Glossary

9.3 Log-linear models for multi-way contingency tables

6 min read•july 30, 2024

Log-linear models are powerful tools for analyzing multi-way contingency tables in biostatistics. They help uncover complex relationships between categorical variables by expressing cell frequencies as linear combinations of and interactions.

These models are crucial for understanding associations in categorical data, a key aspect of this chapter. By examining main effects and interactions, researchers can gain insights into the intricate relationships between variables in biological studies.

Log-linear models for contingency tables

Introduction to log-linear models

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  • Log-linear models are a class of statistical models used to analyze the associations and interactions among multiple categorical variables in a contingency table
  • Multi-way contingency tables are cross-tabulations of three or more categorical variables, where each cell represents the frequency or count of observations falling into a specific combination of categories (gender, age group, and education level)
  • Log-linear models express the logarithm of the expected cell frequencies as a linear combination of main effects and interaction terms, allowing for the examination of the relationships among the variables

Components and assumptions of log-linear models

  • The main effects in a log-linear model represent the independent effects of each variable on the cell frequencies, while the interaction terms capture the dependencies or associations between the variables
  • Log-linear models assume that the cell frequencies follow a Poisson distribution and that the logarithm of the can be modeled as a linear function of the parameters
  • The Poisson distribution is appropriate for modeling count data, such as the number of individuals falling into each cell of a contingency table
  • The logarithmic transformation of the expected frequencies allows for the additive decomposition of the effects and interactions, making the interpretation of the model parameters more straightforward

Constructing log-linear models

Defining variables and model formula

  • To construct a log-linear model, the first step is to define the variables and the possible categories for each variable in the multi-way contingency table
  • For example, in a study examining the relationship between gender, age group, and education level, the variables would be defined as follows:
    • Gender: Male, Female
    • Age group: Young, Middle-aged, Old
    • Education level: Low, Medium, High
  • The model formula specifies the variables and the interaction terms to be included in the log-linear model, using a notation similar to that of analysis of variance (ANOVA) models
  • The formula includes the main effects of each variable and the interaction terms of interest, such as 
    Gender + Age + Education + Gender:Age + Gender:Education + Age:Education + Gender:Age:Education

Hierarchical models and the principle of marginality

  • The saturated log-linear model includes all possible main effects and interaction terms, representing the most complex model that perfectly fits the observed data
  • Hierarchical log-linear models are constructed by systematically removing or adding interaction terms to the model based on the principle of marginality, ensuring that lower-order terms are included before higher-order terms
  • The principle of marginality states that if a higher-order interaction term is included in the model, all lower-order terms that are subsets of the higher-order term must also be included
  • For example, if the 
    Gender:Age:Education
    interaction is included, the main effects of 
    Gender
    , 
    Age
    , and 
    Education
    , as well as the two-way interactions 
    Gender:Age
    , 
    Gender:Education
    , and 
    Age:Education
    , must also be present in the model

Parameter estimation and model fitting

  • The parameters of the log-linear model are estimated using maximum likelihood estimation (MLE) techniques, such as iterative proportional fitting (IPF) or Newton-Raphson algorithms
  • IPF is an iterative algorithm that adjusts the cell frequencies to match the marginal totals of the observed data, converging to the maximum likelihood estimates of the model parameters
  • Newton-Raphson is a general optimization algorithm that iteratively updates the parameter estimates by minimizing the negative log-likelihood function
  • The model fitting process involves estimating the parameters that maximize the likelihood of observing the data given the specified log-linear model

Interpreting log-linear model parameters

Main effects and interaction parameters

  • The parameters of a log-linear model represent the effects of the variables and their interactions on the logarithm of the expected cell frequencies
  • The main effect parameters indicate the independent contribution of each variable to the cell frequencies, while the capture the associations or dependencies among the variables
  • For example, the main effect parameter for 
    Gender
    represents the difference in the logarithm of the expected frequencies between males and females, assuming all other variables are held constant
  • The interaction parameter for 
    Gender:Age
    represents the additional effect on the logarithm of the expected frequencies due to the combination of specific levels of 
    Gender
    and 
    Age
    , beyond their individual main effects

