Decision and choice in transforming data

Statistics are developed for predicting the effect of data transformations on the F statistic when the assumptions of homoscedasticity and normality underlying the AN OVA are not necessarily satisfied. These statistics are useful for determining whether and how to transform, They are developed by partitioning the change in the observed value of the jF-statistic under the transformation, into two expressions, one of which depends on the "truth" of HQ while the other does not. Using this partition, desirable properties are derived for transformations. Criteria are developed defining transformations which tend to preserve the type 1 error while increasing power when needed. Using these criteria, the notion of model robustness is introduced. It is shown that the Box-Cox methodology for selecting a power transform may, under certain conditions, produce a transformation which does not permit inferences to be made about the parent population from the transformed population. An alternative approach suggested here does permit such inferences.     

Recursive location and scale estimators

It is shown that a recursive estimator with the same asymptotic properties as the median has convergence properties in finite samples which depend heavily on the scale of the data. A simple modification which adjusts for the scale is suggested and its application illustrated on simulated data. The modified estimator has much improved properties which are similar to those of the sample (non-recursive) median.     

Some sampling effects of pairwise correlated observations on likelihood ratio tests for the difference between two means

We study the effects of the inclusion of pairs of correlated observations in a sample on likelihood ratio tests for the difference in two means. In particular, we assess how the inclusion of correlated data pairs (e.g., such as data inadvertently collected from sib-pairs) affects the sample size requirements necessary for the implementation of a Likelihood Ratio (LR) test for the difference between two means. Our results suggest that correlated data pairs beneficially or adversely effect sample size requirements for an LR test to a degree functionally related to the mixture parameters dictating their relative frequencies in the larger sample on which the test will be performed, the strength of the correlation between the observations, and the size of imbalances in the sample with respect to the number of observations in each group. The relevance and implications of the results for genetic and epidemiologic research are discussed.     

The asymptotic distribution of a pivotal quantity for a capture-recapture model

A pivotal quantity for a capture-recapture model is introduced and used to construct an asymptotic confidence region for (ε,N), where ε is the capture efficiency and N is the population size. The true confidence levels of certain regions are obtained by simulation. Certain confidence regions for (ε,N) are drawn to show the size of the regions and to show how confidence limits for N depend on ε.     

Practical bayesian approaches to the analystcs of a 2×2 contingency table with incompletely categorized data

To assess the efficacy of a treatment, patients are administered a pre-test, the treatment, and a post-test (identical to the pre-test). These patients are then categorized according to their outcomes observed on both tests,e.g., (S,S), (S,F), etc. Also, we observe "incomplete" information on the pre-tests' outcomes for some patients and the results of only the post-test being known for thers, A Bayesian framework is fit to the problem and Bayes factors, posterior odds ratios, and utility functions are given to evaluate th e treatment, A method of assessing the prior distribution is specified and a numerical example is worked.     

Continuous outcomes stochastically mismeasured as discrete random variables: a maximum likelihood approach

In an economic model of retirement behavior, a continuous dependent variable was required; the variable could only be estimated discretely with error, however. Parameter estimates using this dependent variable and ordinary least squares regression are inefficient. In th is paper, we develop a maximum likelihood procedure which adjusts for this heteroscedasticity.     

A comparison of random balance and two-stage group screening designs: A case study

In this paper we focus on the problem of supersaturated (fewer runs than factors) screening experiments. We consider two major types of designs which have been proposed in this situ¬ation: random balance and two-stage group screening. We discuss the relative merits and demerits of each strategy. In addition, we compare the performance of these strategies by means of a case study in which 100 factors are screened in 20,42,62, and 84 runs.     

Sample size determinations when two binomial proportions are very small

Sample size determination for testing the hypothesis of equality of proportions with a specified type I and type I1 error probabilitiesis of ten based on normal approximation to the binomial distribution. When the proportionsinvolved are very small, the exact distribution of the test statistic may not follow the assumed distribution. Consequently, the sample size determined by the test statistic may not result in the sespecifiederror probabilities. In this paper the author proposes a square root formula and compares it with several existing sample size approximation methods. It is found that with small proportion (p≦.01) the squar eroot formula provides the closest approximation to the exact sample sizes which attain a specified type I and type II error probabilities. Thes quare root formula is simple inform and has the advantage that equal differencesare equally detectable.     

Objective Bayesian Analysis of an Exponential Regression Model with Constrained Parameters Applied to Animal Digestibility

In animal digestibility the proportion of degraded food along the time has usually been modeled as a normal random variable with mean a function of the time and the following three parameters: the proportion of degraded food almost instantaneously, remaining proportion of food to be degraded, and velocity of degradation. The estimation of these parameters has been carried out mainly from a frequentist viewpoint by using the asymptotic distribution of the maximum likelihood estimator. This may give inadmissible estimates, such as values outside of the range of the parameters. This drawback could not appear if a Bayesian approach were adopted. In this article an objective Bayesian analysis is developed and illustrated on real and simulated data.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.