Simple Poisson Model Research Articles

Body Mass Index, Adherence to a Healthy Lifestyle, and Breakfast Consumption Associated with Religious Affiliation in Peruvian University Students: A Cross-Sectional Study.

Despite evidence suggesting a relationship between religiosity and health habits, there is a paucity of studies specifically examining this association in the context of Peruvian university students. This study compared body mass index (BMI), adherence to a healthy lifestyle, and breakfast consumption in Peruvian university students of four religious denominations: Seventh Day Adventists (SDA), Catholics, Baptists, and Evangelicals. A cross-sectional study was conducted online among 4557 students from a Peruvian university. The BMI and the frequency of breakfast consumption were evaluated, and the Diet and Healthy Lifestyle Scale (DEVS) was applied. The variables studied were associated using simple and multiple linear regression and Poisson models with robust variance. Baptist (B = 0.44, 95% CI: 0.10-0.78; p = 0.011), Catholic (B = 0.3, 95% CI: 0.12-0.47; p = 0.001), and Evangelical (B = 0.32, 95% CI: 0.09 to 0.64; p = 0.014) students had a significantly higher BMI compared to SDA. Baptist (B = -0.2, 95% CI: -0.37--0.05; p = 0.017) and Evangelical (B = -0.13, 95% CI: -0.28--0.03; p = 0.012) students exhibited a lower mean score on the measure of healthy lifestyles compared to SDA students. Additionally, Baptist (PR = -0.32, 95% CI: -0.92--0.12; p = 0.035) and Catholic (PR = -0.3, 95% CI: -0.99--0.19; p = 0.016) students exhibited a lower probability of eating breakfast regularly compared to SDA students. Health professionals should consider these findings when designing and implementing health promotion programs that are culturally sensitive and respectful of the beliefs and practices of all religious groups in university settings.

Impact of Maize Production on the Welfare of Small-Scale Famers in Uasin Gishu County; a Case Study of Moiben Constituency

When a community engages in economic activity, they always expect to use it as a resource to elevate their economic status. Residents of Uasin Gishu county have over the years engaged tremendously in maize farming expecting it to aid in elevating their economic status. However, despite the heavy investment in the farming of maize, the majority of small-scale farmers in the region still languish in poverty despite farming maize every season. As a result, the study aimed at determining the impact of maize farming on the economic welfare of small-scale farmers in the region. The study looked at how maize farming influenced the income, food security, and employment status of the farmers. The study was carried out in the Moiben constituency which had a population of 300 maize farmers and it was able to sample a total of 75 farmers from the constituency who were sampled through a simple random sampling technique. The study used a structured questionnaire to interview the farmers. The study used simple linear, Poisson, and binary logistic regression models to determine the effect of maize production on income, food security, and employment of small-scale farmers in the region. The results of the linear regression model showed that maize production had a positive significant effect on the income of the farmers in the region (p<0.05). An increase in the size of land for maize production by 1 ha was determined to increase the income of the farmer by approximately Kshs 55,945/=. Based on the results of the Poisson regression model, maize production was determined to have a positive significant effect on the food security of the farmer (p<0.05). An increase in land size under maize production by 1 ha was determined to increase the number of bags of maize stored for family consumption by 1 bag. Lastly, the logistic regression model showed that maize production had a significant effect on the employment of small-scale farmers (p<0.05). The results showed that a farmer with 1 ha of land more was 3.942 times more likely to only carry out farming as a source of employment compared to a farmer with 1 ha of land under maize farming. In conclusion, the study was able to determine that maize farming was able to increase the income of the farmers, increase food security and provide employment opportunities for small-scale farmers. Therefore, this showed that maize production was able to improve the welfare of maize farmers in the region.

Count (and count-like) data in finance

This paper assesses different econometric approaches to working with count-based outcome variables and other outcomes with similar distributions, which are increasingly common in corporate finance applications. We demonstrate that the common practice of estimating linear regressions of the log of 1 plus the outcome produces estimates with no natural interpretation that can have the wrong sign in expectation. In contrast, a simple fixed-effects Poisson model produces consistent and reasonably efficient estimates under more general conditions than commonly assumed. We also show through replication of existing papers that economic conclusions can be highly sensitive to the regression model employed.

