An extension of an over-dispersion test for count data

While over-dispersion in capture–recapture studies is well known to lead to poor estimation of population size, current diagnostic tools to detect the presence of heterogeneity have not been specifically developed for capture–recapture studies. To address this, a simple and efficient method of testing for over-dispersion in zero-truncated count data is developed and evaluated. The proposed method generalizes an over-dispersion test previously suggested for un-truncated count data and may also be used for testing residual over-dispersion in zero-inflation data. Simulations suggest that the asymptotic distribution of the test statistic is standard normal and that this approximation is also reasonable for small sample sizes. The method is also shown to be more efficient than an existing test for over-dispersion adapted for the capture–recapture setting. Studies with zero-truncated and zero-inflated count data are used to illustrate the test procedures.

Estimating the hidden number of scrapie affected holdings in great Britain using a simple, truncated count model allowing for heterogeneity.

None of the current surveillance streams monitoring the presence of scrapie in Great Britain provide a comprehensive and unbiased estimate of the prevalence of the disease at the holding level. Previous work to estimate the under-ascertainment adjusted prevalence of scrapie in Great Britain applied multiple-list capture–recapture methods. The enforcement of new control measures on scrapie-affected holdings in 2004 has stopped the overlapping between surveillance sources and, hence, the application of multiple-list capture–recapture models. Alternative methods, still under the capture–recapture methodology, relying on repeated entries in one single list have been suggested in these situations. In this article, we apply one-list capture–recapture approaches to data held on the Scrapie Notifications Database to estimate the undetected population of scrapie-affected holdings with clinical disease in Great Britain for the years 2002, 2003, and 2004. For doing so, we develop a new diagnostic tool for indication of heterogeneity as well as a new understanding of the Zelterman and Chao’s lower bound estimators to account for potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a special, locally truncated Poisson likelihood equivalent to a binomial likelihood. This understanding allows the extension of the Zelterman approach by means of logistic regression to include observed heterogeneity in the form of covariates—in case studied here, the holding size and country of origin. Our results confirm the presence of substantial unobserved heterogeneity supporting the application of our two estimators. The total scrapie-affected holding population in Great Britain is around 300 holdings per year. None of the covariates appear to inform the model significantly.

Mixture models for capture-recapture count data

The contribution investigates the problem of estimating the size of a population, also known as the missing cases problem. Suppose a registration system is targeting to identify all cases having a certain characteristic such as a specific disease (cancer, heart disease, ...), disease related condition (HIV, heroin use, ...) or a specific behavior (driving a car without license). Every case in such a registration system has a certain notification history in that it might have been identified several times (at least once) which can be understood as a particular capture-recapture situation. Typically, cases are left out which have never been listed at any occasion, and it is this frequency one wants to estimate. In this paper modelling is concentrating on the counting distribution, e.g. the distribution of the variable that counts how often a given case has been identified by the registration system. Besides very simple models like the binomial or Poisson distribution, finite (nonparametric) mixtures of these are considered providing rather flexible modelling tools. Estimation is done using maximum likelihood by means of the EM algorithm. A case study on heroin users in Bangkok in the year 2001 is completing the contribution.

On the question of proportionality of the count of observed Scrapie cases and the size of holding

Background: The present paper investigates the question of a suitable basic model for the number of scrapie cases in a holding and applications of this knowledge to the estimation of scrapie-ffected holding population sizes and adequacy of control measures within holding. Is the number of scrapie cases proportional to the size of the holding in which case it should be incorporated into the parameter of the error distribution for the scrapie counts? Or, is there a different - potentially more complex - relationship between case count and holding size in which case the information about the size of the holding should be better incorporated as a covariate in the modeling? Methods: We show that this question can be appropriately addressed via a simple zero-truncated Poisson model in which the hypothesis of proportionality enters as a special offset-model. Model comparisons can be achieved by means of likelihood ratio testing. The procedure is illustrated by means of surveillance data on classical scrapie in Great Britain. Furthermore, the model with the best fit is used to estimate the size of the scrapie-affected holding population in Great Britain by means of two capture-recapture estimators: the Poisson estimator and the generalized Zelterman estimator. Results: No evidence could be found for the hypothesis of proportionality. In fact, there is some evidence that this relationship follows a curved line which increases for small holdings up to a maximum after which it declines again. Furthermore, it is pointed out how crucial the correct model choice is when applied to capture-recapture estimation on the basis of zero-truncated Poisson models as well as on the basis of the generalized Zelterman estimator. Estimators based on the proportionality model return very different and unreasonable estimates for the population sizes. Conclusion: Our results stress the importance of an adequate modelling approach to the association between holding size and the number of cases of classical scrapie within holding. Reporting artefacts and speculative biological effects are hypothesized as the underlying causes of the observed curved relationship. The lack of adjustment for these artefacts might well render ineffective the current strategies for the control of the disease.

