11.3 Generalized estimating equations (GEE)
5 min read•august 13, 2024
(GEE) are a powerful tool for analyzing longitudinal and . They extend generalized linear models to account for correlated observations, focusing on rather than subject-specific ones.
GEE offers flexibility in handling various data types and missing values. It provides consistent estimates even with misspecified correlation structures, making it robust and computationally efficient for large datasets. However, it doesn't capture subject-specific effects or include random effects.
Generalized Estimating Equations
Overview and Applications
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- Generalized estimating equations (GEE) extend generalized linear models (GLMs) to account for the correlation between observations in longitudinal or clustered data
- GEE estimates the average response over the population (population-averaged effects) rather than subject-specific effects
- GEE is used when the primary interest lies in the marginal expectation of the response variable, while accounting for the within clusters or subjects
- Applicable to a wide range of data types, including continuous, binary, count, and categorical outcomes
- Can handle missing data under the assumption that the data are missing completely at random (MCAR) or missing at random (MAR)
Advantages and Limitations
- GEE provides consistent estimates of regression coefficients even if the correlation structure is misspecified, as long as the is correctly specified
- Computationally efficient and can handle large datasets with many clusters or subjects
- Allows for the use of , which are valid even if the correlation structure is misspecified
- However, GEE does not provide estimates of subject-specific effects, as it focuses on population-averaged effects
- May not be efficient when the number of clusters is small or when the cluster sizes are highly variable
- Assumes that the data are MCAR or MAR, and violations of these assumptions can lead to biased estimates
- Does not allow for the inclusion of random effects, which may be necessary to capture subject-specific variability
GEE vs Other Methods
Comparison with Mixed Effects Models
- GEE focuses on population-averaged effects, while mixed effects models estimate both population-averaged and subject-specific effects
- Mixed effects models include random effects to capture subject-specific variability, while GEE does not
- GEE is more robust to misspecification of the correlation structure, while mixed effects models rely on correctly specifying the random effects structure
- GEE is computationally more efficient than mixed effects models, especially for large datasets with many clusters or subjects
Comparison with Repeated Measures ANOVA
- GEE can handle a wider range of data types (continuous, binary, count, categorical) compared to repeated measures ANOVA, which is limited to continuous outcomes
- GEE allows for the inclusion of time-varying covariates, while repeated measures ANOVA assumes that covariates are constant over time
- GEE can handle missing data under MCAR or MAR assumptions, while repeated measures ANOVA typically requires complete data or relies on imputation methods
- Repeated measures ANOVA is more sensitive to violations of sphericity assumptions, while GEE is robust to misspecification of the correlation structure
Marginal Models with GEE
Specifying the Mean Structure
- Marginal models specify the mean structure of the response variable as a function of covariates, while accounting for the correlation structure within clusters or subjects
- The mean structure is typically specified using a , such as the identity link for continuous outcomes, the logit link for binary outcomes, or the log link for count outcomes
- Example: In a study of blood pressure over time, the mean structure could be specified as a linear function of time, treatment group, and their interaction using an identity link
Specifying the Correlation Structure
- The correlation structure is specified using a , which can be independent, exchangeable, autoregressive, or unstructured
- Independent: Assumes no correlation between observations within a cluster or subject
- Exchangeable: Assumes a constant correlation between any two observations within a cluster or subject
- Autoregressive: Assumes that the correlation between observations decreases as the time lag between them increases
- Unstructured: Allows for a distinct correlation between any two observations within a cluster or subject
- The choice of the working correlation matrix should be based on the nature of the data and the underlying biological or social processes
Estimating Regression Coefficients
- The regression coefficients are estimated using methods, which involve solving a set of estimating equations that are based on the mean structure and the working correlation matrix
- The sandwich variance estimator is used to obtain robust standard errors for the regression coefficients, which are valid even if the working correlation matrix is misspecified
- Example: In the blood pressure study, the regression coefficients would represent the average change in blood pressure for a one-unit change in time, treatment group, or their interaction
Interpreting GEE Results
Interpreting Regression Coefficients
