Self-modelling with Random Shift and Scale Parameters and a Free-knot Spline Shape Function

The shape invariant model is a semi-parametric approach to estimating a functional relationship from clustered data (multiple observations on each of a number of individuals). The common response curve shape over individuals is estimated by adjusting for individual scaling differences while pooling shape information. In practice, the common response curve is restricted to some flexible family of functions. This paper introduces the use of a free-knot spline shape function and reduces the number of parameters in the shape invariant model by assuming a random distribution on the parameters that control the individual scaling of the shape function. New graphical diagnostics are presented, parameter identifiability and estimation are discussed, and an example is presented.