Purpose: Broken stick analysis is a widely used approach for detecting unknown breakpoints where association between measurements is non-linear. We propose LIMBARE, an advanced li near m ixed-effects b reakpoint a nalysis with r obust e stimation, especially designed for longitudinal ophthalmic studies. LIMBARE accommodates repeated measurements from both eyes and overtime, and effectively address the presence of outliers.
Methods: The model setup of LIMBARE and computing algorithm for point and confidence interval estimates of the breakpoint was introduced. The performance of LIMBARE and other competing methods was assessed via comprehensive simulation studies and application to a longitudinal ophthalmic study with 216 eyes (145 subjects) followed for an average of 3.7±1.3 years to examine the longitudinal association between structural and functional measurements.
Results: In simulation studies, LIMBARE showed the smallest bias and mean squared error (MSE) for estimating the breakpoint, with empirical coverage probability of corresponding CI estimate closest to the nominal level for scenarios with and without outlier data points. In the application to the longitudinal ophthalmic study, LIMBARE detected two breakpoints between visual field mean deviation (MD) and retinal nerve fiber layer thickness (RNFL) and one breakpoint between MD and cup to disc ratio (CDR), while the cross-sectional analysis approach only detected one and none, respectively.
Conclusions: LIMBARE enhances breakpoint estimation accuracy in longitudinal ophthalmic studies, while cross-sectional analysis approach is not recommended for future studies.
Translational relevance: Our proposed method and companion software R package provides a valuable computational tool for advancing longitudinal ophthalmology research and exploring the association relationships between ophthalmic variables.