Equality Of Scale Parameters Research Articles

How to analyze work productivity loss due to health problems in randomized controlled trials? A simulation study

BackgroundAn increasing number of randomized controlled trials (RCTs) have measured the impact of interventions on work productivity loss. Productivity loss outcome is inflated at zero and max loss values. Our study was to compare the performance of five commonly used methods in analysis of productivity loss outcomes in RCTs.MethodsWe conducted a simulation study to compare Ordinary Least Squares (OLS), Negative Binominal (NB), two-part models (the non-zero part following truncated NB distribution or gamma distribution) and three-part model (the middle part between zero and max values following Beta distribution). The main number of observations each arm, Nobs, that we considered were 50, 100 and 200. Baseline productivity loss was included as a covariate.ResultsAll models performed similarly well when baseline productivity loss was set at the mean value. When baseline productivity loss was set at other values and Nobs = 50 with ≤5 subjects having max loss, two-part models performed best if the proportion of zero loss> 50% in at least one arm and otherwise, OLS performed best. When Nobs = 100 or 200, the three-part model performed best if the two arms had equal scale parameters for their productivity loss outcome distributions between zero and max values.ConclusionsOur findings suggest that when treatment effect at any given values of one single covariate is of interest, the model selection depends on the sample size, the proportions of zero loss and max loss, and the scale parameter for the productivity loss outcome distribution between zero and max loss in each arm of RCTs.

Comparing scale parameters in several gamma distributions with known shapes

In this paper, we present eleven approaches for testing the equality of scale parameters in gamma distributions when shape parameters are known. These approaches are applicable to other problems such as testing homogeneity of variances in normal distributions, verifying equality of scale-like parameters in inverse Gaussian distributions, and comparing scale parameters in two-parameter exponential distributions with censored data or K-record values. The performance of the proposed tests is compared in terms of empirical size and power using Monte Carlo simulation. Finally, the proposed methods are illustrated using two real data examples.

Nonparametric multiple sample scale testing using U-statistics

Testing of equality of scale parameters is one of the fundamental problems in Statistics and is useful in many fields such as agriculture, engineering, experimental methods and dose level testing. In this paper, we propose a new class of nonparametric tests, to test whether the scale parameters of all the populations are equal, or there is an umbrella ordering in the scale parameters. The proposed test is based on weighted linear combination of two-sample U-statistics. The optimal choice of weights is determined so that proposed tests have maximum efficiency. The comparison of proposed test with respect to existing tests is made in terms of Pitman asymptotic relative efficiency. A numerical example is provided for illustrative purpose. Power of the proposed test is also assessed using Monte Carlo simulation study.

On stochastic comparisons ofk-out-of-nsystems with Weibull components

AbstractIn this paper we prove that a parallel system consisting of Weibull components with different scale parameters ages faster than a parallel system comprising Weibull components with equal scale parameters in the convex transform order when the lifetimes of components of both systems have different shape parameters satisfying some restriction. Moreover, while comparing these two systems, we show that the dispersive and the usual stochastic orders, and the right-spread order and the increasing convex order are equivalent. Further, some of the known results in the literature concerning comparisons ofk-out-of-nsystems in the exponential model are extended to the Weibull model. We also provide solutions to two open problems mentioned by Balakrishnan and Zhao (2013) and Zhaoet al.(2016).

A general class of non parametric tests for comparing scale parameters

ABSTRACTIn this paper, a general class of non parametric tests is proposed for the two-sample scale problem. Testing of the scale parameter is very useful in real-life situations commonly faced in engineering, trade, cultivation, industries, medicine, etc. In all these fields, one will prefer the method that gives more consistent results. Thus, it is worthwhile to test the equality of scale parameters. The distribution of the proposed test is established. To assess the performance of the proposed test, the asymptotic efficacies are studied for some underlying distributions and the results are interpreted with useful information. To see the working of the proposed test, an illustrative example for the real-life data set is provided. The simulation study is also carried out to find the asymptotic power of the proposed test. An extension of the general class of tests to the multiple-sample problem is also discussed.

Exact inference for the difference of Laplace location parameters

We consider exact procedures for testing the equality of means (location parameters) of two Laplace populations with equal scale parameters based on corresponding independent random samples. The test statistics are based on either the maximum likelihood estimators or the best linear unbiased estimators of the Laplace parameters. By conditioning on certain quantities we manage to express their exact distributions as mixtures of ratios of linear combinations of standard exponential random variables. This allows us to find their exact quantiles and tabulate them for several sample sizes. The powers of the tests are compared either numerically or by simulation. Exact confidence intervals for the difference of the means corresponding to those tests are also constructed. The exact procedures are illustrated via a real data example.

