A Monte Carlo Study to Evaluate the Performance of Selected Tests of Homogeneity in Sparse Contingency Tables

We address the problem of tests of homogeneity in two-way contingency tables in case-control studies when the case category is subdivided into k subcategories. In this situation, we have two cells with large frequencies and 2 X k cells with frequencies that become small as k increases. We propose two ad hoc statistics in which a statistic for the sparse cells is combined with a statistic for the cells with large frequencies. We will study these tests along with the Pearson test (using a chi-square approximation) in a Monte Carlo simulation study. Two sets of null hypothesis models and two sets of alternative hypothesis models are considered. The best test for the models considered is the usual Pearson test (using an approximate chi-square distribution) although the ad hoc models are more powerful under one alternative model considered.     

A New Statistical Goodness-of-Fit Test Based on Graphical Representation

A statistical goodness-of-fit test, based on representing the sample observations by linked vectors, is developed. The direction and the length of the linked vectors are defined as functions of the expected values of the order statistics and sample order statistics, respectively. The underlying method can be used to test distributional assumptions for any location-scale family. A test statistic Q_n is introduced and some of its properties are studied. It is shown that the proposed test can be generalized to test if two or more independent samples come from the same distribution. The test procedure provides a graphical method of identifying the true distribution when the null hypothesis is rejected.     

Measures of Imbalance for Higher Order Designs

In this paper we consider a generalization of the measures of imbalance given by AHRENS and PINCUS (1981) considering the cases: m-fold hierarchical model and m-way classification model in order to quantify the degree of imbalance in an unbalanced design. These measures of imbalance satisfy the same properties as those for the one-way classification model.     

Confidence Intervals on Ratios of Variance Components for the Unbalanced Two Factor Nested Model

The model considered in this article is the two-factor nested unbalanced variance component model: for p = 1, 2, …, P; q = 1, 2, …, Q_p; and r = 1, 2, …, R_pq. The random variables Y_pqr are observable. The constant μ is an unknown parameter, and A_p, B_pq and C_pqr are (unobservable) normal and independently distributed random variables with zero means and finite variances σ²_A, σ²_B, and σ²_C, respectively. Approximate confidence intervals on ?_A and ?_B using unweighted means are derived, where The performance of these approximate confidence intervals are evaluated using computer simulation. The simulated results indicate that these proposed confidence intervals perform satisfactorily and can be used in applied problems.     

On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t‐test is optimum. On the other hand, if the differences are not normal, the t‐test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non‐parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro‐Wilk test of normality to decide whether to use the t‐test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t‐tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy‐tailed, than currently used methods.     

Measures of Imbalance for Unbalanced Models

In this paper we present a procedure to measure the degree of imbalance of an unbalanced data set. The procedure is based on choosing an appropriate loglinear model for the subclass frequencies of the data. A measure of imbalance is then introduced as some function of the chi-squared statistic used in the goodness-of-fit test for the loglinear model. The proposed procedure can also be used to measure departures from certain types of balance, such as proportionality of subclass frequencies, partial balance, and last-stage uniformity.     

Fertility Adjustments in Rectangular Field Trials

Based on Mandel's additive plus multipicative model for ANOVA, this paper presents a procedure to represent fertility differences in field trials arranged in rectangular plans. After fitting any block and treatment effects, the residuals are assigned to their positions in the rectangular planting plan, and then decomposed into orthogonal components, of which the largest ist attributed to fertility. If the variance associated with this component is significantly larger than those remaining, the observed data are then adjusted for it; the process is repeated until there is no significant component. The procedure is illustrated by data from a variety trial.     

S-Sample Smooth Goodness of Fit Testing: Rederivation and Monte Carlo Assessment

S-sample smooth goodness of fit tests may be constructed using components from one sample goodness of fit testing. Each sample could be assessed for consistency with a target distribution using these components, although that is not our objective here. Contrasts in the components may be used to assess consistency of the samples with each other. If all the samples are consistent, we could then conveniently perform a one-sample goodness of fit test for the target distribution. If the samples are not consistent, an LSD-type analysis can be performed on the one-sample components to identify where the differences between occur. This approach gives a detailed and informative scrutiny of the data.     

