Estimation and testing with overdispersed proportions using the beta-logistic regression model of Heckman and Willis

Methods are presented for modeling dose-related effects in proportion data when extra-binomial variability is a concern. Motivation is taken from experiments in developmental toxicology, where similarity among conceptuses within a litter leads to intralitter correlations and to overdispersion in the observed proportions. Appeal is made to the well-known beta-binomial distribution to represent the overdispersion. From this, an exponential function of the linear predictor is used to model the dose-response relationship. The specification was introduced previously for econometric applications by Heckman and Willis; it induces a form of logistic regression for the mean response, together with a reciprocal biexponential model for the intralitter correlation. Large-sample, likelihood-based methods for estimating and testing the joint proportion-correlation response are studied. A developmental toxicity data set illustrates the methods.     

Sensitivity analysis of common statistical models used to study the short-term effects of air pollution on health

The relationship between photochemical air pollutants (nitrogen dioxide and ozone) and emergency room admissions for asthma in Madrid (Spain) for the period 1995-1998 was analysed using the statistical models commonly used to studying the short-term effects of air pollution on health: linear and Cochrane-Orcutt regression, standard Poisson and Poisson corrected by overdispersion, Poisson autoregressive models, and generalised additive models. Linear regression models presented residual autocorrelation, Poisson regression models also showed overdispersion, and generalised additive models did not show residual autocorrelation and overdispersion was substantially reduced. Linear models provided biased estimates because our health outcome is non-normally distributed. Estimates from Poisson regression allowing for overdispersion and autocorrelation did not differ substantially from those reported by generalised additive models, which present the best model fit in terms of the absence of autocorrelation and reduction of overdispersion.     

Increased competition does not lead to increased phylogenetic overdispersion in a native grassland

That competition is stronger among closely related species and leads to phylogenetic overdispersion is a common assumption in community ecology. However, tests of this assumption are rare and field‐based experiments lacking. We tested the relationship between competition, the degree of relatedness, and overdispersion among plants experimentally and using a field survey in a native grassland. Relatedness did not affect competition, nor was competition associated with phylogenetic overdispersion. Further, there was only weak evidence for increased overdispersion at spatial scales where plants are likely to compete. These results challenge traditional theory, but are consistent with recent theories regarding the mechanisms of plant competition and its potential effect on phylogenetic structure. We suggest that specific conditions related to the form of competition and trait conservatism must be met for competition to cause phylogenetic overdispersion. Consequently, overdispersion as a result of competition is likely to be rare in natural communities.     

Some covariance models for longitudinal count data with overdispersion

A family of covariance models for longitudinal counts with predictive covariates is presented. These models account for overdispersion, heteroscedasticity, and dependence among repeated observations. The approach is a quasi-likelihood regression similar to the formulation given by Liang and Zeger (1986, Biometrika 73, 13-22). Generalized estimating equations for both the covariate parameters and the variance-covariance parameters are presented. Large-sample properties of the parameter estimates are derived. The proposed methods are illustrated by an analysis of epileptic seizure count data arising from a study of progabide as an adjuvant therapy for partial seizures.     

Semi‐parametric analysis of overdispersed count and metric data with varying follow‐up times: Asymptotic theory and small sample approximations

Count data are common endpoints in clinical trials, for example magnetic resonance imaging lesion counts in multiple sclerosis. They often exhibit high levels of overdispersion, that is variances are larger than the means. Inference is regularly based on negative binomial regression along with maximum‐likelihood estimators. Although this approach can account for heterogeneity it postulates a common overdispersion parameter across groups. Such parametric assumptions are usually difficult to verify, especially in small trials. Therefore, novel procedures that are based on asymptotic results for newly developed rate and variance estimators are proposed in a general framework. Moreover, in case of small samples the procedures are carried out using permutation techniques. Here, the usual assumption of exchangeability under the null hypothesis is not met due to varying follow‐up times and unequal overdispersion parameters. This problem is solved by the use of studentized permutations leading to valid inference methods for situations with (i) varying follow‐up times, (ii) different overdispersion parameters, and (iii) small sample sizes.     

