A multiple comparisons procedure for detecting differences between treatments and a control in two-factor experiments

In many experiments researchers are interested in comparing several treat¬ment means with a control mean. Their primary interest is to determine whether any treatments are significantly better than the control Several test procedures have been proposed in the literature, but only few of them can pro¬vide simultaneous confidence lower bounds. A new test statistic is proposed to compare treatment means with a control mean in two-factor experiments. Some upper percentage points are tabulated. It yields sharp simultaneous confidence lower bounds for the differences of such means. The new test is forresponding author.     

Bounding sample size projections for the area under a ROC curve

Studies of diagnostic tests are often designed with the goal of estimating the area under the receiver operating characteristic curve (AUC) because the AUC is a natural summary of a test's overall diagnostic ability. However, sample size projections dealing with AUCs are very sensitive to assumptions about the variance of the empirical AUC estimator, which depends on two correlation parameters. While these correlation parameters can be estimated from the available data, in practice it is hard to find reliable estimates before the study is conducted. Here we derive achievable bounds on the projected sample size that are free of these two correlation parameters. The lower bound is the smallest sample size that would yield the desired level of precision for some model, while the upper bound is the smallest sample size that would yield the desired level of precision for all models. These bounds are important reference points when designing a single or multi-arm study; they are the absolute minimum and maximum sample size that would ever be required. When the study design includes multiple readers or interpreters of the test, we derive bounds pertaining to the average reader AUC and the ‘pooled’ or overall AUC for the population of readers. These upper bounds for multireader studies are not too conservative when several readers are involved.     

Test of fit for Marshall–Olkin distributions with applications

Marshall and Olkin [1967. A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44], introduced a bivariate distribution with exponential marginals, which generalizes the simple case of a bivariate random variable with independent exponential components. The distribution is popular under the name ‘Marshall–Olkin distribution’, and has been extended to the multivariate case. L2-type statistics are constructed for testing the composite null hypothesis of the Marshall–Olkin distribution with unspecified parameters. The test statistics utilize the empirical Laplace transform with consistently estimated parameters. Asymptotic properties pertaining to the null distribution of the test statistic and the consistency of the test are investigated. Theoretical results are accompanied by a simulation study, and real-data applications.     

Comparison to control in logistic regression

We are interested in comparing logistic regressions for several test treatments or populations with a logistic regression for a standard treatment or population. The research was motivated by some real life problems, which are discussed as data examples. We propose a step-down likelihood ratio method for declaring differences between the test treatments or populations and the standard treatment or population. Competitors based on the sequentially rejective Bonferroni Wald statistic, sequentially rejective exact Wald statistic and Reiers?l's statistic are also discussed. It is shown that the proposed method asymptotically controls the probability of type I error. A Monte Carlo simulation shows that the proposed method performs well for relatively small sample sizes, outperforming its competitors.     

Combined multiple testing by censored empirical likelihood

We propose a new procedure for combining multiple tests in samples of right-censored observations. The new method is based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. We prove a version of Wilks’ theorem for the multiple constrained censored empirical likelihood ratio, which provides a simple reference distribution for the test statistic of our proposed method. A useful application of the proposed method is, for example, examining the survival experience of different populations by combining different weighted log-rank tests. Real data examples are given using the log-rank and Gehan-Wilcoxon tests. In a simulation study of two sample survival data, we compare the proposed method of combining tests to previously developed procedures. The results demonstrate that, in addition to its computational simplicity, the combined test performs comparably to, and in some situations more reliably than previously developed procedures. Statistical software is available in the R package ‘emplik’.     