Assessing model fit and goodness-of-fit measures

  • Goodness-of-fit measures, such as the likelihood ratio chi-square (G²) and Pearson's chi-square (X²), assess how well the log-linear model fits the observed data
  • A non-significant goodness-of-fit test suggests that the model adequately describes the associations and interactions in the data
    • For example, if the likelihood ratio chi-square test for a log-linear model has a p-value greater than 0.05, it indicates that the model fits the data well and captures the important relationships among the variables
  • A significant goodness-of-fit test indicates that the model does not fit the data well, and additional interaction terms or alternative models should be considered
  • The deviance (G²) and the Akaike information criterion (AIC) are commonly used to compare the fit of nested log-linear models, with lower values indicating better fit
    • Nested models are models where one model is a special case of the other, obtained by setting some parameters to zero or constraining them to be equal

Model selection for log-linear models

Model selection techniques

  • Model selection in log-linear analysis involves choosing the most parsimonious model that adequately describes the associations and interactions among the variables
  • Backward elimination is a model selection technique that starts with the and sequentially removes non-significant interaction terms, based on the or other criteria, until a final model is obtained
    • The process begins with the most complex model and gradually simplifies it by removing higher-order interactions that do not significantly contribute to the model fit
  • Forward selection begins with the simplest model (usually the independence model) and gradually adds interaction terms that significantly improve the model fit
    • The independence model assumes that all variables are independent of each other, and the cell frequencies are determined solely by the main effects of the variables
    • Interaction terms are added one at a time, based on their contribution to the model fit, until no further significant improvements can be made

Assessing the significance of interactions

  • The likelihood ratio test is used to assess the significance of the difference in fit between nested log-linear models, determining whether the inclusion or exclusion of specific interaction terms is justified
    • The test compares the deviance (G²) of the simpler model to that of the more complex model, and a significant result indicates that the additional interaction terms in the complex model significantly improve the fit
  • Partial association tests examine the significance of individual interaction terms by comparing the fit of models with and without the interaction, while controlling for other variables and interactions
    • These tests assess the conditional independence of the variables involved in the interaction, given the other variables in the model
  • The significance of the interaction terms in the selected log-linear model provides insight into the dependencies and associations among the categorical variables, guiding the interpretation of the results
    • Significant interactions suggest that the relationship between two or more variables depends on the levels of other variables, while non-significant interactions indicate that the variables are conditionally independent

Key Terms to Review (19)

