Approximate Critical Values Research Articles

Two-stage designs with small sample sizes

ABSTRACT When applying group-sequential designs in clinical trials with normally distributed outcomes, approximate critical values are often applied. Here, normally distributed test statistics are assumed which, however, are in fact t-distributed. For small sample sizes, the approximation may lead to a serious inflation of the type I error rate. Recently, a method for computing the exact critical boundaries assuring type I error rate control was proposed and the critical boundaries for Pocock- and O’Brien-Fleming-like group-sequential designs were provided. For designs with one interim analysis, we present six alternative designs, which also control the type I error rate and in addition allow flexible design modifications. We compare the characteristics of these 6 two-stage designs. It is shown that considerable sample size savings can be achieved by including futility stopping and by optimizing the designs. Therefore, for clinical trials with small sample sizes as, for example, in the area of rare diseases, optimal two-stage designs with futility stopping may be a valuable alternative to classical group-sequential designs.

Modified likelihood ratio tests for extreme value distributions

The modified Anderson-Darling test statistics have been suggested in testing the highly skewed distributions to detect more possible departures in the right (or left) tail. In this paper, we propose the corresponding modified likelihood ratio test statistics, and compare their performances for the Gumbel distribution with the modified Anderson-Darling test statistics. We also provide the approximate critical values for the generalized extreme value distribution and generalized Pareto distribution where the unknown shape parameter is present.

An Asymptotic Test for A Single Outlier in Linear Regression Models

It is not uncommon to find an outlier in the response variable in linear regression. Such a deviant value needs to be detected and scrutinized to find out why it is not in agreement with its fitted value. Srikantan [1] has developed a test statistic for detecting the presence of an outlier in the response variable in a multiple linear regression model. Approximate critical values of this test statistic are available and are obtained based on the first-order Bonferroni upper bound. The exact critical values are not available and a result of that, tests carried out on the basis of this approximate critical values may not be very accurate. In this paper, we obtained more accurate and precise critical values of this test statistic for large sample sizes (herein called asymptotic critical values) to improve on the tests that use these critical values. The procedure involved using the exact probability density function of this test statistic to obtain its asymptotic critical values. We then compared these asymptotic critical values with the approximate critical values obtained. An application to simulation results for linear regression models was used to examine the power of this test statistic. The asymptotic critical values obtained were found to be more accurate and precise. Also, the test performed better under these asymptotic values (the power performance of this test statistic was found to better when the asymptotic critical values were used).

Direct numerical simulation of confined weak round fountains in homogeneous ambient

Fountains (negatively buoyant jets) are commonly used in numerous engineering applications, such as the natural ventilation in buildings and the smoke spread in compartment fires. In this study, the long-term behavior of weak round fountains in homogeneous ambient confined in a cylindrical container was analyzed using three-dimensional direct numerical simulation over a range of the Froude number (Fr), Reynolds number (Re), dimensionless radius of the container (λ), respectively. The confined weak round fountain behaves in the manner of a ‘fountain filling box’ flow, experiencing five development stages, i.e., the formation of the fountain flow, the intrusion flow, the wall fountain, the reversed flow and the stratification. Three stages of the development of the bulk entrainment rate are identified and the dominant mechanisms involved are analyzed. Fr = 1 and 2 are identified as the approximate critical values to distinguish the behavior of the intrusion, wall fountain and stratification. Re = 200 is determined as the approximate critical value to distinguish the influence of Re. These are consistent with the existing results about the round fountains.

A bivariate test for location based on simplicial depth

We translate the simplicial data depth concept of Liu (1990) into a bivariate test of location that is distribution free under the assumption of angular symmetry about zero, without loss of generality. The test statistic is a count of the number of data triangles that contain zero and rejects the null hypothesis for small values of the test statistic. A straightforward method for computing this statistic is provided. The exact null distribution is computed and tabled for n ≤ 15, the asymptotic null distribution is derived and a formula for approximate critical values is provided. Simulations show that the power of the test is comparable to other bivariate sign type tests. The analogous one dimensional distribution free test is found to be equivalent to a two sided one sample sign test. These ideas are then extended to three dimensions, where we show that, surprisingly, the test is not generally distribution free under the null hypothesis.

