The computational aspects of multiple attribute sampling procedures

One method of controlling the quality of incoming lots is through attribute sampling. To simultaneously control several (possibly dependent) attributes, properly chosen single attribute sampling plans can be merged into a multiple attribute sampling plan. The general form of such a plan is given and various alternatives are discussed. The multinomial distribution is used to develop formulae necessary for an analysis of a multiple attribute plan. Due to the lengthy nature of the calculations involved, a computer algorithm is outlined.     

On the use of curve fitting to model the error of the Cornish-Fisher expansion of the pearson type-VI distribution

The Cornish-Fisher expansion of the Pearson type VI distribution is known to be reasonably accurate when both degrees of freedom are relatively large (say greater than or equal to 5). However, when either or both degrees of freedom are less than 5, the accuracy of the computed percentage point begins to suffer; in some cases severely. To correct for this, the error surface in the degrees of freedom plane is modeled by least squares curve fitting for selected levels of tail probability (.025, .05, and .10) which can be used to adjust the percentage point obtained from the usual Cornish-Fisher expansion. This adjustment procedure produces a computing algorithm that computes percentage points of the Pearson type VI distribution at the above probability levels, accurate to at least + 1 in 3 digits in approximately 11 milliseconds per subroutine call on an IBM 370/145. This adjusted routine is valid for both degrees of freedom greater than or equal to 1.     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

Gaining insight with recursive partitioning of generalized linear models

Recursive partitioning algorithms separate a feature space into a set of disjoint rectangles. Then, usually, a constant in every partition is fitted. While this is a simple and intuitive approach, it may still lack interpretability as to how a specific relationship between dependent and independent variables may look. Or it may be that a certain model is assumed or of interest and there is a number of candidate variables that may non-linearly give rise to different model parameter values. We present an approach that combines generalized linear models (GLM) with recursive partitioning that offers enhanced interpretability of classical trees as well as providing an explorative way to assess a candidate variable's influence on a parametric model. This method conducts recursive partitioning of a GLM by (1) fitting the model to the data set, (2) testing for parameter instability over a set of partitioning variables, (3) splitting the data set with respect to the variable associated with the highest instability. The outcome is a tree where each terminal node is associated with a GLM. We will show the method's versatility and suitability to gain additional insight into the relationship of dependent and independent variables by two examples, modelling voting behaviour and a failure model for debt amortization, and compare it to alternative approaches.     

Confidence Intervals for a Binomial Parameter after Observing No Successes

The easily computed, one-sided confidence interval for the binomial parameter provides the basis for an interesting classroom example of scientific thinking and its relationship to confidence intervals. The upper limit can be represented as the sample proportion from a number of “successes” in a future experiment of the same sample size. The upper limit reported by most people corresponds closely to that producing a 95 percent classical confidence interval and has a Bayesian interpretation.     

Use of the Correlation Coefficient with Normal Probability Plots

The use of the correlation coefficient is suggested as a technique for summarizing and objectively evaluating the information contained in probability plots. Goodness-of-fit tests are constructed using this technique for several commonly used plotting positions for the normal distribution. Empirical sampling methods are used to construct the null distribution for these tests, which are then compared on the basis of power against certain nonnormal alternatives. Commonly used regression tests of fit are also included in the comparisons. The results indicate that use of the plotting position p_i = (i - .375)/(n + .25) yields a competitive regression test of fit for normality.     

On the Use of a Log-Rate Model for Survey-Weighted Categorical Data

For the analysis of survey-weighted categorical data, one recommended method of analysis is a log-rate model. For each cell in a contingency table, the survey weights are averaged across subjects and incorporated into an offset for a loglinear model. Supposedly, one can then proceed with the analysis of unweighted observed cell counts. We provide theoretical and simulation-based evidence to show that the log-rate analysis is not an effective statistical analysis method and should not be used in general. The root of the problem is in its failure to properly account for variability in the individual weights within cells of a contingency table. This results in goodness-of-fit tests that have higher-than-nominal error rates and confidence intervals for odds ratios that have lower-than-nominal coverage.     

Generalized Laplacian Distributions and Autoregressive Processes

Generalized Laplacian distribution is considered. A new distribution called geometric generalized Laplacian distribution is introduced and its properties are studied. First- and higher-order autoregressive processes with these stationary marginal distributions are developed and studied. Simulation studies are conducted and trajectories of the process are obtained for selected values of the parameters. Various areas of application of these models are discussed.     

Analyzing Binomial Data in a Split-Plot Design: Classical Approach or Modern Techniques?

Modeling data that are non-normally distributed with random effects is the major challenge in analyzing binomial data in split-plot designs. Seven methods for analyzing such data using mixed, generalized linear, or generalized linear mixed models are compared for the size and power of the tests. This study shows that analyzing random effects properly is more important than adjusting the analysis for non-normality. Methods based on mixed and generalized linear mixed models hold Type I error rates better than generalized linear models. Mixed model methods tend to have higher power than generalized linear mixed models when the sample size is small.