The Featureless HST/WFC3 Transmission Spectrum of the Rocky Exoplanet GJ 1132b: No Evidence for a Cloud-free Primordial Atmosphere and Constraints on Starspot Contamination

Orbiting an M dwarf 12 pc away, the transiting exoplanet GJ 1132b is a prime target for transmission spectroscopy. With a mass of 1.7 M ⊕ and radius of 1.1 R ⊕, GJ 1132b’s bulk density indicates that this planet is rocky. Yet with an equilibrium temperature of 580 K, GJ 1132b may still retain some semblance of an atmosphere. Understanding whether this atmosphere exists and its composition will be vital for understanding how the atmospheres of terrestrial planets orbiting M dwarfs evolve. We observe five transits of GJ 1132b with the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST). We find a featureless transmission spectrum from 1.1 to 1.7 μm, ruling out cloud-free atmospheres with metallicities <300× solar with >4.8σ confidence. We combine our WFC3 results with transit depths from TESS and archival broadband and spectroscopic observations to find a featureless spectrum across 0.7 to 4.5 μm. GJ 1132b therefore has a high mean molecular weight atmosphere, possesses a high-altitude aerosol layer, or has effectively no atmosphere. Higher-precision observations are required in order to differentiate between these possibilities. We explore the impact of hot and cold starspots on the observed transmission spectrum GJ 1132b, quantifying the amplitude of spot-induced transit depth features. Using a simple Poisson model, we estimate spot temperature contrasts, spot covering fractions, and spot sizes for GJ 1132. These limits, as well as the modeling framework, may be useful for future observations of GJ 1132b or other planets transiting similarly inactive M dwarfs.

Modeling zero inflation is not necessary for spatial transcriptomics

BackgroundSpatial transcriptomics are a set of new technologies that profile gene expression on tissues with spatial localization information. With technological advances, recent spatial transcriptomics data are often in the form of sparse counts with an excessive amount of zero values.ResultsWe perform a comprehensive analysis on 20 spatial transcriptomics datasets collected from 11 distinct technologies to characterize the distributional properties of the expression count data and understand the statistical nature of the zero values. Across datasets, we show that a substantial fraction of genes displays overdispersion and/or zero inflation that cannot be accounted for by a Poisson model, with genes displaying overdispersion substantially overlapped with genes displaying zero inflation. In addition, we find that either the Poisson or the negative binomial model is sufficient for modeling the majority of genes across most spatial transcriptomics technologies. We further show major sources of overdispersion and zero inflation in spatial transcriptomics including gene expression heterogeneity across tissue locations and spatial distribution of cell types. In particular, when we focus on a relatively homogeneous set of tissue locations or control for cell type compositions, the number of detected overdispersed and/or zero-inflated genes is substantially reduced, and a simple Poisson model is often sufficient to fit the gene expression data there.ConclusionsOur study provides the first comprehensive evidence that excessive zeros in spatial transcriptomics are not due to zero inflation, supporting the use of count models without a zero inflation component for modeling spatial transcriptomics.

