An exact test for trend among binomial proportions based on a modified Baumgartner-Weiß-Schindler statistic

The Cochran-Armitage test is the most frequently used test for trend among binomial proportions. This test can be performed based on the asymptotic normality of its test statistic or based on an exact null distribution. As an alternative, a recently introduced modification of the Baumgartner-Weiß-Schindler statistic, a novel nonparametric statistic, can be used. Simulation results indicate that the exact test based on this modification is preferable to the Cochran-Armitage test. This exact test is less conservative and more powerful than the exact Cochran-Armitage test. The power comparison to the asymptotic Cochran-Armitage test does not show a clear winner, but the difference in power is usually small. The exact test based on the modification is recommended here because, in contrast to the asymptotic Cochran-Armitage test, it guarantees a type I error rate less than or equal to the significance level. Moreover, an exact test is often more appropriate than an asymptotic test because randomization rather than random sampling is the norm, for example in biomedical research. The methods are illustrated with an example data set.     

Locally best tests for exponential distributions with interval-censored data

In this article, we derive a locally best test for testing the mean of exponential distributions with interval-censored samples. This locally best test possesses certain optimality. It is of unbiasedness and equivalent to a likelihood ratio test in some circumstances, and it is also a Bayes test for some loss function. For the locally best test, the associated critical values and powers at a nominal level of significance are provided. For a large sample size case, asymptotic critical values and powers are also calculated and tabulated. Moreover, based on the locally best test, a curtailed test is proposed. This curtailed test is equivalent to the locally best test on the acceptance or rejection of the null hypothesis. A Monte Carlo simulation is carried out to illustrate the performance of the curtailed test compared with the locally best test. Numerical results show that the experimental duration time of the curtailed test is substantially smaller than that of the locally best test.     

A new nonparametric bivariate test for two sample location problem

A strictly nonparametric bivariate test for two sample location problem is proposed. The proposed test is easy to apply and does not require the stringent condition of affine-symmetry or elliptical symmetry which is required by some of the major tests available for the same problem. The power function of the proposed test is calculated. The asymptotic distribution of the proposed test statistic is found to be normal. The power of proposed test is compared with some of the well-known tests under various distributions using Monte Carlo simulation technique. The power study shows that the proposed test statistic performs better than most of the test statistics for almost all the distributions considered here. As soon as the underlying population structure deviates from normality, the ability of the proposed test statistic to detect the smallest shift in location increases as compared to its competitors. The application of the test is shown by using a data set.     

Adaptive positive and negative runs test

As a nonparametric randomness test, the positive and negative runs test is widely used in practice due to the simplicity of its procedures. The test can lose efficiency if the alternative distribution is symmetrical at 0.5. In addition, the test can only be applied to test the randomness of a sequence from the uniform distribution. In this paper, we introduce an adaptive positive and negative runs test method to maximize the power function by choosing the optimal cut point. Also, the test is extended to check the randomness of a sequence generated from any other given distributions. Furthermore, we derive the exact distribution and obtain the asymptotical critical values of the proposed test statistics. Compared with the existed test, the efficiency of the proposed adaptive positive and negative runs test is competitive through simulation study.     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.     

Generalizations of Mathisen's median test for comparing several treatments with a control

A distribution-free test for comparing several treatments with a control is proposed for one-way classified data. It is advantageous to use the test in life-testing experiments where testing time is expensive. The proposed test has a shorter expected duration than a previously proposed test by Slivka(1970). The optimal allocation of the experimental units to the treatments for two situations are given. In a simulation study the power of the test is compared with the power of Slivka's test. An extension of the test for two-way classified data is given     

A Modified Knox Test of Space-Time Clustering

Tests of space-time clustering such as the Knox test are used by epidemiologists in the preliminary analysis of datasets where an infectious aetiology is suspected. The Knox test statistic is the number of cases close in both space and time to another case. The test statistic proposed here is the excess number of such cases over that expected under H₀ of no infection. It is argued that this modified test is more powerful than the Knox test, because the test statistic is not heavily tied as is the Knox test statistic. The use of the test is illustrated with examples.     

