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We consider inference for the parameters of a linear model when the covariates are random and the relationship between response and covariates is possibly non-linear. Conventional inference methods such as z intervals perform poorly in these cases. We propose a double bootstrap-based calibrated percentile method, perc-cal, as a general-purpose CI method which performs very well relative to alternative methods in challenging situations such as these. The superior performance of perc-cal is demonstrated by a thorough, full-factorial design synthetic data study as well as a data example involving the length of criminal sentences. We also provide theoretical justification for the perc-cal method under mild conditions. The method is implemented in the R package "perccal", available through CRAN and coded primarily in C++, to make it easier for practitioners to use. 

Given data obtained under two sampling conditions, it is often of interest to identify variables that behave differently in one condition than in the other. We introduce a method for differential analysis of second-order behavior called Differential Correlation Mining (DCM). The DCM method identifies differentially correlated sets of variables, with the property that the average pairwise correlation between variables in a set is higher under one sample condition than the other. DCM is based on an iterative search procedure that adaptively updates the size and elements of a candidate variable set. Updates are performed via hypothesis testing of individual variables, based on the asymptotic distribution of their average differential correlation. We investigate the performance of DCM by applying it to simulated data as well as to recent experimental datasets in genomics and brain imaging. 

The Cusp Catastrophe Model provides a promising approach for health and behavioral researchers to investigate both continuous and quantum changes in one modeling framework. However, application of the model is hindered by unresolved issues around a statistical model fitting to the data. This paper reports our exploratory work in developing a new approach to statistical cusp catastrophe modeling. In this new approach, the Cusp Catastrophe Model is cast into a statistical nonlinear regression for parameter estimation. The algorithms of the delayed convention and Maxwell convention are applied to obtain parameter estimates using maximum likelihood estimation. Through a series of simulation studies, we demonstrate that (a) parameter estimation of this statistical cusp model is unbiased, and (b) use of a bootstrapping procedure enables efficient statistical inference. To test the utility of this new method, we analyze survey data collected for an NIH-funded project providing HIV-prevention education to adolescents in the Bahamas. We found that the results can be more reasonably explained by our approach than other existing methods. Additional research is needed to establish this new approach as the most reliable method for fitting the cusp catastrophe model. Further research should focus on additional theoretical analysis, extension of the model for analyzing categorical and counting data, and additional applications in analyzing different data types.