# Longitudinal Data and Linear Mixed Effects Models

• Ikuko Funatogawa
• Takashi Funatogawa
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

## Abstract

Longitudinal data are measurements or observations taken from multiple subjects repeatedly over time. The main theme of this book is to describe autoregressive linear mixed effects models for longitudinal data analysis. This model is an extension of linear mixed effects models and autoregressive models. This chapter introduces longitudinal data and linear mixed effects models before the main theme in the following chapters. Linear mixed effects models are popularly used for the analysis of longitudinal data of a continuous response variable. They are an extension of linear models by including random effects and variance covariance structures for random errors. Marginal models, which do not include random effects, are also introduced in the same framework. This chapter explains examples of popular linear mixed effects models and marginal models: means at each time point with a random intercept, means at each time point with an unstructured variance covariance, and linear time trend models with a random intercept and a random slope. The corresponding examples of group comparisons are also provided. This chapter also discusses the details of mean structures and variance covariance structures and provides estimation methods based on maximum likelihood and restricted maximum likelihood.

## Keywords

Linear mixed effects model Longitudinal Maximum likelihood Random effect Unbalanced data

## References

1. Diggle PJ (1988) An approach to the analysis of repeated measurements. Biometrics 44:959–971
2. Diggle PJ, Heagerty P, Liang KY, Zeger SL (2002) Analysis of longitudinal data, 2nd edn. Oxford University PressGoogle Scholar
3. Diggle PJ, Liang KY, Zeger SL (1994) Analysis of longitudinal data. Oxford University PressGoogle Scholar
4. Dwyer JH, Feinleib M, Lippert P, Hoffmeister H (eds) (1992) Statistical models for longitudinal studies of health. Oxford University PressGoogle Scholar
5. Fitzmaurice GM, Davidian M, Verbeke G, Molenberghs G (eds) (2009) Longitudinal data analysis. Chapman & Hall/CRC PressGoogle Scholar
6. Fitzmaurice GM, Laird NM, Ware JH (2004) Applied longitudinal analysis. WileyGoogle Scholar
7. Fitzmaurice GM, Laird NM, Ware JH (2011) Applied longitudinal analysis, 2nd edn. WileyGoogle Scholar
8. Funatogawa I, Funatogawa T (2011) Analysis of covariance with pre-treatment measurements in randomized trials: comparison of equal and unequal slopes. Biometrical J 53:810–821
9. Funatogawa I, Funatogawa T, Ohashi Y (2007) An autoregressive linear mixed effects model for the analysis of longitudinal data which show profiles approaching asymptotes. Stat Med 26:2113–2130
10. Funatogawa T, Funatogawa I, Shyr Y (2011) Analysis of covariance with pre-treatment measurements in randomized trials under the cases that covariances and post-treatment variances differ between groups. Biometrical J 53:512–524
11. Funatogawa T, Funatogawa I, Takeuchi M (2008) An autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes. Stat Med 27:6351–6366
12. Gregoire TG, Brillinger DR, Diggle PJ, Russek-Cohen E, Warren WG, Wolfinger RD (eds) (1997) Modelling longitudinal and spatially correlated data. Springer-VerlagGoogle Scholar
13. Hand D, Crowder M (1996) Practical longitudinal data analysis. Chapman & HallGoogle Scholar
14. Heitjan DF (1991) Nonlinear modeling of serial immunologic data: a case study. J Am Stat Assoc 86:891–898
15. Jones RH (1993) Longitudinal data with serial correlation: a state-space approach. Chapman & HallGoogle Scholar
16. Kenward MG (1987) A method for comparing profiles of repeated measurements. Appl Stat 36:296–308
17. Kenward MG, Roger JH (1997) Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53:983–997
18. Laird NM (2004) Analysis of longitudinal & cluster-correlated data. IMSGoogle Scholar
19. Laird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometrics 38:963–974
20. Littell RC, Miliken GA, Stroup WW, Wolfinger RD (1996) SAS system for mixed models. SAS Institute IncGoogle Scholar
21. Littell RC, Miliken GA, Stroup WW, Wolfinger RD, Schabenberger O (2006) SAS for mixed models, 2nd edn. SAS Institute IncGoogle Scholar
22. Satterthwaite FE (1946) An approximate distribution of estimates of variance components. Biometrics Bull 2:110–114
23. Sy JP, Taylor JMG, Cumberland WG (1997) A stochastic model for analysis of bivariate longitudinal AIDS data. Biometrics 53:542–555
24. Tango T (2017) Repeated measures design with generalized linear mixed models for randomized controlled trials. CRC PressGoogle Scholar
25. Taylor JMG, Cumberland WG, Sy JP (1994) A stochastic model for analysis of longitudinal AIDS data. J Am Stat Assoc 89:727–736
26. Taylor JMG, Law N (1998) Does the covariance structure matter in longitudinal modeling for the prediction of future CD4 counts? Stat Med 17:2381–2394
27. Verbeke G, Molenberghs G (eds) (1997) Linear mixed models in practice—a SAS oriented approach. Springer-VerlagGoogle Scholar
28. Verbeke G, Molenberghs G (2000) Linear mixed models for longitudinal data. Springer-VerlagGoogle Scholar
29. Vonesh EF (2012) Generalized linear and nonlinear models for correlated data. Theory and applications using SAS. SAS Institute IncGoogle Scholar
30. Wu H, Zhang J-T (2006) Nonparametric regression methods for longitudinal data analysis. Mixed-effects modeling approaches. WileyGoogle Scholar
31. Zimmerman DL, Núñez-Antón VA (2010) Antedependence models for longitudinal data. CRC PressGoogle Scholar