Statistical Methods for Ordered Categorical Data Based on a Constrained Odds Model

The proportional odds model provides a powerful tool for analysing ordered categorical data and setting sample size, although for many clinical trials its validity is questionable. The purpose of this paper is to present a new class of constrained odds models which includes the proportional odds model. The efficient score and Fisher's information are derived from the profile likelihood for the constrained odds model. These results are new even for the special case of proportional odds where the resulting statistics define the Mann‐Whitney test. A strategy is described involving selecting one of these models in advance, requiring assumptions as strong as those underlying proportional odds, but allowing a choice of such models. The accuracy of the new procedure and its power are evaluated.     

Generalized pairwise comparisons for censored data: An overview

The method of generalized pairwise comparisons (GPC) is an extension of the well-known nonparametric Wilcoxon–Mann–Whitney test for comparing two groups of observations. Multiple generalizations of Wilcoxon–Mann–Whitney test and other GPC methods have been proposed over the years to handle censored data. These methods apply different approaches to handling loss of information due to censoring: ignoring noninformative pairwise comparisons due to censoring (Gehan, Harrell, and Buyse); imputation using estimates of the survival distribution (Efron, Péron, and Latta); or inverse probability of censoring weighting (IPCW, Datta and Dong). Based on the GPC statistic, a measure of treatment effect, the “net benefit,” can be defined. It quantifies the difference between the probabilities that a randomly selected individual from one group is doing better than an individual from the other group. This paper aims at evaluating GPC methods for censored data, both in the context of hypothesis testing and estimation, and providing recommendations related to their choice in various situations. The methods that ignore uninformative pairs have comparable power to more complex and computationally demanding methods in situations of low censoring, and are slightly superior for high proportions (>40%) of censoring. If one is interested in estimation of the net benefit, Harrell's c index is an unbiased estimator if the proportional hazards assumption holds. Otherwise, the imputation (Efron or Peron) or IPCW (Datta, Dong) methods provide unbiased estimators in case of proportions of drop-out censoring up to 60%.     

The Concordance Index C and the Mann–Whitney Parameter Pr(X>Y) with Randomly Censored Data

Harrell's c‐index or concordance C has been widely used as a measure of separation of two survival distributions. In the absence of censored data, the c‐index estimates the Mann–Whitney parameter Pr(X>Y), which has been repeatedly utilized in various statistical contexts. In the presence of randomly censored data, the c‐index no longer estimates Pr(X>Y); rather, a parameter that involves the underlying censoring distributions. This is in contrast to Efron's maximum likelihood estimator of the Mann–Whitney parameter, which is recommended in the setting of random censorship.     

Temporal trends of the airborne fungal spores in Catalonia (NE Spain), 1995–2013

To establish the trends in the period 1995–2013 of the annual fungal spore index of 20 taxa in 8 aerobiological sites of Catalonia, located in 4 phytoclimates, we use the nonparametric Spearman’s Rho and Mann–Kendall tests; when significant, we calculate the magnitude of the change applying the Theil–Sen estimator. We analyze whether the proportional annual change is significantly different from zero, according to phytoclimate, station, and spore taxon, with the Wilcoxon–Mann–Whitney test. Cladosporium, Coprinaceae and Agrocybe are the most prevalent taxa. The proportional annual change analysis shows that 12 taxa present significant increasing trends and 2 decreasing, that the Fresh-Continental Oriental-Humid phytoclimate and Agrocybe show the highest significant increase, while Roquetes-Tortosa (Fresh-Tethyc-Semiarid phytoclimate) and Torula the lowest. The greater significant decreasing proportional annual change corresponds to Drechslera–Helminthosporium and the lesser to Curvularia, and there are no results per phytoclimate and locality. The diversity of characteristics of the sites studied brings the opportunity to evaluate the variability of the fungal values and the magnitude of their change across the study period as depending on the intensity of the land use (urbanization versus agriculture) and the distance to the sea (inland versus littoral), but the effect of the change of the meteorological patterns in the recent years is not negligible. The increasing temperatures and precipitation instability established as effects of the climate change in Catalonia in the last 50 years could be stimulating the sporulation in mountain areas and affecting it in the southern Catalan littoral, thus affecting spore counts.     