Many-to-one comparison of nonlinear growth curves for Washington's Red Delicious apple

In this article, we are interested in comparing growth curves for the Red Delicious apple in several locations to that of a reference site. Although such multiple comparisons are common for linear models, statistical techniques for nonlinear models are not prolific. We theoretically derive a test statistic, considering the issues of sample size and design points. Under equal sample sizes and same design points, our test statistic is based on the maximum of an equi-correlated multivariate chi-square distribution. Under unequal sample sizes and design points, we derive a general correlation structure, and then utilize the multivariate normal distribution to numerically compute critical points for the maximum of the multivariate chi-square. We apply this statistical technique to compare the growth of Red Delicious apples at six locations to a reference site in the state of Washington in 2009. Finally, we perform simulations to verify the performance of our proposed procedure for Type I error and marginal power. Our proposed method performs well in regard to both.     

On inference from Markov chain macro-data using transforms

We consider data that are longitudinal, arising from n individuals over m time periods. Each individual moves according to the same homogeneous Markov chain, with s states. If the individual sample paths are observed, so that ‘micro-data’ are available, the transition probability matrix is estimated by maximum likelihood straightforwardly from the transition counts. If only the overall numbers in the various states at each time point are observed, we have ‘macro-data’, and the likelihood function is difficult to compute. In that case a variety of methods has been proposed in the literature. In this paper we propose methods based on generating functions and investigate their performance.     

A combined p-value test for multiple hypothesis testing

Tests that combine p-values, such as Fisher's product test, are popular to test the global null hypothesis H₀ that each of n component null hypotheses, H₁,…,H_n, is true versus the alternative that at least one of H₁,…,H_n is false, since they are more powerful than classical multiple tests such as the Bonferroni test and the Simes tests. Recent modifications of Fisher's product test, popular in the analysis of large scale genetic studies include the truncated product method (TPM) of Zaykin et al. (2002), the rank truncated product (RTP) test of Dudbridge and Koeleman (2003) and more recently, a permutation based test—the adaptive rank truncated product (ARTP) method of Yu et al. (2009). The TPM and RTP methods require users' specification of a truncation point. The ARTP method improves the performance of the RTP method by optimizing selection of the truncation point over a set of pre-specified candidate points. In this paper we extend the ARTP by proposing to use all the possible truncation points {1,…,n} as the candidate truncation points. Furthermore, we derive the theoretical probability distribution of the test statistic under the global null hypothesis H₀. Simulations are conducted to compare the performance of the proposed test with the Bonferroni test, the Simes test, the RTP test, and Fisher's product test. The simulation results show that the proposed test has higher power than the Bonferroni test and the Simes test, as well as the RTP method. It is also significantly more powerful than Fisher's product test when the number of truly false hypotheses is small relative to the total number of hypotheses, and has comparable power to Fisher's product test otherwise.     

Optimal weighted two-sample t-test with partially paired data in a unified framework

In this paper, we provide a unified framework for two-sample t-test with partially paired data. We show that many existing two-sample t-tests with partially paired data can be viewed as special members in our unified framework. Some shortcomings of these t-tests are discussed. We also propose the asymptotically optimal weighted linear combination of the test statistics comparing all four paired and unpaired data sets. Simulation studies are used to illustrate the performance of our proposed asymptotically optimal weighted combinations of test statistics and compare with some existing methods. It is found that our proposed test statistic is generally more powerful. Three real data sets about CD4 count, DNA extraction concentrations, and the quality of sleep are also analyzed by using our newly introduced test statistic.     

Parametric and permutation testing for multivariate monotonic alternatives

We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Biometrica 46:328–335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465–472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathematica 38:151–161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.     

A Spatio-Temporal Model for Functional Magnetic Resonance Imaging Data – with a View to Resting State Networks

Abstract.  Functional magnetic resonance imaging (fMRI) is a technique for studying the active human brain. During the fMRI experiment, a sequence of MR images is obtained, where the brain is represented as a set of voxels. The data obtained are a realization of a complex spatio-temporal process with many sources of variation, both biological and technical. We present a spatio-temporal point process model approach for fMRI data where the temporal and spatial activation are modelled simultaneously. It is possible to analyse other characteristics of the data than just the locations of active brain regions, such as the interaction between the active regions. We discuss both classical statistical inference and Bayesian inference in the model. We analyse simulated data without repeated stimuli both for location of the activated regions and for interactions between the activated regions. An example of analysis of fMRI data, using this approach, is presented.     

