{ "id": "ep9adrn05k", "slug": "longitudinal-data-analysis", "title": "Longitudinal data analysis", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": 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"https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis" } ] } ], "excerpt": "Longitudinal data analysis is the collection of statistical methods for data measured repeatedly on the same subjects over time, separating within-subject change from between-subject differences.", "snippet": "Longitudinal data analysis is the collection of statistical methods for data measured repeatedly on the same subjects over time, separating within-subject change from between-subject differences.", "node": "physical.math.statistics.stats_statistical_inference.regression-analysis.panel-data-regression", "markdown": "# Longitudinal data analysis\n\nLongitudinal data analysis is the collection of statistical methods for data measured repeatedly on the same subjects over time, used to model within-subject change and the correlation among a subject's observations in medical, social, and behavioral research. Its central advantage over cross-sectional analysis is that it separates change within individuals over time from differences among individuals at a given time, which a single snapshot cannot do.[1](https://scsru.github.io/Modules/introduction-to-longitudinal-data.html) In a linear mixed model, one coefficient (\\( \\beta_{1} \\)) captures the cross-sectional effect, differences between subjects, while another (\\( \\beta_{2} \\)) captures the longitudinal effect, the average change within individuals over time.[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC10507671/) The defining statistical problem is that repeated observations on one person are correlated, and standard regression inference assumes independence.[3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf)\n\n| Key fact | Detail |\n|---|---|\n| What it estimates | Within-subject change over time, separated from between-subject (cohort) differences[1](https://scsru.github.io/Modules/introduction-to-longitudinal-data.html) |\n| Core model families | Linear mixed models, generalized linear mixed models (GLMMs), and generalized estimating equations (GEE)[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) |\n| Why correlation matters | Ignoring positive within-subject correlation inflates type I error for time-independent covariates and type II error for time-dependent covariates[3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf) |\n| Missing data | Full-likelihood mixed models are valid under missing at random (MAR); GEE requires missing completely at random (MCAR)[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) |\n| Sample size | With exchangeable correlation \\( \\rho \\) and \\( n \\) observations per subject, the design effect is \\( D_{\\text{eff}} = \\{1 + (n-1)\\rho\\}/n \\)[5](https://bookdown.org/charlotte_micheloud93/Clinical_Biostatistics/analysis-of-longitudinal-outcomes.html) |\n| Foundational paper | Laird and Ware, \"Random-Effects Models for Longitudinal Data,\" Biometrics, 1982[6](https://doi.org/10.2307/2529876) |\n\n## How it works\n\nLongitudinal models specify two things jointly: a model for the mean of the outcome and a model for the within-subject correlation. Two frameworks dominate. GEE specifies a marginal mean model, \\( g(\\mathrm{E}[Y_{ij} \\mid x_{ij}]) = x_{ij} \\cdot \\beta \\), plus a working correlation model \\( \\mathrm{Corr}[Y_{ij}, Y_{ij0}] = \\rho(\\alpha) \\) for observations j and j₀ on the same subject.[7](https://si.biostat.washington.edu/sites/default/files/modules/11_notes.pdf) A GLMM instead specifies a conditional mean, \\( g(\\mathrm{E}[Y_{ij} \\mid x_{ij}, b_i]) = x_{ij} \\cdot \\beta + z_{ij} \\cdot b_i \\), where the random effects \\( b_i \\sim N(0, D) \\) induce the correlation.[7](https://si.biostat.washington.edu/sites/default/files/modules/11_notes.pdf)\n\nThe two frameworks estimate different quantities. GEE estimates population-averaged contrasts, while GLMMs estimate subject-specific contrasts; the two coincide only for Gaussian outcomes with an identity link.[7](https://si.biostat.washington.edu/sites/default/files/modules/11_notes.pdf) • [3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf) GEE derives its estimating equations without specifying the joint distribution of a subject's observations, and the equations reduce to the usual score equations for multivariate Gaussian outcomes.[8](https://biostat.jhsph.edu/~fdominic/teaching/bio655/references/extra/liang.bka.1986.pdf)\n\nIf within-subject correlation is positive and ignored, for example by fitting ordinary least squares, standard errors are underestimated. This inflates type I error rates for time-independent covariates such as sex or race, and inflates type II error rates for time-dependent covariates.[3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf) [Correlation](https://www.edgepedia.org/correlation) can be handled either by random effects with a specified G structure or by placing a correlation structure directly on the R matrix of within-subject errors.