Mechanical characterization of liver capsule through uniaxial quasi-static tensile tests until failure

Accidentology data showed that liver is often injured in car crashes; three types of injuries occur: hematoma, laceration and vessel failure. This paper focuses on surface laceration, which involves liver capsule and hepatic parenchyma. Liver capsule behavior has been studied but its failure properties are still unclear, particularly on a local point of view. In the present study, tensile quasi-static tests are run on parenchyma and capsule samples until failure to characterize capsule failure. Normalized load as well as failure properties—ultimate load per width unit and ultimate strain—are determined. Digital image correlation is used to measure the full local strain field on the capsule. Mean values of failure characteristics for hepatic capsule are 47±29% for the ultimate local strain and 0.3±0.3 N/mm for the ultimate load per width unit. A comparison between human and porcine tissues is conducted based on Mann–Whitney statistical test; it reveals that capsule characteristics are close between these two species; however, freezing preservation significantly affects porcine capsule failure properties. Therefore using porcine instead of human tissue to determine failure characteristics of liver capsule seems satisfactory only on fresh tissues.     

Prey‐associated head‐size variation in an invasive lizard in the Hawaiian Islands

Biological invasions are recognized as a primary driver of large‐scale changes in global ecosystems. This study addresses ecomorphological variation in head size within and among populations of an ecologically destructive invasive predator, and evaluates the potential roles of environmental components in phenotypic differentiation. We used four size‐corrected measurements of head morphology in Jackson's chameleons, Trioceros jacksonii xantholophus (N = 319), collected from multiple Hawaiian Islands to assess phenotypic variation among and within islands. Results of analysis of variance (ANOVA) comparing chameleon head size (PC1) among islands revealed significant differences (mean difference > 5%) associated with variation in both rainfall and diet composition using Mann–Whitney U‐tests and chi‐squared analyses. These results suggest that morphological differentiation among populations from different islands has occurred over a relatively short ecological timescale, and is likely the result of ecomorphological adaptation to differences in exploited prey hardness. Intra‐island allopatric population variation, however, was also detected in this study. Although we might expect that genetic change is the more likely explanation for differences between islands than within, and that plasticity may be more likely an explanation for the within‐ than the between‐island differences, it is also possible that both within‐ and between‐island patterns are the results of genetic change, or of plasticity.     

Non‐proportional odds multivariate logistic regression of ordinal family data

Methods to examine whether genetic and/or environmental sources can account for the residual variation in ordinal family data usually assume proportional odds. However, standard software to fit the non‐proportional odds model to ordinal family data is limited because the correlation structure of family data is more complex than for other types of clustered data. To perform these analyses we propose the non‐proportional odds multivariate logistic regression model and take a simulation‐based approach to model fitting using Markov chain Monte Carlo methods, such as partially collapsed Gibbs sampling and the Metropolis algorithm. We applied the proposed methodology to male pattern baldness data from the Victorian Family Heart Study.     

STATISTICAL ANALYSES FOR R-INDEX

R‐index is an important statistic for testing and measuring product effects. The validation and merits of R‐index are to a great extent due to the fact that it is closely related to the famous Mann–Whitney U statistic. Based on this fortunate relationship, statistical analyses for R‐index are explored. The statistical analyses include estimations of R‐index and its null and nonconditional variances with and without assuming continuity of data; difference and similarity tests using R‐index; powers and sample sizes for the tests; linking R‐index with Thurstonian δ (or d′). The new techniques developed in the paper extend greatly the original R‐index analysis for categorical ratings data. It is expected that the recognition of the profound theoretical origin of R‐index and the available statistical analyses for R‐index will provide the impetus for the resurgence of interest in using R‐index in sensory and consumer researches.     

