Quiz 8: Extending Loglinear Models 📊

Test your knowledge of the material on extending loglinear models in the following quiz to see how much you learned. This is entirely private for you---no records are kept of your performance.

Questions

1. When one table variable is a binary response, a logit model for that response is:

More complex than the equivalent loglinear model Equivalent to a loglinear model Unrelated to loglinear models Only valid for continuous predictors

Explanation

Each logit model for a binary response C is equivalent to a loglinear model. The loglinear model must include the [AB] association of predictors. The logit model is simpler to interpret because it has fewer parameters and focuses on the effects on the response variable.

2. The main advantage of the GLM (generalized linear model) approach to loglinear models is that GLM:

Only works with categorical predictors Requires larger sample sizes Allows quantitative predictors and can handle ordinal factors Cannot handle overdispersion

Explanation

The GLM approach allows quantitative predictors and special ways of treating ordinal factors. It provides parameter estimates with standard errors and significance tests, can incorporate ordinal variables as linear/polynomial terms, and can handle overdispersion using quasi-Poisson or negative binomial families.

3. The linear-by-linear (L × L) model for ordinal variables assumes:

Different slopes for each category A constant local odds ratio for adjacent categories No association between variables Only main effects, no interaction

Explanation

The L × L model assumes log(m_ij) = μ + λ_i^A + λ_j^B + γ a_ib_j, where γ is a constant local odds ratio. For integer scores, log(θ_ij) = γ. This model uses only 1 more parameter than the independence model, making it very parsimonious.

4. In models for ordered categories, the RC(1) model differs from the L × L model in that RC(1):

Uses more degrees of freedom Assumes independence Estimates optimal row and column scores from the data Only applies to square tables

Explanation

The RC(1) row-and-column effects model estimates optimal category scores α_i and β_j as parameters: λ_ij^AB = γ α_iβ_j. Unlike L × L which assigns fixed integer scores, RC(1) finds the best-fitting scores. The ordering and spacing of categories is estimated from the data, similar to correspondence analysis.

5. For a square table with the same row and column categories, the quasi-independence model:

Assumes complete independence Fits diagonal cells perfectly and tests independence in off-diagonal cells Requires equal marginal totals Only works with 2×2 tables

Explanation

The quasi-independence model adds parameters δ_i for each diagonal cell: log(m_ij) = μ + λ_i^A + λ_j^B + δ_iI(i=j). This fits the diagonal cells (where row = column) perfectly while testing for independence in the off-diagonal cells, which is useful for mobility tables and other square tables.

6. The relationship between symmetry and quasi-symmetry models is:

They are identical models Symmetry is less restrictive than quasi-symmetry Symmetry = quasi-symmetry + marginal homogeneity Quasi-symmetry always fits better than symmetry

Explanation

For square tables, it can be shown that symmetry = quasi-symmetry + marginal homogeneity, and G²(S) = G²(QS) + G²(MH). The symmetry model assumes π_ij = π_ji, but this also implies marginal homogeneity (equal row and column totals). The quasi-symmetry model allows λ_ij = λ_ji without requiring equal margins.

7. The gnm package in R is used to fit:

Only standard loglinear models Only logistic regression models Generalized nonlinear models including RC and UNIDIFF models Only linear regression models

Explanation

The gnm package provides functions to fit generalized nonlinear models (GNMs). RC models are not loglinear—they contain multiplicative terms like γ α_iβ_j. The gnm() function can fit these models using Mult(), Exp(), and other nonlinear functions. The package also provides Diag(), Symm(), and Topo() convenience functions.

8. In the UNIDIFF (uniform difference) model for stratified tables, the layer coefficients γ_k represent:

The main effects of the layer variable Multiplicative effects showing how association strength varies across layers The independence of layers Only the diagonal associations

Explanation

The UNIDIFF model is log(m_ijk) = μ + λ_i^R + λ_j^C + λ_k^L + λ_ik^RL + λ_jk^CL + γ_kδ_ij^RC. The layer coefficients γ_k are multipliers showing how the [RC] association varies across layers (e.g., across countries or time periods), with one layer fixed at 1.0 as reference.

9. Compared to general association models, models for ordinal variables provide:

Less statistical power More parameters to estimate More focused tests with greater power and fewer degrees of freedom Only categorical interpretation

Explanation

Models for ordinal variables give more focused tests → greater power to detect associations. They use fewer df → can fit different models between independence [A][B] and saturated [AB], have fewer parameters → easier interpretation and smaller standard errors. This is similar to using CMH tests or testing linear contrasts in ANOVA.

10. When comparing multiple models using AIC and BIC, which statement is TRUE?

AIC and BIC always select the same model Lower values indicate worse fit BIC penalizes complexity more heavily than AIC, often preferring simpler models Both criteria ignore degrees of freedom

Explanation

BIC (Bayesian Information Criterion) penalizes model complexity more heavily than AIC (Akaike Information Criterion). BIC = -2log-likelihood + log(n)df, while AIC = -2log-likelihood + 2df. As a result, BIC often prefers more parsimonious models. Lower values of both criteria indicate better models, balancing goodness-of-fit against complexity.