Comparison of exact tests for association in unordered contingency tables using standard,mid-p,and randomized test versions

Pearson’s chi-square (Pe), likelihood ratio (LR), and Fisher (Fi)–Freeman–Halton test statistics are commonly used to test the association of an unordered r×c contingency table. Asymptotically, these test statistics follow a chi-square distribution. For small sample cases, the asymptotic chi-square approximations are unreliable. Therefore, the exact p-value is frequently computed conditional on the row- and column-sums. One drawback of the exact p-value is that it is conservative. Different adjustments have been suggested, such as Lancaster’s mid-p version and randomized tests. In this paper, we have considered 3×2, 2×3, and 3×3 tables and compared the exact power and significance level of these test’s standard, mid-p, and randomized versions. The mid-p and randomized test versions have approximately the same power and higher power than that of the standard test versions. The mid-p type-I error probability seldom exceeds the nominal level. For a given set of parameters, the power of Pe, LR, and Fi differs approximately the same way for standard, mid-p, and randomized test versions. Although there is no general ranking of these tests, in some situations, especially when averaged over the parameter space, Pe and Fi have the same power and slightly higher power than LR. When the sample sizes (i.e., the row sums) are equal, the differences are small, otherwise the observed differences can be 10% or more. In some cases, perhaps characterized by poorly balanced designs, LR has the highest power.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Maximum Likelihood Estimation in the Bivariate Binomial (0,1) Distribution:Application TO 2×2 Tables

We consider a 2×2 contingency table, with dichotomized qualitative characters (A,A) and (B,B), as a sample of size n drawn from a bivariate binomial (0,1) distribution. Maximum likelihood estimates p?₁p?₂ and p? are derived for the parameters of the two marginals p₁p₂ and the coefficient of correlation. It is found that p? is identical to Pearson's (1904)?=(χ²/n)½, where ?² is Pearson's usual chi-square for the 2×2 table. The asymptotic variance-covariance matrix of p?_lp?₂and p is also derived.     

Outlier detection with Mahalanobis square distance: incorporating small sample correction factor

Mahalanobis square distances (MSDs) based on robust estimators improves outlier detection performance in multivariate data. However, the unbiasedness of robust estimators are not guaranteed when the sample size is small and this reduces their performance in outlier detection. In this study, we propose a framework that uses MSDs with incorporated small sample correction factor (c) and show its impact on performance when the sample size is small. This is achieved by using two prototypes, minimum covariance determinant estimator and S-estimators with bi-weight and t-biweight functions. The results from simulations show that distribution of MSDs for non-extreme observations are more likely to fit to chi-square with p degrees of freedom and MSDs of the extreme observations fit to F distribution, when c is incorporated into the model. However, without c, the distributions deviate significantly from chi-square and F observed for the case with incorporated c. These results are even more prominent for S-estimators. We present seven distinct comparison methods with robust estimators and various cut-off values and test their outlier detection performance with simulated data. We also present an application of some of these methods to the real data.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Mixture Distributions Based Methods of Calibration for the Empirical Log-Likelihood Ratio

Empirical likelihood ratio confidence regions based on the chi-square calibration suffer from an undercoverage problem in that their actual coverage levels tend to be lower than the nominal levels. The finite sample distribution of the empirical log-likelihood ratio is recognized to have a mixture structure with a continuous component on [0, + ∞) and a point mass at + ∞. The undercoverage problem of the Chi-square calibration is partly due to its use of the continuous Chi-square distribution to approximate the mixture distribution of the empirical log-likelihood ratio. In this article, we propose two new methods of calibration which will take advantage of the mixture structure; we construct two new mixture distributions by using the F and chi-square distributions and use these to approximate the mixture distributions of the empirical log-likelihood ratio. The new methods of calibration are asymptotically equivalent to the chi-square calibration. But the new methods, in particular the F mixture based method, can be substantially more accurate than the chi-square calibration for small and moderately large sample sizes. The new methods are also as easy to use as the chi-square calibration.     