CAMCR: Computer assisted mixture model analysis for capture-recapture count data

Population size estimation with discrete or nonparametric mixture models is considered, and reliable ways of construction of the nonparametric mixture model estimator are reviewed and set into perspective. Construction of the maximum likelihood estimator of the mixing distribution is done for any number of components up to the global nonparametric maximum likelihood bound using the EM algorithm. In addition, the estimators of Chao and Zelterman are considered with some generalisations of Zelterman’s estimator. All computations are done with CAMCR, a special software developed for population size estimation with mixture models. Several examples and data sets are discussed and the estimators illustrated. Problems using the mixture model-based estimators are highlighted.

Estimating the hidden number of scrapie affected holdings in great Britain using a simple, truncated count model allowing for heterogeneity

None of the current surveillance streams monitoring the presence of scrapie in Great Britain provide a comprehensive and unbiased estimate of the prevalence of the disease at the holding level. Previous work to estimate the under-ascertainment adjusted prevalence of scrapie in Great Britain applied multiple-list capture-recapture methods. The enforcement of new control measures on scrapie-affected holdings in 2004 has stopped the overlapping between surveillance sources and, hence, the application of multiple-list capture-recapture models. Alternative methods, still under the capture-recapture methodology, relying on repeated entries in one single list have been suggested in these situations. In this article, we apply one-list capture-recapture approaches to data held on the Scrapie Notifications Database to estimate the undetected population of scrapie-affected holdings with clinical disease in Great Britain for the years 2002, 2003, and 2004. For doing so, we develop a new diagnostic tool for indication of heterogeneity as well as a new understanding of the Zelterman and Chao's lower bound estimators to account for potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a special, locally truncated Poisson likelihood equivalent to a binomial likelihood. This understanding allows the extension of the Zelterman approach by means of logistic regression to include observed heterogeneity in the form of covariates-in case studied here, the holding size and country of origin. Our results confirm the presence of substantial unobserved heterogeneity supporting the application of our two estimators. The total scrapie-affected holding population in Great Britain is around 300 holdings per year. None of the covariates appear to inform the model significantly.

Superovulation and embryo collection in nulliparous Boer goat does immunized against a recombinant ovine alpha-subunit inhibin

With the purpose of eliciting a superovulatory response, 12 adult nulliparous Boer goat does were actively immunized against a recombinant a-subunit of ovine inhibin (roIHN-alpha; two injections of 100 mg 4 weeks apart). Another 12 control Boer goat does were treated with physiological saline and acted as controls. One year later the immunized animals were boostered by the administration of another dose (100 mg) of the immunogen. Following treatment, blood samples were collected twice weekly for the periods of 16 and 12 weeks, respectively, to monitor the inhibin binding ability with the aid of a radio-tracer binding assay. Throughout the experiment, estrus detection was conducted twice daily with the aid of an aproned intact buck. From the first day after treatment to 48 h after standing estrus, ovarian activity was monitored daily by transrectal ultrasonography. On alternate estrous cycles, does were mated and 6 days later flushed transcervically to recover embryos. All goats treated with the roIHN-alpha produced antibodies reactive to the native bovine inhibin tracer-the titre increasing from 2.9 +/- 0.4 to a maximum of 21.9 +/- 2.9% binding after the second injection. The antibody titre gradually subsided over the next 16 weeks. The booster injection restored an elevated antibody titre (11.7 +/- 0.4%), which was maintained until the end of the sampling period 12 weeks later. In the control goats only trace amounts of antibody were recorded throughout the trial. In the roIHN-alpha-immunized goats the number of follicles reaching a diameter of > 4 mm was 14.6 +/- 1.2 per doe. A positive correlation was recorded between the follicle number and antibody titre (r=0.61; P < 0.01). The number of follicles ovulating per doe (6.9 +/- 0.7) followed the same tendency-however, the proportion decreased with increasing follicle numbers. A relatively weak correlation was recorded between the inhibin binding ability and number of ovulations (r=0.27; P < 0.05). In the control goats the majority (92%) of follicles exceeding 4 mm in diameter ovulated (2.5 +/- 0.1 follicles/doe). Embryo collection proved unsatisfactory (42% versus 39% recovery for immunized and control animals, respectively)-presumably because the uterine lumen of the nulliparous does was too narrow to permit effective flushing. In the group of immunized goats the occurrence of short estrous cycles (< 15 days) recorded was 34% versus only 6% in the controls. Overall, immunization of goats against roIHN-alpha led to an almost six-fold increase in number of ovarian follicles, a three-fold increase in ovulations and, despite the low recovery rate, a more than three-fold increase in ova or embryos recovered. It may be concluded that treatment of female goats with roIHN-alpha leads to an inhibin antibody response, accompanied by enhanced ovarian activity. The response was, however, accompanied by a large proportion of retained follicles and a high incidence of short estrous cycles. These problems need to be further investigated before rendering the method fit for application in embryo transfer programs in goats. (C) 2007 Elsevier B.V. All rights reserved.