Universal Count Correction for High-Throughput Sequencing

We show that existing RNA-seq, DNase-seq, and ChIP-seq data exhibit overdispersed per-base read count distributions that are not matched to existing computational method assumptions. To compensate for this overdispersion we introduce a nonparametric and universal method for processing per-base sequencing read count data called Fixseq. We demonstrate that Fixseq substantially improves the performance of existing RNA-seq, DNase-seq, and ChIP-seq analysis tools when compared with existing alternatives.     

Radiation induced chromosome aberrations and the poisson distribution

Summary Data on the distribution of dicentrics and acentrics observed when human lymphocytes are cultured for 48 h after irradiation by X-rays,-rays, and neutrons are presented. Analysis shows that for dicentrics, the observed distribution for X-rays,-rays, and fission neutrons may be described by Poisson statistics but for higher energy neutrons overdispersion is observed. The phenomenon of overdispersion is also observed for acentrics irrespective of the radiation used. The possibility that overdispersion results from the variations of dose in sensitive sites leads to the conclusion that for dicentrics the site size is considerably larger than the 1–2 µm diameter derived by applying the dual action theory to the dose effect relationships. This larger site may well be the cell nucleus.     

A novel method for quantifying overdispersion in count data and its application to farmland birds

The statistical modelling of count data permeates the discipline of ecology. Such data often exhibit overdispersion compared with a standard Poisson distribution, so that the variance of the counts is greater than that of the mean. Whereas modelling to reveal the effects of explanatory variables on the mean is commonplace, overdispersion is generally regarded as a nuisance parameter to be accounted for and subsequently ignored. Instead, we propose a method that models the overdispersion as a biologically interesting property of a data set and show how novel inference is provided as a result. We adapted the double hierarchical generalized linear model approach to create an easily extendible model structure that quantifies the influence of explanatory variables on the overdispersion of count data, and apply it to farmland birds. These data were from a study within Irish agricultural ecosystems, in which total bird species abundance and the abundance of farmland indicator species were compared on dairy and non‐dairy farms in the winter and breeding seasons. In general, overdispersion in bird counts was greater on dairy farms than on non‐dairy farms, and for total bird numbers, overdispersion was greatest on dairy farms in winter. Our code is fitted using the Bayesian package Rstan, and we make all code and data available in a GitHub repository. Within a Bayesian framework, this approach facilitates a meaningful quantification of the effects of categorical explanatory variables on any response variable with a tendency to overdispersion that has a meaningful biological or ecological explanation.     

Evaluating Dependence Among Mule Deer Siblings in Fetal and Neonatal Survival Analyses

Abstract: The assumption of independent sample units is potentially violated in survival analyses where siblings comprise a high proportion of the sample. Violation of the independence assumption causes sample data to be overdispersed relative to a binomial model, which leads to underestimates of sampling variances. A variance inflation factor, c, is therefore required to obtain appropriate estimates of variances. We evaluated overdispersion in fetal and neonatal mule deer (Odocoileus hemionus) datasets where more than half of the sample units were comprised of siblings. We developed a likelihood function for estimating fetal survival when the fates of some fetuses are unknown, and we used several variations of the binomial model to estimate neonatal survival. We compared theoretical variance estimates obtained from these analyses with empirical variance estimates obtained from data-bootstrap analyses to estimate the overdispersion parameter, c. Our estimates of c for fetal survival ranged from 0.678 to 1.118, which indicate little to no evidence of overdispersion. For neonatal survival, 3 different models indicated that ĉ ranged from 1.1 to 1.4 and averaged 1.24–1.26, providing evidence of limited overdispersion (i.e., limited sibling dependence). Our results indicate that fates of sibling mule deer fetuses and neonates may often be independent even though they have the same dam. Predation tends to act independently on sibling neonates because of dam-neonate behavioral adaptations. The effect of maternal characteristics on sibling fate dependence is less straightforward and may vary by circumstance. We recommend that future neonatal survival studies incorporate additional sampling intensity to accommodate modest overdispersion (i.e., ĉ = 1.25), which would facilitate a corresponding ĉ adjustment in a model selection analysis using quasi-likelihood without a reduction in power. Our computational approach could be used to evaluate sample unit dependence in other studies where fates of individually marked siblings are monitored.     