Improving Power for Testing Genetic Association in Case–Control Studies by Reducing the Alternative Space

Summary .  To detect association between a genetic marker and a disease in case–control studies, the Cochran–Armitage trend test is typically used. The trend test is locally optimal when the genetic model is correctly specified. However, in practice, the underlying genetic model, and hence the optimal trend test, are usually unknown. In this case, Pearson's chi-squared test, the maximum of three trend test statistics (optimal for the recessive, additive, and dominant models), and the test based on genetic model selection (GMS) are useful. In this article, we first modify the existing GMS method so that it can be used when the risk allele is unknown. Then we propose a new approach by excluding a genetic model that is not supported by the data. Using either the model selection or exclusion, the alternative space is reduced conditional on the observed data, and hence the power to detect a true association can be increased. Simulation results are reported and the proposed methods are applied to the genetic markers identified from the genome-wide association studies conducted by the Wellcome Trust Case–Control Consortium. The results demonstrate that the genetic model exclusion approach usually performs better than existing methods under its worst situation across scientifically plausible genetic models we considered.     

Chop-Lump Tests for Vaccine Trials

Summary This article proposes new tests to compare the vaccine and placebo groups in randomized vaccine trials when a small fraction of volunteers become infected. A simple approach that is consistent with the intent‐to‐treat principle is to assign a score, say W, equal to 0 for the uninfecteds and some postinfection outcome X > 0 for the infecteds. One can then test the equality of this skewed distribution of W between the two groups. This burden of illness (BOI) test was introduced by Chang, Guess, and Heyse (1994, Statistics in Medicine 13 , 1807–1814). If infections are rare, the massive number of 0s in each group tends to dilute the vaccine effect and this test can have poor power, particularly if the X's are not close to zero. Comparing X in just the infecteds is no longer a comparison of randomized groups and can produce misleading conclusions. Gilbert, Bosch, and Hudgens (2003, Biometrics 59 , 531–541) and Hudgens, Hoering, and Self (2003, Statistics in Medicine 22 , 2281–2298) introduced tests of the equality of X in a subgroup—the principal stratum of those “doomed” to be infected under either randomization assignment. This can be more powerful than the BOI approach, but requires unexaminable assumptions. We suggest new “chop‐lump” Wilcoxon and t‐tests (CLW and CLT) that can be more powerful than the BOI tests in certain situations. When the number of volunteers in each group are equal, the chop‐lump tests remove an equal number of zeros from both groups and then perform a test on the remaining W's, which are mostly >0. A permutation approach provides a null distribution. We show that under local alternatives, the CLW test is always more powerful than the usual Wilcoxon test provided the true vaccine and placebo infection rates are the same. We also identify the crucial role of the “gap” between 0 and the X's on power for the t‐tests. The chop‐lump tests are compared to established tests via simulation for planned HIV and malaria vaccine trials. A reanalysis of the first phase III HIV vaccine trial is used to illustrate the method.     

The Effect of Design Imbalance on the Power of the F‐test of a Variance Component in the One‐Way Random Model

The ANOVA‐based F‐test used for testing the significance of the random effect variance component is a valid test for an unbalanced one‐way random model. However, it does not have an uniform optimum property. For example, this test is not uniformly most powerful invariant (UMPI). In fact, there is no UMPI test in the unbalanced case (see Khuri , Mathew , and Sinha , 1998). The power of the F‐test depends not only on the design used, but also on the true values of the variance components. As Khuri (1996) noted, we can gain a better insight into the effect of data imbalance on the power of the F‐test using a method for modelling the power in terms of the design parameters and the variance components. In this study, generalized linear modelling (GLM) techniques are used for this purpose. It is shown that GLM, in combination with a method of generating designs with a specified degree of imbalance, is an effective way of studying the behavior of the power of the F‐test in a one‐way random model.     

Changes in the realized niche of the invasive succulent CAM plant Furcraea foetida

Furcraea foetida (Asparagaceae) is a native plant of Central America and northern South America but there is no information about its country of origin. The species was introduced into Brazil and is now considered invasive, particularly in coastal ecosystems. To date, nothing is known about the environmental factors that constrain its distribution and there is only inconclusive information about its location of origin. We used reciprocal distribution models (RDM) to assess invasion risk of F. foetida across Brazil and to identify source regions in its native range. We also tested the niche conservatism hypothesis using Principal Components Analyses and statistical tests of niche equivalency and similarity between its native and invaded ranges. For RDM analysis, we built two models using maximum entropy, one using records in the native range to predict the invaded distribution (forward‐Ecological Niche Model or forward‐ENM) and one using records in the invaded range to predict the native distribution (reverse‐ENM). Forward‐ENM indicated invasion risk in the Cerrado region and the innermost region of the Atlantic Forest, however, failed to predict the current occurrence in southern Brazil. Reverse‐ENM supported an existing hypothesis that F. foetida originated in the Orinoco river basin, Amazon basin and Caribbean islands. Prediction errors in the RDM and multivariate analysis indicated that the species expanded its realized niche in Brazil. The niche similarity test further suggested that the niche differences are because of differences in habitat availability between the two ranges, not because of evolutionary changes. We hypothesize that physiological pre‐adaptation (especially, the crassulacean acid metabolism), human‐driven propagule pressure and high competitive ability are the main factors determining the current spatial distribution of the species in Brazil. Our study highlights the need to include F. foetida in plant invasion monitoring programs, especially in priority conservation areas where the species has still not been introduced.     

