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Richard A. Bilonick, Kyle C. McKenna; Modeling the Correlation Structure of Longitudinal Data Using the Variogram. Invest. Ophthalmol. Vis. Sci. 2011;52(14):2804.
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Longitudinal studies are based on repeated measures over time which typically are positively autocorrelated (AC), i.e., data closer together tend to be more alike. Regression for each subject is inappropriate as the standard errors (SEs) tend to be too small and overstate statistical significance. Mixed effects models (MEMs) accommodate AC. Variograms (VG) measure the variance (inverse of correlation) between time points even with irregularly spaced data. There is a simple relation between variance and correlation so that one can be converted into the other. The purpose of this study was to use the VG to model the AC for a MEM.
Data from 38 patients with primary uveal melanoma were used. % CD11b+15+ granulocytes in blood were measured for a maximum of 386 days starting on the day of surgery. Patients were divided into two groups: low granulocytes (< 10% before surgery) and high granulocytes (>= 10% before surgery). MEMs with a time-by-group interaction were fitted via maximum likelihood using the nlme package in the R statistical language.
The initial MEM assumed no AC (AIC=326.4). The VG pattern (Figure, panel a) contradicted this assumption showing that the variance tended to increase over time and then leveled off. This suggested a spherical VG model with a range (the point at which the AC reaches zero) of about 50 days. A revised MEM (AIC=321.9) with a spherical VG model was fitted and lowered the AIC implying a better model. The VG range was estimated to be 38.6 days. The VG for the normalized residuals from this model (Figure, panel b) indicated a lack of AC in the residuals so that the SEs can be trusted.
Longitudinal data almost always exhibit AC which must be accounted for by the statistical model in order to correctly estimate SEs and assess statistical significance. The variogram is an effective tool for detecting, modeling, and removing AC in the model residuals, especially when the data are irregularly spaced.
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