Hierarchical Bayesian analysis of censored microbiological contamination data for use in risk assessment and mitigation

Inferring the distribution of norovirus in individual oysters below the limit of quantification by pooled sampling

Norovirus (NoV) in oysters is a food safety risk of much concern. In order to assess the risk of the exposure, the distribution of the number of NoV copies contained in each oyster should be acquired first for comprehensively quantifying the associated risks. However, the part of the distribution below the limit of quantification cannot be obtained directly by laboratory detecting methods, which hampers accurate assessment. To tackle this challenging problem, a systematic method (Distribution Inference Method by Pooled Sampling) is proposed to infer the unobservable part of distribution based upon all measurements of the pooled samples with n = 2. Using convolutional integrals and real-coded genetic algorithm for inferring, this method has neither requirements for the type or properties of the original distribution, nor requirements for historical data, even nor requirements for the relationship between observable and unobservable parts of the distribution. A series of experiments were conducted on simulated datasets of a variety of types, including normal distribution, uniform distribution, gamma distribution, lognormal distribution, zero-inflated Poisson distribution, their combinations, and even their splicing, covering common distribution types in oyster NoV scenario and more general scenarios. The results show that almost all inferred simulation data and their original counterparts passed Kolmogorov-Smirnov tests, which implies that they are essential of the same distribution. Based on this method, a ready-to-use web system was developed for researchers to infer their original distribution with pooled-sampling measurements from the detection of NoV or even other substances.

Ready-to-eat Meat Plant Characteristics Associated with Food Safety Deficiencies During Regulatory Compliance Audits, Ontario, Canada

Food safety deficiencies in ready-to-eat (RTE) meat processing plants can increase foodborne disease risks. The purpose of this study was to identify common deficiencies and factors related to improved food safety performance in RTE meat plants in Ontario. Routine food safety audit records for licensed provincial free-standing meat processing plants (FSMPs) and abattoirs that process RTE meats were obtained and analyzed in Ontario, Canada, from 2015 to 2019. A Bayesian regression analysis was conducted to examine the association between selected plant characteristics and two outcomes: overall audit rating (pass vs. conditional pass or fail) and individual audit item fail rate. The audit rating was examined in a logistic model, while the audit item fail rate was evaluated in a negative binomial model. The majority (87.7%, n = 800/912) of audits resulted in a pass rating (compared to conditional pass or fail). The mean number of employees per plant, among 200/204 plants with employee data available, was 11.6 (SD = 20.6, range = 1-200). For the logistic regression model, FSMPs were predicted to have a much higher probability of passing audits than abattoirs (32.0% on average, with a 95% credible interval [CI] of 13.8-52.8%). The number of plant employees, water source (municipal vs. private), and types of RTE meat products produced had little to no consistent association with this outcome. The negative binomial model predicted a -0.009 points lower fail rate, on average, for audit items among FSMPs than abattoirs (95% CI: -0.001, -0.018). Meat plants producing jerky had a higher audit item fail rate compared to those that did not produce such products. The other investigated variables had little to no association with this outcome. The results found in this study can support and guide future inspection, audit and outreach efforts to reduce foodborne illness risks associated with RTE meats.

Bayesian inference of heavy metals exposure in crayfish for assessing human non-carcinogenic health risk

Contaminant concentrations often presented left censorship that below the limit of detection (LOD), which may contain true zero values because of no residue. In this study, we analyzed the concentrations of lead (Pb), cadmium (Cd), mercury (Hg), arsenic (As), and chromium (Cr) in 391 crayfish samples collected from 24 provinces of China, modelled the concentrations with two different models in a Bayesian framework, and evaluated health risk by estimating the target hazard quotient (THQ). The highest concentration was associated with Cr in Northwest China (0.1753 ± 0.1520 mg/kg w. w.), and the minimum concentration was Cd in Southwest China (0.0052 ± 0.0144 mg/kg w. w.), all heavy metal concentrations were below their safety limits. The posterior means of not detect rates P₀ of Pb, Cd, Hg, As, and Cr obtained with two models were both nearly equal to the observed not detect rates (51.15%, 36.83%, 27.37%, 64.71%, and 43.73%, respectively). The posterior probabilistic density lines for Pb, Hg, and Cd concentrations obtained with two models were similar, and fitted the empirical distributions well. The posterior density lines of THQs showed that the non-carcinogenic risk of As and Hg were significant high. Moreover, Bayesian approach presented a better understanding of the percentage of population exposed to potential risk.