A Bounds Approach to Inference Using the Long Run Multiplier

Pesaran, Shin, and Smith (2001) (PSS) proposed a bounds procedure for testing for the existence of long run cointegrating relationships between a unit root dependent variable ($y_{t}$) and a set of weakly exogenous regressors$\boldsymbol{x}_{t}$when the analyst does not know whether the independent variables are stationary, unit root, or mutually cointegrated processes. This procedure recognizes the analyst’s uncertainty over the nature of the regressors but not the dependent variable. When the analyst is uncertain whether$y_{t}$is a stationary or unit root process, the test statistics proposed by PSS are uninformative for inference on the existence of a long run relationship (LRR) between$y_{t}$and$\boldsymbol{x}_{t}$. We propose the long run multiplier (LRM) test statistic as a means of testing for LRRs without knowing whether the series are stationary or unit roots. Using stochastic simulations, we demonstrate the behavior of the test statistic given uncertainty about the univariate dynamics of both$y_{t}$and$\boldsymbol{x}_{t}$, illustrate the bounds of the test statistic, and generate small sample and approximate asymptotic critical values for the upper and lower bounds for a range of sample sizes and model specifications. We demonstrate the utility of the bounds framework for testing for LRRs in models of public policy mood and presidential success.

An outlier-resistant indicator of anomalies among inter-laboratory comparison data with associated uncertainty

A new robust pairwise statistic, the pairwise median scaled difference (MSD), is proposed for the detection of anomalous location/uncertainty pairs in heteroscedastic interlaboratory study data with associated uncertainties. The distribution for the IID case is presented and approximate critical values for routine use are provided. The determination of observation-specific quantiles and p-values for heteroscedastic data, using parametric bootstrapping, is demonstrated by example. The theoretical breakdown point is high due to the use of a median, offering strong resistance to extreme secondary location/uncertainty pairs. Short simulations are used to illustrate test power and robustness for some simple cases; these suggest good power and very good resistance to the effects of secondary outlying values.

Approximate Non-Similar critical values based tests vs Maximized Monte Carlo tests

Testing in the presence of nuisance parameters is a problem often faced by researchers; consequently, a number of ways are suggested in the literature to manage this situation. Among these, Maximized Monte Carlo (MMC) tests or asymptotically valid MMC (AMMC) tests are becoming popular. The MMC type tests have certain advantages as well as disadvantages. This paper introduces a simple way to obtain Approximate Non-Similar (ANS) critical values using a global optimizer called Simulated Annealing (SA). All three methods are applied in the dynamic linear regression model context. As expected the AMMC approach is certainly less time consuming than the MMC approach. Overall the AMMC approach seems best in terms of power properties; however the ANS approach takes negligible time compared to its competitors. Though the ANS approach controls the sizes well it can be slightly less powerful than its competitors.

Detecting non-simultaneous changes in means of vectors

Likelihood ratio type test statistics are suggested for detecting changes in means of coordinates of observed random vectors. It is supposed that changes in different coordinates need not to occur at the same time. Under the assumption of no change, asymptotic distributions of the proposed test statistics are given by distributions of maxima of \(\chi ^2\) random fields. High-level exceedance probabilities of non-homogeneous \(\chi ^2\) fields may be applied to get approximate asymptotic critical values.

Specification testing in nonstationary time series models

Summary In this paper, we consider a specification testing problem in nonlinear time series models with nonstationary regressors, and we propose using a nonparametric kernel-based test statistic. The null asymptotics for the proposed nonparametric test statistic have been well developed in the existing literature. In this paper, we study the local asymptotics of the test statistic (i.e. the asymptotic properties of the test statistic under a sequence of general nonparametric local alternatives) and show that the asymptotic distribution depends on the asymptotic behaviour of the distance function, which is the local deviation from the parametrically specified model in the null hypothesis. In order to implement the proposed test in practice, we introduce a bootstrap procedure to approximate the critical values of the test statistic and establish a new Edgeworth expansion, which is used to justify the use of such an approximation. Based on the approximate critical values, we develop a bandwidth selection method, which chooses the optimal bandwidth that maximizes the local power of the test while its size is controlled at a given significance level. The local power is defined as the power of the proposed test for a given sequence of local alternatives. Such a bandwidth selection is made feasible by an approximate expression for the local power of the test as a function of the bandwidth. A Monte Carlo simulation study is provided to illustrate the finite sample performance of the proposed test.

