[This is a very early draft]

Longitudinal data are data where each patient is observed on multiple occasions over time. Analysis of longitudinal data are challenging because measurements on the same subject are correlated. Another way to think about this is that two measurements on the same subject will have less variation than two measurements on different subjects.

A closely related concept is the cluster design. A cluster design is one where the researcher selects clusters of patients rather than selects patients individually. For example, a researcher might randomly select several families and evaluate all children in that family. As another example, a researcher might randomly select several clinical practices and then evaluate a random group of patients at each practice. In a cluster design, two measurements on patients within the same cluster will have less variations than measurements of two patients in differing clusters. . In genetics, this correlation is of great interest, and can help you understand concepts like heritability.

Many of the methods described below for longitudinal designs would also be useful for cluster designs. For simplicity, I will discuss these methods solely from the perspective of a longitudinal design.

If your data are continuous, then there are several "classical" approaches such as multivariate analysis of variance and repeated measures analysis of variance. These approaches work well for simple well structured longitudinal data.

An alternative is to use mixed linear models. These models handle missing data well and can handle situations where the times of measurement vary from one patient to another.

In a mixed linear model, you specify a particular structure for the correlations. For example, an autoregressive structure is commonly used to represent structure where correlations are strongest for measurements close in time and which become weaker for measurements that are further separated in time.

In many situations, the correlations are not of direct interest, but we only account for them because failure to do so will lead to incorrect inferences.

When you are examining the correlation structure, a statistic called the Akaike Information Criteria (AIC). This statistic measures how closely the model fits the data, but it includes a penalty for overly complex models.

Unfortunately, there are two different formulas for AIC. For one formula, a large value of AIC is good, and for the other formula, a small value is good.

AIC values should only be compared for models where the only change is in the correlation structure. It would not make sense to compare an AIC from a model with linear relationships to a model with quadratic relationships.

What if your data is not continuous? L. Fang discussed some of the approaches commonly used when the data represents binomial counts.

- Mixed models
- Mixed model after arcsin transformation
- GEE
- GLIMM

**Further reading**

**A comparison of different approaches for fitting centile curves to birthweight data.**Bonellie SR, Raab GM. Statistics in Medicine 1996: 15(24); 2657-67. [Medline]**Longitudinal methods for evaluating therapy.**Clemens JD, Horwitz RI. Biomed Pharmacother 1984: 38(9-10); 440-3. [Medline]**Exact Tests for Correlated Data [pdf]**. Corcoran C, Senchaudhuri P, Cytel Software. Accessed on 2003-12-26. www.cytel.com/papers/cytel_newsletter_chris_new.pdf**Power for Simple Mixed Models**. Dickson P, School of Nursing, The University of Texas at Austin. Accessed on 2003-08-28. www.nur.utexas.edu/Dickson/stats/mxpower.html**Analysis of Longitudinal Data.**Diggle PJ (1994) Oxford: Clarendon Press.**Lecture Notes in Statistics: Probabilistic Causality in Longitudinal Studies.**Eerola M (1994) New York: Springer-Verlag.**Extension of the gauss-markov theorem to include the estimation of random effects.**Harville D. The Annals of Statistics 1976: 4(2); 384-95.**Random-effects regression models for clustered data with an example from smoking prevention research.**Hedeker D, Gibbons RD, Flay BR. J Consult Clin Psychol 1994: 62(4); 757-65.**A discussion of the two-way mixed model.**Hocking R. The American Statistician 1973: 27(4); 148-52.**Approximations for standard errors of estimators of fixed and random effects in mixed linear models.**Kackar R. Journal of the American Statistical Association 1984: 79(388); 853-62.**Statistical Tests for Mixed Linear Models.**Khuri A, Mathew T, Sinha B (1998) New York: John Wiley & Sons, Inc.**Does practice really make perfect?**Laine C, Sox HC. Ann Intern Med 2003: 139(8); 696-8. [Medline] [Full text] [PDF]**Models for Repeated Measurements.**Lindsey JK (1993) Oxford: Clarendon Press.**SAS System for Mixed Models.**Littell RC, Ph.D., Milliken GA, Ph.D., Stroup WW, Ph.D., Wolfinger RD, Ph.D. (1996) Cary, North Carolina: SAS Institute Inc.**Random Coefficient Models.**Longford NT (1993) Oxford: Claredon Press.**Assessing change with longitudinal and clustered binary data.**Neuhaus JM. Annu Rev Public Health 2001: 22; 115-28.**The effect of clustering of outcomes on the association of procedure volume and surgical outcomes.**Panageas KS, Schrag D, Riedel E, Bach PB, Begg CB. Ann Intern Med 2003: 139(8); 658-65. [Medline] [Abstract] [Full text] [PDF]**Further statistics in dentistry. Part 7: repeated measures.**Petrie A, Bulman JS, Osborn JF. Br Dent J 2003: 194(1); 17-21. [Medline] [Abstract]**Mixed-Effects Models in S and S-PLUS.**Pinheiro JC, Bates DM (2000) New York: Springer-Verlag.**Comparing personal trajectories and drawing causal inferences from longitudinal data.**Raudenbush SW. Annu Rev Psychol 2001: 52; 501-25.- Linear mixed-effects modeling in SPSS: An introduction to the MIXED procedure. [[Link temporarily misplace.]]
**Using SAS PROC MIXED to Fit Multi-Level Models, Hierarchical Models, and Individual Growth Models.**Singer JD. Journal of Educational and Behavioral Statistics 1998: 24(4); 323-355.**Statistics notes: Analysing controlled trials with baseline and follow up measurements.**Vickers AJ, Altman DG. Bmj 2001: 323(7321); 1123-4.**Longitudinal Data Analysis for Discrete and Continuous Outcomes.**Zeger SL, Liang K-L. Biometrics 1986: 42(1); 121-130.**An overview of methods for the analysis of longitudinal data.**Zeger SL, Liang KY. Stat Med 1992: 11(14-15); 1825-39.

You can find an earlier version of this page on my original website.