A common tendency for phylogenetic overdispersion in mammalian assemblages

Competition has long been proposed as an important force in structuring mammalian communities. Although early work recognized that competition has a phylogenetic dimension, only with recent increases in the availability of phylogenies have true phylogenetic investigations of mammalian community structure become possible. We test whether the phylogenetic structure of 142 assemblages from three mammalian clades (New World monkeys, North American ground squirrels and Australasian possums) shows the imprint of competition. The full set of assemblages display a highly significant tendency for members to be more distantly related than expected by chance (phylogenetic overdispersion). The overdispersion is also significant within two of the clades (monkeys and squirrels) separately. This is the first demonstration of widespread overdispersion in mammal assemblages and implies an important role for either competition between close relatives where traits are conserved, habitat filtering where distant relatives share convergent traits, or both.     

Overdispersion in marine fish parasites

A modification of Taylor's Power law was used to compare the degree of overdispersion in frequency distributions from 38 datasets of marine parasites, data that had originally been collected for fish stock discrimination. The results strongly indicate that the overriding factor contributing to overdispersion in these helminths and crustaceans is the number of hosts in the life cycle. This was particularly well shown by juveniles of Anisakis 1 from different fish species. Data on the cestode Otobothrium cysticum and the monogenean Pricea multae appear anomalous and lead to conclusions about their biology not at first evident from the literature.     

THE BETA-BINOMIAL MODEL: ACCOUNTING FOR INTER-TRIAL VARIATION IN REPLICATED DIFFERENCE AND PREFERENCE TESTS

Binomial tests are commonly used in sensory difference and preference testing under the assumptions that choices are independent and choice probabilities do not vary from trial to trial. This paper addresses violations of the latter assumption (often referred to as overdispersion) and accounts for variation in inter-trial choice probabilities following the Beta distribution. Such variation could arise as a result of differences in test substrate from trial to trial, differences in sensory acuity among subjects or the existence of latent preference segments. In fact, it is likely that overdispersion occurs ubiquitously in product testing. Using the Binomial model for data in which there is inter-trial variation may lead to seriously misleading conclusions from a sensory difference or preference test. A simulation study in this paper based on product testing experience showed that when using a Binomial model for overdispersed Binomial data, Type I error may be 0.44 for a Binomial test specification corresponding to a level of 0.05. Underestimation of Type I error using the Binomial model may seriously undermine legal claims of product superiority in situations where overdispersion occurs. The Beta-Binomial (BB) model, an extension of the Binomial distribution, was developed to fit overdispersed Binomial data. Procedures for estimating and testing the parameters as well as testing for goodness of fit are discussed. Procedures for determining sample size and for calculating estimate precision and test power based on the BB model are given. Numerical examples and simulation results are also given in the paper. The BB model should improve the validity of sensory difference and preference testing.     

Regression models for overdispersed jejunal surviving crypts data

The dose-response model concerns to establish a relationship between a dose and the magnitude of the response produced by the dose. A common complication in the dose-response model for jejunal crypts cell surviving data is overdispersion, where the observed variation exceeds that predicted from the binomial distribution. In this study, two different methods for analyzing jejunal crypts cell survival after regimens of several fractions are contrasted and compared. One method is the logistic regression approach, where the numbers of surviving crypts are predicted by the logistic function of a single dose of radiation. The other one is the transform-both-sides approach, where the arcsine transformation family is applied based on the first-order variance-stabilizing transformation. This family includes the square root, arcsine, and hyperbolic arcsine transformations, which have been used for Poisson, binomial, and negative binomial count data, as special cases. These approaches are applied to a data set from radiobiology. Simulation study indicates that the arcsine transformation family is more efficient than the logistic regression when there exists moderate overdispersion.     