Multivariate generalisations of k-sample rank tests for umbrella alternatives

We consider the problem of multivariate generalisation of k-sample rank tests for umbrella alternatives. That is, testing the null hypothesis of equality of medians against the alternatives of interest of the form for all x and g=1, …, p; with at least one strict inequality for at least one g when ℓ is known. This is known as umbrella alternatives and from now on ℓ will be referred to as the peak of the umbrella. Sometimes when comparing populations, we are interested in whether the medians of the groups are increasing up to some point then decreasing as the treatment levels are increasing. We propose a multivariate generalisation of umbrella alternatives based on coordinate-wise approach when the peak of the umbrella ℓ is known. Approximate critical values for the small sample null distributions are also discussed. Furthermore, we study the power of the test under location-family alternatives using small sample simulations.

Testing for multiple change points

In this paper we concentrate on testing for multiple changes in the mean of a series of independent random variables. Suggested method applies a maximum type test statistic. Our primary focus is on an effective calculation of critical values for very large sample sizes comprising (tens of) thousands of observations and a moderate to large number of segments. To that end, Monte Carlo simulations and a modified Bellman's principle of optimality are used. It is shown that, indisputably, computer memory becomes a critical bottleneck in solving a problem of such a size. Thus, minimization of the memory requirements and appropriate order of calculations appear to be the keys to success. In addition, the formula that can be used to get approximate asymptotic critical values using the theory of exceedance probability of Gaussian fields over a high level is presented.

Bootstrap Point Optimal Unit Root Tests

In this article, we investigate and compare the behaviour of some bootstrap unit root tests in finite ARMA models with a constant and/or a trend and use them to obtain asymptotic results for the point optimal (hereafter PO) test, in terms of both size and power. We demonstrate the asymptotic validity of bootstrapping the PO test. We provide a feasible method for obtaining approximate critical values for the PO unit root test. Through simulations, we investigate how effective the bootstrap is in different sample sizes, correlative coefficients and close unity autoregressive roots in two different models. Our main objective is to show that the bootstrap PO test can be used in regression models with AR and MA errors and trending regressors. The results reported here provide an analytical investigation of the use of the bootstrap for PO tests with dependent data.The main contribution of this article has two features. First, we choose the PO test and make this powerful but unfeasible procedure both powerful and feasible, by plugging in a consistent estimation of the coefficient structure, and we show that the bootstrap PO test provides asymptotically valid critical values. Second, through simulation, our numerical results suggest that the bootstrap PO test performs well in having the correct size properties and retaining good power in the parametric (and semi-parametric) bootstrap procedure.

Simultaneous testing for the successive differences of exponential location parameters under heteroscedasticity

In this paper, the design-oriented two-stage and data-analysis one-stage multiple comparison procedures for successive comparisons of exponential location parameters under heteroscedasticity are proposed. One-sided and two-sided simultaneous confidence intervals are also given. We also extend these simultaneous confidence intervals for successive differences to a larger class of contrasts of the location parameters. Upper limits of critical values are obtained using the recent techniques given in Lam [Lam, K., 1987. Subset selection of normal populations under heteroscedasticity. In: Proceedings of the Second International Advanced Seminar/Workshop on Inference Procedures Associated with Statistical Ranking and Selection, Sydney, Australia; Lam, K., 1988. An improved two-stage selection procedure. Communications in Statistics Simulation and Computation. 17 (3), 995–1006]. These approximate critical values are shown to have better results than the approximate critical values using the Bonferroni inequality developed in this paper. Finally, the application of the proposed procedures is illustrated with an example.

