K. Ruben Gabriel

A Simple Method of Multiple Comparisons of Means

Abstract A simple procedure for multiple comparisons of means is proposed for unequally sized samples. The procedure is attractive for its simplicity and for a graphical display which allows the experimenter to evaluate all comparisons of interest at a glance. The critical points used are those of the Studentized Maximum Modulus and two inequalities are involved: a conservative probability inequality and a radical algebraic inequality. Apparently, the joint effect of the two inequalities is conservative unless the imbalance in sample sizes is very large. For equally sized samples one can use critical points of the Studentized Range and the procedure becomes equivalent to Tukeys well-known T-method.

The Biplot as a Diagnostic Tool for Models of Two-Way Tables

The biplot graphical display of matrices is proposed as a data analytic tool for diagnosing the type of model to fit the data. Rows and columns of the data matrix are represented by markers—the diagnosis proceeds by examining these markers for collinearity. Thus, for example. if set I of row markers are collinear and a set J of column markers are also collinear. a Tukey degree-of-freedom-for-non-additivity is diagnosed for the submatrix of these I-rows and J-columns: if the two lines are at 90°. an additive model is diagnosed. Mandel type models are similarly diagnosed when either the row or the column markers are collinear but not both. Illustrations from physical science data show how simple biplot diagnostic modeling is and how many features of the data are revealed. Of particular interest are biplots of three-way layouts which allow visual inspection of main effects and interactions.

Alcohol Dependence and Domestic Violence as Sequelae of Abuse and Conduct Disorder in Childhood.

OBJECTIVES To examine in the Navajo population: (1) the importance of childhood abuse as a risk factor for conduct disorder; (2) the importance of each form of abuse and conduct disorder as risk factors for alcohol dependence; and (3) the relative importance of each form of abuse, conduct disorder, and alcohol dependence as risk factors for being a perpetrator and/or victim of domestic violence. METHOD The study is based on a case-control design. Cases (204 men and 148 women) between the ages of 21 and 65 were interviewed in alcohol treatment program and matched to community controls. There were two groups of controls: alcohol dependent (374 men, 60 women) and nonalcohol dependent (157 men, 143 women). When adjusted for stratification by age, community of residence, and sex, the combined control groups comprise a representative sample of the Navajo male and female population 21-65 years of age. RESULTS The prevalence of physical and sexual abuse before age 15 is within limits observed in other populations. Each form of abuse is a risk factor for conduct disorder. Along with conduct disorder, physical abuse is a risk factor for alcohol dependence. Physical abuse and alcohol dependence are independent risk factors for being involved in domestic violence as both perpetrator and victim. There appears to have been no secular trend in the incidence of childhood abuse over the past several generations, but there is suggestive evidence that domestic violence has become more common. CONCLUSIONS Physical abuse is a significant risk factor for alcohol dependence as well as for domestic violence independent of the effects of alcohol abuse. The effects of sexual abuse with regard to both domestic violence and alcohol dependence do not appear to be significant.

Closure of the Newman-Keuls Multiple Comparisons Procedure

Abstract Closure of the family of hypotheses tested by the Newman-Keuls procedure has been proposed in order to avoid excessive Type I errors. This proposal is examined in some detail, and a practical procedure for carrying it out is described and programmed in FORTRAN. The relation of such closure to other multiple comparison procedures is discussed, and its relatively high power is pointed out.

Analysis of Meteorological Data by Means of Canonical Decomposition and Biplots

Abstract The biplot is a graphical display of a two-dimensional approximation to a matrix. Its usefulness in the display and analysis of matrices of meteorological data is demonstrated by a detailed exposition of two illustrations based on Israeli rainfall. In the first illustration, the biplot for the sample matrix for monthly rainfall averages is shown to provide a visual display of patterns existing in the data. In the second illustration, data from a rainmaking experiment is the basis for the generation of a biplot which is a graphical approximation to simultaneous tests of a variety of sub-hypotheses in the multivariate analysis of variance (MA.NOVA) one-way layout. The biplot is useful whenever the two largest characteristic roots of the matrix times its transpose account for most of the variance. When this is the case, relationships and trends in the data are displayed which may be difficult to obtain by common analytic methods.

A permutation test of association between configurations by means of the RV coefficient

The RV coefficient is known to be suitable for measuring association between homologous configurations. The randomness of such association can be tested by permutational significance. In order to approximate the p-value, we derive the first two moments of the population permutation distribution of the RV coefficient. The permutational distributions of the RV coefficients are explored through several examples. Their permutation distributions appear to be markedly skewed to the right, regardless of the magnitudes of the observed RV coefficients herein. Log-transformation of the first two moments is suggested for a better approximation to the permutational p-value.

Multiple Comparisons by Rerandomization Tests

Abstract Classical methods of simultaneous inference cannot be used with rerandomization tests without risking excessive type I errors. This is illustrated by examples that show some published methods to be invalid for multiple comparisons. However, the use of stepwise procedures is shown to remain valid with rerandomization tests. In particular, this article advocates an adaptation of Begun and Gabriels (1981) technique and provides the algorithm for using it.

Ratio Statistics for Randomized Experiments in Precipitation Stimulation

Abstract A variety of ratio statistics has been used in the design and evaluation of weather modification experiments and their significance has usually been estimated by rerandomization. These ratios, and especially their logarithms, are asymptotically normal with null expectations and variances that can be readily calculated. This paper reviews and generalizes several useful ratio statistics and derives their variances. The variances presented here should make it easier for users of these ratios statistics in large experiments, 100 or more units, to assess significance without going through a large number of rerandomizations. It also shows how these formulas can be used to evaluate power and the required sample sizes. Some illustrations from Israel and from Puglia, Italy, are given.

The Second Israeli Rainfall Stimulation Experiment: Analysis of Precipitation on Both Targets

Abstract This paper presents the results of the second Israeli randomized rainfall experiment for both alternative targets (earlier analyses were confined to the north target). It shows that apparently significant positive seeding effects in the north were accompanied by equally strong apparently negative effects in the south. Various statistical breakdowns of the data are tried, but they do not allow an unequivocal conclusion as to whether there were real and opposite effects in the two areas, or the apparent negative effect was a chance occurrence, or both apparent effects were due to chance. A comparison with Israel I is made and possible explanations are given in a companion paper.

Temporal features in thunder days in the United States

Abstract1901–80 data for the contiguous U.S. show that secular variability of thunder days was very much less than that of precipitation or of frequency of extra tropical cyclones. Overall, there may have been a slight decline, but more evident was an increase to the thirties followed by a falling off, broken only by a peak in the seventies. These up-and-down movements were evident in most months of the year and regions of the U.S. The general decrease, however, was clear only in the South East and replaced by an increase in the Upper Great Lakes region. Secular variation in thunder day frequency was slightly correlated positively with that of extra tropical cyclone frequency and negatively with sea level pressure. The analysis also confirmed well known seasonal and regional patterns of thunder activity.