A note on overdispersion as an index of behavioural synchrony: a pilot study in dairy cows

We developed a method for studying the synchrony of behaviour based on calculations of overdispersion of a binomial process. The lying behaviour of cows was investigated under two different housing units inside the same barn. The first unit housed 30 cows undergoing conventional milking and the second unit housed 27 cows undergoing automatic milking. The lying behaviour of the cows was observed over 3 weeks in 12 periods of 6 h each. Every 5 min, we counted the number of cows lying down in the cubicles. As external cues, like feeding and conventional milking, can promote synchrony in dairy cows, we expected that cows conventionally milked would show more behavioural synchrony than automatically milked cows. Cows lied down synchronously in both units (overdispersion 1.67, P < 0.01). Lying synchrony tended to be slightly bigger in automatically than in conventionally milked cows (overdispersion 1.76 v. 1.58, P = 0.09), although the proportion of cows lying down was on average greater in conventionally than in automatically milked cows (60.7% v. 45.6%). This suggests that synchronized lying behaviour in cows is a constant phenomenon that depends on social facilitation rather than on external cues. The overdispersion index appears to be a useful tool for studying the synchrony of animal behaviour when observations are made at the group level.     

Overdispersion of the molecular clock varies between yeast, Drosophila and mammals

Although protein evolution can be approximated as a "molecular evolutionary clock," it is well known that sequence change departs from a clock-like Poisson expectation. Through studying the deviations from a molecular clock, insight can be gained into the forces shaping evolution at the level of proteins. Generally, substitution patterns that show greater variance than the Poisson expectation are said to be "overdispersed." Overdispersion of sequence change may result from temporal variation in the rate at which amino acid substitutions occur on a phylogeny. By comparing the genomes of four species of yeast, five species of Drosophila, and five species of mammals, we show that the extent of overdispersion shows a strong negative correlation with the effective population size of these organisms. Yeast proteins show very little overdispersion, while mammalian proteins show substantial overdispersion. Additionally, X-linked genes, which have reduced effective population size, have gene products that show increased overdispersion in both Drosophila and mammals. Our research suggests that mutational robustness is more pervasive in organisms with large population sizes and that robustness acts to stabilize the molecular evolutionary clock of sequence change.     

Testing approaches for overdispersion in poisson regression versus the generalized poisson model

Overdispersion is a common phenomenon in Poisson modeling, and the negative binomial (NB) model is frequently used to account for overdispersion. Testing approaches (Wald test, likelihood ratio test (LRT), and score test) for overdispersion in the Poisson regression versus the NB model are available. Because the generalized Poisson (GP) model is similar to the NB model, we consider the former as an alternate model for overdispersed count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes a score test for overdispersion based on the GP model and compares the power of the test with the LRT and Wald tests. A simulation study indicates the score test based on asymptotic standard Normal distribution is more appropriate in practical application for higher empirical power, however, it underestimates the nominal significance level, especially in small sample situations, and examples illustrate the results of comparing the candidate tests between the Poisson and GP models. A bootstrap test is also proposed to adjust the underestimation of nominal level in the score statistic when the sample size is small. The simulation study indicates the bootstrap test has significance level closer to nominal size and has uniformly greater power than the score test based on asymptotic standard Normal distribution. From a practical perspective, we suggest that, if the score test gives even a weak indication that the Poisson model is inappropriate, say at the 0.10 significance level, we advise the more accurate bootstrap procedure as a better test for comparing whether the GP model is more appropriate than Poisson model. Finally, the Vuong test is illustrated to choose between GP and NB2 models for the same dataset.     

Optimal experimental design and sample size for the statistical evaluation of data from somatic mutation and recombination tests (SMART) in Drosophila

In genetic toxicology it is important to know whether chemicals should be regarded as clearly hazardous or whether they can be considered sufficiently safe, which latter would be the case from the genotoxicologist's view if their genotoxic effects are nil or at least significantly below a predefined minimal effect level. A previously presented statistical decision procedure which allows one to make precisely this distinction is now extended to the question of how optimal experimental sample size can be determined in advance for genotoxicity experiments using the somatic mutation and recombination tests (SMART) of Drosophila. Optimally, the statistical tests should have high power to minimise the chance for statistically inconclusive results. Based on the normal test, the statistical principles are explained, and in an application to the wing spot assay, it is shown how the practitioner can proceed to optimise sample size to achieve numerically satisfactory conditions for statistical testing. The somatic genotoxicity assays of Drosophila are in principle based on somatic spots (mutant clones) that are recovered in variable numbers on individual flies. The underlying frequency distributions are expected to be of the Poisson type. However, some care seems indicated with respect to this latter assumption, because pooling of data over individuals, sexes, and experiments, for example, can (but need not) lead to data which are overdispersed, i.e, the data may show more variability than theoretically expected. It is an undesired effect of overdispersion that in comparisons of pooled totals it can lead to statistical testing which is too liberal, because overall it yields too many seemingly significant results. If individual variability considered alone is not contradiction with Poisson expectation, however, experimental planning can help to minimise the undesired effects of overdispersion on statistical testing of pooled totals. The rule for the practice is to avoid disproportionate sampling. It is recalled that for optimal power in statistical testing, it is preferable to use equal total numbers of flies in the control and treated series. Statistical tests which are based on Poisson expectations are too liberal if there is overdispersion in the data due to excess individual variability. In this case we propose to use the U test as a non-parametric two-sample test and to adjust the estimated optimal sample size according to (i) the overdispersion observed in a large historical control and (ii) the relative efficiency of the U test in comparison to the t test and related parametric tests.     