Evaluation of needle injection practices contributing to Campylobacter contamination in New Zealand chicken and chicken products

One hypothesis for the higher rate of campylobacteriosis in New Zealand (NZ) is that secondary poultry processing practices increase chicken contamination. Chicken marination with needle injection may introduce pathogenic bacteria from the surface deep into the interior muscle tissue. The survival of Campylobacter in/on multi-needle injected chicken products was performed at the processing plant and retail. The 'reduced salts' marinade was not effective in reducing Campylobacter contamination level as the 'high salt' marinade. At the plant, every tested single injected drumstick with 'reduced salt' marinade was contaminated with Campylobacter with up to 3.5 log per drumstick where only 30% of the injected drumsticks with the 'high salt' marinade were contaminated on the surface. At retail, chicken products injected with the 'low salt', the contamination was very low or undetectable as all the products were sold frozen, but the chicken products injected with 'high salt' marinade were sold fresh, and the contamination level varies and can marginally exceed the target Campylobacter contamination limit (3.78 log CFU/carcass) set by The NZ Authority. The multi-needle injection practice tested in this study did not indicate that the marination process could increase the contamination level on chicken or chicken products.

On Bayesian modeling of censored data in JAGS

Background

Just Another Gibbs Sampling (JAGS) is a convenient tool to draw posterior samples using Markov Chain Monte Carlo for Bayesian modeling. However, the built-in function dinterval() for censored data misspecifies the default computation of deviance function, which limits likelihood-based Bayesian model comparison.

Results

To establish an automatic approach to specifying the correct deviance function in JAGS, we propose a simple and generic alternative modeling strategy for the analysis of censored outcomes. The two illustrative examples demonstrate that the alternative strategy not only properly draws posterior samples in JAGS, but also automatically delivers the correct deviance for model assessment. In the survival data application, our proposed method provides the correct value of mean deviance based on the exact likelihood function. In the drug safety data application, the deviance information criterion and penalized expected deviance for seven Bayesian models of censored data are simultaneously computed by our proposed approach and compared to examine the model performance.

Conclusions

We propose an effective strategy to model censored data in the Bayesian modeling framework in JAGS with the correct deviance specification, which can simplify the calculation of popular Kullback–Leibler based measures for model selection. The proposed approach applies to a broad spectrum of censored data types, such as survival data, and facilitates different censored Bayesian model structures.

Microbial-Maximum Likelihood Estimation Tool for Microbial Quantification in Food From Left-Censored Data Using Maximum Likelihood Estimation for Microbial Risk Assessment

In food microbial measurements, when most or very often bacterial counts are below to the limit of quantification (LOQ) or the limit of detection (LOD) in collected food samples, they are either ignored or a specified value is substituted. The consequence of this approach is that it may lead to the over or underestimation of quantitative results. A maximum likelihood estimation (MLE) or Bayesian models can be applied to deal with this kind of censored data. Recently, in food microbiology, an MLE that deals with censored results by fitting a parametric distribution has been introduced. However, the MLE approach has limited practical application in food microbiology as practical tools for implementing MLE statistical methods are limited. We therefore developed a user-friendly MLE tool (called “Microbial-MLE Tool”), which can be easily used without requiring complex mathematical knowledge of MLE but the tool is designated to adjust log-normal distributions to observed counts, and illustrated how this method may be implemented for food microbial censored data using an Excel spreadsheet. In addition, we used two case studies based on food microbial laboratory measurements to illustrate the use of the tool. We believe that the Microbial-MLE tool provides an accessible and comprehensible means for performing MLE in food microbiology and it will also be of help to improve the outcome of quantitative microbial risk assessment (MRA).

Comparison of Statistical Methods for Identifying Risk Factors for Salmonella Contamination of Whole Chicken Carcasses

ABSTRACT

The complexity of the food system makes analyzing microbiological data from food studies challenging because many of the assumptions (e.g., linear relationship between independent and dependent variable and independence of observations) associated with common analytical approaches (e.g., analysis of variance) are violated. Repeated sampling within an establishment introduces longitudinal correlation that must be accounted for during analyses. In this study, statistical methods for clustered or correlated data were used to determine how correlation impacts conclusions and to compare how assumptions associated with statistical methods impact the appropriateness of these methods within the context of food safety. Risk factor analyses for Salmonella contamination of whole chicken carcasses were conducted as a case study with regulatory data collected by the U.S. Department of Agriculture Food Safety and Inspection Service between May 2015 and December 2019 from 203 regulated establishments. Three models, generalized estimating equation, random effects, and logistic, were fit to Salmonella presence or absence data with establishment demographics and inspection history included as potential covariates. Beta parameter estimates and their standard errors and odds ratios and their 95% confidence intervals were compared across models. Conclusions drawn from the three models differed with respect to geographic region, whether the chicken establishment also slaughters turkeys, and establishment noncompliance with 9 CFR §417.4 (hazard analysis critical control point system validation, verification, and reassessment) in the 84 days leading up to sample collection. The results of this study reveal the need to consider clustering and correlation when analyzing food microbiological data, provide context for selecting a statistical method, and suggest that generalized estimating equation and random effects models are preferrable over logistic regression when analyzing correlated food data. These results support a renewed focus on statistical methodology in food safety.

