Lack of Fit in Self Modeling Regression: Application to Pulse Waveforms

Self modeling regression (SEMOR) is an approach for modeling sets of observed curves that have a common shape (or sequence of features) but have variability in the amplitude (y-axis) and/or timing (x-axis) of the features across curves. SEMOR assumes the x and y axes for each observed curve can be separately transformed in a parametric manner so that the features across curves are aligned with the common shape, usually represented by non-parametric function. We show that when the common shape is modeled with a regression spline and the transformational parameters are modeled as random with the traditional distribution (normal with mean zero), the SEMOR model may surprisingly suffer from lack of fit and the variance components may be over-estimated. A random effects distribution that restricts the predicted random transformational parameters to have mean zero or the inclusion of a fixed transformational parameter improves estimation. Our work is motivated by arterial pulse pressure waveform data where one of the variance components is a novel measure of short-term variability in blood pressure.