Parametric Bootstrap Test Research Articles

An Approximate Hotelling T<sup>2</sup>-Test for Heteroscedastic One-Way MANOVA

In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling T2 distribution with one parameter estimated from the data, resulting in the so-called approximate Hotelling T2 (AHT) test. The AHT test is shown to be invariant under affine transformation, different choices of the contrast matrix specifying the same hypothesis, and different labeling schemes of the mean vectors. The AHT test can be simply conducted using the usual F-distribution. Simulation studies and real data applications show that the AHT test substantially outperforms the test of [1] and is comparable to the parametric bootstrap (PB) test of [2] for the multivariate k-sample Behrens-Fisher problem which is a special case of the GLHT problem in heteroscedastic one-way MANOVA.

A modified Bartlett test for heteroscedastic one-way MANOVA

In this paper, we investigate tests of linear hypotheses in heteroscedastic one-way MANOVA via proposing a modified Bartlett (MB) test. The MB test is easy to conduct via using the usual χ2-table. It is shown to be invariant under affine transformations, different choices of the contrast matrix used to define the same hypothesis and different labeling schemes of the mean vectors. Simulation studies and real data applications demonstrate that the MB test performs well and is generally comparable to Krishnamoorthy and Lu’s (J Statist Comput Simul 80(8):873–887, 2010) parametric bootstrap test in terms of size controlling and power.

Finite-sample bootstrap inference in GARCH models with heavy-tailed innovations

A general method is proposed for the construction of valid simultaneous confidence sets in the context of stationary GARCH models. The proposed method proceeds by numerically inverting the conventional likelihood ratio test. In order to hedge against the risk of a spurious rejection, candidate points that are rejected by the conventional test undergo a finite-sample parametric bootstrap test. A projection technique is then exploited to produce conservative confidence sets for general functions of the parameters. A simulation study illustrates the performance of the parametric bootstrap approach in the context of a GARCH model with heavy-tailed and skewed innovations. That model is then used in an empirical application to construct simultaneous confidence intervals for multi-step ahead volatility forecasts for the returns on a major stock market index.

Testing for space–time interaction in conditional autoregressive models

Data on disease incidence or mortality over a set of contiguous regions have been commonly used to describe geographic patterns of disease, helping epidemiologists and public health researchers to identify possible etiologic factors. Nowadays, the availability of historical mortality registers offers the possibility of going further, describing the spatio‐temporal distribution of risks. The literature on spatio‐temporal modeling of risks is very rich, and it is mainly focused on the use of conditional autoregressive models from a fully Bayesian perspective. The complexity of the estimation procedure makes the Empirical Bayes approach a plausible alternative. In this context, it is of interest to test for interaction between space and time, as an absence of space–time interactions simplifies modeling and interpretation. In this work, a score test is derived as well as a bootstrap approximation of its null distribution. A parametric bootstrap test is also provided for comparison purposes. Results are illustrated using brain cancer mortality data from Spain in the period 1996–2005. Copyright © 2011 John Wiley & Sons, Ltd.

A Holarctic Biogeographical Analysis of the Collembola (Arthropoda, Hexapoda) Unravels Recent Post-Glacial Colonization Patterns.

We aimed to describe the main Arctic biogeographical patterns of the Collembola, and analyze historical factors and current climatic regimes determining Arctic collembolan species distribution. Furthermore, we aimed to identify possible dispersal routes, colonization sources and glacial refugia for Arctic collembola. We implemented a Gaussian Mixture Clustering method on species distribution ranges and applied a distance- based parametric bootstrap test on presence-absence collembolan species distribution data. Additionally, multivariate analysis was performed considering species distributions, biodiversity, cluster distribution and environmental factors (temperature and precipitation). No clear relation was found between current climatic regimes and species distribution in the Arctic. Gaussian Mixture Clustering found common elements within Siberian areas, Atlantic areas, the Canadian Arctic, a mid-Siberian cluster and specific Beringian elements, following the same pattern previously described, using a variety of molecular methods, for Arctic plants. Species distribution hence indicate the influence of recent glacial history, as LGM glacial refugia (mid-Siberia, and Beringia) and major dispersal routes to high Arctic island groups can be identified. Endemic species are found in the high Arctic, but no specific biogeographical pattern can be clearly identified as a sign of high Arctic glacial refugia. Ocean currents patterns are suggested as being an important factor shaping the distribution of Arctic Collembola, which is consistent with Antarctic studies in collembolan biogeography. The clear relations between cluster distribution and geographical areas considering their recent glacial history, lack of relationship of species distribution with current climatic regimes, and consistency with previously described Arctic patterns in a series of organisms inferred using a variety of methods, suggest that historical phenomena shaping contemporary collembolan distribution can be inferred through biogeographical analysis.

