory category \(j\)
• \(C_E\), \(C_F\) — number of categories for \(E\) and \(F\)

Base AIC (no explanatory model)

\[ AIC(E; \phi) = -2 \sum_i n_E(i) \ln \frac{n_E(i)}{n} + 2 (C_E - 1) \]

ΔAIC (the value pycatdap reports)

\[ \Delta AIC = AIC(E; F) - AIC(E; \phi) \]

Interpretation:

• \(\Delta AIC < 0\) → \(F\) is informative about \(E\)
• \(\Delta AIC \geq 0\) → \(F\) adds no information beyond model complexity

Zero-frequency convention

CATDAP follows the convention \(0 \cdot \ln(0) = 0\). The internal _safe_xlogy helper uses scipy.special.xlogy when available and falls back to a vectorized np.where implementation otherwise. This guarantees numerical stability even on sparse contingency tables.

Why AIC over other association measures

Unlike Cramér's V or mutual information, AIC explicitly trades information gain against model complexity (the \(2(C_E - 1)C_F\) penalty). This makes it especially well-suited to:

• Variable selection — you don't need a separate p-value threshold
• Comparing models with different numbers of parameters
• Continuous-variable pooling — the AIC of different bin counts can be directly compared

References

• Sakamoto, Y., & Katsura, K. (1980). Categorical Data Analysis by AIC. Mathematical Sciences.
• Akaike, H. (1973). Information theory as an extension of the maximum likelihood principle.