A Bayesian predictive inference for small area means incorporating covariates and sampling weights

The main goal in small area estimation is to use models to ‘borrow strength’ from the ensemble because the direct estimates of small area parameters are generally unreliable. However, model-based estimates from the small areas do not usually match the value of the single estimate for the large area. Benchmarking is done by applying a constraint, internally or externally, to ensure that the ‘total’ of the small areas matches the ‘grand total’. This is particularly useful because it is difficult to check model assumptions owing to the sparseness of the data. We use a Bayesian nested error regression model, which incorporates unit-level covariates and sampling weights, to develop a method to internally benchmark the finite population means of small areas. We use two examples to illustrate our method. We also perform a simulation study to further assess the properties of our method.     

Sums of weighted spacings and a test for the logistic distribution

A simple proof for a theorem of Csörgö and Révész (1981b and 1984) concerning sums of weighted spacings is given, The conditions of the theorem are relaxed. As an application, a goodness-of-fit test for the logistic distribution is proposed. The percentage points of the proposed test statistic are obtained by a simulation experiment.     

Testing of two sample proportional intensity assumption for non-homogeneous Poisson processes

For two independent non-homogeneous Poisson processes with unknown intensities we propose a test for testing the hypothesis that the ratio of the intensities is constant versus it is increasing on (0,t]. The existing test procedures for testing such relative trends are based on conditioning on the number of failures observed in (0,t] from the two processes. Our test is unconditional and is based on the original time truncated data which enables us to have meaningful asymptotics. We obtain the asymptotic null distribution (as t becomes large) of the proposed test statistic and show that the proposed test is consistent against several large classes of alternatives. It was observed by Park and Kim (IEEE. Trans. Rehab. 40 (1), 1992, 107–111) that it is difficult to distinguish between the power-law and log-linear processes for certain parameter values. We show that our test is consistent for such alternatives also.     

An extreme null distribution approach to the problem of paired comparisons

David (1963) and Davidson & Farquhar(1976) contain extensive bibliographies of proposed approaches to problems involving paired comparisons. However, each of the discussed methods that is based on a hypothesis test, relies heavily on the assumption that all paired comparisons are made independently. In this paper we eliminate this assumption and develop a new procedure based on an adaptation of a statistic considered by Kendall & Babington Smith (1940). We show that their original test procedure substantially underestimates the true significance level if the comparisons are not made independently. Our modification utilizes the approach developed in Costello & Wolfe (1985) for the problem of agreement between two groups of judges and relies heavily on computer-generated tables.     

Computation of the percentage points and the power for the two-sided Kolmogorov-Smirnov one sample test

Two recursive schemes are presented for the calculation of the probabilityP(g(x)≤S _n (x)≤h(x) for allx∈®), whereS _n is the empirical distribution function of a sample from a continuous distribution andh, g are continuous and isotone functions. The results are specialized for the calculation of the distribution and the corresponding percentage points of the test statistic of the two-sided Kolmogorov-Smirnov one sample test. The schemes allow the calculation of the power of the test too. Finally an extensive tabulation of percentage points for the Kolmogorov-Smirnov test is given.     

On reciprocal symmetry

On the positive half line, there are two natural, and complementary, analogues of the single notion of symmetry of distributions on the real line. One is the R-symmetry recently proposed and investigated by Mudholkar and Wang [2007. IG-symmetry and R-symmetry: interrelations and applications to the inverse Gaussian theory. J. Statist. Plann. Inference 137, 3655–3671]; the other is the ‘log-symmetry’ investigated here. Log-symmetry can be thought of either in terms of a random variable having the same distribution as its reciprocal or as ordinary symmetry of the distribution of the logged random variable. Various properties, analogies, comparisons and consequences are investigated.