Comparing multiple process overall yields from multiple manufacturing lines

A manufacturing process often uses multiple manufacturing lines to produce the same product for the productivity purpose. The capabilities of multiple manufacturing lines are often non-identical in practice. The task of process evaluation needs to measure the overall yield. This research compares the overall yield among I processes from K manufacturing lines, I ? 2, K ? 2. An unconstrained multiple comparisons with the best (UMCB) method based on the overall yield index is proposed to determine the best process with the highest overall yield. The simultaneous confidence intervals of each difference from the unknown best process in terms of the overall yield index are provided at a confidence level of at least 1 ?α. To demonstrate the effectiveness of the UMCB method, the method was applied to choose the best process producing the pre-amplifier of microelectromechanical systems (MEMS) sensors in the production environment with multiple manufacturing lines.     

Detecting multiple confounders

This paper proposes an approach for detecting multiple confounders which combines the advantages of two causal models, the potential outcome model and the causal diagram. The approach need not use a complete causal diagram as long as it is known that a known covariate set Z$Z$ contains the parent set of the exposure E  . On the other hand, whether a covariate is or not a confounder may depend on its categorization. We introduce uniform non-confounding which implies non-confounding in any subpopulation defined by the interval of a covariate (or any pooled level for a discrete covariate). We show that the conditions in Miettinen and Cook's criteria for non-confounding also imply uniform non-confounding. Further we present an algorithm for deleting non-confounders from the potential confounder set Z$Z$, which extends Greenland et al.'s [1999a. Causal diagrams for epidemiologic research. Epidemiology 10, 37–48] approach by splitting Z$Z$ into a series of potential confounder subsets. We also discuss conditions for non-confounding bias in the subpopulations in which we are interested, where the subpopulations may be defined by non-confounders.     

Robust multiple comparisons

The use of several robust estimators of location with their associated variance estimates in a modified T-method for pairwise multiple comparisons between treatment means was compared with the sample mean and variance and with the k-sample rank sum test. The methods were compared with respect to the stability of their experimentwise error rates under a variety of non-normal situations (robustness of validity) and their average confidence interval lengths (robustness of efficiency).     

Testing multiple slippages

A class of invariant Bayes rules is derived for testing homogeneity of k (≥2) different populations against (^k_t) slippage alternatives that some (unknown) subset of size t of the given populations has parameter larger than the remaining k-t, where t is a given integer between 1 and k-1. For a similar problem in nonparametric situations, locally best tests based on ranks are derived.     

Inferences on missing information under multiple imputation and two-stage multiple imputation

In the presence of missing values, researchers may be interested in the rates of missing information. The rates of missing information are (a) important for assessing how the missing information contributes to inferential uncertainty about, Q, the population quantity of interest, (b) are an important component in the decision of the number of imputations, and (c) can be used to test model uncertainty and model fitting. In this article I will derive the asymptotic distribution of the rates of missing information in two scenarios: the conventional multiple imputation (MI), and the two-stage MI. Numerically I will show that the proposed asymptotic distribution agrees with the simulated one. I will also suggest the number of imputations needed to obtain reliable missing information rate estimates for each method, based on the asymptotic distribution.     

MIMCA: multiple imputation for categorical variables with multiple correspondence analysis

We propose a multiple imputation method to deal with incomplete categorical data. This method imputes the missing entries using the principal component method dedicated to categorical data: multiple correspondence analysis (MCA). The uncertainty concerning the parameters of the imputation model is reflected using a non-parametric bootstrap. Multiple imputation using MCA (MIMCA) requires estimating a small number of parameters due to the dimensionality reduction property of MCA. It allows the user to impute a large range of data sets. In particular, a high number of categories per variable, a high number of variables or a small number of individuals are not an issue for MIMCA. Through a simulation study based on real data sets, the method is assessed and compared to the reference methods (multiple imputation using the loglinear model, multiple imputation by logistic regressions) as well to the latest works on the topic (multiple imputation by random forests or by the Dirichlet process mixture of products of multinomial distributions model). The proposed method provides a good point estimate of the parameters of the analysis model considered, such as the coefficients of a main effects logistic regression model, and a reliable estimate of the variability of the estimators. In addition, MIMCA has the great advantage that it is substantially less time consuming on data sets of high dimensions than the other multiple imputation methods.     

Nonparametric multiple testing procedures

Assume that we have n_i independent observations from each of k independent populations. Each population has the same distribution except for a translation parameter. We are interested in specific pairwise differences of the parameters in various settings, such as treatment vs. control, change point or all pairwise differences. We propose new multiple testing procedures for the pairwise differences. The new procedures are based on ranks and they have desirable practical properties not shared by existing procedures. These include tests that satisfy the interval property. Furthermore, the test method provides an interval that serves as an estimate of the difference in the parameters of interest.     

Rank-based multiple change-point detection

Abstract

A nonparametric procedure is proposed to estimate multiple change-points of location changes in a univariate data sequence by using ranks instead of the raw data. While existing rank-based multiple change-point detection methods are mostly based on sequential tests, we treat it as a model selection problem. We derive the corresponding Schwarz’s information criterion for rank-statistics, theoretically prove the consistency of the change-point estimator and use a pruned dynamic programing algorithm to achieve the change-point estimator. Simulation studies show our method’s robustness, effectiveness and efficiency in detecting mean-changes. We also apply the method to a gene dataset as an illustration.     

Bootstrap multiple test procedure

A method for multiple comparisons is presented. It is based on the bootstrap and takes care of the logical structure among the hypotheses to be tested. The test procedure is very general according to number of hypotheses, sample distribution and number of observations.     