Predicting the distribution of four species of raptors (Aves: Accipitridae) in southern Spain: statistical models work better than existing maps

Aim To test the effectiveness of statistical models based on explanatory environmental variables vs. existing distribution information (maps and breeding atlas), for predicting the distribution of four species of raptors (family Accipitridae): common buzzard Buteo buteo (Linnaeus, 1758), short‐toed eagle Circaetus gallicus (Gmelin, 1788), booted eagle Hieraaetus pennatus (Gmelin, 1788) and black kite Milvus migrans (Boddaert, 1783). Location Andalusia, southern Spain. Methods Generalized linear models of 10 × 10 km squares surveyed for the presence/absence of the species by road census. Statistical models use as predictors variables derived from topography, vegetation and land‐use, and the geographical coordinates (to take account of possible spatial trends). Predictions from the models are compared with current distribution maps from the national breeding atlas and leading reference works. Results The maps derived from statistical models for all four species were more predictive than the previously published range maps and the recent national breeding atlas. The best models incorporated both topographic and vegetation and land‐use variables. Further, in three of the four species the inclusion of spatial coordinates to account for neighbourhood effects improved these models. Models for the common buzzard and black kite were highly predictive and easy to interpret from an ecological point of view, while models for short‐toed eagle and, particularly, booted eagle were not so easy to interpret, but still predicted better than previous distribution information. Main conclusions It is possible to build accurate predictive models for raptor distribution with a limited field survey using as predictors environmental variables derived from digital maps. These models integrated in a geographical information system produced distribution maps that were more accurate than previously published ones for the study species in the study area. Our study is an example of a methodology that could be used for many taxa and areas to improve unreliable distribution information.     

Parapatric species and the implications for climate change studies: a case study on hares in Europe

Parapatry is a biogeographical term used to refer to organisms whose ranges do not overlap, but are immediately adjacent to each other; they only co‐occur – if at all – in a narrow contact zone. Often there are no environmental barriers in the contact zones, hence competitive interaction is usually advocated as the factor that modulates species distribution ranges. Even though the effects of climate change on species distribution have been widely studied, few studies have explored these effects on the biogeographical relationships between closely related, parapatric, species. We modelled environmental favourability for three parapatric hare species in Europe – Lepus granatensis, L. europaeus and L. timidus – using ecogeographical variables and projected the models into the future according to the IPCC A2 emissions scenario. Favourabilities for present and future scenarios were combined using fuzzy logic with the following aims: (i) to determine the biogeographical relationships between hare species in parapatry, that is L. granatensis/L. europaeus and L. europaeus/L. timidus and (ii) to assess the effects of climate change on each species as well as on their interspecific interactions. In their contact area L. granatensis achieved higher favourability values than L. europaeus, suggesting that if both species have a similar population status, the former species may have some advantages over the latter if competitive relationships are established. Climate change had the most striking effect on the distribution of L. timidus, especially when interspecific interactions with L. europaeus were taken into account, which may compromise the co‐existence of L. timidus. The results of this study are relevant not only for understanding the distribution patterns of the hares studied and the effects of climate change on these patterns, but also for improving the general application of species distribution models to the prediction of the effects of climate change on biodiversity.     

Generalizing Double and Triple Sampling for Repeated Surveys and Partial Verification

Population composition is often estimated by double sampling in which the value of a covariate is noted on each of a large number of randomly selected units and the value of the covariate and the exact class to which the unit belongs is noted for a smaller sample. The cross‐classified sample can be used to estimate the classification rates and these, in turn, can be used in conjunction with the estimated distribution of the covariate to obtain an improved estimate of the population composition over that obtained by direct observation of the identity of the individuals in a small sample. There are two approaches to this problem characterized by the way in which the classification rates are defined. The simplest approach uses estimates of the probability P(i | j) that the unit is actually in class i given that the covariate is in class j. The more complicated approach uses estimates of the probability Pi | j) that the covariate falls in class j given that the unit is actually in class i. The latter approach involves estimating more parameters than the former but avoids the necessity for the two samples to be drawn from the same population. We show the two approaches can be combined when there are multiple surveys. For example, one might conduct a disease survey for several years; in each year the accurate and/or error‐prone techniques may be applied to samples. The sensitivities and specificities of the error‐prone test are assumed constant across surveys. Generalizations allow for more than one error‐prone classifier and partial verification (estimation of misclassification rates by application of the accurate technique to fixed subsamples from each error‐prone category). The general approach is illustrated by considering a repeated survey for malaria.     

