Log-linear model

Log-linear models are statistical models that are used to analyze the relationships between categorical variables. They are a type of generalized linear model that is particularly useful for examining the patterns of frequency distributions in contingency tables. Log-linear models can be applied in various fields, including biostatistics, social sciences, and market research, to explore how different factors interact with each other.

Overview

A log-linear model expresses the logarithm of expected cell frequencies in a contingency table as a linear function of parameters. These models are called "log-linear" because they model the natural logarithm of the expected frequencies. The general form of a log-linear model for a two-way table is:

\[ \log(\mu_{ij}) = \lambda + \lambda_i^A + \lambda_j^B + \lambda_{ij}^{AB}, \]

where \(\mu_{ij}\) is the expected frequency for cell \((i, j)\), \(\lambda\) is the overall mean effect, \(\lambda_i^A\) and \(\lambda_j^B\) are the effects of the individual factors (A and B, respectively), and \(\lambda_{ij}^{AB}\) is the interaction effect between factors A and B.

Applications

Log-linear models are widely used to analyze data in the form of contingency tables. Some common applications include:

• Epidemiological studies to explore the relationship between diseases and potential risk factors.
• Sociological research to study the association between various social factors.
• Market research to understand consumer behavior and preferences.

Model Fitting

Fitting a log-linear model typically involves estimating the parameters that best describe the observed data. This is usually done through maximum likelihood estimation (MLE). The likelihood function for a log-linear model is a function of the parameters that measures the probability of observing the given data. The MLE process finds the parameter values that maximize this likelihood function.

Hypothesis Testing

In the context of log-linear models, hypothesis testing is often used to assess the significance of factors and their interactions. This involves comparing a full model, which includes all terms, to a reduced model that omits some terms. The Likelihood ratio test is commonly used for this purpose.

Advantages and Limitations

Log-linear models offer several advantages, including the ability to handle multi-way tables and to assess interactions between factors. However, they also have limitations. For instance, they require sufficient data to provide reliable estimates, and interpreting the results can be complex, especially in models with many factors and interactions.

See Also

• Multinomial distribution
• Poisson regression
• Chi-squared test
• Factor analysis