Loglinear models for multivariate discrete data#

Part 4 · Stage 13 · 🔗 Generalized Linear Models · Lesson 112 of 144 · advanced

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Important

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Modelling a table of counts#

Cross-classify a sample by several categorical variables — sex by education by vote — and the data become a contingency table of counts. Loglinear models describe such tables by modelling the log expected count in each cell as a linear function of the variables and their interactions, turning questions about association into questions about coefficients.

The model#

For a two-way table with factors \(A\) and \(B\), the log expected cell count is

\[\log \mu_{ij} = \lambda + \lambda^A_i + \lambda^B_j + \lambda^{AB}_{ij},\]

with main effects for each margin and an interaction term. The interaction is the object of interest: \(\lambda^{AB}_{ij} = 0\) for all \(i, j\) is exactly the statement that \(A\) and \(B\) are independent. Testing independence becomes testing whether an interaction batch is negligible — and, Bayesianly, estimating how large it is.

import pymc as pm
# loglinear model for a two-way table of counts
with pm.Model():
    lam = pm.Normal("lam", 0, 5)
    aA = pm.Normal("aA", 0, 2, shape=I)                  # margin A effects
    aB = pm.Normal("aB", 0, 2, shape=J)                  # margin B effects
    s_ab = pm.HalfNormal("s_ab", 1)
    aAB = pm.Normal("aAB", 0, s_ab, shape=(I, J))        # interaction (association), pooled
    log_mu = lam + aA[:, None] + aB[None, :] + aAB
    pm.Poisson("y", pm.math.exp(log_mu).flatten(), observed=counts.flatten())

The connections#

Loglinear models tie together several threads. Their equivalence to Poisson regression with categorical predictors is exact — a table of counts is count data with factor predictors — so everything from the overdispersion lesson applies, including the caution that cell counts may vary more than Poisson allows. For binary outcomes they connect directly to logistic regression: the loglinear interaction involving the response reproduces the logistic coefficient. And treating the interaction terms as an exchangeable batch with a shared scale — the batching idea one final time — gives a hierarchical loglinear model that pools sparse cells, taming the zeros and small counts that make large contingency tables notoriously unstable.

Where it sits#

For high-dimensional discrete data — many categorical variables at once — loglinear models with hierarchical interaction terms are a principled tool, letting the data determine which associations are real while pooling away the noise. They close the generalized linear model stage by handling the last data type on the list: whole tables of categorical counts. Part IV has taken the linear model from continuous responses through binary, count, ordered, multivariate and tabular data; Part V now abandons the linear predictor itself, for functions and infinite-dimensional models.

Hint