Counting Datasets Research Articles

Statistical models for autocorrelated count data

A generalized linear mixed model is an increasingly popular choice for the modelling of correlated, non-normal responses in a regression setting. A number of methods are currently available for fitting a generalized linear mixed model including Monte-Carlo Markov-Chain maximum likelihood algorithms, approximate maximum likelihood (PQL), iterative bias correction, and others. Of interest in this paper is to compare the parameter estimation of the various methods in the modelling of a count data set, the incidence of polio in the USA over the period 1970-1983, using a longlinear generalized linear mixed model with an autoregressive correlation structure. Despite the fact that all of these methods are considered valid modelling techniques, we find that parameter estimates and standard errors differ substantially between analyses, particularly in the estimation of the parameters describing the random effects distribution. A small simulation study is helpful in understanding some of these differences. The methods lead to reasonably similar predictions for future observations, with small differences observed in some monthly counts.

Effects of Reducing Sample Size on Density Estimates of Citrus Rust Mite (Acari: Eriophyidae) on Citrus Fruit: Simulated Sampling

The consequence of reducing sample size on the accuracy and precision of estimates of citrus rust mite, Phyllocoptruta oleivora (Ashmead), densities on oranges was investigated. The sample unit was a 1-cm2 surface area on fruit. Sampling plans consisting of 360, 300, 200, 160, 80, 48, 36, or 20 samples per 4 ha were evaluated through computer simulations by using real count data from 32 data sets of 600 sample units per 4 ha. The original and reduced sampling plans were hierarchical with different numbers of sample areas per 4 ha, trees per area, fruit per tree, and samples per fruit. Individual estimates (n=100 simulations per data set) using each plan were sometimes considerably below or above target densities. In an original set of count data with a mean of six mites per cm2, simulations of 36 samples per 4 ha produced individual estimates ranging from one to 16 mites per cm2, whereas 80 samples per 4 ha produced estimates ranging from two to 10 mites per cm2. The plans consisting of 36 or more samples were projected to provide precision levels of 0.25 (SEM/mean) or better at densities of five or more mites per cm2 based on log-data, a projection that needs to be verified under real-grove situations. Each plan consistently provided mite detection in these sampling simulations except those consisting of 20 or 36 samples, which sometimes failed to detect mites when the target density was less than five mites per cm2. The study provided insight into the probable precision, accuracy and detection thresholds for eight candidate sampling plans varying from relatively low to high resource input.

Efficient Semiparametric Marginal Estimation for Longitudinal/Clustered Data

We consider marginal generalized semiparametric partially linear models for clustered data. Lin and Carroll derived the semiparametric efficient score function for this problem in the multivariate Gaussian case, but they were unable to construct a semiparametric efficient estimator that actually achieved the semiparametric information bound. Here we propose such an estimator and generalize the work to marginal generalized partially linear models. We investigate asymptotic relative efficiencies of the estimators that ignore the within-cluster correlation structure either in nonparametric curve estimation or throughout. We evaluate the finite-sample performance of these estimators through simulations and illustrate it using a longitudinal CD4 cell count dataset. Both theoretical and numerical results indicate that properly taking into account the within-subject correlation among the responses can substantially improve efficiency.

Model-based geostatistical interpolation of the annual number of ozone exceedance days in the Netherlands

This paper discusses two model-based geostatistical methods for spatial interpolation of the number of days that ground level ozone exceeds a threshold level. The first method assumes counts to approximately follow a Poisson distribution, while the second method assumes a log-Normal distribution. First, these methods were compared using an extensive data set covering the Netherlands, Belgium and Germany. Second, the focus was placed on only the Netherlands, where only a small data set was used. Bayesian techniques were used for parameter estimation and interpolation. Parameter estimates are comparable due to the log-link in both models. Incorporating data from adjacent countries improves parameter estimation. The Poisson model predicts more accurately (maximum kriging standard deviation of 2.16 compared to 2.69) but shows smoother surfaces than the log-Normal model. The log-Normal approach ensures a better representation of the observations and gives more realistic patterns (an RMSE of 2.26 compared to 2.44). Model-based geostatistical procedures are useful to interpolate limited data sets of counts of ozone exceedance days. Spatial risk estimates using existing prior information can be made relating health effects to environmental thresholds.