Multiple Comparisons with More than One Control for Exponential Location Parameters Under Heteroscedasticity

In this article, the design-oriented two-stage and data analysis one-stage multiple comparison procedures for comparing several two-parameter exponential populations with more than one control population under heteroscedasticity are proposed. One-sided and two-sided simultaneous confidence intervals are also given. Upper limits of critical values are obtained using the recent techniques given in Lam (1987, 1988). These approximate critical values are shown to have better results than the approximate critical values using the Bonferroni inequality developed in this article. The multiple comparisons of test treatments with the best control are also developed. Finally, the application of the proposed procedures is illustrated with an example.

Log-likelihood ratio test for detecting transient change

The limit behaviour of scan statistics in a form of maxima of moving sum processes with changing window size for detecting a transient change is studied. Approximate critical values obtained from tail behaviour of the limit variables that are maxima of some locally homogeneous Gaussian fields are compared with critical values obtained by simulation.

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.

Testing the Equality of Multivariate Distributions Using the Bootstrap and Integrated Empirical Processes

We introduce some projected integrated empirical processes for testing the equality of two multivariate distributions. The bootstrap is used for determining the approximate critical values. We show that the bootstrap test is consistent. A number-theoretic method is used for efficient computation of the bootstrap critical values. Some simulation results are also given.

Two stage multiple comparisons with the average for exponential location parameters under heteroscedasticity

In this article, two-stage procedures for multiple comparisons with the average for location parameters of two-parameter exponential distributions under heterocedasity including one- and two-sided confidence intervals are proposed. These intervals can be used to identify a subset which includes all no-worse-than-the-average treatments in an experimental design and to identify better-than-the-average, worse-than-the-average and not-much-different-from-the-average products in agriculture, stock market, medical research, and auto models. An upper limit of critical values are obtained using the recent techniques given in Lam (Proceedings of the Second International Advanced Seminar/Workshop on Inference Procedures Associated with Statistical Ranking and Selection, Sydney, Australia, August 1987; Comm. Statist. Simulation Comput. B17(3) (1988) 55). These approximate critical values are shown to have better results than the approximate critical values using the Bonferroni inequality developed in this paper. An example of comparing four drugs in the treatment of leukemia is given to demonstrate the proposed methodology.

Split-plot design optimization for trace determination of lead by anodic stripping voltammetry in a homogeneous ternary solvent system

A split-plot design has been used to simultaneously optimize reagent conditions and solvent medium for Pb 2+ determination by anodic stripping voltammetry (ASV). Three mixture components, N, N-dimethylformamide (DMF), ethanol and water, and two process variable levels, ammonium acetate (supporting electrolyte) and hydrochloric acid concentrations, were varied. The calculations of main-plot, subplot and main-subplot interaction ANOVA sums of squares for regression and lack of fit are illustrated. These values are shown to be useful for model development. Six different models were evaluated. The bilinear-special cubic model has only a very slight lack of fit and is preferred. Standard error estimates were calculated using a method described in the literature by Cornell, and approximate critical values for t-tests on the model coefficients were investigated. Normal probability graphs seem appropriate for this analysis. The significant terms in this model are capable of describing how the mixture response surfaces change as process level conditions are varied. Optimized mixture proportions for each factorial design level are determined. Optimum conditions in the (− −) main-plot quadrant for the determination of lead by ASV in the homogeneous ternary solvent system were achieved with a solvent composition of 8.0 g of N, N-dimethylformamide (DMF), 7.0 g of ethanol and 5.0 g of water, corresponding to 40% m/m of DMF, 35% n/m of ethanol and 25% m/m of water. Water is essential since the sample and the buffer (0.1 mol kg −1 ammonium acetate containing 0.00880 mol kg −1 HCl) are added to the system as aqueous solutions. The other three main-plot quadrants presented optimum analytical signals for 95% aqueous solutions. The split-plot design seems to be especially appropriate for simultaneously optimizing process and mixture variables of this chemical system facilitating operational procedures by means of block randomization.