Rank Tests for Complete Block Designs

In this article a general univariate K-sample rank test for complete block designs with proportional cell frequencies is derived. It is shown that the test statistic has under H₀ and for arbitrary scores asymptotically a X²-distribution with K — 1 degrees of freedom. Special cases of this test are the Kruskal-Wallis test and the Friedman test. The test is compared with the Benard-van-Elteren test, the Mack-Skillings test and a test proposed by Downton. Finally the application of the test is illustrated by two examples.     

On Testing Equality of Means of Correlated Variates with Missing Data on One Response

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. The distribution of the statistic is approximated by a normal distribution under the null hypothesis. The empirical powers of the statistic are computed and compared with those of the conventional paired t and the other known statistics. The power comparisons support the use of the proposed test.     

Change and stasis in marine hybrid zones in response to climate warming

Aim We test the prediction that hybrid zones between warm‐ and cold‐adapted species will move towards the territory formerly occupied by the cold‐adapted species in response to a warming climate. We use multiple tests of this prediction to distinguish amongst potential mechanistic hypotheses of responses to climate change. Location We sampled 97 locations on the Atlantic coast of Spain and France and the English Channel that span three hybrid zones formed between two species of marine mussels (Mytilus galloprovincialis and M. edulis). Methods Mussels were sampled in 2005–07 and analysed at a nuclear gene (Glu‐5′) that is diagnostically differentiated between the subject species. Results were compared to those of studies made in the same region over the past two decades. Historical change in sea surface temperature (SST) was analysed using National Oceanic and Atmospheric Administration (NOAA) Optimum Interpolation Daily SST. Species distribution models (random forest and maximum entropy) of the current distribution of mussels were constructed and validated by hindcasting the historical distributions of these species. Validated models were used in combination with forecasts of SST to predict changes in mussel distribution to 2050 and 2100. Results We show that over the past two decades two of the hybrid zones in France have not changed in either position or shape. The third hybrid zone, however, has shifted in the predicted direction, c. 100 km eastward into the warming English Channel. Species distribution modelling strongly implicates changes in winter cold SST as driving this change in the position of one of the hybrid zones. Forecasts of future SST indicate that rapid changes in distribution will occur over the next century. Main conclusions Hybrid zones can be used to conduct repeated tests of ecological responses to climate change and can be valuable in sorting among prospective mechanistic hypotheses that underlie that change. Winter temperatures, but not seasonal high temperature, appear to control the distribution of both species. Species distribution modelling indicates that the collapse of these hybrid zones is imminent, with the rapid expansion of the subtropical species in response to continuing SST warming.     

Topo‐climatic microrefugia explain the persistence of a rare endemic plant in the Alps during the last 21 millennia

Ongoing rapid climate change is predicted to cause local extinction of plant species in mountain regions. However, some plant species could have persisted during Quaternary climate oscillations without shifting their range, despite the limited evidence from fossils. Here, we tested two candidate mechanisms of persistence by comparing the macrorefugia and microrefugia (MR) hypotheses. We used the rare and endemic Saxifraga florulenta as a model taxon and combined ensembles of species distribution models (SDMs) with a high‐resolution paleoclimatic and topographic dataset to reconstruct its potential current and past distribution since the last glacial maximum. To test the macrorefugia hypothesis, we verified whether the species could have persisted in or shifted to geographic areas defined by its realized niche. We then identified potential MR based on climatic and topographic properties of the landscape and applied refined scenarios of MR dynamics and functions over time. Last, we quantified the number of known occurrences that could be explained by either the macrorefugia or MR model. A consensus of two or three SDM techniques predicted absence between 14–10, 3–4 and 1 ka bp , which did not support the macrorefugia model. In contrast, we showed that S. florulenta could have contracted into MR during periods of absence predicted by the SDMs and later re‐colonized suitable areas according to the macrorefugia model. Assuming a limited and realistic seed dispersal distance for our species, we explained a large number of the current occurrences (61–96%). Additionally, we showed that MR could have facilitated range expansions or shifts of S. florulenta. Finally, we found that the most recent and the most stable MR were the ones closest to current occurrences. Hence, we propose a novel paradigm to explain plant persistence by highlighting the importance of supporting functions of MR when forecasting the fate of plant species under climate change.