Equivalence of truncated count mixture distributions and mixtures of truncated count distributions

This article is about modeling count data with zero truncation. A parametric count density family is considered. The truncated mixture of densities from this family is different from the mixture of truncated densities from the same family. Whereas the former model is more natural to formulate and to interpret, the latter model is theoretically easier to treat. It is shown that for any mixing distribution leading to a truncated mixture, a (usually different) mixing distribution can be found so. that the associated mixture of truncated densities equals the truncated mixture, and vice versa. This implies that the likelihood surfaces for both situations agree, and in this sense both models are equivalent. Zero-truncated count data models are used frequently in the capture-recapture setting to estimate population size, and it can be shown that the two Horvitz-Thompson estimators, associated with the two models, agree. In particular, it is possible to achieve strong results for mixtures of truncated Poisson densities, including reliable, global construction of the unique NPMLE (nonparametric maximum likelihood estimator) of the mixing distribution, implying a unique estimator for the population size. The benefit of these results lies in the fact that it is valid to work with the mixture of truncated count densities, which is less appealing for the practitioner but theoretically easier. Mixtures of truncated count densities form a convex linear model, for which a developed theory exists, including global maximum likelihood theory as well as algorithmic approaches. Once the problem has been solved in this class, it might readily be transformed back to the original problem by means of an explicitly given mapping. Applications of these ideas are given, particularly in the case of the truncated Poisson family.

Comparison of dietary fat and fatty acid intake estimated by the duplicate diet collection technique and estimated dietary records

Introduction A high saturated fatty acid intake is a well recognized risk factor for coronary heart disease development. More recently a high intake of n-6 polyunsaturated fatty acids (PUFA) in combination with a low intake of the long chain n-3 PUFA, eicosapentaenoic acid and docosahexaenoic acid has also been implicated as an important risk factor. Aim To compare total dietary fat and fatty acid intake measured by chemical analysis of duplicate diets with nutritional database analysis of estimated dietary records, collected over the same 3-day study period. Methods Total fat was analysed using soxhlet extraction and subsequently the individual fatty acid content of the diet was determined by gas chromatography. Estimated dietary records were analysed using a nutrient database which was supplemented with a selection of dishes commonly consumed by study participants. Results Bland & Altman statistical analysis demonstrated a lack of agreement between the two dietary assessment techniques for determining dietary fat and fatty acid intake. Conclusion The lack of agreement observed between dietary evaluation techniques may be attributed to inadequacies in either or both assessment techniques. This study highlights the difficulties that may be encountered when attempting to accurately evaluate dietary fat intake among the population.

CAMCR: Computer-Assisted Mixture model analysis for Capture–Recapture count data

Population size estimation with discrete or nonparametric mixture models is considered, and reliable ways of construction of the nonparametric mixture model estimator are reviewed and set into perspective. Construction of the maximum likelihood estimator of the mixing distribution is done for any number of components up to the global nonparametric maximum likelihood bound using the EM algorithm. In addition, the estimators of Chao and Zelterman are considered with some generalisations of Zelterman’s estimator. All computations are done with CAMCR, a special software developed for population size estimation with mixture models. Several examples and data sets are discussed and the estimators illustrated. Problems using the mixture model-based estimators are highlighted.

Stubs versus swabs? A comparison of gunshot residue collection techniques

The collection efficiency of two widely used gunshot residue (GSR) collection techniques—carbon-coated adhesive stubs and alcohol swabs—has been compared by counting the number of characteristic GSR particles collected from the firing hand of a shooter after firing one round. Samples were analyzed with both scanning electron microscopy and energy dispersive X-rays by an experienced GSR analyst, and the number of particles on each sample containing Pb, Ba, and Sb counted. The adhesive stubs showed a greater collection efficiency as all 24 samples gave positive results for GSR particles whereas the swabs gave only positive results for half of the 24 samples. Results showed a statistically significant collection efficiency for the stub collection method and likely reasons for this are considered.