11.2 Mixed-effects models
4 min read•august 13, 2024
Mixed-effects models are powerful tools for analyzing longitudinal and clustered data. They handle hierarchical structures, incorporating both fixed and to model variability at different levels. This approach is particularly useful for repeated measures and grouped observations.
These models offer advantages like handling unbalanced data, accounting for within-subject and between-subject variability, and modeling correlation structures. They're widely applied in fields such as healthcare, education, and psychology, providing more accurate and efficient estimates of effects and their standard errors.
Mixed-effects models for longitudinal data
Principles and advantages
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- Handle data with hierarchical or nested structure (repeated measures within individuals, observations clustered within groups)
- Incorporate both (constant parameters across individuals or groups) and random effects (randomly varying parameters across individuals or groups)
- Allows modeling variability at different levels of data hierarchy
- Advantages include ability to handle unbalanced or missing data, account for within-subject and between-subject variability, and model correlation structure of data
- Particularly useful for analyzing longitudinal data (repeated measurements on same individuals over time) and clustered data (observations grouped within higher-level units like students within schools or patients within hospitals)
- Capture heterogeneity and dependence among observations within the same individual or group by incorporating random effects
- Leads to more accurate and efficient estimates of fixed effects and their standard errors
Applications and examples
- (repeated measures over time)
- Tracking changes in health outcomes, such as blood pressure or weight, in patients undergoing treatment
- Assessing cognitive development in children across different ages
- Clustered data (observations grouped within higher-level units)
- Evaluating educational interventions on student performance, accounting for clustering within schools or classrooms
- Analyzing patient outcomes in multi-center clinical trials, considering hospital-level variability
Fixed vs Random effects
Specifying fixed and random effects
- Fixed effects represent average effects of predictor variables on response variable across all individuals or groups in population, assuming constant effects
- Random effects capture variability or heterogeneity in effects of predictor variables across individuals or groups, allowing for subject-specific or group-specific deviations from fixed effects
- Specification depends on research question, data structure, and assumed distribution of random effects (normal distribution)
- Fixed effects specified as coefficients of predictor variables in linear predictor
- Random effects specified as coefficients of grouping variables (individual or cluster identifiers) and their assumed covariance structure
Estimating fixed and random effects
- Estimation commonly performed using maximum likelihood (ML) or restricted maximum likelihood (REML) methods
- Provide estimates of fixed effects coefficients, variance components of random effects, and their standard errors
- Choice between ML and REML depends on focus of inference (fixed effects vs. variance components) and sample size
- REML generally preferred for small sample sizes or when focus is on estimating variance components
Interpreting mixed-effects models
Interpreting fixed and random effects
- Fixed effects interpretation similar to standard regression models
- Represent average change in response variable associated with one-unit change in predictor variable, holding other variables constant
- Random effects interpretation focuses on variability or heterogeneity in effects of predictor variables across individuals or groups
- Quantified by variance components and their standard deviations
- Assess statistical significance of fixed effects using t-tests or F-tests, based on estimated coefficients and standard errors
- Assess significance of random effects using likelihood ratio tests or Wald tests
- Construct confidence intervals for fixed effects and variance components to quantify precision of estimates and make inferences about population parameters
Assessing model fit and diagnostics
- Assess model fit using criteria like Akaike Information Criterion (), Bayesian Information Criterion (), or
- Balance goodness of fit with model complexity
- Use residual diagnostics (plots of residuals against fitted values or predictor variables) to check assumptions of linearity, , and normality of errors
- Quantify proportion of variance explained by fixed and random effects using (ICC) or marginal and conditional -squared measures
- Provides insights into relative importance of different sources of variability in data
Applying mixed-effects models to data
Software and implementation
- Various statistical software packages provide tools for fitting and analyzing mixed-effects models
- R (lme4 or nlme packages), (MIXED procedure), Stata (mixed command)
- Application involves data preparation (handling missing data, transforming variables), model specification (defining fixed and random effects structures), model estimation (using ML or REML), and model interpretation and inference
- Choose appropriate mixed-effects model based on research question, data structure, and assumptions about distribution of random effects and residual errors
Model selection and presentation
- Use model selection techniques (backward elimination, forward selection) to identify most parsimonious and informative model among candidate models with different fixed and random effects structures
- Present results in tabular or graphical form, including estimated fixed effects coefficients, standard errors, p-values, variance components of random effects, and model fit statistics
- Interpret results considering substantive meaning of fixed and random effects, magnitude and precision of estimates, and limitations and assumptions of model
- Conduct sensitivity analyses to assess robustness of results to different model specifications, estimation methods, or distributional assumptions, and explore impact of influential observations or outliers on estimates
Key Terms to Review (20)
AIC: AIC, or Akaike Information Criterion, is a measure used for model selection that evaluates how well a model fits the data while penalizing for complexity. It helps in comparing different statistical models, where a lower AIC value indicates a better fit with fewer parameters. This criterion is widely used in various regression techniques, including logistic regression, robust estimation, mixed-effects models, and regression diagnostics.
BIC: BIC, or Bayesian Information Criterion, is a statistical measure used for model selection among a finite set of models. It balances model fit with complexity, penalizing models that are too complex while rewarding those that explain the data well. The goal is to identify the model that best describes the underlying data structure while avoiding overfitting.
