A note on successive comparisons between several ordered variances

ABSTRACT

In this article, a procedure for comparisons between k (k ? 3) successive populations with respect to the variance is proposed when it is reasonable to assume that variances satisfy simple ordering. Critical constants required for the implementation of the proposed procedure are computed numerically and selected values of the computed critical constants are tabulated. The proposed procedure for normal distribution is extended for making comparisons between successive exponential populations with respect to scale parameter. A comparison between the proposed procedure and its existing competitor procedures is carried out, using Monte Carlo simulation. Finally, a numerical example is given to illustrate the proposed procedure.     

A revision of the tukey multiple comparisons procedure to control the probability of committing at most one type i error

Halperin et al. (1988) suggested an approach which allows for k Type I errors while using Scheffe's method of multiple comparisons for linear combinations of p means. In this paper we apply the same type of error control to Tukey's method of multiple pairwise comparisons. In fact, the variant of the Tukey (1953) approach discussed here defines the error control objective as assuring with a specified probability that at most one out of the p(p-l)/2 comparisons between all pairs of the treatment means is significant in two-sided tests when an overall null hypothesis (all p means are equal) is true or, from a confidence interval point of view, that at most one of a set of simultaneous confidence intervals for all of the pairwise differences of the treatment means is incorrect. The formulae which yield the critical values needed to carry out this new procedure are derived and the critical values are tabulated. A Monte Carlo study was conducted and several tables are presented to demonstrate the experimentwise Type I error rates and the gains in power furnished by the proposed procedure     

Comparing Several Variances with a Control in Direction-mixed Families of Hypotheses

This article deals with simultaneous comparisons of k(k ? 1) populations with a control population with respect to variance in direction-mixed families of hypotheses. Computation of critical constants required for the implementation of the proposed procedures is discussed and selected values of the critical constants are tabulated. Power comparison of the proposed procedures with their existing competitors is carried out using Monte Carlo simulation. An example is given to illustrate the implementation of proposed procedures.     

On a Multiple Three-Decision Problem for Comparing Several Treatments with the Best Control

In this article, a multiple three-decision procedure is proposed to classify p (≥2) treatments as better or worse than the best of q (≥2) control treatments in one way layout. Critical constants required for the implementation of the proposed procedure are tabulated for some pre-specified values of probability of no misclassification. Power function of the proposed procedure is defined and a common sample size necessary to guarantee various pre-specified power levels are tabulated under two optimal allocation schemes. Finally the implementation of the proposed methodology is demonstrated through numerical examples based on real life data.     

Mutiple three-decision procedure for comparing several exponential treatments with the best control

In this paper, the three-decision procedures to classify p treatments as better than or worse than one control, proposed for normal/symmetric probability models [Bohrer, Multiple three-decision rules for parametric signs. J. Amer. Statist. Assoc. 74 (1979), pp. 432–437; Bohrer et al., Multiple three-decision rules for factorial simple effects: Bonferroni wins again!, J. Amer. Statist. Assoc. 76 (1981), pp. 119–124; Liu, A multiple three-decision procedure for comparing several treatments with a control, Austral. J. Statist. 39 (1997), pp. 79–92 and Singh and Mishra, Classifying logistic populations using sample medians, J. Statist. Plann. Inference 137 (2007), pp. 1647–1657]; in the literature, have been extended to asymmetric two-parameter exponential probability models to classify p(p≥1) treatments as better than or worse than the best of q(q≥1) control treatments in terms of location parameters. Critical constants required for the implementation of the proposed procedure are tabulated for some pre-specified values of probability of no misclassification. Power function of the proposed procedure is also defined and a common sample size necessary to guarantee various pre-specified power levels are tabulated. Optimal allocation scheme is also discussed. Finally, the implementation of the proposed methodology is demonstrated through numerical example.     