On the bivariate and multivariate weighted generalized exponential distributions

This article proposes a particular member of the weighted bivariate distribution, namely, bivariate weighted generalized exponential distribution. This distribution is obtained via conditioning, starting from three independent generalized exponential distributions with different shape but equal scale parameters. Several structural properties of the proposed bivariate weighted generalized exponential distribution including total positivity of order two, marginal moments, reliability parameter and estimation of the model parameters are studied. A multivariate extension of the proposed model is discussed with some properties. Small simulation experiments have been performed to see the behavior of the maximum likelihood estimators, and one data analysis has been presented for illustrative purposes.

Estimation and inference in sets of Weibull samples

Menon’s method of estimation of the Weibull parameters is extended to the analysis of multiple samples of a common size. A methodology is developed for testing the equality of shape parameters among the populations from which the samples are drawn. When the shape parameters are judged to be equal, point and interval estimates of the common shape parameter are constructed. An analysis of variance approach is shown to be valid for testing the equality of scale parameters when a common shape parameter is hypothesized. A post hoc multiple comparison test is developed for assessing which scale parameters differ. Confidence intervals on a percentile may be set for any of the sample populations using the estimated common shape parameter. Tables of the percentage points of the distributions needed for testing hypotheses and constructing interval estimates were obtained by Monte Carlo sampling and are given for a useful range of sample sizes and number of samples. The result is a comprehensive, easily computed, method of analyzing uncensored sets of Weibull data such as occurs when the flexural strength of brittle materials is observed under different conditions of formulation, finishing or test environment. An example analysis of data of this type is included.

Distribution of products of independently distributed pathway random variables

A lot of work has been done on products and ratios of random variables by Provost and his co-workers, see, for example, Provost [S.B. Provost, The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables, Canad. J. Statist. 14 (1986), pp. 61–67; S.B. Provost, The distribution function of a statistic for testing the equality of scale parameters in two gamma populations, Metrika 36 (1989), pp. 337–345]. Here, we extend this idea by introducing a pathway model. The exact density functions of the products of pathway random variables are obtained using the Mellin transform technique. Their computable series forms are derived. The particular cases of the derived results are shown to be associated with the thermonuclear functions and reaction rate probability integral in the theory of nuclear reaction rate, Krätzel integral in applied analyses and inverse Gaussian density in stochastic processes. Graphical representations of the density functions of the product of random variables for the different values of the pathway parameters are shown. The new probability model is fitted to revenue data.

Optimal designs for contingent response models with application to toxicity–efficacy studies

We describe a general family of contingent response models. These models have ternary outcomes constructed from two Bernoulli outcomes, where one outcome is only observed if the other outcome is positive. This family is represented in a canonical form which yields general results for its Fisher information. A bivariate extreme value distribution illustrates the model and optimal design results. To provide a motivating context, we call the two binary events that compose the contingent responses toxicity and efficacy. Efficacy or lack thereof is assumed only to be observable in the absence of toxicity, resulting in the ternary response (toxicity, efficacy without toxicity, neither efficacy nor toxicity). The rate of toxicity, and the rate of efficacy conditional on no toxicity, are assumed to increase with dose. While optimal designs for contingent response models are numerically found, limiting optimal designs can be expressed in closed forms. In particular, in the family of four parameter bivariate location-scale models we study, as the marginal probability functions of toxicity and no efficacy diverge, limiting D optimal designs are shown to consist of a mixture of the D optimal designs for each failure (toxicity and no efficacy) univariately. Limiting designs are also obtained for the case of equal scale parameters.

Testing equality of inverse Gaussian means under heterogeneity, based on generalized test variable

The inverse Gaussian (IG) distribution is an ideal candidate for modelling positive, right-skewed data due to the fact that its inference theory and methodology bear a striking resemblance to the normal theory and methods. For testing equality of several IG means under the assumption of equal scale parameters, there exists the ANORE (analysis of reciprocal) F test, which is analogous to the ANOVA F test for the normal distribution. In this article, the concept of generalized P -value, introduced by Tsui and Weerahandi [1989. Generalized P -values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84, 602–607], is applied for testing equality of several IG means for the general cases without the assumption of homogeneity. Simulation results indicate that the proposed test has excellent type I error control under both heterogeneity and homogeneity, whereas the type I error probabilities of the ANORE test can be much larger than the nominal level under heterogeneity. The proposed procedure is illustrated using two examples.