Quaternary basaltic volcanic fields of the American Southwest

AbstractThe southwestern United States contains numerous monogenetic basaltic volcanoes distributed in intraplate volcanic fields. We review, on a regional scale, our current understanding of the Quaternary basalts with a focus on aspects pertinent to hazard assessment, such as physical volcanology and geochronology, while also summarizing the several petrogenetic conceptual models that have been proposed for the range of local tectonic settings in the region. We count 2229 volcanoes in 37 volcanic fields (including the Pinacate volcanic field, which is mostly in northern Sonora, Mexico). Volcanic landforms are dominantly scoria cones and ramparts with attendant lava fields that have a spectrum of ‘a'ā and blocky to pāhoehoe morphologies, while a small percentage of the volcanoes are maars and tuff cones. Explosive eruption styles that were driven mainly by magmatic volatiles, where they have been studied in detail, included Hawaiian, Strombolian, violent Strombolian, and sub-Plinian activity. The latter two have resulted in substantial fallout deposits that can be traced tens of kilometers from source vents. Phreatomagmatic styles have produced pyroclastic current (mainly pyroclastic surges), ballistic, and fallout deposits. These eruption styles pose hazards to humans when they occur in populated areas and to air travel and regional infrastructure even in sparsely populated areas. All but one of the major volcanic fields (fields that contain ∼100 or more Quaternary volcanoes) together form a northwest-southeast–trending band, which we suggest may reflect an influence of plate-boundary-related shearing on melt segregation in the upper mantle along with other factors; this view is consistent with recent global positioning system (GPS) and structural geologic data indicating the influence of dextral motion along the North America-Pacific plate boundary deep inside the Southwest. Of the 2229 Quaternary volcanoes identified, ∼548 (25%) have been dated, and only ∼15% have been dated with methods such as 40Ar/39Ar and cosmogenic surface exposure methods that are considered optimal for young basalts. Acknowledging the large uncertainty due to the poor geochronological data coverage, we use a simple Poisson model to provide a first-order estimate of recurrence rates of monogenetic volcanoes on the scale of the region as a whole; recurrence rates using our compiled age data set range from 3.74 × 10−4 yr−1 to 8.63 × 10−4 yr−1. These values are only based on dated and mapped volcanoes, respectively, and do not account for undated and buried volcanoes or other uncertainties in the volcano count. The time between monogenetic eruptions in the Southwest is similar to the repose times of some polygenetic volcanoes, which suggests that the regional hazard is potentially commensurate with the hazard from a reawakening stratovolcano such as those in the Cascade Range. Notable in our review is that only a few volcanoes have been the subject of physical volcanological characterization, interpretation, and detailed petrologic study that may elucidate factors such as magma generation, ascent (including time scales), and controls on eruption style.

Count Data in Finance

This paper assesses different econometric approaches to working with count-based outcome variables and other outcomes with similar distributional properties, which are increasingly common in corporate finance applications. We demonstrate that the common practice of adding a constant to the outcome and then estimating log-linear regressions produces estimates with no natural interpretation that can easily have the wrong sign. In contrast, a simple fixed-effects Poisson model produces consistent and reasonably efficient estimates under more general conditions than commonly assumed. We also show through replication of existing papers that economic conclusions are frequently sensitive to the econometric approach employed.

Evaluating the impact of a small number of areas on spatial estimation.

BackgroundThere is an expanding literature on different representations of spatial random effects for different types of spatial correlation structure within the conditional autoregressive class of priors for Bayesian spatial models. However, little is known about the impact of these different priors when the number of areas is small. This paper aimed to investigate this problem both in the context of a case study of spatial analysis of dengue fever and more generally through a simulation study.MethodsBoth the simulation study and the case study considered count data aggregated to a small area level in a region. Five different conditional autoregressive priors for a simple Bayesian Poisson model were considered: independent, Besag-York-Mollié, Leroux, and two variants of a localised clustering model. Data were simulated with eight different sizes of areal grids, ranging from 4 to 2500 areas, and two different levels of both spatial autocorrelation and disease counts. Model goodness-of-fit measures and model estimates were compared. A case study involving dengue fever cases in 14 local areas in Makassar, Indonesia, was also considered.ResultsThe simulation study showed that model performance varied under different scenarios. When areas had low autocorrelation and high counts, and the number of areas was at most 25, the BYM, Leroux and localised G = 2 models performed similarly and better than the independent and localised G = 3 models. However, when the number of areas were at least 100, all models performed differently, and the Leroux model performed the best. Overall, the Leroux model performed the best for every scenario especially when there were at least 16 areas. Based on the case study, the comparative performance of spatial models may also vary for a small number of areas, especially when the data have a relatively large mean and variance over areas. In this case, the localised model with G = 3 was a better choice.ConclusionDetecting spatial patterns can be difficult when there are very few areas. Understanding the characteristics of the data and the relative influence of alternative conditional autoregressive priors is essential in selecting an appropriate Bayesian spatial model.

Differential expression analysis for RNAseq using Poisson mixed models.