[3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf)\n\n## How it is done\n\nDesign comes first: the investigator decides the study duration, the frequency of visits, and the sample size, power, and number of observations per subject.[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC10507671/) Adding equally spaced measurements across a fixed treatment period does not generally increase the probability of detecting a true treatment effect, so visit frequency should serve the model, not the other way around.[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC10507671/)\n\nA recommended fitting workflow is to clean the data, check assumptions, model the time trend, select the covariance structure, and then perform variable selection, iterating the middle steps as needed.[1](https://scsru.github.io/Modules/introduction-to-longitudinal-data.html) For linear mixed models, restricted maximum likelihood (REML) is the estimation method of choice because standard errors are biased downward under maximum likelihood (ML).[3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf) Misspecifying the covariance structure generally biases standard errors and hypothesis tests rather than the fixed-effect estimates themselves, so AIC or likelihood-ratio tests are used to compare candidate structures.[19](https://www.tandfonline.com/doi/abs/10.1080/00273170701540537) • [3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf)\n\nFor GEE, the regression coefficient estimates remain broadly valid as sample size grows regardless of the chosen working correlation, but model-based standard errors require the correlation model to be correct; empirical (sandwich) standard errors, reported as a standard feature, provide valid uncertainty estimates.[9](https://faculty.washington.edu/heagerty/Courses/VA-longitudinal/private/LDAchapter.pdf) By Gauss–Markov optimal estimation theory, the most efficient choice of working correlation is the true correlation structure.[9](https://faculty.washington.edu/heagerty/Courses/VA-longitudinal/private/LDAchapter.pdf)\n\nSample size for a two-group longitudinal comparison is adjusted by a design effect. With exchangeable correlation \\( \\rho \\) and \\( n \\) observations per patient,\n\n\\[ D_{\\text{eff}} = \\{1 + (n-1)\\rho\\}/n, \\]\n\nand the required sample size is the standard cross-sectional sample size multiplied by this factor.[5](https://bookdown.org/charlotte_micheloud93/Clinical_Biostatistics/analysis-of-longitudinal-outcomes.html) For an arbitrary correlation matrix R, the design factor is \\( D_{\\text{eff}} = \\{\\boldsymbol{1}^\\top \\cdot \\boldsymbol{R}^{-1} \\cdot \\boldsymbol{1}\\}^{-1} \\).[5](https://bookdown.org/charlotte_micheloud93/Clinical_Biostatistics/analysis-of-longitudinal-outcomes.html) Standard errors for longitudinal effects are smaller when observations are spread out over time, while standard errors for intercept and cross-sectional terms are smaller with sequential visits; all decline as the number of subjects and observations grows.[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC10507671/)\n\n## Origin\n\nThe two-stage random-effects framework for longitudinal data was set out in Nan M. Laird and [James H. Ware](https://www.edgepedia.org/james-h-ware)'s paper \"Random-Effects Models for Longitudinal Data,\" published in [Biometrics](https://www.edgepedia.org/biometrics) in 1982.[6](https://doi.org/10.2307/2529876) • [10](https://people.stat.sc.edu/hansont/stat740/LairdWare1982.pdf) The paper presented a general family of models that includes both growth models and repeated-measures models as special cases, suited to highly unbalanced data where multivariate models with general covariance structure are difficult to apply, and proposed a unified fitting approach combining empirical Bayes and maximum likelihood estimation using the EM algorithm.[10](https://people.stat.sc.edu/hansont/stat740/LairdWare1982.pdf) Multilevel models, a variant name for the same family, appear in the literature from 1995.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/)\n\nGEE models were developed during the 1980s as an extension of generalized linear models to correlated data[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/); the 1986 [Biometrika](https://www.edgepedia.org/biometrika) paper that proposed the approach modeled the marginal rather than the conditional mean and discussed independence, m-dependence, and exchangeable working correlation structures.[8](https://biostat.jhsph.edu/~fdominic/teaching/bio655/references/extra/liang.bka.1986.pdf) Before these methods, analysis relied on repeated-measures ANOVA and MANOVA-type growth curve models.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/)\n\n## Variants\n\nMixed-effects regression models have been developed under several names, including random-effects models, variance component models, multilevel models, two-stage models, random coefficient models, and hierarchical linear models.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) Extensions include the generalized linear mixed model for non-Gaussian outcomes and nonlinear mixed-effects models.