Testing a linear relationship in varying coefficient spatial autoregressive models

In this article, we propose a test to check a linear relationship in varying coefficient spatial autoregressive models, in which a residual-based bootstrap procedure is suggested to approximate the null distribution of the resulting test statistic. We conduct simulation studies to assess the performance of the test, including the validity of the bootstrap approximation to the null distribution of the test statistic and the power of the test. The simulation results demonstrate that the residual-based bootstrap procedure gives very accurate estimate of the null distribution of the test statistic and the test is of satisfactory power. Furthermore, a real example is given to demonstrate the application of the proposed test.     

Test procedures in coyariamce analysis assuming a general correlation pattern

The analysis of covariance procedure is considered when the observations in each cell are equicorrelated. A correction procedure is given, A computationally easier conservative test statistic is also given. The conservative test statistic allows one to more readily determine the consequences of ignoring correlations, even slight correlations, in the analysis of covariance procedure. The difference of the corrected test and the conservative test is shown to converge in probability to zero. This conservative test is easy to implement on statistical computer packages, It is shown, that for the general correlation pattern, any test involving the regression coefficients of the covariables is an exact test. An example illustrates the procedure     

Maximum Test versus Adaptive Tests for the Two-Sample Location Problem

For the non-parametric two-sample location problem, adaptive tests based on a selector statistic are compared with a maximum and a sum test, respectively. When the class of all continuous distributions is not restricted, the sum test is not a robust test, i.e. it does not have a relatively high power across the different possible distributions. However, according to our simulation results, the adaptive tests as well as the maximum test are robust. For a small sample size, the maximum test is preferable, whereas for a large sample size the comparison between the adaptive tests and the maximum test does not show a clear winner. Consequently, one may argue in favour of the maximum test since it is a useful test for all sample sizes. Furthermore, it does not need a selector and the specification of which test is to be performed for which values of the selector. When the family of possible distributions is restricted, the maximin efficiency robust test may be a further robust alternative. However, for the family of t distributions this test is not as powerful as the corresponding maximum test.     

Testing the equality of the variances of two linear models

Testing the equality of variances of two linear models with common β-parameter is considered. A test based on least squares residuals (ASR test) is proposed, and it is shown that this test is invariant under the group of scale and translation changes. For some special cases, it is also proved that this test has a monotone power function. Finding the exact critical values of this test is not easy; an approximation is given to facilitate the computation of these. The powers of the BLUS test, the F-test and the new test are computed for various alternatives and compared in a particular case. The proposed test seems to be locally more powerful than the alternative tests.     

Book Reviews

The Levene test is a widely used test for detecting differences in dispersion. The modified Levene transformation using sample medians is considered in this article. After Levene's transformation the data are not normally distributed, hence, nonparametric tests may be useful. As the Wilcoxon rank sum test applied to the transformed data cannot control the type I error rate for asymmetric distributions, a permutation test based on reallocations of the original observations rather than the absolute deviations was investigated. Levene's transformation is then only an intermediate step to compute the test statistic. Such a Levene test, however, cannot control the type I error rate when the Wilcoxon statistic is used; with the Fisher–Pitman permutation test it can be extremely conservative. The Fisher–Pitman test based on reallocations of the transformed data seems to be the only acceptable nonparametric test. Simulation results indicate that this test is on average more powerful than applying the t test after Levene's transformation, even when the t test is improved by the deletion of structural zeros.     

A sequential testing procedure for outliers and structural change

In this paper a test for outliers based on externally studentized residuals is shown to be related to a test for predictive failure. The relationships between a test for outliers, a test for a correlated mean shift and a test for an intercept shift are developed. A sequential testing procedure for outliers and structural change is shown to be independent, so that the overall size of the joint test can be determined exactly. It is established that a joint test for outliers and constancy of variances cannot be performed.     