Identifying differentially expressed (DE) genes from RNA sequencing (RNAseq) studies is among the most common analyses in genomics. However, RNAseq DE analysis presents several statistical and computational challenges, including over-dispersed read counts and, in some settings, sample non-independence. Previous count-based methods rely on simple hierarchical Poisson models (e.g. negative binomial) to model independent over-dispersion, but do not account for sample non-independence due to relatedness, population structure and/or hidden confounders. Here, we present a Poisson mixed model with two random effects terms that account for both independent over-dispersion and sample non-independence. We also develop a scalable sampling-based inference algorithm using a latent variable representation of the Poisson distribution. With simulations, we show that our method properly controls for type I error and is generally more powerful than other widely used approaches, except in small samples (n <15) with other unfavorable properties (e.g. small effect sizes). We also apply our method to three real datasets that contain related individuals, population stratification or hidden confounders. Our results show that our method increases power in all three data compared to other approaches, though the power gain is smallest in the smallest sample (n = 6). Our method is implemented in MACAU, freely available at www.xzlab.org/software.html.

Estimating mating rates in wild Drosophila melanogaster females by decay rates of male reproductive proteins in their reproductive tracts.

Female Drosophila melanogaster frequently mate with multiple males in nature as shown through parentage analysis. Although polyandry is well documented, we know little about the timing between mating events in wild Drosophila populations due to the challenge of following behaviours of individual females. In this study, we used the presence of a male reproductive protein that is transferred to the female during mating (Sex Peptide, SP) to determine whether she had recently mated. We sampled females throughout the day, conducted control matings to determine the decay rate of SP within the female reproductive tract and performed computer simulations to fit the observed proportion of mated females to a nonhomogenous Poisson process that defined the expected time between successive matings for a given female. In our control matings, 100% of mated females tested positive for SP 0.5h after the start of mating (ASM), but only 24% tested positive 24h ASM. Overall, 35% of wild-caught females tested positive for the presence of SP. Fitting our observed data to our simple nonhomogenous Poisson model provided the inference that females are mating, on average, approximately every 27h (with 95% credibility interval 23-31h). Thus, it appears that females are mating a bit less frequently that once per day in this natural population and that mating events tend to occur either early in the morning or late in the afternoon.

Regular Expressions: Mixed Effects Models.

Regular Expressions: Mixed Effects Models.

Testing for zero inflation and overdispersion in INAR(1) models

The marginal distribution of count data processes rarely follows a simple Poisson model in practice. Instead, one commonly observes deviations such as overdispersion or zero inflation. To express the extend of such deviations from a Poisson model, one can compute an appropriately defined dispersion index or zero index. In this article, we develop several tests based on such indexes, including joint tests being based on an index combination. The asymptotic distribution of the resulting test statistics under the null hypothesis of a Poisson INAR(1) model is derived, and the finite-sample performance of the resulting tests is analyzed. Real data examples illustrate the application of these tests in practice.

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.

Quantifying the impact of inter-site heterogeneity on the distribution of ChIP-seq data.

Chromatin Immunoprecipitation followed by sequencing (ChIP-seq) is a valuable tool for epigenetic studies. Analysis of the data arising from ChIP-seq experiments often requires implicit or explicit statistical modeling of the read counts. The simple Poisson model is attractive, but does not provide a good fit to observed ChIP-seq data. Researchers therefore often either extend to a more general model (e.g., the Negative Binomial), and/or exclude regions of the genome that do not conform to the model. Since many modeling strategies employed for ChIP-seq data reduce to fitting a mixture of Poisson distributions, we explore the problem of inferring the optimal mixing distribution. We apply the Constrained Newton Method (CNM), which suggests the Negative Binomial - Negative Binomial (NB-NB) mixture model as a candidate for modeling ChIP-seq data. We illustrate fitting the NB-NB model with an accelerated EM algorithm on four data sets from three species. Zero-inflated models have been suggested as an approach to improve model fit for ChIP-seq data. We show that the NB-NB mixture model requires no zero-inflation and suggest that in some cases the need for zero inflation is driven by the model's inability to cope with both artifactual large read counts and the frequently observed very low read counts. We see that the CNM-based approach is a useful diagnostic for the assessment of model fit and inference in ChIP-seq data and beyond. Use of the suggested NB-NB mixture model will be of value not only when calling peaks or otherwise modeling ChIP-seq data, but also when simulating data or constructing blacklists de novo.