Effect of Tooth Removal on Recaptures of Raccoons

Abstract: Researchers have extensively used mark—recapture techniques to obtain information on demographic parameters of wildlife populations. However, researchers have recognized that a number of factors can influence capture probabilities of wildlife species, which in turn can bias mark—recapture estimates of demographic parameters. Tooth extraction, which is a commonly used technique in studies of mesopredator species to obtain precise age estimates and to monitor the use of vaccine baits, is an aspect of animal handling that clearly might affect the recapture probability of individuals. However, the effect that tooth removal has on the individual recapture probabilities of wildlife species is unknown. During 2005, we trapped and marked 91 raccoons (Procyon lotor) in northern Indiana, USA, as part of a mark—recapture study designed specifically to determine if tooth extractions have an effect on recapture probabilities of individuals. We performed tooth extractions on 50% of the raccoons at the time of capture, and we attempted to balance tooth extractions with respect to sex and age of raccoons. We used logistic regression to model the effects of sex, age, and tooth removal on recapture probabilities, and we used Mann—Whitney U-tests to examine the effect of tooth removal on the number of times we recaptured individuals. The probability of recapture differed between sexes but did not differ as a function of tooth removal or among age classes. In addition, we failed to detect any difference in the mean number of times that we recaptured raccoons between the tooth removed and non—tooth-removed groups. Our results suggest that managers can use tooth extractions as an effective management tool without biasing population estimates or compromising other management objectives.     

Effects of pruning on the concentration of trace elements in Colophospermum mopane leaves

Despite the nutritional value of Colophospermum mopane to browser's diets, there is still insufficient knowledge on the effect of browsers on concentrations of these trace elements. A field experiment was conducted in Musina Nature Reserve, Limpopo Province, South Africa, to determine the effect of pruning on the concentration of trace elements in mopane leaves. Samples were analysed for iron (Fe), manganese (Mn), boron (B), molybdenum (Mo), copper (Cu), zinc (Zn), cobalt (Co), fluoride (F) and selenium (Se) using the inductively coupled plasma atomic emission spectrometry technique. The effect of pruning was tested using the two‐tailed t‐test: two‐sample assuming equal variance and two‐tailed Mann–Whitney U‐test. Results showed that the concentration of trace elements in the control and pruned trees varies slightly through the year. Fe, Mn, Mo, Cu, Zn and Se are higher during leaf flush, but declined as the leaves matured and aged. This study concluded that simulated browsing had no significant effect on the concentration of trace elements in the mopane leaves. Seasonal variation in the amount of trace elements has implications on the distribution of browsers in the mopane woodland.     

Combining choice modeling estimates and stochastic simulations to assess the potential of new crops—The case of lignocellulosic perennials in Southwestern Germany

In the future, the lignocellulosic perennial crops short rotation coppice (SRC) and miscanthus are supposed to provide renewable raw materials for a bio‐based economy. To assess the potential regional supply of these crops, which are not yet widespread in Baden–Wuerttemberg (Southwest Germany), we used a two‐step approach. In a first step, we conducted a Discrete Choice Experiment (DCE) in regions of Baden–Wuerttemberg that—given their site conditions—are suitable for SRC or miscanthus. The respondents were characterized by significant preference heterogeneity for both (negatively valued) perennial crops and for all presented choice attributes. Thus, it was appropriate to estimate a random parameter logit model (RPL). The attributes average yearly contribution margin, long‐term purchase guarantee and cultivation by colleagues in the neighborhood had a significantly positive effect on the likelihood of cultivation, whereas the attributes contribution margin variability and initial investment need showed a significantly negative effect. In a second step, assuming realistic values for the levels of the attributes considered in the DCE, in stochastic simulations, we randomly draw part‐worth utilities from the multivariate normal distribution of these parameters according to the RPL results. This way, for alternative biomass prices, we derived shares of farmers’ willing to engage in perennial crop production and produced related regional supply functions. Under moderate yield and realistic input and farmland opportunity cost assumptions, the full regional miscanthus potential can only be achieved when farmers are offered either subsidies or price‐risk‐reducing long‐term contracts. Based on empirically determined heterogeneous farmer preferences, our two‐step approach is suitable to yield realistic estimations of any not yet implemented farming practices. We finally note caveats related to our analysis and discuss some policy implications of the major findings.     