Detection of anthelmintic resistance: a comparison of mathematical techniques

Anthelmintic resistance has become an increasing problem particularly to gastrointestinal tract nematodes and appropriate methods are required to detect this phenomenon so the correct action can be taken. This paper compares a number of mathematical techniques that are used to analyse data. The negative binomial distribution is a mathematical distribution used to model aggregated data and hence is suitable to model the intensity of parasite burden and the magnitude of the faecal egg counts. Maximum likelihood techniques are utilised to exploit this mathematical distribution to analyse the magnitude of the faecal egg count reduction and decline in the worm burden in response to anthelmintic treatment. Data from experimental groups of sheep described in the accompanying paper are used. In addition, simulated data sets of faecal egg counts were created using a random number generator following appropriate negative binomial distributions. The results demonstrate this statistical model can detect evidence of anthelmintic resistance with a faecal egg reduction test that otherwise might require a slaughter trial to demonstrate. In addition, the simulated data sets confirm that there is a significant probability of failure to detect low anthelmintic efficacy with commonly used mathematical techniques. Consequently, the use of maximum likelihood mathematical techniques with a negative binomial statistical model would aid in the early detection of anthelmintic resistance using faecal egg count reductions and result in a lower probability of inappropriately assigning an anthelmintic as effective.

Analysis of low count time series data by poisson autoregression

Abstract. This study provides new methods of assessing the adequacy of the Poisson autoregressive time‐series model for count data. New expressions are given for the score function and the information matrix and these lead to the construction of new types of residuals for this model. However, these residuals often need to be supplemented by formal statistical procedures and an overall test of the model adequacy is given via the information matrix equality that holds for correctly specified models. The techniques are applied to a monthly count data set of claimants for wage loss benefit, in order to estimate the the expected duration of claimants in the system.

Dealing with large particle counting data sets

A particle count survey of 21 South African water treatment plants over a period of 15 months presented the authors with challenges similar to those experienced by other workers in the field. The amount of data generated was staggering and had to be dealt with in an orderly and structured way to derive the maximum benefit from the survey. Furthermore, the number of significant data points involved if the entire count is to be taken into consideration complicated interpretation of particle counting data. This led to the application of several data reduction techniques to reduce the number of significant parameters that had to be considered during analysis. The most common conventional method is the use of the total particle count larger than a given size, for example total count per ml > 2 μm. This method, however, negates one of the most powerful abilities of the particle counter, namely the ability to indicate particle size distribution. The application of the power law, a common alternative, provides a more detailed description but has its flaws. In this paper the authors illustrate how many particle counts were successfully handled in a purpose-designed database and how the power law concept was improved to provide a better particle counting data-reduction methodology.

Estimating the number of species in a stochastic abundance model.

Consider a stochastic abundance model in which the species arrive in the sample according to independent Poisson processes, where the abundance parameters of the processes follow a gamma distribution. We propose a new estimator of the number of species for this model. The estimator takes the form of the number of duplicated species (i.e., species represented by two or more individuals) divided by an estimated duplication fraction. The duplication fraction is estimated from all frequencies including singleton information. The new estimator is closely related to the sample coverage estimator presented by Chao and Lee (1992, Journal of the American Statistical Association 87, 210-217). We illustrate the procedure using the Malayan butterfly data discussed by Fisher, Corbet, and Williams (1943, Journal of Animal Ecology 12, 42-58) and a 1989 Christmas Bird Count dataset collected in Florida, U.S.A. Simulation studies show that this estimator compares well with maximum likelihood estimators (i.e., empirical Bayes estimators from the Bayesian viewpoint) for which an iterative numerical procedure is needed and may be infeasible.

A new method to measure spatial association for ecological count data

A new method is introduced to assess the spatial association between two sets of count data. This features a measure of local association for counts, defined for each sample unit. The new measure is based on a comparison of the spatial SADIE clustering index of the two sets at each sample unit; the mean of the measure is represented by the simple correlation coefficient between the clustering indices of the two sets. The randomization method allows the construction of a test and critical values. For the first time, spatial association may be mapped for count data; clusters of units with positive association or negative dissociation may be identified. The method is exemplified by analysis of spatial pattern and spatial association of counts of male and female tupelo trees from three plots in a South Carolina swamp forest. In addition, methods are presented to distinguish larger-scale apparent association between the sexes, caused by indirect effects, from direct smaller-scale association. No tendency was found for the sexes to occur together at the small-scale, only an apparent affinity caused through their co-location in particular subareas of each plot. The conversion from mapped to count data requires a choice of unit size; the conclusions of these analyses were not affected greatly by changes in unit size.

