Bad Probability,Good Statistics,and Group Testing for Binomial Estimation

This article examines some improperly stated but often used textbook probability problems. Moving from a probabilistic to a statistical setting provides insight into group testing (i.e., observing only whether one or more of a group responds and not the response of each individual). Exact methods are used to construct tables showing (i) that group testing n times to estimate p can be more efficient than n individual tests even for small n and large p, (ii) optimal grouping strategies for various (n, p) combinations, and (iii) the efficiencies and biases achieved.     

In Search of the Optimum Harvey Wallbanger Recipe via Mixture Experiment Techniques

A Harvey Wallbanger is an alcoholic drink composed of three ingredients: orange juice, vodka, and Galliano. One of the authors (Sahrmann), a connoisseur of the libation, was perplexed upon discovering that different bartender handbooks listed different recipes for the drink. Determined to shed some light on this irksome problem, he issued invitations for an evening workshop session (party) to former and current attendees of an experimental-design training course taught within his company. A mixture experiment was designed and data collected using the class participants. This article describes the experimental design and the analysis of the data. The results of the experiment indicate that reducing the proportion of vodka produces better-flavored drinks. The results of the experiment were not entirely conclusive, however, so recommendations are made for future experimentation.     

LogXact 4.1 for Windows

Box's paper helicopter has been used to teach experimental design for more than a decade. It is simple, inexpensive, and provides real data for an involved, multifactor experiment. Unfortunately it can also further an all-too-common practice that Professor Box himself has repeatedly cautioned against, namely ignoring the fundamental science while rushing to solve problems that may not be sufficiently understood. Often this slighting of the science so as to get on with the statistics is justified by referring to Box's oft-quoted maxim that “All models are wrong, however some are useful.” Nevertheless, what is equally true, to paraphrase both Professor Box and George Orwell, is that “All models are wrong, but some are more wrong than others.” To experiment effectively it is necessary to understand the relevant science so as to distinguish between what is usefully wrong, and what is dangerously wrong.

This article presents an improved analysis of Box's helicopter problem relying on statistical and engineering knowledge and shows that this leads to an enhanced paper helicopter, requiring fewer experimental trails and achieving superior performance. In fact, of the 20 experimental trials run for validation—10 each of the proposed aerodynamic design and the conventional full factorial optimum—the longest 10 flight times all belong to the aerodynamic optimum, while the shortest 10 all belong to the conventional full factorial optimum. I further discuss how ancillary engineering knowledge can be incorporated into thinking about—and teaching—experimental design.     

Quantile plots for mean effects in the presence of variance effects for 2k-p designs

In robust parameter design, variance effects and mean effects in a factorial experiment are modelled simultaneously. If variance effects are present in a model, correlations are induced among the naive estimators of the mean effects. A simple normal quantile plot of the mean effects may be misleading because the mean effects are no longer iid under the null hypothesis that they are zero. Adjusted quantiles are computed for the case when one variance effect is significant and examples of 8-run and 16-run fractional factorial designs are examined in detail. We find that the usual normal quantiles are similar to adjusted quantiles for all but the largest and smallest ordered effects for which they are conservative. Graphically, the qualitative difference between the two sets of quantiles is negligible (even in the presence of large variance effects) and we conclude that normal probability plots are robust in the presence of variance effects.     

Change-Point Estimation in Long Memory Nonparametric Models with Applications

We consider the estimation of a change point or discontinuity in a regression function for random design model with long memory errors. We provide several change-point estimators and investigate the consistency of the estimators. Using the fractional ARIMA process as an example of long memory process, we report a small Monte Carlo experiment to compare the performance of the estimators in finite samples. We finish by applying the method to a climatological data example.     

Double Sampling Designs in Multivariate Linear Models with Missing Variables

We provide a method for finding the optimal double sampling plan for estimating the mean value of a continuous outcome. It is assumed that the fallible and true outcome data are related by a multivariate linear regression model where only some of the explanatory variables are sampled. Conditions under which double sampling is preferred over standard sampling plans are determined. An application of the method to a well-known data set on air pollution is presented.     

Sequential Designs for Binary Data with the Purpose to Maximize the Probability of Response

Two kinds of sequential designs are proposed for finding the point that maximizes the probability of response assuming a binary response variable and a quadratic logistic regression model. One is a parametric optimal design approach, and the other one is a nonparametric stochastic approximation approach. The suggested sequential designs are evaluated and compared in a simulation study. In summary, the parametric approach performed very well whereas its competitor failed in some cases.     

Evaluating the Effective Degrees of Freedom of the Delete-a-Group Jackknife

The delete-a-group jackknife is sometimes used when estimating the variances of statistics based on a large sample. We investigate heavily poststratified estimators for a population mean and a simple regression coefficient, where both full-sample and domain estimates are of interest. The delete-a-group (DAG) jackknife employing 30, 60, and 100 replicates is found to be highly unstable, even for large sample sizes. The empirical degrees of freedom of these DAG jackknives are usually much less than their nominal degrees of freedom. This analysis calls into question whether coverage intervals derived from replication-based variance estimators can be trusted for highly calibrated estimates.     

A generating algorithm for linear trend-free and nearly linear trend-free block designs

The theory and properties of trend-free (TF) and nearly trend-free (NTF) block designs are wel1 developed. Applications have been hampered because a methodology for design construction has not been available.

This article begins with a short review of concepts and properties of TF and NTF block designs. The major contribution is provision of an algorithm for the construction of linear and nearly linear TF block designs. The algorithm is incorporated in a computer program in FORTRAN 77 provided in an appendix for the IBM PC or compatible microcomputer, a program adaptable also to other computers. Three sets of block designs generated by the program are given as examples.

A numerical example of analysis of a linear trend-free balanced incomplete block design is provided.