Testing methods for the one-way fixed effects ANOVA models of log-normal samples

For one-way fixed effects of log-normal data with unequal variance, the present study proposes a method to deal with heterogeneity. An appropriate hypothesis testing is demonstrated; and one of the approximate tests, such as the Alexander-Govern test, Welch test or James second-order test, is applied to control Type I error rate. Monte Carlo simulation is used to investigate the performance of the F test for log-scale, the F test for original scale, the James second-order test, the Welch test, and the Alexander-Govern test. The simulated results and real data analysis show that the proposed method is valid and powerful.     

Accurate tests for the equality of coefficients of variation

ABSTRACT

A frequently encountered statistical problem is to determine if the variability among k populations is heterogeneous. If the populations are measured using different scales, comparing variances may not be appropriate. In this case, comparing coefficient of variation (CV) can be used because CV is unitless. In this paper, a non-parametric test is introduced to test whether the CVs from k populations are different. With the assumption that the populations are independent normally distributed, the Miller test, Feltz and Miller test, saddlepoint-based test, log likelihood ratio test and the proposed simulated Bartlett-corrected log likelihood ratio test are derived. Simulation results show the extreme accuracy of the simulated Bartlett-corrected log likelihood ratio test if the model is correctly specified. If the model is mis-specified and the sample size is small, the proposed test still gives good results. However, with a mis-specified model and large sample size, the non-parametric test is recommended.     

Statistical Inference on the Parametric Component in Partially Linear Spatial Autoregressive Models

A statistical test procedure is proposed to check whether the parameters in the parametric component of the partially linear spatial autoregressive models satisfy certain linear constraint conditions, in which a residual-based bootstrap procedure is suggested to derive the p-value of the test. Some simulations are conducted to assess the performance of the test and the results show that the bootstrap approximation to the null distribution of the test statistic is valid and the test is of satisfactory power. Furthermore, a real-world example is given to demonstrate the application of the proposed test.     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

The multisample Cucconi test

The multisample version of the Cucconi rank test for the two-sample location-scale problem is proposed. Even though little known, the Cucconi test is of interest for several reasons. The test is compared with some Lepage-type tests. It is shown that the multisample Cucconi test is slightly more powerful than the multisample Lepage test. Moreover, its test statistic can be computed analytically whereas several others cannot. A practical application example in experimental nutrition is presented. An R function to perform the multisample Cucconi test is given.     

A pooled Bayes test of independence for sparse contingency tables from small areas

We study the association between bone mineral density (BMD) and body mass index (BMI) when contingency tables are constructed from the several U.S. counties, where BMD has three levels (normal, osteopenia and osteoporosis) and BMI has four levels (underweight, normal, overweight and obese). We use the Bayes factor (posterior odds divided by prior odds or equivalently the ratio of the marginal likelihoods) to construct the new test. Like the chi-squared test and Fisher's exact test, we have a direct Bayes test which is a standard test using data from each county. In our main contribution, for each county techniques of small area estimation are used to borrow strength across counties and a pooled test of independence of BMD and BMI is obtained using a hierarchical Bayesian model. Our pooled Bayes test is computed by performing a Monte Carlo integration using random samples rather than Gibbs samples. We have seen important differences among the pooled Bayes test, direct Bayes test and the Cressie-Read test that allows for some degree of sparseness, when the degree of evidence against independence is studied. As expected, we also found that the direct Bayes test is sensitive to the prior specifications but the pooled Bayes test is not so sensitive. Moreover, the pooled Bayes test has competitive power properties, and it is superior when the cell counts are small to moderate.     

A Nonparametric Test for Granger Causality in Distribution With Application to Financial Contagion

This article introduces a kernel-based nonparametric inferential procedure to test for Granger causality in distribution. This test is a multivariate extension of the kernel-based Granger causality test in tail event. The main advantage of this test is its ability to examine a large number of lags, with higher-order lags discounted. In addition, our test is highly flexible because it can be used to identify Granger causality in specific regions on the distribution supports, such as the center or tails. We prove that the test converges asymptotically to a standard Gaussian distribution under the null hypothesis and thus is free of parameter estimation uncertainty. Monte Carlo simulations illustrate the excellent small sample size and power properties of the test. This new test is applied to a set of European stock markets to analyze spillovers during the recent European crisis and to distinguish contagion from interdependence effects.