Estimation and inference for the causal effect of receiving treatment on a multinomial outcome

Summary .  This article considers the analysis of two-arm randomized trials with noncompliance, which have a multinomial outcome. We first define the causal effect in these trials as some function of outcome distributions of compliers with and without treatment (e.g., the complier average causal effect, the measure of stochastic superiority of treatment over control for compliers), then estimate the causal effect with the likelihood method. Next, based on the likelihood-ratio (LR) statistic, we test those functions of or the equality of the outcome distributions of compliers with and without treatment. Although the corresponding LR statistic follows a chi-squared  (χ²)  distribution asymptotically when the true values of parameters are in the interior of the parameter space under the null, its asymptotic distribution is not  χ²  when the true values of parameters are on the boundary of the parameter space under the null. Therefore, we propose a bootstrap/double bootstrap version of a LR test for the causal effect in these trials. The methods are illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

The diversity field of New World leaf‐nosed bats (Phyllostomidae)

Aim To analyse how the patterns of species richness for the whole family Phyllostomidae determine the structure of diversity fields (sets of species‐richness values) within the ranges of individual bat species. Location The range of the family Phyllostomidae in North and South America. Methods We generated a database of the occurrence of 143 phyllostomid bat species in 6794 quadrats, analysing the species‐richness frequency distribution for all sites, and for subsets of sites defined by the geographic ranges of species. Range–diversity plots, depicting simultaneously the size and the mean species richness of ranges, were built to explore the patterns of co‐occurrence in widespread and restricted species. We compared the empirical patterns against two null models: (1) with scattered (non‐cohesive) ranges, and (2) with cohesive ranges modelled with the spreading‐dye algorithm. Diversity fields were analysed with richness maps for individual species and with comparisons of species‐richness frequency distributions. Results Overall richness frequency distribution showed a multimodal pattern, whereas simulated distributions showed lower values of variance, and were unimodal (for model 1) and bimodal (for model 2). Range–diversity plots for the empirical data and for the cohesive‐ranges simulation showed a strong tendency of species to co‐occur in high‐diversity sites. The scattered‐ranges simulation showed no such tendency. Diversity fields varied according to idiosyncratic features of species generating particular geographic patterns and richness frequency distributions. Main conclusions Phyllostomid bats show a higher level of co‐occurrence than expected from null models. That tendency in turn implies a higher variance in species richness among sites, generating a wider species‐richness frequency distribution. The diversity field of individual species results from the size, shape and location of ranges, but also depends on the general pattern of richness for the whole family.     

The Nonparametric Behrens‐Fisher Problem: Asymptotic Theory and a Small‐Sample Approximation

A generalization of the Behrens‐Fisher problem for two samples is examined in a nonparametric model. It is not assumed that the underlying distribution functions are continuous so that data with arbitrary ties can be handled. A rank test is considered where the asymptotic variance is estimated consistently by using the ranks over all observations as well as the ranks within each sample. The consistency of the estimator is derived in the appendix. For small samples (n₁, n₂ ≥ 10), a simple approximation by a central t‐distribution is suggested where the degrees of freedom are taken from the Satterthwaite‐Smith‐Welch approximation in the parametric Behrens‐Fisher problem. It is demonstrated by means of a simulation study that the Wilcoxon‐Mann‐Whitney‐test may be conservative or liberal depending on the ratio of the sample sizes and the variances of the underlying distribution functions. For the suggested approximation, however, it turns out that the nominal level is maintained rather accurately. The suggested nonparametric procedure is applied to a data set from a clinical trial. Moreover, a confidence interval for the nonparametric treatment effect is given.     