[11](https://www.ncbi.nlm.nih.gov/books/NBK385371/)\n\nGrowth curve models fall into two classes. The latent-curve (LC) approach treats repeated measures as multivariate (\"wide\" format) and is fitted with structural equation modeling software; the mixed-effect (ME) approach treats them as univariate (\"long\" format) and is fitted with regression software, modeling the intercept and time coefficients as random effects.[12](https://link.springer.com/article/10.3758/s13428-017-0976-5) The ME approach suits straightforward models with complex data structures, such as small samples, time-unstructured data, or multiple levels of nesting; the LC approach suits complex models with straightforward data structures, such as model-fit assessment, time-varying covariates, and complex variance functions.[12](https://link.springer.com/article/10.3758/s13428-017-0976-5) Similarly, the linear mixed-effects model is preferable when data are unbalanced or incomplete with a common change function and error covariance, while the latent curve model is preferable with mostly complete data, complex change functions, and flexible error structures.[11](https://www.ncbi.nlm.nih.gov/books/NBK385371/)\n\nA random intercept plus a random slope for time induces within-individual correlations that decrease in magnitude as measurements are further apart in time.[3](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf) Common working structures include unstructured, exchangeable (a single parameter \\( \\rho \\)), AR(1) with entries \\( \\rho^{|j-l|} \\) for evenly spaced observations, and exponential correlation, \\( \\rho_{jl} = \\exp(-\\phi|t_{ij} - t_{il}|) \\), which collapses to AR(1) when observations are equally spaced.[1](https://scsru.github.io/Modules/introduction-to-longitudinal-data.html)\n\n## Applications\n\nLongitudinal methods are standard in clinical trials, where repeated outcome measures track treatment effects over a defined period, and in cohort studies, where design choices about visit duration and frequency must be balanced against sample size and power.[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC10507671/) In life-course research, both linear mixed-effects and latent curve models handle incomplete data without imputation or complete-case restriction, and support multivariate, multiple-group, and latent-class extensions.[11](https://www.ncbi.nlm.nih.gov/books/NBK385371/) In genetic epidemiology, longitudinal genome-wide association methods fall into four groups: mixed-effect or [random regression models](https://www.edgepedia.org/random-regression-model), Bayesian approaches, latent class trajectory models, and multi-variant or multi-trait methods.[13](https://www.annualreviews.org/content/journals/10.1146/annurev-biodatasci-092724-035434) Joint models link a longitudinal sub-model to a survival sub-model, using the longitudinal process as a time-varying covariate in survival risk.[14](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-112723-034334) Bayesian estimation with [Markov chain Monte Carlo](https://www.edgepedia.org/markov-chain-monte-carlo), using Gibbs and Metropolis–Hastings samplers, underpins dynamic structural equation models for intensive longitudinal data.[15](https://www.statmodel.com/download/DSEM.pdf)\n\n## Limitations and alternatives\n\nMissingness is classified as missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR, also called informative or non-ignorable).[7](https://si.biostat.washington.edu/sites/default/files/modules/11_notes.pdf) The validity split follows the likelihood: under MCAR both GEE and mixed-effects models are valid; under MAR only full-likelihood mixed models are valid, because GEE is a partial-likelihood ([quasi-likelihood](https://www.edgepedia.org/quasi-likelihood)) method; under MNAR neither is valid without further modeling of the missingness mechanism.[7](https://si.biostat.washington.edu/sites/default/files/modules/11_notes.pdf) • [4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) Laird and Ware's 1982 framework showed that mixed-effects regression can analyze all available data under MAR, replacing approaches such as last-observation-carried-forward and change-score analyses.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/)\n\nThe failure mode under non-ignorable dropout is quantified by simulation: weighted least squares and particularly random-effects estimates tended to underestimate the average rate of marker change by about 10 percent, while unweighted least squares, conditional likelihood, and joint-model random-effects estimators showed bias of only 3 to 5 percent.[16](https://onlinelibrary.wiley.com/doi/10.1002/sim.1114) This is an unresolved tension in the literature, since mixed models are promoted as valid under MAR while simulations show substantial bias when missingness is actually non-ignorable.[16](https://onlinelibrary.wiley.com/doi/10.1002/sim.1114) • [4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) In small samples with missing data, modified covariance estimators help; Mancl and DeRouen's estimator performs best among modified estimators, followed by Fay and Graubard's, with performance nearly equivalent to the Kenward–Roger method with an unstructured covariance.[17](https://onlinelibrary.wiley.com/doi/10.1111/insr.12447) For time-structured designs, simulation work argues that single-level (wide-format) multiple imputation is more flexible than multilevel imputation because it requires fewer assumptions and accommodates a wider range of analysis models.