AIC Criteria: AIC criteria, or Akaike Information Criterion, is a statistical measure used for model selection that estimates the quality of each model relative to others. It helps determine how well a model fits the data while penalizing for complexity, thus preventing overfitting. In the context of log-linear models for multi-way contingency tables, AIC assists in selecting the most appropriate model from a set of candidate models by balancing goodness-of-fit with the number of parameters.
Cell probabilities: Cell probabilities refer to the likelihood of observing a specific combination of categorical variables in a multi-way contingency table. These probabilities help in understanding the association between different variables, allowing for the analysis of patterns and interactions within the data. They serve as a foundation for log-linear models, which are used to model the relationships among multiple categorical variables simultaneously.
Chi-square statistic: The chi-square statistic is a measure used in statistics to determine if there is a significant association between categorical variables. It compares the observed frequencies of events with the expected frequencies under the null hypothesis, helping to identify whether any deviations are due to chance or indicate a real relationship between the variables involved.
Contingency coefficient: The contingency coefficient is a statistical measure used to assess the strength of association between two categorical variables in a contingency table. It helps quantify the degree of dependence between the variables, providing insight into how changes in one variable relate to changes in another. This coefficient ranges from 0 to 1, with values closer to 1 indicating a stronger relationship, and is particularly useful in the context of log-linear models for analyzing multi-way contingency tables.
Cramér's V: Cramér's V is a statistical measure used to assess the strength of association between two categorical variables. Ranging from 0 to 1, with 0 indicating no association and 1 indicating perfect association, it provides insights into the degree to which the presence of one variable affects the other. In the context of log-linear models for multi-way contingency tables, Cramér's V helps evaluate the relationships among multiple categorical variables simultaneously, revealing underlying patterns and dependencies.
Epidemiology: Epidemiology is the study of how diseases affect the health and illness of populations. It examines the distribution and determinants of health-related states or events in specified populations, using this knowledge to control health problems. By understanding the patterns and causes of diseases, epidemiology informs public health strategies and interventions aimed at improving population health.
Expected frequencies: Expected frequencies are the theoretical counts of occurrences in each category of a contingency table under the assumption of independence between the variables. These values are calculated based on the overall total and the distribution of the marginal totals, serving as a foundation for various statistical tests, particularly in log-linear models. In multi-way contingency tables, expected frequencies help to assess the fit of the model by comparing them to observed frequencies.
Frequency table: A frequency table is a statistical tool that displays the number of occurrences of different values in a dataset. This table helps to summarize and organize data, making it easier to identify patterns and trends. In the context of log-linear models for multi-way contingency tables, frequency tables provide essential information about the relationships among categorical variables and facilitate the analysis of interactions between them.
Independence Assumption: The independence assumption is a fundamental concept in statistical modeling, particularly in the context of multi-way contingency tables. It posits that the occurrence of one event does not affect the probability of another event occurring, implying that the variables are statistically independent of one another. This assumption simplifies the analysis of categorical data and is crucial for the validity of log-linear models, allowing researchers to examine relationships without assuming direct dependencies among variables.
Interaction parameters: Interaction parameters are numerical values that describe how the effect of one variable on an outcome changes depending on the level of another variable in statistical models. In the context of log-linear models for multi-way contingency tables, these parameters help capture the relationship between categorical variables, revealing whether the association between them is influenced by their interactions. Understanding these parameters is essential for interpreting complex data and making informed conclusions about relationships among multiple factors.
Likelihood Ratio Test: A likelihood ratio test is a statistical method used to compare the goodness of fit of two competing models based on the ratio of their likelihoods. This test helps determine whether the evidence supports a more complex model over a simpler one by evaluating the ratio of the maximum likelihood estimates from both models. It is widely applied in various fields, especially in biostatistics, to assess hypotheses and make inferences about biological phenomena, multi-way contingency tables, and survival analysis.
Main Effects: Main effects refer to the direct influence of an independent variable on a dependent variable in a statistical model, showing how changes in one factor affect outcomes without considering interactions with other factors. Understanding main effects is crucial as it helps in identifying the primary impacts of each factor, making it easier to interpret results from experiments and observational studies.
Marginal Model: A marginal model is a statistical framework that describes the relationship between the response variable and predictors while focusing on the marginal distributions of the data rather than the joint distribution. This approach is particularly useful for analyzing categorical data, where it allows for the evaluation of association patterns between variables without the need for specifying a full joint model, which can be complex and computationally intensive. Marginal models are commonly employed in log-linear models for multi-way contingency tables, enabling researchers to interpret effects of predictors on the marginal distribution of response variables.
Poisson distribution assumption: The Poisson distribution assumption refers to the requirement that events occur independently and at a constant average rate over a fixed interval of time or space. This assumption is critical in log-linear models for multi-way contingency tables, as it helps to determine the expected frequency of events in different categories, allowing for proper modeling and inference.
Public health research: Public health research refers to the systematic investigation of health-related issues and determinants affecting populations to improve health outcomes and inform policy decisions. This type of research often involves analyzing data from various sources, including surveys, experiments, and observational studies, to understand trends, risk factors, and the effectiveness of interventions. The insights gained from public health research play a crucial role in shaping health policies and practices aimed at promoting health and preventing disease across communities.
R: In statistics, 'r' typically refers to the correlation coefficient, a measure that quantifies the strength and direction of a relationship between two variables. It plays a crucial role in understanding how variables are related in biological research, helping researchers to identify patterns and make predictions based on data.
SAS: SAS, which stands for Statistical Analysis System, is a software suite used for advanced analytics, multivariate analysis, business intelligence, data management, and predictive analytics. It's widely used in biostatistics to analyze complex datasets and is essential for applying various statistical methods and models in biological research, including survival analysis and regression techniques.
Saturated Model: A saturated model is a statistical model that includes all possible interactions and main effects for a given set of variables, effectively capturing all variability in the data. This model serves as a benchmark in the analysis of multi-way contingency tables, providing a comprehensive understanding of the relationships between categorical variables. It allows researchers to see how well the data fits the model since it accounts for every possible combination of variable levels.
SPSS: SPSS (Statistical Package for the Social Sciences) is a comprehensive software tool used for statistical analysis, data management, and graphical representation of data. Its user-friendly interface allows researchers and students to easily perform complex statistical tests and interpret results, making it a vital resource for analyzing data in various fields including biology, psychology, and social sciences.
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