Relationship between subjective assessment of oral health and medical expenses in community-dwelling elderly persons

doi: 10.1111/j.1741‐2358.2011.00459.x
Relationship between subjective assessment of oral health and medical expenses in community‐dwelling elderly persons Objectives: The increasing medical expenses of elderly persons in Japan’s rapidly ageing society have become a major concern. It is therefore important to elucidate the factors associated with such escalation. Here, we focused on the relationship between subjective self‐assessment of oral health, as an index of general health, and medical expenses (excluding dental repair) under the hypothesis that oral health contributes to general medical expenses. Several studies have shown that oral health status is correlated with general health status among elderly persons. We speculated that oral health status might show a relation with medical costs among elderly persons. However, few studies have investigated this relationship to date. Materials and Methods: Participants were 259 elderly subjects (range: 65–84 years; 120 men, 139 women) residing independently. Subjective assessment of oral health was evaluated by their responses (‘Good’, ‘Not good’ and ‘Not at all good’) on a survey questionnaire. The correlation between subjective assessment of oral health and medical expenditure was analysed using Spearman’s rank method, the Mann–Whitney U‐test and the Kruskal–Wallis test. Medical expenses were used as the dependent variable in multinomial logistic regression analysis with background and intraoral factors as independent variables. Results: A slight yet statistically significant correlation was observed between subjective assessment of oral health and outpatient treatment fees. Conclusion: The findings revealed that subjective assessment of oral health is significantly and independently related to the medical expenses of community‐dwelling elderly persons after adjusting for social background, living environment and physical factors.     

Merits of Modelling Multivariate Survival Data Using Random Effects Proportional Odds Model

Recently, there has been a great deal of interest in the analysis of multivariate survival data. In most epidemiological studies, survival times of the same cluster are related because of some unobserved risk factors such as the environmental or genetic factors. Therefore, modelling of dependence between events of correlated individuals is required to ensure a correct inference on the effects of treatments or covariates on the survival times. In the past decades, extension of proportional hazards model has been widely considered for modelling multivariate survival data by incorporating a random effect which acts multiplicatively on the hazard function. In this article, we consider the proportional odds model, which is an alternative to the proportional hazards model at which the hazard ratio between individuals converges to unity eventually. This is a reasonable property particularly when the treatment effect fades out gradually and the homogeneity of the population increases over time. The objective of this paper is to assess the influence of the random effect on the within‐subject correlation and the population heterogeneity. We are particularly interested in the properties of the proportional odds model with univariate random effect and correlated random effect. The correlations between survival times are derived explicitly for both choices of mixing distributions and are shown to be independent of the covariates. The time path of the odds function among the survivors are also examined to study the effect of the choice of mixing distribution. Modelling multivariate survival data using a univariate mixing distribution may be inadequate as the random effect not only characterises the dependence of the survival times, but also the conditional heterogeneity among the survivors. A robust estimate for the correlation of the logarithm of the survival times within a cluster is obtained disregarding the choice of the mixing distributions. The sensitivity of the estimate of the regression parameter under a misspecification of the mixing distribution is studied through simulation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

CALIBRATED DIVERSITY,TREE TOPOLOGY AND THE MOTHER OF MASS EXTINCTIONS: THE LESSON OF TEMNOSPONDYLS

Abstract: Three family‐level cladistic analyses of temnospondyl amphibians are used to evaluate the impact of taxonomic rank, tree topology, and sample size on diversity profiles, origination and extinction rates, and faunal turnover. Temnospondyls are used as a case study for investigating replacement of families across the Permo‐Triassic boundary and modality of recovery in the aftermath of the end‐Permian mass extinction. Both observed and inferred (i.e. tree topology‐dependent) values of family diversity have a negligible effect on the shape of the diversity curve. However, inferred values produce both a flattening of the curve throughout the Cisuralian and a less pronounced increase in family diversity from Tatarian through to Induan than do observed values. Diversity curves based upon counts of genera and species display a clearer distinction between peaks and troughs. We use rarefaction techniques (specifically, rarefaction of the number of genera and species within families) to evaluate the effect of sampling size on the curve of estimated family‐level diversity during five time bins (Carboniferous; Cisuralian; Guadalupian–Lopingian; Early Triassic; Middle Triassic–Cretaceous). After applying rarefaction, we note that Cisuralian and Early Triassic diversity values are closer to one another than they are when the observed number of families is used; both values are also slightly higher than the Carboniferous estimated diversity. The Guadalupian–Lopingian value is lower than raw data indicate, reflecting in part the depauperate land vertebrate diversity from the late Cisuralian to the middle Guadalupian (Olson’s gap). The time‐calibrated origination and extinction rate trajectories plot out close to one another and show a peak in the Induan, regardless of the tree used to construct them. Origination and extinction trajectories are disjunct in at least some Palaeozoic intervals, and background extinctions exert a significant role in shaping temnospondyl diversity in the lowermost Triassic. Finally, species‐, genus‐, and family trajectories consistently reveal a rapid increase in temnospondyl diversity from latest Permian to earliest Triassic as well as a decline near the end of the Cisuralian. However, during the rest of the Cisuralian family diversity increases slightly and there is no evidence for a steady decline, contrary to previous reports.     