A semi‐parametric regression analysis of CD4 cell counts

This paper considers the regression analysis of CD4 cell counts, a commonly used indicator and prognostic factor of AIDS progression. For this purpose, a number of methods have been proposed and most of them are based on random effects models. We present an alternative that is based on a mean function regression model. The new method is more natural and particularly appropriate if population parameters such as treatment comparison are mainly of interest. To estimate regression parameters, we present an estimating equation-based approach, which does not require estimation of the baseline mean function. This makes the implementation of the approach and the study of the properties of the estimated regression parameters much easier than those based on random effects models. The method is illustrated with a set of CD4 cell count data arising from an AIDS clinical trial and simulated data. Copyright © 2001 John Wiley & Sons, Ltd.

A semi-parametric regression analysis of CD4 cell counts.

This paper considers the regression analysis of CD4 cell counts, a commonly used indicator and prognostic factor of AIDS progression. For this purpose, a number of methods have been proposed and most of them are based on random effects models. We present an alternative that is based on a mean function regression model. The new method is more natural and particularly appropriate if population parameters such as treatment comparison are mainly of interest. To estimate regression parameters, we present an estimating equation-based approach, which does not require estimation of the baseline mean function. This makes the implementation of the approach and the study of the properties of the estimated regression parameters much easier than those based on random effects models. The method is illustrated with a set of CD4 cell count data arising from an AIDS clinical trial and simulated data.

Applications of quasi-periodic oscillation models to seasonal small count time series

Quasi-periodic oscillation models for analysis of small count time series are considered within a framework of a generalized state space model (GSSM). In particular, we focus on the analysis of seasonal count data. The Monte Carlo filter (MCF) is fully employed in this study to handle a generalized state space model with higher state dimensions. To illustrate, we study three seasonal count data sets: polio incidence time series, the monthly number of drivers killed in road accidents, and the monthly number of the sun's spotless days. In addition, we demonstrate an application of the model proposed to the yearly occurrence of intense hurricanes with a quasi-periodic component associated with solar cycle activity.

Predictions in Overdispersed Series of Counts Using an Approximate Predictive Likelihood

The generalized autoregression model or GARM, originally used to model series of non-negative data measured at irregularly spaced time points (Lambert, 1996a), is considered in a count data context. It is first shown how the GARM can be expressed as a GLM in the special case of a linear model for some transform of the location parameter. The Butler approximate predictive likelihood (Butler, 1986, Rejoinder) is then used to define likelihood prediction envelopes. The width of these intervals is shown to be slightly wider than the Fisher (1959, pp. 128-33) and Lejeune and Faulkenberry (1982) predictive likelihood-based envelopes which assume that the parameters have fixed known values (equal to their maximum likelihood estimates). The method is illustrated on a small count data set showing overdispersion.

Evaluation of the Power Law and Patchiness Regressions with Regression Diagnostics

We used regression diagnostics to evaluate the robustness of the least-squares regression method for estimating the power law and patchiness regression parameters for 3 data sets of insect counts, specifically for the Bemisia argentifolii Bellows &; Perring, and the squash bug, Anasa tristis (De Geer). Extreme values in the independent variable, x, and dependent variable, y, were detected with the leverage term, h, and standardized residuals, e, respectively. The assumption of homogeneity of variances was evaluated with plots of e, against x for all regressions, and significant autocorrelations were tested with the Durbin-Watson statistic. For both techniques, we compared least-squares regression results for all data with regressions obtained after outlier data points were removed. We also calculated power law regressions excluding means (m) <2 and variances (S2) <4 to reduce possible bias resulting from small mean densities. Outlier data points did not have a significant effect on the power law regressions, but they had a strong influence on some patchiness regressions. The distribution of standardized residuals of some power law regressions were biased toward positive values for low mean densities, indicating underestimation of variances. Additionally, least-squares regression estimates for m ≥2, s2 ≥4 indicated a general increase in slopes for the power law. The distribution of standardized residuals for patchiness regressions indicated strong heteroscedasticity; therefore, the assumption of constant variance for y was not fulfilled. Our results show that suitability of least-squares regression assumptions should be considered whenever pest management decisions are based on the power law or patchiness regressions.

Estimating Taylor's power law parameters for weeds and the effect of spatial scale

SummaryThe spatial variability of weeds within fields was studied for six sets of count data. Heterogeneity for a given mean population density was measured using the variance of the counts between sample units at different locations; relatively large values of sample variance imply aggregation. The dependence of variance on mean was measured using the relationship known as Taylor's power law, ubiquitous in animal ecology but seldom used for plant populations. This was fitted to an extensive set of plant counts and 69 estimates of its parameters b, an index of aggregation, and log10a were computed. Estimates were corrected for bias when the number of samples was small. Overall, b varied between 1.32 and 2.61, and log10a varied between ‐0.85 and 1.58. agreeing well with previous estimates for both plant and animal populations. Parameter estimates varied with sample size and spatial sample scale, but unpredictably. Parameter values when species counts were combined were compared with individual species analyses. Knowledge of the likely range of these parameters for weed populations provides an important basis for future modelling of the relationship between weed density and crop yield loss.

