Determining the dimension of the central subspace and central mean subspace

The central subspace and central mean subspace are two importanttargets of sufficient dimension reduction. We propose a weightedchi-squared test to determine their dimensions based on matriceswhose column spaces are exactly equal to the central subspaceor the central mean subspace. The asymptotic distribution ofthe test statistic is obtained. Simulation examples are usedto demonstrate the performance of this test.     

Dimension reduction strategies for analyzing global gene expression data with a response

The analysis of global gene expression data from microarrays is breaking new ground in genetics research, while confronting modelers and statisticians with many critical issues. In this paper, we consider data sets in which a categorical or continuous response is recorded, along with gene expression, on a given number of experimental samples. Data of this type are usually employed to create a prediction mechanism for the response based on gene expression, and to identify a subset of relevant genes. This defines a regression setting characterized by a dramatic under-resolution with respect to the predictors (genes), whose number exceeds by orders of magnitude the number of available observations (samples). We present a dimension reduction strategy that, under appropriate assumptions, allows us to restrict attention to a few linear combinations of the original expression profiles, and thus to overcome under-resolution. These linear combinations can then be used to build and validate a regression model with standard techniques. Moreover, they can be used to rank original predictors, and ultimately to select a subset of them through comparison with a background 'chance scenario' based on a number of independent randomizations. We apply this strategy to publicly available data on leukemia classification.     

Implications of influence function analysis for sliced inverse regression and sliced average variance estimation

Sliced inverse regression, sliced inverse regression II andsliced average variance estimation are three related dimension-reductionmethods that require relatively mild model assumptions. As anapproximation for the relative influence of single observationsfrom large samples, the influence function is used to comparethe sensitivity of the three methods to particular observationaltypes. The analysis carried out here helps to explain why thereis a lack of agreement concerning the preferability of thesedimension-reduction procedures in general. An efficient sampleversion of the influence function is also developed and evaluated.     

Optimal sufficient dimension reduction for the conditional mean in multivariate regression

The aim of this article is to develop optimal sufficient dimensionreduction methodology for the conditional mean in multivariateregression. The context is roughly the same as that of a relatedmethod by Cook & Setodji (2003), but the new method hasseveral advantages. It is asymptotically optimal in the sensedescribed herein and its test statistic for dimension alwayshas a chi-squared distribution asymptotically under the nullhypothesis. Additionally, the optimal method allows tests ofpredictor effects. A comparison of the two methods is provided.     

A confidence interval for the unknown value of the regressor with given expectation of the regressand in model I of linear regression

In the case of model I of linear regression there is derived a confidence interval for that x_o where the “true line” will reach a given value y_o. The interval can be given by the intersections between the line y = y_o and the hyperbolas providing pointwise confidence intervals of the expectations of y.     

Estimating Growth Reduction Dose for Polynomial Response Relationships

Mutation breeders often estimate an irradiation dose which causes a specified reduction in growth of a biological material. In this paper, we estimate the growth reduction dose (GRD) and its confidence interval, when the dose-response relationship is of a general polynomial form. We make a realistic assumption that the response at the control dose is a random variable. We compare the estimates, standard error and confidence intervals of GRD with those obtained under a situation where response at control dose is a known constant. We illustrate the procedure with data on the effect of gamma irradiation and Ethylmethane sulphate on shoot lengths of chickpea genotypes.