- The regression coefficients in GEE represent the average change in the response variable for a one-unit change in the corresponding covariate, while holding all other covariates constant
- For continuous outcomes, the coefficients directly represent the change in the mean response
- For binary outcomes, the exponentiated coefficients (odds ratios) represent the change in the odds of the response
- For count outcomes, the exponentiated coefficients (rate ratios) represent the change in the rate of the response
- Example: In the blood pressure study, a coefficient of -2.5 for the treatment group would indicate that, on average, the treatment group has a 2.5 mmHg lower blood pressure compared to the control group, holding time constant
Assessing Model Fit and Diagnostics
- The quasi-likelihood information criterion (QIC) can be used to compare the fit of different marginal models, with lower values indicating better fit
- QIC is an extension of the Akaike information criterion (AIC) for GEE models
- Example: Comparing QIC values for models with different mean structures or working correlation matrices can help select the most appropriate model
- Residual plots and other diagnostic tools can be used to assess the adequacy of the mean structure and the correlation structure, and to identify outliers or influential observations
- Residual plots can reveal patterns or trends that suggest misspecification of the mean structure or the presence of outliers
- Influence diagnostics, such as Cook's distance or leverage, can identify observations that have a disproportionate impact on the estimated coefficients
- Example: A residual plot showing a clear non-linear trend would suggest that the mean structure should be modified to include non-linear terms or transformations of the covariates
Key Terms to Review (18)
Clustered data: Clustered data refers to a type of data structure where observations are grouped together based on certain characteristics or shared traits, often reflecting a hierarchical or nested design. This setup commonly arises in fields such as social sciences and healthcare, where measurements are collected from subjects within distinct groups, like patients in hospitals or students in classrooms. Understanding clustered data is crucial for accurately analyzing relationships and variations within these groups, especially when using methods like generalized estimating equations (GEE).
Correlation structure: Correlation structure refers to the pattern of relationships among variables within a dataset, indicating how changes in one variable relate to changes in another. Understanding correlation structure is essential for analyzing data, especially when dealing with clustered or longitudinal data where observations may not be independent. This concept plays a key role in modeling and estimating relationships, particularly in methods that account for the dependencies among observations.
Fitzmaurice et al.: Fitzmaurice et al. refers to a group of researchers who made significant contributions to the development and application of Generalized Estimating Equations (GEE), which are used to analyze correlated data. Their work is particularly important in the context of longitudinal data analysis, where observations are not independent, and they provide methods to account for within-subject correlation while obtaining valid statistical inferences.
Flexibility with correlation structures: Flexibility with correlation structures refers to the ability of a statistical model, particularly in the context of Generalized Estimating Equations (GEE), to accommodate various types of correlation patterns among repeated measures or clustered data. This concept is important as it allows researchers to specify different correlation structures that can better reflect the underlying relationships in the data, leading to more accurate estimations and inferences.
Generalized Estimating Equations: Generalized estimating equations (GEE) are a statistical method used for analyzing correlated data, especially in longitudinal studies where repeated measurements are taken from the same subjects. This approach allows researchers to account for the correlation between observations while providing robust estimates of parameters, making it particularly useful in situations where traditional methods may fall short due to non-independence of observations.
Handling missing data: Handling missing data refers to the various strategies and techniques used to address gaps in datasets where some values are absent. Missing data can arise for multiple reasons, including non-response in surveys, data entry errors, or equipment malfunction. Proper handling of missing data is crucial, particularly when using methods like Generalized Estimating Equations (GEE), as it can significantly impact the validity and reliability of statistical inferences drawn from the analysis.
Health outcomes analysis: Health outcomes analysis is the systematic evaluation of the effects of healthcare interventions and treatments on patient health status and quality of life. It involves assessing various health indicators to determine the effectiveness and efficiency of healthcare services, guiding clinical decision-making, policy development, and resource allocation.
