Longitudinal data analysis (no date)
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[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
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 many situations, the correlations are not of direct interest, but we
only account for them because failure to do so will lead to incorrect
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
- A comparison of different approaches for fitting centile curves to
birthweight data. Bonellie SR, Raab GM. Statistics in Medicine 1996:
- Longitudinal methods for evaluating therapy. Clemens JD, Horwitz
RI. Biomed Pharmacother 1984: 38(9-10); 440-3.
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
- 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.
- Models for Repeated Measurements. Lindsey JK (1993) Oxford:
- 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
- Random Coefficient Models. Longford NT (1993) Oxford: Claredon
- 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.
- Further statistics in dentistry. Part 7: repeated measures. Petrie
A, Bulman JS, Osborn JF. Br Dent J 2003: 194(1); 17-21.
- 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.