Effect of non‐normality on test statistics for one‐way independent groups designs

The data obtained from one-way independent groups designs is typically non-normal in form and rarely equally variable across treatment populations (i.e., population variances are heterogeneous). Consequently, the classical test statistic that is used to assess statistical significance (i.e., the analysis of variance F test) typically provides invalid results (e.g., too many Type I errors, reduced power). For this reason, there has been considerable interest in finding a test statistic that is appropriate under conditions of non-normality and variance heterogeneity. Previously recommended procedures for analysing such data include the James test, the Welch test applied either to the usual least squares estimators of central tendency and variability, or the Welch test with robust estimators (i.e., trimmed means and Winsorized variances). A new statistic proposed by Krishnamoorthy, Lu, and Mathew, intended to deal with heterogeneous variances, though not non-normality, uses a parametric bootstrap procedure. In their investigation of the parametric bootstrap test, the authors examined its operating characteristics under limited conditions and did not compare it to the Welch test based on robust estimators. Thus, we investigated how the parametric bootstrap procedure and a modified parametric bootstrap procedure based on trimmed means perform relative to previously recommended procedures when data are non-normal and heterogeneous. The results indicated that the tests based on trimmed means offer the best Type I error control and power when variances are unequal and at least some of the distribution shapes are non-normal.

Notes on the Goodness-of-Fit Tests for the Ordinal Response Model

In this paper we discuss some cautionary notes in using the Pearson chi-squared test statistic for the goodness-of-fit of the ordinal response model. If a model includes continuous type explanatory variables, the resulting table from the fit of a model is not a regular one in the sense that the cell boundaries are not fixed but randomly determined by some other criteria. The chi-squared statistic from this kind of table does not have a limiting chi-square distribution in general and we need to be very cautious of the use of a chi-squared type goodness-of-fit test. We also study the limiting distribution of the chi-squared type statistic for testing the goodness-of-fit of cumulative logit models with ordinal responses. The regularity conditions necessary to the limiting distribution will be reformulated in the framework of the cumulative logit model by modifying those of Moore and Spruill (1975). Due to the complex limiting distribution, a parametric bootstrap testing procedure is a good alternative and we explained the suggested method through a practical example of an ordinal response dataset.

Comparison of several means: A fiducial based approach

To study the equality of several normal means when the variances are unknown and unequal, we propose a fiducial based test, and theoretically examine the frequentist property of the proposed test. We numerically compare the performance of the proposed approach with several tests, recently developed such as the generalized F-test by Weerahandi (1995a), the trimmed mean-based test by Lix and Keselman (1998), and the parametric bootstrap test by Krishnamoorthy et al. (2007). The simulation results indicate that the proposed approach can provide a reasonable p -value via a few straightforward simulation steps. Finally, the proposed approach is applied to an analysis of two real data sets for illustration.

Male vocalizations, female discrimination and molecular phylogeny: multiple perspectives on the taxonomic status of a critically endangered Caribbean frog

Until recently, morphology has been the predominant basis on which taxonomic decisions have been made. Now, many sources of data inform decisions in taxonomy, yet few studies are available that directly compare the conclusions made on the basis of different datasets. The difficulty of reaching clear taxonomic decisions is further complicated by the existence of allopatric populations, which may differ from other populations in notable ways yet not be distinct evolutionary units. We analyzed differences at the molecular level based on sequences of two mitochondrial genes, analyzed acoustic differences in male vocalizations (nine variables) and conducted a phonotaxis experiment with females to assess the taxonomic status of two putative Caribbean frog species (Mannophryne olmonae and Mannophryne trinitatis, Aromobatidae), which some authors have indicated as conspecific. A 16S gene tree (75 sequences of 15 putative species, 530 bp), a parametric bootstrap test, and the results of acoustic comparisons suggested that these entities were evolutionarily distinct. However, in the phonotaxis experiment, females of either species did not display significant preference among the male vocalizations presented. On the basis of the bioacoustic data and the 16S gene tree, we conclude that these taxa are distinct and suggest that lack of selection for pre-mating isolation in allopatry explains the lack of discrimination shown by females. Phonotaxis experiments in taxa with acoustic means of mate attraction should continue to be useful in assessing the evolutionary independence of putative sympatric entities, but our results suggest that they should be employed and interpreted cautiously when applied to allopatric populations. To most accurately assess the boundaries of evolutionary lineages, a pluralistic approach, utilizing as many sources of data as possible, is desirable.