Link function: A link function is a crucial component in generalized linear models (GLMs) that connects the linear predictor to the mean of the distribution function of the response variable. It transforms the expected value of the response variable into a form that is more amenable for analysis, often ensuring that predicted values remain within valid ranges, such as probabilities between 0 and 1. The choice of link function influences the model's interpretation and the relationship between the predictor variables and the response variable.
Longitudinal data: Longitudinal data refers to data collected over time from the same subjects, allowing researchers to observe changes and trends within the same individuals or units. This type of data is crucial for understanding dynamics, as it captures the temporal evolution of variables, providing insights that cross-sectional data cannot offer. It is particularly important for studies examining developmental, social, and health-related changes.
Mean structure: Mean structure refers to the mathematical representation of the expected value of the response variable in a statistical model. It outlines how the mean of the response variable is related to the predictors or independent variables in the model, establishing the foundation for understanding relationships between variables in various contexts, especially when dealing with correlated data, such as in longitudinal studies or clustered data.
Population-averaged effects: Population-averaged effects refer to the overall impact or relationship of a treatment or exposure on a population rather than on individual subjects. This concept is particularly relevant in the context of statistical methods like generalized estimating equations (GEE), where the aim is to estimate the average response in the population, accounting for correlated observations within clusters or repeated measures. Understanding population-averaged effects is crucial for making inferences about the generalizability of study findings beyond individual-level outcomes.
Quasi-likelihood: Quasi-likelihood is a statistical concept that extends the notion of likelihood by allowing for the estimation of parameters in models where the distribution of the data is not fully specified. It provides a way to analyze data that may exhibit correlations or other complexities, particularly in the context of generalized estimating equations, which focus on estimating population-averaged effects in clustered or correlated data. This approach leverages robust variance estimates to make inferences without needing a full likelihood specification.
R: In statistics, 'r' represents the correlation coefficient, a numerical measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values close to 1 indicate a strong positive correlation, values close to -1 indicate a strong negative correlation, and values around 0 suggest no linear correlation. Understanding 'r' is crucial for interpreting relationships in data across various analyses.
Robust standard errors: Robust standard errors are a statistical technique used to provide more reliable estimates of the standard errors of coefficients in regression models, especially when the assumptions of homoscedasticity are violated. They help in making valid inferences by adjusting for potential heteroscedasticity or other forms of model misspecification that can lead to biased results. This adjustment is crucial in generalized estimating equations (GEE), which deal with correlated observations often encountered in longitudinal or clustered data.
SAS: SAS, which stands for Statistical Analysis System, is a software suite used for advanced analytics, multivariate analysis, business intelligence, and data management. This powerful tool enables researchers and statisticians to conduct complex statistical analyses and visualize data effectively, making it integral to a variety of statistical techniques and methodologies.
Social Sciences Studies: Social sciences studies encompass a broad range of academic disciplines that explore human behavior, societal structures, and cultural norms. This field of study utilizes both qualitative and quantitative research methods to analyze how individuals and groups interact within various contexts, such as economics, psychology, sociology, and anthropology. The insights gained from social sciences studies are crucial for understanding complex social phenomena and informing policy-making and practice.
Working Correlation Matrix: A working correlation matrix is a matrix used in the context of Generalized Estimating Equations (GEE) to specify the correlation structure of the repeated measurements or clustered data. It allows researchers to account for the potential correlation between observations, which can enhance the accuracy of parameter estimates and standard errors when analyzing data with such dependencies.
Zeger and Liang: Zeger and Liang refer to a foundational paper that introduced the concept of Generalized Estimating Equations (GEE), which are used for analyzing correlated data typically arising from longitudinal studies or clustered samples. Their work established a framework for estimating parameters in such complex data structures while accommodating the correlation between observations, making it a significant contribution to the field of biostatistics and epidemiology.