Multiple Comparisons Controlling Expected Number of False Discoveries

In this paper, it is put forward that the task of designing a procedure for a set of multiple comparisons should be considered as a decision-making under uncertainty. Due to this motivation, for the problem of multiple comparisons, we considered another error rate to be controlled, called PFER (per-family error rate), which requests that the expected number of false rejections of a test procedure should be bounded no more than a prespecified level k. Although PFER was proposed by Tukey in 1953, there is not much studying about it so far. We first present Bonferroni procedure (single-step) and then build two step-up procedures with one having generic critical values and another using critical values in BH (Benjamini and Hochberg) type. These procedures are compared through simulations.     

Computing Critical Values of Exact Tests by Incorporating Monte Carlo Simulations Combined with Statistical Tables

Various exact tests for statistical inference are available for powerful and accurate decision rules provided that corresponding critical values are tabulated or evaluated via Monte Carlo methods. This article introduces a novel hybrid method for computing p‐values of exact tests by combining Monte Carlo simulations and statistical tables generated a priori. To use the data from Monte Carlo generations and tabulated critical values jointly, we employ kernel density estimation within Bayesian‐type procedures. The p‐values are linked to the posterior means of quantiles. In this framework, we present relevant information from the Monte Carlo experiments via likelihood‐type functions, whereas tabulated critical values are used to reflect prior distributions. The local maximum likelihood technique is employed to compute functional forms of prior distributions from statistical tables. Empirical likelihood functions are proposed to replace parametric likelihood functions within the structure of the posterior mean calculations to provide a Bayesian‐type procedure with a distribution‐free set of assumptions. We derive the asymptotic properties of the proposed nonparametric posterior means of quantiles process. Using the theoretical propositions, we calculate the minimum number of needed Monte Carlo resamples for desired level of accuracy on the basis of distances between actual data characteristics (e.g. sample sizes) and characteristics of data used to present corresponding critical values in a table. The proposed approach makes practical applications of exact tests simple and rapid. Implementations of the proposed technique are easily carried out via the recently developed STATA and R statistical packages.     

A modification of the partial sequential probability ratio test

For the problem of discriminating between two simple hypoth¬eses concerning a Koopman - Darmois parameter, a modification of the partial sequential probability ratio test is proposed where instead of drawing only one fixed sample, two fixed samples are drawn and then Wald's SPRT is started. The OC and the ASN func¬tions are derived. Numerical comparisons are made with Wald's and Read's procedures for testing the normal mean with known variance. For some parameter values, the test procedure has a lower ASN than that of Read's procedure.     

A closed testing procedure for comparing successive exponential populations with respect to location parameter

Consider k independent random samples such that i^th sample is drawn from a two-parameter exponential population with location parameter μ_i and scale parameter θ_i,?i = 1, …, k. For simultaneously testing differences between location parameters of successive exponential populations, closed testing procedures are proposed separately for the following cases (i) when scale parameters are unknown and equal and (ii) when scale parameters are unknown and unequal. Critical constants required for the proposed procedures are obtained numerically and the selected values of the critical constants are tabulated. Simulation study revealed that the proposed procedures has better ability to detect the significant differences and has more power in comparison to exiting procedures. The illustration of the proposed procedures is given using real data.     

Testing equality of several exponential distributions

Roy's union-intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions. Upper five and one percent values of the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.     

A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

A multiple directional decision procedure for successive comparisons of treatment effects

Suppose that the k treatments under comparison are ordered in a certain way. For example, there may be a sequence of increasing dose levels of a drug. It is interesting to look directly at the successive differences between the treatment effects μ_i's, namely the set of differences μ₂−μ₁,μ₃−μ₂,…,μ_k−μ_k−1. In particular, directional inferences on whether μ_i<μ_i+1 or μ_i>μ_i+1 for i=1,…,k−1 are useful. Lee and Spurrier (J. Statist. Plann. Inference 43 (1995) 323) present a one- and a two-sided confidence interval procedures for making successive comparisons between treatments. In this paper, we develop a new procedure which is sharper than both the one- and two-sided procedures of Lee and Spurrier in terms of directional inferences. This new procedure is able to make more directional inferences than the two-sided procedure and maintains the inferential sensitivity of the one-sided procedure. Note however this new procedure controls only type III error, but not type I error. The critical point of the new procedure is the same as that of Lee and Spurrier's one-sided procedure. We also propose a power function for the new procedure and determine the sample size necessary for a guaranteed power level. The application of the procedure is illustrated with an example.     

