Robert E. Odeh

Sample size choice : charts for experiments with linear models

A guide to testing statistical hypotheses for readers familiar with the Neyman-Pearson theory of hypothesis testing including the notion of power, the general linear hypothesis (multiple regression) problem, and the special case of analysis of variance. The second edition (date of first not mentione

On Jonckheere's k-Sample Test Against Ordered Alternatives

In this paper a k-sample non-parametric test for trend is considered. Given a sample of size ni , i = 1, …, k respectively from each of k populations, the test rejects the hypothesis that the k populations are identical if S = Σ k i=2 Si ≥ Si . Here Si is the Mann-Whitney statistic computed when each observation in the i-th sample is compared with the combined observations from the first (i – 1) populations. A recurrence formula is derived for computing the exact distribution of S. Tables of exact probabilities and critical values are given for nominal values of α = 0.5, 0.2, 0.1, 0.05, 0.025, 0.01, and 0.005 for k = 3 and all possible sample sizes from 2 to 8, and for equal sample sizes for values of n = 2(1)6, k = 4(1)6.

Tables of two-sided tolerance factors for a normal distribution

Given a random sample of size N from a normal distribution, we consider tolerance intervals of the form X − ks to X + ks, where X is the sample mean and s is the sample standard deviation. The value of k is chosen so that the interval covers a given proportion P of the population with confidence γ. Exact values of k, computed from numerical integration, are given for N = 2(1)100; P = 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.999; and γ = 0.5, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995. The exact values are compared with the values obtained from an approximation developed by Wald and Wolfowitz (1946).

Two-sided prediction intervals to contain at least k out of m future observations from a normal distribution

Prediction intervals are frequently used when it is necessary to use past data to make a statistical statement about a small number of future observations. Given the sample mean and the sample standard deviation s. computed from a random sample of size n from a normal distribution, this article provides factors r = r(k, m, n; γ) such that the two-sided prediction interval ± rs will contain at least k out of m future observations with (1ooγ)% confidence. The original n observations and the additional m observations are assumed to be independent samples from the same normal population. Values of r are given for γ = .90, .95, .99; n = 8(1)12, 1.5, 20, 25, 30, 40, 60, 120, 240, 480, X m = l(l)9 for k = m; m = 10, 15, 20(10)60, 80, 100, for k = m – j, where j = 0(l)8.

Extended tables of the distribution of friedman’ s s-statistic in the two-way layout

Friedman’s (1937, 1940) S-statistic is designed to test the hypothesis that there is no treatment effect in a randomized-block design with k treatments and n blocks. In this paper we give tables of the null distribution of S for k = 5, n = 6(1)8, and for k = 6, n = 2(1)6. Computational details are discussed.

The correlation coefficient between the smallest and largest observations when (N-1) of the n observations are iid exponentially distributed

A random sample of size n is obtained from two exponential populations in the following manner: (n-1) observations are selected from one of these populations and a single observation is selected from the other population. The correlation coefficient for Y1and Yn, the first and n-th order statistics is obtained. More ngenerally, it is shown in the course of the derivation that Yj-Yi is independent of YK-Y1 for all i, j, l and k such that I < j < l < k. This result is weil known when all the observations are iid exponentially distributed.

Tables of percentage points of the distribution of the maximum absolute value of equally correlated normal random variables

Let X1,…,X2,…,XN be multinomial with zero means, unitvariances, and equal correlations ρ>0. For i=l,…,N let Yi =|Xi| and let Y(1) &…Yρ) the probability that YNyρ)=l−α for a=0.25, 0.10, 0.05, 0.025, 0.01, 0.005, 0.001; N=2(1)40(2)50; and p=0.100, 0.125, 0.200, 0.250, 0.300, 1/3, 0.375, 0.400, 0.500, 0.600, 0.625, 2/3, 0.700, 0.750, 0.800, 0.875, 0.900,1/(1+√N).

Sample-size determination for two-sided b-expectation tolerance intervals for a normal distribution

In statistical practice, tolerance limits are constructed to contain a specified proportion of a population. When only sample data are available, the actual proportion contained in the interval is random and unknown but controlled by a statistical criterion. In this article, we consider some properties of two-sided β-expectation tolerance intervals for a normal distribution based on the sample mean and the sample standard deviation S computed from a random sample of size n. The tolerance interval is given by ± kS, where k = (l/n + l)½t and t is an appropriate quantile of a Student-t distribution. In repeated sampling, such intervals will, on the average, contain l0O0β% of the sampled distribution. These intervals provide a useful description of a population if the spread in the actual proportion contained in the interval is controlled or evaluated. We consider the effect that sample size has on the proportion of the population contained in the interval, using two different criteria for measuring the varia...

Some inconsistencies in judging problems

Abstract Suppose that I individuals are ordered on the basis of the sums of ranks assigned independently by J judges, and that there is a unique winner. If the winner is deleted and the ranks assigned to the remaining individuals are adjusted, then an ordering of the reduced set is obtained. This ordering is said to be consistent with the original ordering if the relative positions of all remaining individuals are unchanged. An examination is made of conditions under which an individual, who was beaten by the winner and at least one other person in the original ranking, emerges as the winner in the reduced ranking. Such an inconsistency is called an interchange.

Critical values of the sample product-moment correlation coefficient in the bivariate normal distribution

Let R be the sample product-moment correlation coefficient computed from a random sample of n pairs of observations from a bivariate normal distribution with population correlation coefficient ρ. For −1 < r < 1, define fn (r,p) to be the density function for R, and Fn (r,p)=Pr[R <=r] to be the cumulative distribution function. Extensive tables of fn(r,p) and Fn (r,p) are given by David (1954) . Tables of upper and lower confidence limits on p are given by Odeh and Owen (1980). The major purpose of this paper is to give extensive tables of the critical values of the Distribution of R. In particular we give values of ry=r(y,n,ρ) to 5 decimal places which satisfy Fn(ry,ρ)= y. Tables are given for values of ρ=0.0(0.10)0.90, 0.95; n = 4(1)30(2)40(5)50(10)100(20)200(100)1000; Y=0.25, 0.10, 0.05, 0.025, 0.01, 0.005;Y=0.75, 0.90, 0.95, 0.975, 0.99, 0.995. We also show how critical values for ρ < 0 can be obtained from the tables.