Clustered Data Analysis: Clustered data analysis refers to statistical methods used to analyze data that is grouped or clustered in a specific way, often because of the nature of the data collection process. This approach helps account for the correlation among observations within clusters, which is important when making inferences about the population. The methods applied in this analysis are crucial in accurately estimating the effects of predictors while recognizing the hierarchical or nested structure of data.
Fixed effects: Fixed effects refer to a statistical technique used in models that accounts for individual-specific characteristics that do not change over time. This approach allows researchers to control for variables that could bias results by focusing on changes within individuals or entities rather than between them. It is particularly useful in mixed-effects models and hierarchical linear modeling, as it helps isolate the impact of independent variables while holding constant the unobserved heterogeneity among subjects.
Gary H. McLennan: Gary H. McLennan is a notable figure in the field of mixed-effects models, particularly recognized for his contributions to the understanding and application of these statistical methods in various research contexts. His work often emphasizes the importance of hierarchical data structures and the use of mixed-effects models to account for both fixed and random effects, allowing for more nuanced interpretations of complex data sets.
Generalized linear mixed model: A generalized linear mixed model (GLMM) is a statistical model that combines the principles of generalized linear models and mixed effects models to analyze data that exhibit both fixed and random effects. This approach is particularly useful for handling non-normal response variables and hierarchical or grouped data, allowing for greater flexibility in modeling complex relationships while accounting for random variations across groups.
Hierarchical data: Hierarchical data refers to a structured format where data is organized in a tree-like structure, with levels of parent-child relationships. This organization allows for the representation of data that has multiple levels of categories, making it easy to understand relationships and dependencies between different data points.
Homoscedasticity: Homoscedasticity refers to the property of a dataset in which the variance of the residuals, or errors, is constant across all levels of the independent variable(s). This characteristic is crucial for valid inference in regression analysis, as it ensures that the model's predictions are reliable. When homoscedasticity holds, the spread of the residuals is uniform, leading to better model fit and accurate hypothesis testing. Violation of this assumption can impact the results, causing inefficiencies and biased estimates.
Intraclass correlation coefficient: The intraclass correlation coefficient (ICC) is a statistical measure used to assess the reliability or agreement of measurements made by different observers measuring the same quantity. It evaluates how strongly units in the same group resemble each other, making it especially relevant in studies that involve repeated measures, like mixed-effects models and hierarchical linear modeling. The ICC ranges from 0 to 1, with higher values indicating greater reliability or consistency among the measurements.
Julian Besag: Julian Besag is a renowned statistician known for his significant contributions to the fields of spatial statistics and mixed-effects models. His work has influenced the development of statistical methodologies that account for both fixed and random effects, allowing for more accurate analysis of data that exhibit correlation in space or time. Besag's methods are particularly valuable in fields like ecology, epidemiology, and geography, where understanding the underlying structure of spatial data is crucial.
Likelihood ratio test: A likelihood ratio test is a statistical method used to compare the fit of two competing models to determine which model better explains the data. It is based on the ratio of the maximum likelihoods of the two models, allowing researchers to assess the strength of evidence against a null hypothesis. This test is particularly useful in scenarios where robust estimation and mixed-effects models are employed, as it provides a way to make inferences about parameters while considering model complexity and data variability.
Linear mixed-effects model: A linear mixed-effects model is a statistical method that combines fixed effects, which are constant across individuals, and random effects, which vary among individuals or groups. This model is particularly useful for analyzing data that has multiple levels of variability, such as repeated measurements from the same subjects or nested data structures. By accounting for both types of effects, linear mixed-effects models provide a more accurate representation of the relationships within complex data sets.
Longitudinal studies: Longitudinal studies are research methods that involve repeated observations of the same variables over long periods of time, allowing researchers to track changes and developments within a population or individual. This approach is particularly valuable for examining how variables influence one another over time and is essential in understanding trends, causality, and the dynamics of change in various fields such as psychology, medicine, and social sciences.
Nested data: Nested data refers to data structures where observations are organized within higher-level groups or clusters, such as students within classrooms or patients within hospitals. This type of organization often reflects the hierarchical nature of data, where lower-level units (like individuals) are grouped into higher-level units (like groups), and it is crucial for understanding relationships and variances in mixed-effects models.
Normality assumption: The normality assumption is a statistical premise that assumes the residuals or errors of a model are normally distributed. This assumption is crucial because many statistical methods, including mixed-effects models, rely on the accuracy of this condition to produce valid results and inferences. If the normality assumption is violated, it can lead to biased estimates and invalid conclusions, affecting the overall reliability of the model.
Python: Python is a high-level programming language known for its readability and versatility, widely used in data analysis, machine learning, and statistical modeling. Its simplicity allows users to focus on problem-solving rather than complex syntax, making it popular among both beginners and experts in quantitative methods.
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.
Random effects: Random effects refer to variables in statistical models that account for variability across different groups or clusters, allowing for the analysis of hierarchical or clustered data structures. These effects capture the influence of unobserved factors that vary randomly across levels of a grouping variable, making them essential for accurately estimating relationships within complex data. By incorporating random effects, models can account for the non-independence of observations within groups, leading to more robust statistical inferences.
Random intercepts: Random intercepts are a feature of mixed-effects models that allow for varying intercepts across different groups or clusters in the data. This means that each group can have its own unique baseline level, which captures the inherent differences between groups while still considering overall trends in the data. By incorporating random intercepts, researchers can account for the hierarchical structure of data, where observations within the same group are more similar to each other than to those in different groups.
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.