Permutation and Parametric Bootstrap Tests for Gene-Gene and Gene-Environment Interactions

Permutation tests are widely used in genomic research as a straightforward way to obtain reliable statistical inference without making strong distributional assumptions. However, in this paper we show that in genetic association studies it is not typically possible to construct exact permutation tests of gene-gene or gene-environment interaction hypotheses. We describe an alternative to the permutation approach in testing for interaction, a parametric bootstrap approach. Using simulations, we compare the finite-sample properties of a few often-used permutation tests and the parametric bootstrap. We consider interactions of an exposure with single and multiple polymorphisms. Finally, we address when permutation tests of interaction will be approximately valid in large samples for specific test statistics.

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.

Accuracy of p-values of approximate tests in testing for equality of means under unequal variances

Abstract Seemingly, testing for fixed effects in linear models with variance-covariance components has been solved for decades. However, even in simple situations such as in fixed one-way model with heteroscedastic variances (a multiple means case of the Behrens-Fisher problem) the questions of statistical properties of various approximations of test statistics are still alive. Here we present a brief overview of several approaches suggested in the literature as well as those available in statistical software, accompanied by a simulation study in which the accuracy of p-values is studied. Our interest is limited here to the Welch’s test, the Satterthwaite-Fai-Cornelius test, the Kenward-Roger test, the simple ANOVA F-test, and the parametric bootstrap test. We conclude that for small sample sizes, regardless the number of compared means and the heterogeneity of variance, the ANOVA F-test p-value performs the best. For higher sample sizes (at least 5 per group), the parametric bootstrap performs well, and the Kenward-Roger test also performs well.

Heteroscedastic ANOVA: old p values, new views

The generalization of the Behrens–Fisher problem to comparing more than two means from nonhomogeneous populations has attracted the attention of statisticians for many decades. Several approaches offer different approximations to the distribution of the test statistic. The question of statistical properties of these approximations is still alive. Here, we present a brief overview of several approaches suggested in the literature and implemented in software with a focus on investigating the accuracy of p values as well as their dependence on nuisance parameters and on the underlying assumption of normality. We illustrate by simulation the behavior of p values. In addition to the Satterthwaite–Fai–Cornelius test, the Kenward–Roger test, the simple ANOVA F test, the parametric bootstrap test, and the generalized F test will be briefly discussed.

Superstars without Talent? The Yule Distribution Controversy

Chung and Cox (1994) provided an intuitively appealing stochastic model indicating that superstars may exist regardless of talent, giving rise to the Yule distribution. We adopt a different empirical approach and test its goodness of fit using a parametric bootstrap and several powerful test statistics. Just like the discrete Pareto distribution, it is overwhelmingly rejected: it is a fairly accurate approximation of the lower quantiles of the superstar distribution but overestimates the snowball effect that makes consumers purchase records of the most successful artists. In other words, the Yule distribution captures stardom, but not superstardom. A generalization of the Yule distribution provides an excellent fit in two of the three data sets.

Goodness-of-Fit Tests for the Ordinal Response Models with Misspecified Links

The Pearson chi-squared statistic or the deviance statistic is widely used in assessing the goodness-of-fit of the generalized linear models. But these statistics are not proper in the situation of continuous explanatory variables which results in the sparseness of cell frequencies. We propose a goodness-of-fit test statistic for the cumulative logit models with ordinal responses. We consider the grouping of a dataset based on the ordinal scores obtained by fitting the assumed model. We propose the Pearson chi-squared type test statistic, which is obtained from the cross-classified table formed by the subgroups of ordinal scores and the response categories. Because the limiting distribution of the chi-squared type statistic is intractable we suggest the parametric bootstrap testing procedure to approximate the distribution of the proposed test statistic.

Homogeneity tests for several Poisson populations

In this paper we compare the size distortions and powers for Pearson’s χ 2 -statistic, likelihood ratio statistic L R , score statistic S C and two statistics, which we call U T and V T here, proposed by [Potthoff, R.F., Whittinghill, M., 1966. Testing for homogeneity: II. The Poisson distribution. Biometrika 53, 183–190] for testing the equality of the rates of K Poisson processes. Asymptotic tests and parametric bootstrap tests are considered. It is found that the asymptotic U T test is too conservative to be recommended, while the other four asymptotic tests perform similarly and their powers are close to those of their parametric bootstrap counterparts when the observed counts are large enough. When the observed counts are not large, Monte Carlo simulation suggested that the asymptotic test using S C , L R and U T statistics are discouraged; none of the parametric bootstrap tests with the five statistics considered here is uniformly best or worst, and the asymptotic tests using Pearson’s χ 2 and V T always have similar powers to their bootstrap counterparts. Thus, the asymptotic Pearson’s χ 2 and V T tests have an advantage over all five parametric bootstrap tests in terms of their computational simplicity and convenience, and over the other four asymptotic tests in terms of their powers and size distortions.

