Monotonic regression based on Bayesian P-splines: an application to estimating price response functions from store-level scanner data

Generalized additive models have become a widely used instrument for flexible regression analysis. In many practical situations, however, it is desirable to restrict the flexibility of nonparametric estimation in order to accommodate a presumed monotonic relationship between a covariate and the response variable. For example, consumers usually will buy less of a brand if its price increases, and therefore one expects a brand's unit sales to be a decreasing function in own price. We follow a Bayesian approach using penalized B-splines and incorporate the assumption of monotonicity in a natural way by an appropriate specification of the respective prior distributions. We illustrate the methodology in an empirical application modeling demand for a brand of orange juice and show that imposing monotonicity constraints for own- and cross-item price effects improves the predictive validity of the estimated sales response function considerably

Degrees of freedom and model selection in semiparametric additive monotone regression

The degrees of freedom of semiparametric additive monotone models are derived using results about projections onto sums of order cones. Two important related questions are also studied, namely, the de nition of estimators for the parameter of the error term and the formulation of speci c Akaike Information Criteria statistics. Several alternatives are proposed to solve both problems and simulation experiments are conducted to compare the behavior of the di erent candidates. A new selection criterion is proposed that combines the ability to guess the model but also the e ciency to estimate the variance parameter. Finally, the criterion is used to select the model in a regression problem from a well known data set.Ministerio de Ciencia e Innovación grant (MTM2012-37129

Semiparametric and nonparametric methods for evaluating risk prediction markers in case-control studies

The performance of a well calibrated risk model, Risk(Y)=P(D=1|Y), can be characterized by the population distribution of Risk(Y) and displayed with the predictiveness curve. Better performance is characterized by a wider distribution of Risk(Y), since this corresponds to better risk stratification in the sense that more subjects are identified at low and high risk for the outcome D=1. Although methods have been developed to estimate predictiveness curves from cohort studies, most studies to evaluate novel risk prediction markers employ case-control designs. Here we develop semiparametric and nonparametric methods that accommodate case-control data and assume apriori knowledge of P(D=1). Large and small sample properties are investigated. The semiparametric methods are flexible, substantially more efficient than the nonparametric counterparts and naturally generalize methods previously developed for cohort data. Applications to prostate cancer risk prediction markers illustrate the methods

Isotonic propensity score matching

We propose a one-to-many matching estimator of the average treatment effect based on propensity scores estimated by isotonic regression. The method relies on the monotonicity assumption on the propensity score function, which can be justified in many applications in economics. We show that the nature of the isotonic estimator can help us to fix many problems of existing matching methods, including efficiency, choice of the number of matches, choice of tuning parameters, robustness to propensity score misspecification, and bootstrap validity. As a by-product, a uniformly consistent isotonic estimator is developed for our proposed matching method

Generalized additive and index models with shape constraints

We study generalised additive models, with shape restrictions (e.g. monotonicity, convexity, concavity) imposed on each component of the additive prediction function. We show that this framework facilitates a nonparametric estimator of each additive component, obtained by maximising the likelihood. The procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals. More generally, our methodology can be applied to generalised additive index models. Here again, the procedure can be justified on theoretical grounds and, like the original algorithm, possesses highly competitive finite-sample performance. Practical utility is illustrated through the use of these methods in the analysis of two real datasets. Our algorithms are publicly available in the \texttt{R} package \textbf{scar}, short for \textbf{s}hape-\textbf{c}onstrained \textbf{a}dditive \textbf{r}egression.Both authors are supported by the second author’s Engineering and Physical Sciences Research Fellowship EP/J017213/1.This is the final version of the article. It first appeared from Wiley via http://dx.doi.org/10.1111/rssb.1213