Bias and size corrections in extreme value modeling

Extreme value theory models have found applications in myriad fields. Maximum likelihood (ML) is attractive for fitting the models because it is statistically efficient and flexible. However, in small samples, ML is biased to O(N^?1) and some classical hypothesis tests suffer from size distortions. This paper derives the analytical Cox–Snell bias correction for the generalized extreme value (GEV) model, and for the model's extension to multiple order statistics (GEVr). Using simulations, the paper compares this correction to bootstrap-based bias corrections, for the generalized Pareto, GEV, and GEVr. It then compares eight approaches to inference with respect to primary parameters and extreme quantiles, some including corrections. The Cox–Snell correction is not markedly superior to bootstrap-based correction. The likelihood ratio test appears most accurately sized. The methods are applied to the distribution of geomagnetic storms.     

Gap-ratio goodness of fit tests for weibull or extreme value distribution assumptions with left or right censored data

With data collection in environmental science and bioassay, left censoring because of nondetects is a problem. Similarly in reliability and life data analysis right censoring frequently occurs. There is a need for goodness of fit tests that can adapt to left or right censored data and be used to check important distributional assumptions without becoming too difficult to regularly implement in practice. A new test statistic is derived from a plot of the standardized spacings between the order statistics versus their ranks. Any linear or curvilinear pattern is evidence against the null distribution. When testing the Weibull or extreme value null hypothesis this statistic has a null distribution that is approximately F for most combinations of sample size and censoring of practical interest. Our statistic is compared to the Mann-Scheuer-Fertig statistic which also uses the standardized spacings between the order statistics. The results of a simulation study show the two tests are competitive in terms of power. Although the Mann-Scheuer-Fertig statistic is somewhat easier to compute, our test enjoys advantages in the accuracy of the F approximation and the availability of a graphical diagnostic.     

Encompassing and indirect inference

Summary In the literature on encompassing [see e.g. Mizon-Richard (1986), Hendry-Richard (1990), Florens-Hendry-Richard (1987)] there is a basic contradiction: on the one hand it is said that it is not possible to assume that the true distribution belongs to one of two competing modelM ₁ andM ₂, but, on the other hand, this assumption is made in the study of encompassing tests. In this paper we first propose a formal definition of encompassing, we then briefly examine the properties of this notion and we propose encompassing tests which do not assume that the true distribution belongs toM ₁ orM ₂; these tests are based on simulations. Finally, generalizing an idea used in the definition of an encompassing test (the GET test) we propose a new kind of inference, called indirect inference, which allows for estimation and test procedures when the model is too complicated to be treated by usual methods (for instance maximum likelihood methods); the only assumption made on the model is that it can be simulated, which seems to be a minimal requirement. This new class of inference methods can be used in a large number of domains and some examples are given. The present paper is based on Gouriéroux-Monfort (1992), and Gouriéroux-Monfort-Renault (1993), respectively GM and GMR hereafter. Invited paper at the Conference on ?Statistical Tests: Methodology and Econometric Applications?, held in Bologna, Italy, 27–28 May 1993.     

Homogeneity Test of Risk Differences of Marginal and Conditional Probabilities in Several Incomplete Correlated 2 × 2 Tables

This article considers K pairs of incomplete correlated 2 × 2 tables in which the interesting measurement is the risk difference between marginal and conditional probabilities. A Wald-type statistic and a score-type statistic are presented to test the homogeneity hypothesis about risk differences across strata. Powers and sample size formulae based on the above two statistics are deduced. Figures about sample size against risk difference (or marginal probability) are given. A real example is used to illustrate the proposed methods.     

Admissible tests for exponential families with finite support 2

We state some necessary and sufficient conditions for admissibility of tests for a simple and a composite null hypotheses against ”one-sided” alternatives on multivariate exponential distributions with discrete support.

Admissibility of the maximum likelihood test for “one –sided” alternatives and z χ²test for the independence hypothesis in r× scontingency tables is deduced among others.     

Analysis of ordered categorical data with score averaging: with applications to case-control genetic associations

The trend test is often used for the analysis of 2×K ordered categorical data, in which K pre-specified increasing scores are used. There have been discussions on how to assign these scores and the impact of the outcomes on different scores. The scores are often assigned based on the data-generating model. When this model is unknown, using the trend test is not robust. We discuss the weighted average of a trend test over all scientifically plausible choices of scores or models. This approach is more computationally efficient than a commonly used robust test MAX when K is large. Our discussion is for any ordered 2×K table, but simulation and applications to real data are focused on case-control genetic association studies. Although there is no single test optimal for all choices of scores, our numerical results show that some score averaging tests can achieve the performance of MAX.     