Inferring multiple graphical structures

Gaussian Graphical Models provide a convenient framework for representing dependencies between variables. Recently, this tool has received a high interest for the discovery of biological networks. The literature focuses on the case where a single network is inferred from a set of measurements. But, as wetlab data is typically scarce, several assays, where the experimental conditions affect interactions, are usually merged to infer a single network. In this paper, we propose two approaches for estimating multiple related graphs, by rendering the closeness assumption into an empirical prior or group penalties. We provide quantitative results demonstrating the benefits of the proposed approaches. The methods presented in this paper are embeded in the R package simone from version 1.0-0 and later.     

A multiple warm standby δ-shock system with a repairman having multiple vacations

In this article, a warm standby n-unit system is studied. The system is operational as long as there is one unit normal. The unit online, which has a lifetime distribution governed by a phase-type distribution, is also attacked by a shock from some external causes. Assume that shocks arrive according to a Poisson process. Whenever an interarrival time of shock is less than a threshold, the unit online fails. The lifetimes of the units in warm standby is exponentially distributed. A repairman who can take multiple vacations repairs the failed units based on the “first-in-first-out” rule. The repair times and the vacation times of repairman are governed by different phase-type distributions. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime; the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated. A numerical application is performed to illustrate the calculations.     

Joint impact of multiple observations on a subset of variables in multiple linear regression

In multiple linear regression analysis, each observation affects the fitted regression equation differently and has varying influences on the regression coefficients of the different variables. Chatterjee & Hadi (1988) have proposed some measures such as DSSE_ij (Impact on Residual Sum of Squares of simultaneously omitting the ith observation and the jth variable), F_j (Partial F-test for the jth variable) and F_j(i) (Partial F-test for the jth variable omitting the ith observation) to show the joint impact and the interrelationship that exists among a variable and an observation. In this paper we have proposed more extended form of those measures DSSE_IJ, F_J and F_J(I) to deal with the interrelationships that exist among the multiple observations and a subset of variables by monitoring the effects of the simultaneous omission of multiple variables and multiple observations.     

Interpolation models with multiple hyperparameters

A traditional interpolation model is characterized by the choice of regularizer applied to the interpolant, and the choice of noise model. Typically, the regularizer has a single regularization constant , and the noise model has a single parameter . The ratio / alone is responsible for determining globally all these attributes of the interpolant: its complexity, flexibility, smoothness, characteristic scale length, and characteristic amplitude. We suggest that interpolation models should be able to capture more than just one flavour of simplicity and complexity. We describe Bayesian models in which the interpolant has a smoothness that varies spatially. We emphasize the importance, in practical implementation, of the concept of conditional convexity when designing models with many hyperparameters. We apply the new models to the interpolation of neuronal spike data and demonstrate a substantial improvement in generalization error.     

Extensions of multiple linear regression

Many statistical methods are linked together through their connection with weighted least squares and hence regression. This article reviews these connections, emphasising the iteratively weighted least squares algorithm.     

Fitting a multiple regression function

Consider the p-dimensional unit cube [0,1]^p, p≥1. Partition [0, 1]^p into n regions, R_1,n,…,R_n,n such that the volume Δ(R_j,n) is of order n^?1,j=1,…,n. Select and fix a point in each of these regions so that we have x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_n. Suppose that associated with the j-th predictor vector x⁽ⁿ⁾_j there is an observable variable Y⁽ⁿ⁾_j, j=1,…,n, satisfying the multiple regression model $\text{Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{=g(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{)+e}{}^{\mathrm{(n)}}{}_{\mathrm{j}}$, where g is an unknown function defined on [0, 1]^pand {e⁽ⁿ⁾_j} are independent identically distributed random variables with Ee⁽ⁿ⁾₁=0 and Var e⁽ⁿ⁾₁=σ²<∞. This paper proposes as an estimator of g(x), where k(u) is a known p-dimensional bounded density and {a_n} is a sequence of reals converging to 0 asn→∞. Weak and strong consistency of g_n(x) and rates of convergence are obtained. Asymptoticnormality of the estimator is established. Also proposed is $\text{σ}{}^2{}_{\mathrm{n}}\text{=n}{}^{\mathrm{?1}}\text{Σ}{}^{\mathrm{n}}{}_{\mathrm{j=1}}\text{(Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{?g}{}_{\mathrm{n}}\text{(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{))}{}^2$ as a consistent estimate of σ².     

A multiple model discrimination procedure

The case of selecting between a set of fixed models is considered. The true model is assumed to be contained in the set of proposed models and errors are taken to be normally distributed. A sequential procedure which yeilds probabilities of incorrect selections is proposed. The procedure is shown to have optimal properties and is extended to the estimated model case by a bootstrap procedure.     

Randomization tests for multiple regression

Four generic means of conducting randomization tests in the context of multiple regression are analysed. Based on their performance in traditional repeated samples, three of these are shown to be inappropriate or applicable only in special circumstances; their shortcomings are illustrated via Monte Carlo studies     

Intrinsic autoregressions at multiple resolutions

We consider intrinsic autoregression models at multiple resolutions. Firstly, we describe a method to construct a class of approximately coherent Markov random fields (MRF) at different scales, overcoming the problem that the marginal Gaussian MRF is not, in general, a MRF with respect to any non-trivial neighbourhood structure. This is based on the approximation of non-Markov Gaussian fields as Gaussian MRFs and is optimal according to different theoretic notions such as Kullback–Leibler divergence. We extend the method to intrinsic autoregressions providing a novel multi-resolution framework.     

On multiple constraint skewed models

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.