Bootstrap tests for variance components in generalized linear mixed models

AbstractIn many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada

Drawing inferences about the coancestry coefficient

The coancestry coefficient, also known as the population structure parameter, is of great interest in population genetics. It can be thought of as the intraclass correlation of pairs of alleles within populations and it can serve as a measure of genetic distance between populations. For a general class of evolutionary models it determines the distribution of allele frequencies among populations. Under more restrictive models it can be regarded as the probability of identity by descent of any pair of alleles at a locus within a random mating population. In this paper we review estimation procedures that use the method of moments or are maximum likelihood under the assumption of normally distributed allele frequencies. We then consider the problem of testing hypotheses about this parameter. In addition to parametric and non-parametric bootstrap tests we present an asymptotically-distributed chi-square test. This test reduces to the contingency-table test for equal sample sizes across populations. Our new test appears to be more powerful than previous tests, especially for loci with multiple alleles. We apply our methods to HapMap SNP data to confirm that the coancestry coefficient for humans is strictly positive.

Exact and asymptotic tests for possibly non-regular hypotheses on stochastic volatility models

Abstract We study the problem of testing hypotheses on the parameters of one- and two-factor stochastic volatility models (SV), allowing for the possible presence of non-regularities such as singular moment conditions and unidentified parameters, which can lead to non-standard asymptotic distributions. We focus on the development of simulation-based exact procedures–whose level can be controlled in finite samples–as well as on large-sample procedures which remain valid under non-regular conditions. We consider Wald-type, score-type and likelihood-ratio-type tests based on a simple moment estimator, which can be easily simulated. We also propose a C ( α ) -type test which is very easy to implement and exhibits relatively good size and power properties. Besides usual linear restrictions on the SV model coefficients, the problems studied include testing homoskedasticity against a SV alternative (which involves singular moment conditions under the null hypothesis) and testing the null hypothesis of one factor driving the dynamics of the volatility process against two factors (which raises identification difficulties). Three ways of implementing the tests based on alternative statistics are compared: asymptotic critical values (when available), a local Monte Carlo (or parametric bootstrap) test procedure, and a maximized Monte Carlo (MMC) procedure. The size and power properties of the proposed tests are examined in a simulation experiment. The results indicate that the C ( α ) -based tests (built upon the simple moment estimator available in closed form) have good size and power properties for regular hypotheses, while Monte Carlo tests are much more reliable than those based on asymptotic critical values. Further, in cases where the parametric bootstrap appears to fail (for example, in the presence of identification problems), the MMC procedure easily controls the level of the tests. Moreover, MMC-based tests exhibit relatively good power performance despite the conservative feature of the procedure. Finally, we present an application to a time series of returns on the Standard and Poor’s Composite Price Index.

Phylogeography of the Pygmy Rain Frog (Pristimantis ridens) across the lowland wet forests of isthmian Central America

We used a phylogeographic approach to elucidate the evolutionary history of a lineage of frogs, known as Pristimantis (formerly Eleutherodactylus) ridens (Anura: Brachycephalidae), restricted to the wet forests occurring along the Caribbean versant of isthmian Central America as well as the disjunct wet forest on the Pacific slope of Costa Rica. We placed our phylogeographic study of P. ridens within a larger molecular phylogenetic analysis of Central American Pristimantis. All phylogenetic inferences were based on a 1455 base pair fragment of mitochondrial DNA, containing the complete ND2 gene and five flanking tRNA genes. Our reconstruction of the intraspecific phylogeny of P. ridens yielded a basal trichotomy dating to an estimated 12+ million years ago (Ma), consisting of central Panama, western Panama, and Costa Rica plus Honduras. Thus, the presence of P. ridens appears to predate the completion of the Isthmus 3.1Ma. Using a parametric bootstrap (SOWH) test, we evaluated four a priori zoogeographic hypotheses for the origin and spread of P. ridens. This analysis suggested that the P. ridens populations on the Caribbean versant of Costa Rica were established by Pacific versant ancestors only recently, in contrast to the very old lineages found in Panama. Our results support a model of Miocene colonization, long-term geographic stasis, followed by rapid dispersal across the Caribbean lowlands during the Pliocene or Pleistocene.