Constrained penalized splines

The penalized spline is a popular method for function estimation when the assumption of “smoothness” is valid. In this paper, methods for estimation and inference are proposed using penalized splines under additional constraints of shape, such as monotonicity or convexity. The constrained penalized spline estimator is shown to have the same convergence rates as the corresponding unconstrained penalized spline, although in practice the squared error loss is typically smaller for the constrained versions. The penalty parameter may be chosen with generalized cross‐validation, which also provides a method for determining if the shape restrictions hold. The method is not a formal hypothesis test, but is shown to have nice large‐sample properties, and simulations show that it compares well with existing tests for monotonicity. Extensions to the partial linear model, the generalized regression model, and the varying coefficient model are given, and examples demonstrate the utility of the methods. The Canadian Journal of Statistics 40: 190–206; 2012 © 2012 Statistical Society of Canada     

Optimal Method for Realizing Two-Sided Inferences About a Linear Combination of Two Proportions

In order to reach the inference about a linear combination of two independent binomial proportions, various procedures exist (Wald's classic method, the exact, approximate, or maximized score methods, and the Newcombe-Zou method). This article defines and evaluates 25 different methods of inference, and selects the ones with the best behavior. In general terms, the optimal method is the classic Wald method applied to the data to which z ² _α/2/4 successes and z ² _α/2/4 failures are added (≈1 if α = 5%) if no sample proportion has a value of 0 or 1 (otherwise the added increase may be different).

Supplemental materials are available for this article. Go to the publisher's online edition of Communications in Statistics - Simulation and Computation to view the free supplemental file.     

Robust Lagrange multiplier test for detecting ARCH/GARCH effect using permutation and bootstrap

The Lagrange Multiplier (LM) test is one of the principal tools to detect ARCH and GARCH effects in financial data analysis. However, when the underlying data are non‐normal, which is often the case in practice, the asymptotic LM test, based on the χ²‐approximation of critical values, is known to perform poorly, particularly for small and moderate sample sizes. In this paper we propose to employ two re‐sampling techniques to find critical values of the LM test, namely permutation and bootstrap. We derive the properties of exactness and asymptotically correctness for the permutation and bootstrap LM tests, respectively. Our numerical studies indicate that the proposed re‐sampled algorithms significantly improve size and power of the LM test in both skewed and heavy‐tailed processes. We also illustrate our new approaches with an application to the analysis of the Euro/USD currency exchange rates and the German stock index. The Canadian Journal of Statistics 40: 405–426; 2012 © 2012 Statistical Society of Canada     

On limiting distribution laws of statistics analogous to pearson's chi-square

Two test-statistics analogous to Pearson's chi-square test function - given in (1.6) and (1.7) - are investigated. These statistics utilize, apart from the number of sample elements lying in the respective intervals of the partition, their positions within the intervals too. It is shown that the test-statistics are asymptotically distributed - as the sample size N tends to infinity - according to the x ²distribution with parameter r, i.e. the number of intervals chosen. The limiting distribution of the test statistics under the null-hypothesis when N tends to the infinity and r =O(N ^α) (0<α<1), further the consistency of the tests based on these statistics is considered. Some remarks are made concerning the efficiency of the corresponding goodness of fit tests also; the authors intend to return to a more detailed treatment of the efficiency later.     

Testing Inference in Inflated Beta Regressions under Model Misspecification

We consider testing inference in inflated beta regressions subject to model misspecification. In particular, quasi-z tests based on sandwich covariance matrix estimators are described and their finite sample behavior is investigated via Monte Carlo simulations. The numerical evidence shows that quasi-z testing inference can be considerably more accurate than inference made through the usual z tests, especially when there is model misspecification. Interval estimation is also considered. We also present an empirical application that uses real (not simulated) data.     

Weighted product index and its two-independent-sample comparison based on weighted sensitivity and specificity

Background: On the basis of statistical methods about index S (S = SEN × SPE), we develop a new weighted ways (weighted product index S_w) of combining sensitivity and specificity with user-defined weights. Methods: The new weighted product index S_w is defined as S_w = (SEN) (Youden 1950)^2w × (SPE) (Youden 1950) ^2(1?w) Results: For the large sample, the test statistics Z of two-independent-sample weighted product indices can either be a monotonous increasing/decreasing function or a no-monotonous function of weight w. Type I error of this statistics can be guaranteed close to the nominal level of 5%, which is more conservative than the weighted Youden index from simulation.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.