A covariate adjustment for zero-truncated approaches to estimating the size of hidden and elusive populations

In this paper we consider the estimation of population size from onesource capture–recapture data, that is, a list in which individuals can potentially be found repeatedly and where the question is how many individuals are missed by the list. As a typical example, we provide data from a drug user study in Bangkok from 2001 where the list consists of drug users who repeatedly contact treatment institutions. Drug users with 1, 2, 3, . . . contacts occur, but drug users with zero contacts are not present, requiring the size of this group to be estimated. Statistically, these data can be considered as stemming from a zero-truncated count distribution.We revisit an estimator for the population size suggested by Zelterman that is known to be robust under potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a locally truncated Poisson likelihood which is equivalent to a binomial likelihood. This result allows the extension of the Zelterman estimator by means of logistic regression to include observed heterogeneity in the form of covariates. We also review an estimator proposed by Chao and explain why we are not able to obtain similar results for this estimator. The Zelterman estimator is applied in two case studies, the first a drug user study from Bangkok, the second an illegal immigrant study in the Netherlands. Our results suggest the new estimator should be used, in particular, if substantial unobserved heterogeneity is present.

Extending Zelterman’s approach for robust estimation of population size to zero-truncated clustered data

Estimation of population size with missing zero-class is an important problem that is encountered in epidemiological assessment studies. Fitting a Poisson model to the observed data by the method of maximum likelihood and estimation of the population size based on this fit is an approach that has been widely used for this purpose. In practice, however, the Poisson assumption is seldom satisfied. Zelterman (1988) has proposed a robust estimator for unclustered data that works well in a wide class of distributions applicable for count data. In the work presented here, we extend this estimator to clustered data. The estimator requires fitting a zero-truncated homogeneous Poisson model by maximum likelihood and thereby using a Horvitz-Thompson estimator of population size. This was found to work well, when the data follow the hypothesized homogeneous Poisson model. However, when the true distribution deviates from the hypothesized model, the population size was found to be underestimated. In the search of a more robust estimator, we focused on three models that use all clusters with exactly one case, those clusters with exactly two cases and those with exactly three cases to estimate the probability of the zero-class and thereby use data collected on all the clusters in the Horvitz-Thompson estimator of population size. Loss in efficiency associated with gain in robustness was examined based on a simulation study. As a trade-off between gain in robustness and loss in efficiency, the model that uses data collected on clusters with at most three cases to estimate the probability of the zero-class was found to be preferred in general. In applications, we recommend obtaining estimates from all three models and making a choice considering the estimates from the three models, robustness and the loss in efficiency. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Nitrogen fertilizer and seed rate effects on Hagberg failing number of hybrid wheats and their parents are associated with alpha-amylase activity, grain cavity size and dormancy

Field experiments were carried out to assess the effects of nitrogen fertilization and seed rate on the Hagberg falling number (HFN) of commercial wheat hybrids and their parents. Applying nitrogen (200 kg N ha(-1)) increased HFN in two successive years. The HFN of the hybrid Hyno Esta was lower than either of its parents (Estica and Audace), particularly when nitrogen was not applied. Treatment effects on HFN were negatively associated with a-amylase activity. Phadebas grain blotting suggested two populations of grains with different types of a-amylase activity: Estica appeared to have a high proportion of grains with low levels of late maturity endosperm a-amylase activity (LMEA); Audace had a few grains showing high levels of germination amylase; and the hybrid, Hyno Esta, combined the sources from both parents to show heterosis for a-amylase activity. Applying nitrogen reduced both apparent LMEA and germination amylase. The effects on LMEA were associated with the size and disruption of the grain cavity, which was greater in Hyno Esta and Estica and in zero-nitrogen treatments. External grain morphology failed to explain much of the variation in LMEA and cavity size, but there was a close negative correlation between cavity size and protein content. Applying nitrogen increased post-harvest dormancy of the grain. Dormancy was greatest in Estica and least in Audace. It is proposed that effects of seed rate, genotype and nitrogen fertilizer on HFN are mediated through factors affecting the size and disruption of the grain cavity and therefore LMEA, and through factors affecting dormancy and therefore germination amylase. (c) 2004 Society of Chemical Industry.

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.

A bagging-based correction for the mixture model estimator of population size

Estimation of a population size by means of capture-recapture techniques is an important problem occurring in many areas of life and social sciences. We consider the frequencies of frequencies situation, where a count variable is used to summarize how often a unit has been identified in the target population of interest. The distribution of this count variable is zero-truncated since zero identifications do not occur in the sample. As an application we consider the surveillance of scrapie in Great Britain. In this case study holdings with scrapie that are not identified (zero counts) do not enter the surveillance database. The count variable of interest is the number of scrapie cases per holding. For count distributions a common model is the Poisson distribution and, to adjust for potential heterogeneity, a discrete mixture of Poisson distributions is used. Mixtures of Poissons usually provide an excellent fit as will be demonstrated in the application of interest. However, as it has been recently demonstrated, mixtures also suffer under the so-called boundary problem, resulting in overestimation of population size. It is suggested here to select the mixture model on the basis of the Bayesian Information Criterion. This strategy is further refined by employing a bagging procedure leading to a series of estimates of population size. Using the median of this series, highly influential size estimates are avoided. In limited simulation studies it is shown that the procedure leads to estimates with remarkable small bias.

Writing in French in secondary schools in England and Germany: are the British really 'bad language learners’?

It is widely assumed that the British are poorer modern foreign language (MFL) learners than their fellow Europeans. Motivation has often been seen as the main cause of this perceived disparity in language learning success. However, there have also been suggestions that curricular and pedagogical factors may play a part. This article reports a research project investigating how German and English 14- to 16-year-old learners of French as a first foreign language compare to one another in their vocabulary knowledge and in the lexical diversity, accuracy and syntactic complexity of their writing. Students from comparable schools in Germany and England were set two writing tasks which were marked by three French native speakers using standardised criteria aligned to the Common European Framework of Reference (CEF). Receptive vocabulary size and lexical diversity were established by the X_lex test and a verb types measure respectively. Syntactic complexity and formal accuracy were respectively assessed using the mean length of T-units (MLTU) and words/error metrics. Students' and teachers' questionnaires and semi-structured interviews were used to provide information and participants' views on classroom practices, while typical textbooks and feedback samples were analysed to establish differences in materials-related input and feedback in the two countries. The German groups were found to be superior in vocabulary size, and in the accuracy, lexical diversity and overall quality – but not the syntactic complexity – of their writing. The differences in performance outcomes are analysed and discussed with regard to variables related to the educational contexts (e.g. curriculum design and methodology).