[18](https://www.tandfonline.com/doi/full/10.1080/00273171.2026.2710728)\n\nCompared with alternatives, the univariate repeated-measures ANOVA assumes compound symmetry, equal variances and covariances across time, and breaks down for unbalanced designs with subject discontinuation; based on these limitations it should no longer be used for longitudinal data.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) MANOVA growth curve models allow general correlation but require complete data, and removing incomplete subjects risks bias.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/) How longitudinal analysis compares with time-series analysis, and a detailed treatment of transition models, are not covered here.\n\n## References\n\n1. [Introduction to Longitudinal Data | Topics in Statistical Consulting (SCSRU)](https://scsru.github.io/Modules/introduction-to-longitudinal-data.html)\n2. [Design Issues in Longitudinal Studies (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC10507671/)\n3. [Core Guide: Correlation Structures in Longitudinal Data Analysis (Duke Global Health RDAC)](https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Correlation-Structures-in-Longitudinal-Data-Analysis_09-19-17.pdf)\n4. [Advances in Analysis of Longitudinal Data (Hedeker et al., review)](https://pmc.ncbi.nlm.nih.gov/articles/PMC2971698/)\n5. [Chapter 14: Analysis of longitudinal outcomes | Clinical Biostatistics](https://bookdown.org/charlotte_micheloud93/Clinical_Biostatistics/analysis-of-longitudinal-outcomes.html)\n6. [Nan M. Laird, James H. Ware (1982). Random-Effects Models for Longitudinal Data. Biometrics.](https://doi.org/10.2307/2529876)\n7. [Module 11: Mixed-effects Models for Longitudinal Data Analysis (Fitzmaurice course notes, UW Biostatistics)](https://si.biostat.washington.edu/sites/default/files/modules/11_notes.pdf)\n8. [Longitudinal data analysis using generalized linear models (Liang & Zeger, 1986, Biometrika)](https://biostat.jhsph.edu/~fdominic/teaching/bio655/references/extra/liang.bka.1986.pdf)\n9. [Longitudinal Data Analysis (Heagerty, chapter)](https://faculty.washington.edu/heagerty/Courses/VA-longitudinal/private/LDAchapter.pdf)\n10. [Random-Effects Models for Longitudinal Data (Laird & Ware, Biometrics 1982)](https://people.stat.sc.edu/hansont/stat740/LairdWare1982.pdf)\n11. [Chapter 8: Linear Mixed-Effects and Latent Curve Models for Longitudinal Life Course Analyses (NCBI Bookshelf)](https://www.ncbi.nlm.nih.gov/books/NBK385371/)\n12. [Differentiating between mixed-effects and latent-curve approaches to growth modeling (Behavior Research Methods)](https://link.springer.com/article/10.3758/s13428-017-0976-5)\n13. [Statistical Methods for Understanding Trajectories in Genetic Epidemiology (Annual Review of Biomedical Data Science)](https://www.annualreviews.org/content/journals/10.1146/annurev-biodatasci-092724-035434)\n14. [Joint Modeling of Longitudinal and Survival Data (Annual Review of Statistics and Its Application)](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-112723-034334)\n15. [Dynamic Structural Equation Models (Mplus documentation)](https://www.statmodel.com/download/DSEM.pdf)\n16. [Impact of missing data due to drop-outs on estimators for rates of change in longitudinal studies: a simulation study (Statistics in Medicine)](https://onlinelibrary.wiley.com/doi/10.1002/sim.1114)\n17. [Practical Review and Comparison of Modified Covariance Estimators for Linear Mixed Models in Small-sample Longitudinal Studies with Missing Data (International Statistical Review)](https://onlinelibrary.wiley.com/doi/10.1111/insr.12447)\n18. [Flexible Multiple Imputation of Missing Data in Time-Structured Longitudinal Designs (Multivariate Behavioral Research, 2026)](https://www.tandfonline.com/doi/full/10.1080/00273171.2026.2710728)\n19. [tandfonline.com](https://www.tandfonline.com/doi/abs/10.1080/00273170701540537)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Regression analysis › Panel data regression*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://biostat.jhsph.edu/~fdominic/teaching/bio655/references/extra/liang.bka.1986.pdf" ], "url": "https://www.edgepedia.org/longitudinal-data-analysis", "markdown_url": "https://www.edgepedia.org/longitudinal-data-analysis.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Longitudinal data analysis\", Edgepedia (EdgeChat), https://www.edgepedia.org/longitudinal-data-analysis. Edgepedia Community License 1.0.", "credit_md": "\"[Longitudinal data analysis](https://www.edgepedia.org/longitudinal-data-analysis)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/longitudinal-data-analysis](https://www.edgepedia.org/longitudinal-data-analysis). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Longitudinal data analysis\", Edgepedia (EdgeChat), https://www.edgepedia.org/longitudinal-data-analysis. Edgepedia Community License 1.0.", "speakable": "Longitudinal data analysis is the collection of statistical methods for data measured repeatedly on the same subjects over time, separating within-subject change from between-subject differences." }