Log-Linear Models for Contingency Tables

Log-linear models describe the cell counts of a multiway contingency table with a Poisson model on the log scale, where interaction terms represent associations among the categorical variables.

로그선형 모형은 다원 분할표의 칸 빈도를 로그 척도의 포아송 모형으로 설명하며, 상호작용 항으로 범주형 변수 사이의 연관을 나타내는 방법입니다.

I have a three-way table of counts; which pairs of variables are associated once the third is taken into account?

세 범주형 변수로 만든 빈도표가 있는데, 셋째 변수를 고려해도 어떤 두 변수가 연관되어 있는지 알고 싶어요.

A log-linear model fits the expected count of each cell as a product of main effects and interaction terms, and its deviance G² measures how far the fitted counts are from the observed ones. Dropping one interaction at a time and comparing deviances tests that association, including conditional independence of two variables given the others.

When to use Log-Linear Models for Contingency Tables

• You have a table cross-classifying three or more categorical variables and no single response variable.
• You want to test conditional independence or homogeneous association in a multiway table.
• You want to find the simplest hierarchical model that reproduces the observed counts.

When to use something else

• One variable is clearly the outcome; use logistic or multinomial regression.
• The table has only two variables; a chi-square test of independence answers the question.

Assumptions and requirements

• Counts are independent Poisson or multinomial observations.
• Most expected cell counts are 5 or more for the chi-square approximation.
• Models are hierarchical: every lower-order term of an included interaction is also included.

What the analysis reports

• Likelihood-ratio G² and Pearson X² goodness of fit with degrees of freedom
• Likelihood-ratio test for each highest-order term in the model
• Conditional independence test for each pair of variables given the rest
• Fitted counts and adjusted residuals by cell
• Optional backward elimination path

How to report it

A hierarchical log-linear model [model terms] fit the data well, G²([df], N = [value]) = [value], p = [value]. Removing the [A]×[B] association worsened fit, ΔG²([df]) = [value], p = [value].

Related methods

• Chi-Square Test of Independence
• Poisson Regression
• Cochran-Mantel-Haenszel Test

Run Log-Linear Models for Contingency Tables on Quorant