Waste Not, Want Not: Why Rarefying Microbiome Data Is Inadmissible

Current practice in the normalization of microbiome count data is inefficient in the statistical sense. For apparently historical reasons, the common approach is either to use simple proportions (which does not address heteroscedasticity) or to use rarefying of counts, even though both of these approaches are inappropriate for detection of differentially abundant species. Well-established statistical theory is available that simultaneously accounts for library size differences and biological variability using an appropriate mixture model. Moreover, specific implementations for DNA sequencing read count data (based on a Negative Binomial model for instance) are already available in RNA-Seq focused R packages such as edgeR and DESeq. Here we summarize the supporting statistical theory and use simulations and empirical data to demonstrate substantial improvements provided by a relevant mixture model framework over simple proportions or rarefying. We show how both proportions and rarefied counts result in a high rate of false positives in tests for species that are differentially abundant across sample classes. Regarding microbiome sample-wise clustering, we also show that the rarefying procedure often discards samples that can be accurately clustered by alternative methods. We further compare different Negative Binomial methods with a recently-described zero-inflated Gaussian mixture, implemented in a package called metagenomeSeq. We find that metagenomeSeq performs well when there is an adequate number of biological replicates, but it nevertheless tends toward a higher false positive rate. Based on these results and well-established statistical theory, we advocate that investigators avoid rarefying altogether. We have provided microbiome-specific extensions to these tools in the R package, phyloseq.

Insular Organization of Gene Space in Grass Genomes

Wheat and maize genes were hypothesized to be clustered into islands but the hypothesis was not statistically tested. The hypothesis is statistically tested here in four grass species differing in genome size, Brachypodium distachyon, Oryza sativa, Sorghum bicolor, and Aegilops tauschii. Density functions obtained under a model where gene locations follow a homogeneous Poisson process and thus are not clustered are compared with a model-free situation quantified through a non-parametric density estimate. A simple homogeneous Poisson model for gene locations is not rejected for the small O. sativa and B. distachyon genomes, indicating that genes are distributed largely uniformly in those species, but is rejected for the larger S. bicolor and Ae. tauschii genomes, providing evidence for clustering of genes into islands. It is proposed to call the gene islands “gene insulae” to distinguish them from other types of gene clustering that have been proposed. An average S. bicolor and Ae. tauschii insula is estimated to contain 3.7 and 3.9 genes with an average intergenic distance within an insula of 2.1 and 16.5 kb, respectively. Inter-insular distances are greater than 8 and 81 kb and average 15.1 and 205 kb, in S. bicolor and Ae. tauschii, respectively. A greater gene density observed in the distal regions of the Ae. tauschii chromosomes is shown to be primarily caused by shortening of inter-insular distances. The comparison of the four grass genomes suggests that gene locations are largely a function of a homogeneous Poisson process in small genomes. Nonrandom insertions of LTR retroelements during genome expansion creates gene insulae, which become less dense and further apart with the increase in genome size. High concordance in relative lengths of orthologous intergenic distances among the investigated genomes including the maize genome suggests functional constraints on gene distribution in the grass genomes.

A fast GIS-based risk assessment for tephra fallout: the example of Cotopaxi volcano, Ecuador

We present a comprehensive probabilistic hazard assessment for tephra fallout of Cotopaxi volcano (Ecuador), a quiescent but active stratovolcano known for its highly explosive behaviour. First, we developed a set of possible eruptive scenarios based on thorough field investigations, literature studies and using the Global Volcanism Program (GVP) database. Five eruption scenarios were identified, including two based on large pre-historical sub-Plinian/Plinian eruptions with eruptive parameters constrained from field investigations (One Eruption Scenario; OES) and three Eruption Range Scenarios (ERS) based on the Volcanic Explosivity Index (VEI) classification, for which eruptive parameters (i.e. erupted volume, plume height and median grainsize) were stochastically sampled within boundaries defined by VEI 3, 4 and 5. Second, the modelling was performed using the advection-diffusion model TEPHRA2 in combination with wind profiles from the NOAA NCEP/NCAR Reanalysis 1 database. We performed 1,000 runs for each eruption scenario, stochastically sampling a wind profile (OES and ERS) and a set of eruptive parameters (ERS only) at each run. Using the conditional probabilities of occurrence of eruption of VEI 3, 4 and 5 calculated from the GVP catalogue, we assessed the probability of tephra accumulation in a given time window. Based on the GVP database, a simple Poisson model shows that an eruption of VEI ≥3 has a 36 % probability of occurrence in the next 10 years. Finally, the hazard assessment was compiled based on three different outputs, including (i) probability maps for a given tephra accumulation, (ii) isomass maps for a given probability value and (iii) hazard curves for a given location. We conclude that the area west of Cotopaxi is exposed to light to severe tephra fallout for the smallest eruption magnitude considered (i.e. VEI 3). This area comprises a main communication axis (Panamerican Highway) topographically constrained at the bottom of the Interandean Valley, as well as the capital Quito and the town of Latacunga. In a companion paper, Biass et al. (this volume) propose a method for a rapid risk assessment for tephra fallout using global and easily accessible data and the hazard assessment described here.