Coefficients of lee-gurland two-sample test on normal means

The Behrens-Fisher problem in comparing means of two normal populations is revisited Lee and Gurland (1975) suggested a solution to the problem and provided the set of coefficients required in computing critical values for the case α=005, where α is the nominal level of significance This solution, called the Lee-Guiland Test in this article, has proven to be practical as far as calculation is involved, and more importantly, it maintains the actual size very close to α= 0.05 for possible values of the ratio of population variances This merit has not been attained by most of the Behrens-Fisher solutions in the literature. In this article, the coefficients for other values of α, namely 0 025, 0 01 and 0.005 are provided for wider applications of the test Moreover, careful and detailed comparisons are made in terms of size and power with the other practical solution:the Welch's Approximate t I est Due to a possible drawback of the Welch's Approximate t I est in controlling the actual size, especially for small a and small sample sizes, the Lee-Gurland lest presents itself as a slightly better' alternative in testing equality of two normal population means I he coefficients mentioned above are also fitted by the functions of the reciprocals of the degrees of freedom, so that the substantial amount of table-looking can be avoided Some discussions are also made in regarding the recent “Welch vs Gosset” argument: Should the Student's t Test be dispensed off’from the routine use in testing the equality of two normal means?.     

Repetitive variable acceptance sampling plan for one-sided specification

In this paper, a variable repetitive group sampling plans based on one-sided process capability indices is proposed to deal with lot sentencing for one-sided specifications. The parameters of the proposed plans are tabulated for some combinations of acceptance quality levels with commonly used producer's risk and consumer's risk. The efficiency of the proposed plan is compared with the Pearn and Wu [Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives. Omega – Int J Manag Sci. 2006;34(1):90–101] plan in terms of sample size and the power curve. One example is given to illustrate the proposed methodology.     

TESTING NEW BETTER THAN RENEWAL USED LIFE DISTRIBUTIONS BASED ON U-TEST

In this paper, the problem of testing exponentiality against new better (worse) than renewal used in expectation is investigated and similarly for the case of nuharmonic new better than renewal used in expectation. For each of these two aging properties, a nonparametric procedure (U-statistic) is presented. Selected critical values are tabulated for sample sizes n = 5(1)30(10)50. The Pitman asymptotic relative efficiency to the test relative to other classes are studied. A real example is given to elucidate the use of the proposed test statistics for the reliability analysis.     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.     

Tables for pairwise multiple comparisons using shaffer's modifiedsequentially-rejective procedure

A sequentially rejective (SR) testing procedure introduced by Holm (1979) and modified (MSR) by Shaffer (1986) is considered for testing all pairwise mean comparisons.For such comparisons, both the SR and MSR methods require that the observed test statistics be ordered and compared, each in turn, to appropriate percentiles on Student's t distribution.For the MSR method these percentiles are based on the maximum number of true null hypotheses remaining at each stage of the sequential procedure, given prior significance at previous stages, A function is developed for determining this number from the number of means being tested and the stage of the test.For a test of all pairwise comparisons, the logical implications which follow the rejection of a null hypothesis renders the MSR procedure uniformly more powerful than the SR procedure.Tables of percentiles for comparing K means, 3 < K < 6, using the MSR method are presented.These tables use Sidak's (1967) multiplicative inequality and simplify the use of t he MSR procedure.Several modifications to the MSR are suggested as a means of further increasing the power for testing the pairwise comparisons.General use of the MSR and the corresponding function for testing other parameters besides the mean is discussed.     

Change-point analysis with bathtub shape for the exponential distribution

Likelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.     

Multiple comparisons of exponential location parameters with a control based on doubly censored sample under heteroscedasticity

In this article, a one-sample procedure for multiple comparisons of exponential location parameters with a control under heteroscedasticity is proposed. The observations are obtained by doubly censored samples. A one-sided and two-sided confidence intervals are used to perform such multiple comparisons. Statistical tables of critical values and an example of comparing four drugs in treating leukemia are provided.