A geometric interpretation of Mallows' Cp statistic and an alternative plot in variable selection

Mallows' Cp plot is a useful tool for variable selection in linear regression. Though not as popular as the Cp plot, Spjotvoll's Fp and Pp plots are also used in the variable selection procedure. The Cp, Fp and Pp plots are useful in their own right. If the interest is the direct measure of the amount of bias of the submodels and a distributional assumption is not made about the error term, a Cp or Fp plot is used. If a formal testing procedure is to be performed, then a Pp plot is employed. A geometrical approach is used in order to propose an alternative plot that unifies all the information in these three plots, and that has some advantages over them. A Mathematica package has been written to implement the approach.

Visual, Motor, and Visual-Motor Integration Difficulties in Students with Autism Spectrum Disorders

Autism spectrum disorders (ASDs) affect 1 in every 88 U.S. children. ASDs have been described as neurological and developmental disorders impacting visual, motor, and visual-motor integration (VMI) abilities that affect academic achievement (CDC, 2010). Forty-five participants (22 ASD and 23 Typically Developing [TD]) 8 to 14 years old completed the Bender-Gestalt Test, Second Edition (BG II), Beery-Buktenica Developmental Test of Visual-Motor Integration, 5th Edition (VMI-V), NEPSY Second Edition (NEPSY-II), Test of Visual Perceptual Skills-3 (TVPS-3), Navon Task, Kaufman Test of Educational Achievement, Second Edition, Kaufman Brief Intelligence Test, Second Edition, Behavior Assessment Scale for Children, Second Edition, and Autism Spectrum Screening Questionnaire. Three hypotheses examined whether students with ASDs were more likely than TD peers to have: (1) a visual processing bias; (2) fine motor difficulties; and (3) VMI difficulties. Additional hypotheses analyzed the relationship between (4) local processing bias and fine motor difficulties on VMI ability and (5) local processing bias, fine motor difficulties, and VMI difficulties on academic achievement. A series oft-tests indicated the TVPS-3 (p=.72), Navon Task (p= .78), BG-II (p = .39), and VMI-V (p = .14) were not significantly different between groups. Students with ASDs demonstrated increased difficulty compared to TD students on the NEPSY-II (p = .01) and slower completion time on the Navon Task (p = .01). Regression analyses for VMI indicated the best predictors for the BG-II (p \u3c .001) were the TVPS-3 and Navon Completion Time; the best predictor for the VMI-V (p\u3c .001) was the TVPS-3. Regression analyses indicated that VMI-V predicted all domains of academic achievement. In addition to VMI-V, fine motor skills related to writing achievement, and BG-II related to math achievement. Based on the results, the speed of processing plays an important role on VMI skills and academic achievement, more so than the local processing bias. Although this study may have been impacted by homogeneity in the participants, it investigates a relationship between visual processing biases, fine motor difficulties, visual-motor integration and academic achievement that has received little attention in the literature. Findings can inform the development of more effective interventions for academic functioning for students with ASDs

Improved monotone polynomial fitting with applications and variable selection

We investigate existing and new isotonic parameterisations for monotone polynomials, the latter which have been previously unconsidered in the statistical literature. We show that this new parameterisation is faster and more flexible than its alternatives enabling polynomials to be constrained to be monotone over either a compact interval or a semi-compact interval of the form [a;∞), in addition to over the whole real line. Due to the speed and efficiency of algorithms based on our new parameterisation the use of standard bootstrap methodology becomes feasible. We investigate the use of the bootstrap under monotonicity constraints to obtain confidence and prediction bands for the fitted curves and show that an adjustment by using either the ‘m out of n’ bootstrap or a post hoc symmetrisation of the confidence bands is necessary to achieve more uniform coverage probabilities. However, the same such adjustments appear unwarranted for prediction bands. Furthermore, we examine the model selection problem, not only for monotone polynomials, but also in a general sense, with a focus on graphical methods. Specifically, we describe how to visualize measures of description loss and of model complexity to facilitate the model selection problem. We advocate the use of the bootstrap to assess the stability of selected models and to enhance our graphical tools and demonstrate which variables are important using variable inclusion plots, showing that these can be invaluable plots for the model building process. We also describe methods for using the ‘m out of n’ bootstrap to select the degree of the fitted monotone polynomial and demonstrate it’s effectiveness in the specific constrained regression scenario. We demonstrate the effectiveness of all of these methods using numerous case studies, which highlight the necessity and usefulness of our techniques. All algorithms discussed in this thesis are available in the R package MonoPoly (version 0.3-6 or later)

A quantitative assessment of the impacts of water status and chemical bioassays on structural attributes of Eucalyptus clones and plantation soil nitrogen using hyperspectral data.

Ph.D. University of KwaZulu-Natal, Durban 2012.Abstract available in PDF file