High Conservation Value of Forest Fragments for Plant and Frugivore Communities in a Fragmented Forest Landscape in South Africa

Fragmentation is a major threat factor for plant–frugivore communities in tropical and subtropical forests. Resulting changes in the distribution of traits within these communities, e.g., a loss in large‐bodied frugivores, may lead to strong changes in plant–frugivore interactions in fragmented forests. Yet, we still lack a thorough understanding of the interplay between forest fragmentation, the trait‐composition of communities and resulting plant–frugivore interactions on a community‐scale. In a fragmented South African landscape comprising different forest categories—i.e., continuous natural forest, forest fragments surrounded by natural grassland, and forest fragments surrounded by sugarcane—we investigated the relationship between communities of fruiting plants and their frugivore visitors in response to forest fragmentation, as well as the interactive effects of forest fragmentation and fruit size of the plants on the number of frugivore visitors and their body size. Neither the fruit size of plant nor the body mass of frugivore communities differed between natural forest sites and forest fragments. Moreover, in‐depth analyses of frugivore assemblages visiting plant species revealed no effect of forest category on the number of frugivore visits or their mean body mass. The number of visits and body mass of frugivores were merely determined by the crop and fruit size of the focal plant species. Overall, our results suggest that frugivory of plant species with differently sized fruits was not reduced in forest fragments. Thus, fragments with high fruit availability may be key elements maintaining the functional connectivity of a heterogeneous forest landscape.     

Nonparametric Analysis of Ordered Categorical Data in Designs with Longitudinal Observations and Small Sample Sizes

For designs with longitudinal observations of ordered categorical data, a nonparametric model is considered where treatment effects and interactions are defined by means of the marginal distributions. These treatment effects are estimated consistently by ranking methods. The hypotheses in this nonparametric setup are formulated by means of the distribution functions. The asymptotic distribution of the estimators for the nonparametric effects are given under the hypotheses. For small samples, a rather accurate approximation is suggested. A clinical trial with ordered categorical data is used to motivate the ideas and to explain the procedures which are extensions of the Wilcoxon‐Mann‐Whitney test to factorial designs with longitudinal observations. The application of the procedures requires only some trivial regularity assumptions.     

Caught in the middle: Asymmetric competition causes high variance in intermediate trait abundances

In asymmetric competition between two individuals of the same or different species, one individual has a distinct advantage over the other due to a particular beneficial trait. An important trait that induces asymmetric competition is size (body size in animals, height in plants). There is usually a trade-off between fecundity and the trait that leads to competitive superiority (e.g. seed number vs seed size), enabling coexistence of populations with different trait values. These predictions on coexistence are based on classic deterministic models. Here, we explore the behaviour of a stochastic model of asymmetric competition where stochasticity is assumed to be demographic. We derive approximations for the temporal variance and covariance of the population sizes of the coexisting species. The derivations highlight that the variability of the population size of a species strongly depends on the stochastic fluctuations of species with higher trait values, while they are less influenced by species with lower trait values. Particularly, species with intermediate trait values are strongly affected resulting in relatively high variability. As a result these species have a relative high probability of extinction even though they have a larger population size than species with high trait values. We confirm these approximations with individual-based simulations. Thus, our analysis can explain gaps in size distributions as an emergent property of systems with a fecundity–competition trade-off.