Decoding in the dark: extracting information from spontaneous activity in primary visual cortex

Understanding the role of spontaneous activity in cortex is a crucial challenge, since it might be involved in sampling cortical states and in the preparation of the network for rapid responses to external stimuli. However, whether there is also information processing during spontaneous activity is still debated. We asked how much information it is possible to extract from V1 spontaneous activity following visual stimulation. We analyzed multi-electrode (100) recordings in an experiment setup where a drifting grating with 8 possible motion directions was presented for 1s and was followed by a blank period of around 1s. Using maximum likelihood estimation (MLE) on a simple independent Poisson model, we found than spontaneous activity during a blank period allowed us to predict the stimulus orientation previously used with an accuracy of around 25% (100/8=12.5% chance level). This result shows that spontaneous activity has in fact information about the preceding stimulus and establishes an information lower bound. Another crucial question is to know by how much we can improve the information that can be decoded. We observed that spontaneous activity is characterized by periods of high activity interleaving periods of low activity. Surprisingly, the average population vector during high activity periods was almost perfectly a scaled version of the population vector at low activity periods. This suggests a model where neurons are Poisson-like but their rates are modulated by a common mode. Applying this model to the data we could predict correctly 40% of the stimulus orientations. Our results not only show that on average approximately half of the information provided during stimulation is still present in the subsequent blank period, but also suggest that global slow oscillations in spontaneous activity -well described by a multiplicative factor akin to the effect of attention to neuronal activity- are a critical component for information processing. Understanding the role of spontaneous activity in cortex is a crucial challenge, since it might be involved in sampling cortical states and in the preparation of the network for rapid responses to external stimuli. However, whether there is also information processing during spontaneous activity is still debated.

A parametric model fitting time to first event for overdispersed data: application to time to relapse in multiple sclerosis

In this article, we propose a parametric model for the distribution of time to first event when events are overdispersed and can be properly fitted by a Negative Binomial distribution. This is a very common situation in medical statistics, when the occurrence of events is summarized as a count for each patient and the simple Poisson model is not adequate to account for overdispersion of data. In this situation, studying the time of occurrence of the first event can be of interest. From the Negative Binomial distribution of counts, we derive a new parametric model for time to first event and apply it to fit the distribution of time to first relapse in multiple sclerosis (MS). We develop the regression model with methods for covariate estimation. We show that, as the Negative Binomial model properly fits relapse counts data, this new model matches quite perfectly the distribution of time to first relapse, as tested in two large datasets of MS patients. Finally we compare its performance, when fitting time to first relapse in MS, with other models widely used in survival analysis (the semiparametric Cox model and the parametric exponential, Weibull, log-logistic and log-normal models).

POISSON VERSUS BINOMIAL: APPOINTMENT OF JUDGES TO THE U.S. SUPREME COURT

Summary The problem of discriminating between the Poisson and binomial models is discussed in the context of a detailed statistical analysis of the number of appointments of the U.S. Supreme Court justices from 1789 to 2004. Various new and existing tests are examined. The analysis shows that both simple Poisson and simple binomial models are equally appropriate for describing the data. No firm statistical evidence in favour of an exponential Poisson regression model was found. Two attendant results were obtained by simulation: firstly, that the likelihood ratio test is the most powerful of those considered when testing for the Poisson versus binomial and, secondly, that the classical variance test with an upper-tail critical region is biased.