Our results show that the relationship between the holding size and the number of confirmed scrapie cases followed a curved line for the years 2002, 2003 and 2004 of SND data. The scrapie count increases with holding size with a peak around holding sizes of 100 adult sheep after which the scrapie case count decreases again. This pattern not only appears consistently for the SND in the 3 years, it also occurs in the same way for the CSFS data set. This similarity between these two sets of analysis was expected due to the large weight of the SND data (70.6%) in the CSFS dataset.
In broad terms, the observed curved relationship between the case count and the holding size can be a manifestation of the dynamics of infection within the holding or an artefact related to the reporting and/or testing of cases; or more likely, a combination of both. We can only speculate about the processes potentially responsible for the observed relationship. Under a biological approach, only the exhaustion of all susceptible animals would stop the progression of the epidemic within holding. For scrapie, where the outcome is always fatal, the exhaustion of the susceptible population is only achieved by means of depopulation or selection of resistant genotypes to the disease . To relate either intervention to our results, their effects would have to be different depending on the holding size. Larger holdings might have pursued genotyping of their animals at a greater rate than smaller ones. Or, similarly, the increased number of movements into larger holdings might have facilitated the shift in their genotype profiles towards resistant types more rapidly than in smaller, more static holdings. Either option might explain the reduced number of scrapie cases observed at the right end of our data.
The effect of reporting artefacts appears plausible and allows meaningful interpretation of our SND results. We could hypothesize a proportional relationship between the holding size and the level of supervision, defined in this setting as the regular observation of the holding by the farmer. It follows that with greater supervision, all other things equal (e.g. no biological effects associated with the occurrence of disease as suggested above), we should expect, from a larger holding, a greater case count. Our results on SND data show different indicating that some of the steps in the logic above do not hold. To test the rationale above, for SND data, is not straightforward as there is no clear definition of "supervision" and even less clear approach to its quantification and measurement. Our analyses of the CSFS data could provide some proxy indication of the relationship between supervision and case count. Within the CSFS, the TSE-testing equates to the supervision in SND holdings. More specifically, our results show the proportional relationship between the number of samples tested within holding and the number of cases detected in the CSFS. This is different from the curved pattern observed with the SND data and would indicate the lack of proportionality between holding size and supervision in the field. This explanation would fit well with the large body of evidence that support the occurrence of reporting artefacts affecting the ascertainment of scrapie [3, 4, 9, 10, 12, 19]. It would appear intuitive to think that larger holdings may have more difficulty in identifying all the cases of scrapie. This reduced detection capability would account for the significant quadratic relationship in the case of the SND datasets.
It appears difficult to reconcile our results with those of previous works [2, 3, 20] where larger holdings appeared at greater risk of having scrapie. On the other hand, it was shown that the risk of scrapie followed a quadratic shape with increasing holding size for Great Britain and SND data from 1993 to 2002 . Note that this previous work compares counts of holdings stratified by holding size with the occurrence of scrapie regardless of the number of cases within holding. The results presented here are not directly comparable as we focus on the count of confirmed scrapie cases within holding and hence, we are not informing any measure of risk of scrapie. However, our present results and those of  are consistent with the occurrence of reporting artefacts: a non-proportional supervision in larger holdings will result in fewer of them reporting suspect cases.
Our results on the zero-inflated CSFS dataset inform a pre-clinical stage and, by definition, one free from human-related artefacts. They also refer to a different disease stage, infection, from that, clinical disease, shown by the analyses on SND data. Unfortunately, two sources of biases are still pertinent to our CSFS results. The first one originates from the sampling scheme performed within the CSFS, as per EU requirements, by which sampling stops at 150 animals regardless of the holding size . This sampling strategy is aimed at detecting, with an assumed within-holding prevalence of 2%, at least one infected animal. Sampling for detection of disease does not favour case counts in larger holdings . The second source of biases, a less certain one, would affect the representativeness of our results on the general scrapie-affected population. In  a large proportion of CSFS holdings with just one confirmed case of scrapie is reported, the index case. This might reflect the active search of scrapie cases by farmers in order to benefit from the compensation schemes introduced with the CSFS in 2004. Indeed, a significant increase in the number of cases in some areas in Great Britain around the introduction of the CSFS was reported recently .