Excess Zeros in Count Models for Recreational Trips

This article develops a modeling approach for a count dataset for recreational boating trips that shows a frequency of zero counts significantly higher than that expected for Poisson-distributed data. We consider several parametric and semiparametric mixed and modified Poisson models as alternatives to the Poisson regression. The analysis suggests that the negative binomial hurdles model, which allows for overdispersion and also accommodates the presence of excess zeros, is the most satisfactory of all those considered

Normalizing Counts and Cerebral Blood Flow Intensity in Functional Imaging Studies of the Human Brain

Image intensity normalization is frequently applied to eliminate or adjust for subject or injection global blood flow (gCBF) and other sources of nuisance variation. Normalization has several other positive effects on the analysis of PET images. However, the choice of an intensity normalization technique affects the statistical and psychometric properties of the image data. We compared three normalization procedures, the ratio approach (regional (r)CBF/gCBF), histogram equalization, and ANCOVA, on both PET count and flow data sets. The ratio method presents the proportional increase of regions, the histogram equalization method offers the relative ranking of intensities over the image, and the ANCOVA method provides statistical deviations from an expected linear model of regional values from the subject's gCBF. The original study used 33 normal subjects in a standard subtraction paradigm. The normalization methods were evaluated on their ability to remove extraneous error variation, induce homogeneity of intersubject variation, and remove unwanted dependencies. In general, the normalization modified the subtraction image more than the individual condition images. All three methods worked well at removing the dependency of rCBF on gCBF in count and flow images. For count data, the three methods also reduced the amount of error variation equally well, improving the signal to noise ratio. For flow data, the histogram equalization and ratio methods worked best at reducing statistical error. All three methods dramatically stabilized the variance over the image.

The Hard X-ray Telescope (HXT) for the SOLAR-A mission

The Hard X-ray Telescope (HXT) is a Fourier-synthesis imager; a set of spatially-modulated photon count data are taken from 64 independent subcollimators and are Fourier-transformed into an image by using procedures such as the maximum entropy method (MEM) or CLEAN. The HXT takes images of solar flares simultaneously in four energy bands, nominally 15 (or 19)–24, 24–35, 35–57, and 57–100 keV, with an ultimate angular resolution as fine as ∼ 5 arc sec and a time resolution 0.5 s. Each subcollimator has a field of view wider than the solar disk. The total effective area of the collimator/detector system reaches ∼ 70 cm2, about one order of magnitude larger than that of the HINOTORI hard X-ray imager. Thanks to these improvements, HXT will for the first time enable us to take images of flares at photon energies above ∼ 30 keV. These higher-energy images will be compared with lower-energy ones, giving clues to the understanding of nonthermal processes in solar flares, i.e., the acceleration and confinement of energetic electrons. It is of particular importance to specify the acceleration site with regard to the magnetic field figuration in a flaring region, which will be achieved by collaborative observations between HXT and the Soft X-ray Telescope on board the same mission.

Asymptotic Normality of Goodness-Of-Fit Statistics for Sparse Product Multinomials

SUMMARY The asymptotic distribution of the Pearson X 2 and likelihood ratio G 2 goodness-of-fit statistics is derived for k independent, non-identically distributed multinomials, as k approaches infinity, with expected cell frequencies bounded. The application is to large datasets of counts with many zero cells. Conditions are given under which the limiting distributions are normal for originating probabilities (i) known and (ii) dependent on a common, finite-dimensional parameter. A logistic regression model is given which satisfies (ii). Finally, skewness is discussed as a measure of closeness of moderate upper percentage points of X 2 to normal percentiles.

Marginal Distribution Estimators for the Gamma-Prior Parameters for a Group of Poisson Processes

Multiple data-sets of experimental times and failure counts from Poisson processes are used to estimate the parameters of the gamma distributions which are assumed appropriate for the failure rates. The experimental data are combined in two ways to estimate the failure rates; they are called unweighted and time weighted. These lead in turn to two different sets of gamma-parameter estimates. Marginal maximum likelihood estimates (MLE) are also considered. The concept of linkage is introduced, wherein some of the data sets are associated with priors which have common values of either scale parameters or shape parameters or both. A numerical example is presented with real data, showing the three sets of estimators for scale and shape parameters (unweighted, time-weighted, and MLE) for each type of linkage.