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R. A. Bilonick, L. Kagemann, G. Wollstein, H. Ishikawa, J. S. Schuman; Optimally Determining the Change Point When Measurements Are Repeated. Invest. Ophthalmol. Vis. Sci. 2009;50(13):5070.
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Spline regression or piecewise linear regression (referred to as "broken stick" regression) can be used with non-clustered data. Repeated measurements on one eye or measurements from both eyes result in data clustering, violating the assumption of statistical independence employed by ordinary linear regression and invalidating the usual methods for determining confidence intervals and p-values. In these cases, a mixed effects model needs to be used. The purpose of this study was to demonstrate the optimal estimation of the change point in a "broken stick" mixed effects model for clustered structure/function data.
If visual field mean deviation (MD) is essentially flat above a single change point for retinal nerve fiber layer (RNFL) thickness (obtained from time-domain optical coherence tomography) and declines linearly below this point, then a "broken stick" model can be a reasonably accurate description of the relationship. This model can be fitted to the data to provide an initial estimate of the change point. (Alternatively, Davies Test can be used.) A linear mixed effects model is then computed using a first degree spline with the knot equal to the the initial change point estimate. By varying the change point by small amounts, the optimal value of the change point can be ascertained using the likelihood or alternatively the Akaike information criterion (AIC).
MD is essentially flat above a global RNFL thickness of 73 microns while it declines linearly and rapidly below this point (see figure). The initial change point estimate of 73.09 was close to the optimal estimate of 72.34 microns. Similar results were achieved for the quadrants. The effects of clustering on the change point estimate will vary from case to case.
Using a simple "broken stick" segmented regression model (or alternatively, the Davies Test) will provide a good initial estimate. The optimal change point estimate can then be quickly computed by incrementing or decrementing the initial estimate and noting when the likelihood is maximized (or the AIC is minimized).
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