It would seem natural to expect that the case count rose linearly if not proportional with the holding size and, had this been correct, it could be used beneficially in estimating the hidden burden of the scrapie epidemic. The offset-model (proportionality hypothesis) was considered in  and needs to be revised on the basis of these findings here. For example, for SND data and 2002 and 2004, the proportionality model returns estimates of the scrapie-affected population twice as large as those returned by the model with the best fit, that with the quadratic term. Clearly, the appropriate choice of a model is crucial for deriving an appropriate estimate of the population size of scrapie affected holdings. It appears founded then that for the surveillance schemes of SND and CSFS the hypothesis of proportionality does not hold. It was also shown that the robust generalised Zelterman estimation based upon fitting a logistic model  with quadratic terms for the year-specific SND data will lead to a more realistic value of the population size. Note that the appropriate choice of the model is not only crucial for avoiding a spurious population size estimate of scrapie (see Table 2), it is also important for achieving valid variance estimates leading to trustworthy confidence intervals. As Table 3 shows, all confidence intervals computed on the basis of the – inappropriate – model 1 are meaningless since they are too large. For the robust regression model the inclusion of the quadratic term is less consequential: for the years 2002 and 2004 the population size estimates based upon model 2 and model 3 are quite close (see Table 3) whereas for the year 2003 model 2 will be preferred. We can conclude from this analysis that it is less important to consider quadratic terms in the robust generalized Zelterman approach. This can easily be explained since the robust approach ignores higher scrapie counts in the holdings and, hence, is less sensitive to model changes affecting only larger scrapie counts.
Our efforts to prevent the effect of reporting artefacts on the relationship between holding size and the case count concentrated on the analyses of CSFS data after the removal of the index case. This is also a more realistic scenario as the accumulation of clinical cases within holding is no longer possible after the introduction of the CSFS in 2004 . For the latter dataset, and to increase the number of observations (holdings) in our lists, we joined the two years of data, 2005 and 2006. The joint list prevented the estimation of a biologically meaningful measure of frequency from this dataset and, hence, comparisons with previous works .
In general terms, our results have two applications: 1) for the correct adjustment of population estimates of scrapie-affected holdings and 2) to advice on the correct relationship between holding size and the count of scrapie cases within holding. The former constitutes one of the first and most basic parameters in surveillance planning. The latter will help in the development of predictive models and, on a more practical side, in the refinement and increased understanding of targeted surveillance approaches to classical scrapie. It is important to stress at this point that our results are applicable only to the classical form of scrapie. Clinical atypical scrapie was first diagnosed in Great Britain in 2005 , after the introduction of the CSFS and, so far, only one holding has shown multiple cases of atypical scrapie (Del Rio Vilas, personal communication). It is worth mentioning that the testing regime applied under the CSFS scheme allows the discrimination of the two types of scrapie .
The implications of our results are large, if not for what they are, for what they show: either a biased, for the SND, or a truncated, for the CSFS, picture of the effects of holding size in the case count within holding. This limitation affects our understanding of the results derived from the scrapie surveillance. The current active surveillance for scrapie is a targeted one: i) at the individual level by targeting sheep older than 18 months of age where the likelihood of detecting infection is greater and ii) at the holding level by, inadvertently, targeting large holdings . The latter requires careful consideration after our results. The occurrence of under-ascertainment in larger holdings by means of reduced supervision will affect the detection capability of one of the surveys, the fallen stock. The lack of opportunity to spot disease can easily be extrapolated to insufficient supervision to identify and promptly report the dead-on-farm sheep. This might contribute to explain the reduced average size of scrapie-affected holdings detected by this survey relative to those detected by the abattoir survey .
Reporting artefacts should not be a problem for the abattoir survey due to its random nature. Farmers can do little to influence the sampling approach of the survey. Only a biological effect, e.g. a greater proportion of resistant genotypes in larger holdings, might affect the results from this survey. In fact, if there was such a biological effect, the natural tendency of the survey to over-sample larger holdings might result in the underestimation of the prevalence of scrapie. This effect might also explain the reduced sensitivity of the abattoir survey relative to the fallen stock .