Population trends from count data: Handling environmental bias,overdispersion and excess of zeroes

The assessment of population trends is a key point in wildlife conservation. Survey data collected over long period may not be comparable due to the presence of environmental biases (i.e. inadequate representation of the variability of environmental covariates in the study area). Moreover, count data may be affected by both overdispersion (i.e. the variance is larger than the mean) and excess of zero counts (potentially leading to zero inflation). The aim of this study was to define a modelling procedure to assess long-term population trends that addressed these three issues and to shed light on the effects of environmental bias, overdispersion, and zero inflation on trend estimates. To test our procedure, we used six bird species whose data were collected in northern Italy from 1992 to 2019. We designed a multi-step approach. First, using generalised additive models (GAMs), we implemented a full factorial design of models (eight models per species) taking or not into account the environmental bias (including or not including environmental covariates, respectively), overdispersion (using a negative binomial distribution or a Poisson distribution, respectively), and zero inflation (using or not using zero-inflated models, respectively). Models were ranked according to the Akaike Information Criterion. Second, annual population indices (median and 95% confidence interval of the number of breeding pairs per point count) were predicted through a parametric bootstrap procedure. Third, long-term population trends were assessed and tested for significance fitting weighted least square linear regression models to the predicted annual indices. To evaluate the effect of environmental bias, overdispersion, and zero inflation on trend estimates, an average discrepancy index was calculated for each model group. The results showed that environmental bias was the most important driver in determining different trend estimates, although overlooking overdispersion and zero inflation could lead to misleading results. For five species, zero-inflated GAMs resulted the best models to predict annual population indices. Our findings suggested a mutual interaction between zero inflation and overdispersion, with overdispersion arising in non-zero-inflated models. Moreover, for species having flocking foraging and/or colonial breeding behaviours, overdispersed and zero-inflated models may be more adequate. In conclusion, properly handling environmental bias, which may affect several data sets coming from long-term monitoring programs, is crucial to obtain reliable estimates of population trends. Furthermore, the extent to which overdispersion and zero inflation may affect trend estimates should be assessed by comparing different models, rather than presumed using statistical assumption.     

Bottleneck Effect on Evolutionary Rate in the Nearly Neutral Mutation Model

Variances of evolutionary rates among lineages in some proteins are larger than those expected from simple Poisson processes. This phenomenon is called overdispersion of the molecular clock. If population size N is constant, the overdispersion is observed only in a limited range of 2Nσ under the nearly neutral mutation model, where σ represents the standard deviation of selection coefficients of new mutants. In this paper, we investigated effects of changing population size on the evolutionary rate by computer simulations assuming the nearly neutral mutation model. The size was changed cyclically between two numbers, N(1) and N(2) (N(1) > N(2)), in the simulations. The overdispersion is observed if 2N(2)σ is less than two and the state of reduced size (bottleneck state) continues for more than ~0.1/u generations, where u is the mutation rate. The overdispersion results mainly because the average fitnesses of only a portion of populations go down when the population size is reduced and only in these populations subsequent advantageous substitutions occur after the population size becomes large. Since the fitness reduction after the bottleneck is stochastic, acceleration of the evolutionary rate does not necessarily occur uniformly among loci. From these results, we argue that the nearly neutral mutation model is a candidate mechanism to explain the overdispersed molecular clock.