A Bayesian Test for Equality of Scale Parameters of Several Exponential Distributions

This article develops a Bayesian test for equality of scale parameters of several exponential distributions. The null distribution of the test statistic is approximated by the chi-square distribution using heuristic reasoning in conjunction with the Wilson-Hilferty transfomation for the chi-square random variable. The accuracy of the chi-square approximation of the test statistic and the modified likelihood ratio statistic is examined and their powers compared using Monte Carlo simulation. The proposed test is found to be comparable to Bartlett's modified likelihood ratio test in terms of accuracy and power. A numerical example is included to illustrate the applications of these tests. Journal of Agriculture, Science and Technology Vol.4(1) 2002: 75-82

Runs in superpositions of renewal processes with applications to discrimination

Wald and Wolfowitz [Ann. Math. Statist. 11 (1940) 147–162] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let ( S n ) n ⩾ 1 and ( T n ) n ⩾ 1 be the arrival processes for channel 1 and channel 2, respectively, and ( W n ) n ⩾ 1 their be superposition with counting process N ( t ) = def sup { n ⩾ 1 : W n ⩽ t } . Let further R n * be the number of runs in W 1 , … , W n and R t = R N ( t ) * the number of runs observed up to time t. We study the asymptotic behavior of R n * and R t , first for the case where ( S n ) n ⩾ 1 and ( T n ) n ⩾ 1 have exponentially distributed increments with parameters λ 1 and λ 2 , and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level α tests for testing λ 1 = λ 2 against λ 1 ≠ λ 2 in the first case, and for testing for equal scale parameters in the second.

Testing equality of scale parameters against restricted alternatives for m≥3 gamma distributions with unknown common shape parameter

Shiue and Bain (1983) proposed an approximate F-test for the equality of the scale parameters of two gamma distributions with equal but unknown shape parameters. In this article, we propose a simple procedure to test equality of scale parameters of m≥3 gamma distributions against nonincreasing order. The test is based on Fisher's method of combining p-values. The actual size of the resulting test is investigated through Monte Carlo studies. Also asymptotic results are derived for the nominal test size. These can be used to obtain a test which achieves the desired size. The case of more general partial orders is discussed.

Multisample tests for scale based on kernel density estimation

We propose test statistics based on kernel density estimation for testing the equality of scale parameters. The statistics are compared with other statistics with respect to the asymptotic relative efficiency. The statistics are more efficient than the c-sample analogs of the two-sample Mood test and the two-sample Ansari–Bradley test for the normal distribution and the Cauchy distribution. We also give a comparison of Type I error and power by simulation.

Interval estimation of scale parameters following a pre-test for two exponential distributions

If two censored samples come from the exponential distributions with identical scale parameters, it is advantageous to pool the two samples to construct a confidence interval for the scale parameter. In practice, when it is uncertain whether these two samples come from the distributions with identical scale parameters, a preliminary test may be used to partially resolve the uncertainty. A confidence interval for the scale parameter following a preliminary test concerning the equality of scale parameters is investigated. This interval is referred to as a pre-test confidence interval. The expected length of the pre-test confidence interval is shorter than that of the usual confidence interval. The coverage probability of the pre-test confidence interval is less than the nominal level. However, at some selected levels of significance, the coverage probability of the pre-test confidence interval is only slightly less than the nominal level but the maximum reduction in expected length is substantial.

Estimation of Ordered Location Parameters: The Exponential Distribution

The problem of component wise estimation of ordered location parameters θ 1 (θ 1 ≤ θ 2) of two independent exponential distributions is investigated. The scale parameters are assumed to be unequal but known. Independent random samples of unequal sample sizes are drawn from two populations and the estimators admissible among the mixed estimators of θ 1 and θ 2 are obtained. It is shown that the minimum risk estimators (MREs) of θ 1 and θ 2 without assuming θ 1 ≤ θ 2 are inadmissible when one does assume that θ 1 ≤ θ 2. The efficiencies of mixed estimators relative to MREs (without assuming θ 1 ≤ θ 2) are tabulated for equal sample sizes and equal scale parameters.

On a modified test of equality of scale parameters of exponential distributions

In this paper, an exact distribution of a modifier likelihood ratio criterion for testing the equality of scale parameters of several two parameter exponential distributions is obtained for the case of unequal sample size in a computational form. A short table of critical values of the proposed statistic is also presented.

The distribution function of a statistic for testing the equality of scale parameters in two gamma populations

The exact distribution function of the ratio of two sums of gamma variates is derived in this paper. The result applies to ratios of quadratic forms and to a statistic used for testing the equality of scale parameters in two gamma populations.

Asymmetric mixtures and catastrophes

The topology of a mixture of two expected loss functions from a rich family with equal scale parameters has long been shown to be that of the cusp catastrophe. In this paper a more general problem is tackled. When the scale factors are not equal the resulting mixture is called asymmetric. It has a more complicated structure, but it can be shown that nothing worse than a cusp point appears in its topology. Some consequences of this fact for statistical modelling involving mixtures are discussed.