NONPARAMETRIC AND SEMIPARAMETRIC MODELS IN COMPARISON OF OBSERVATIONS OF A PARTICLE-SIZE DISTRIBUTION(Statistical Models for Biomedical Research)

Testing hypotheses about pairs of unnormalized histograms motivates this paper. The histograms contain particle counts for particle-size intervals. The analysis involves generalized-linear-model fitting of cubic splines with irregularly-spaced knots. Of interest is testing the null hypothesis that two sets of particle counts correspond to intensity functions that differ only by a scale factor and a constant shift in horizontal registration. An unknown smooth function is common to the two intensities. The alternative hypothesis is that in addition, the difference between the two intensities is also an unknown smooth function. We consider three approaches to knot placement. First is specification of so many knots that adequate representations of the unknown functions cannot be doubted. Second is data-driven choice of knots. Third is choice of knots based on prior knowledge of what intensity differences are plausible. For the data at hand, we show that specification of too many knots leads to tests with too little power and that data-driven knot selection can lead to false rejection of the null hypothesis. The data at hand seem to call for use of prior knowledge to construct a semiparametric model that incorporates the distinction between the two hypotheses in the parametric part.