HIGHLIGHTS

BIKE: Dietary Exposure Model for Foodborne Microbiological and Chemical Hazards

BIKE is a Bayesian dietary exposure assessment model for microbiological and chemical hazards. A graphical user interface was developed for running the model and inspecting the results. It is based on connected Bayesian hierarchical models, utilizing OpenBUGS and R in tandem. According to occurrence and consumption data given as inputs, a specific BUGS code is automatically written for running the Bayesian model in the background. The user interface is based on shiny app. Chronic and acute exposures are estimated for chemical and microbiological hazards, respectively. Uncertainty and variability in exposures are visualized, and a few optional model structures can be used. Simulated synthetic data are provided with BIKE for an example, resembling real occurrence and consumption data. BIKE is open source and available from github.

Attempt of Bayesian Estimation from Left-censored Data Using the Markov Chain Monte Carlo Method: Exploring Cr(VI) Concentrations in Mineral Water Products

Hexavalent chromium (Cr(VI)) is toxic, carcinogenic, and mutagenic substances. Oral exposure to Cr(VI) is thought to be primarily from drinking water. However, under the certain reporting limit (~0.1 µg/L), percentage of Cr(VI) concentration in mineral water products under the reporting limit were estimated higher than 50%. Data whose values are below certain limits and thus cannot be accurately determined are known as left-censored. The high censored percentage leads to estimation of Cr(VI) exposure uncertain. It is well known that conventional substitution method often used in food analytical science cause severe bias. To estimate appropriate summary statistics on Cr(VI) concentration in mineral water products, parameter estimation using the Markov chain Monte Carlo (MCMC) method under assumption of a lognormal distribution was performed. Stan, a probabilistic programming language, was used for MCMC. We evaluated the accuracy, coverage probability, and reliability of estimates with MCMC by comparison with other estimation methods (discard nondetects, substituting half of reporting limit, Kaplan-Meier, regression on order statistics, and maximum likelihood estimation) using 1000 randomly generated data subsets (n = 150) with the obtained parameters. The evaluation shows that MCMC is the best estimation method in this context with greater accuracy, coverage probability, and reliability over a censored percentage of 10-90%. The mean concentration, which was estimated with MCMC, was 0.289×10-3 mg/L and this value was sufficiently lower than the regulated value of 0.05 mg/L stipulated by the Food Sanitation Act.

Listeria monocytogenes risk assessment on cold smoked and salt-cured fishery products in Finland - A repeated exposure model

Listeria monocytogenes causes severe consequences especially for persons belonging to risk groups. Finland is among the countries with highest number of listeriosis cases in the European Union. Although most reported cases appear to be sporadic and the maximum bacterial concentration of 100 cfu/g is not usually exceeded at retail, cold smoked and salt-cured fish products have been noted as those products with great risk especially for the elderly. In order to investigate the listeriosis risk more carefully, an exposure assessment was developed, and laboratory results for cold smoked and salt-cured salmon products were exploited. L. monocytogenes exposure was modeled for consumers in two age groups, the elderly population as a risk group and the working-age population as a reference. Incidence was assessed by estimating bacterial growth in the food products at three temperatures. Bayesian estimation of the risk was based on bacterial occurrence and product consumption data and epidemiological population data. The model builds on a two-state Markov chain describing repeated consumption on consecutive days. The cumulative exposure is probabilistically governed by the daily decreasing likelihood of continued consumption and the increasing bacterial concentrations due to growth. The population risk was then predicted with a Poisson distribution accounting for the daily probabilities of purchasing a contaminated product and the cumulative total probability of infection from its use. According to the model presented in this article, elderly Finns are at a greater risk of acquiring listeriosis than healthy adults. The risk for the elderly does not fully diminish even if the products have been stored at the recommended temperature (between 0 and 3 °C). It can be concluded that the stage after retail, i.e. food handling and storage by consumer or professional kitchens, is essential to protection against listeriosis. The estimation model provides means for assessing the joint impacts of these effects.

Toward improved analysis of concentration data: Embracing nondetects

Various statistical tests on concentration data serve to support decision-making regarding characterization and monitoring of contaminated media, assessing exposure to a chemical, and quantifying the associated risks. However, the routine statistical protocols cannot be directly applied because of challenges arising from nondetects or left-censored observations, which are concentration measurements below the detection limit of measuring instruments. Despite the existence of techniques based on survival analysis that can adjust for nondetects, these are seldom taken into account properly. A comprehensive review of the literature showed that managing policies regarding analysis of censored data do not always agree and that guidance from regulatory agencies may be outdated. Therefore, researchers and practitioners commonly resort to the most convenient way of tackling the censored data problem by substituting nondetects with arbitrary constants prior to data analysis, although this is generally regarded as a bias-prone approach. Hoping to improve the interpretation of concentration data, the present article aims to familiarize researchers in different disciplines with the significance of left-censored observations and provides theoretical and computational recommendations (under both frequentist and Bayesian frameworks) for adequate analysis of censored data. In particular, the present article synthesizes key findings from previous research with respect to 3 noteworthy aspects of inferential statistics: estimation of descriptive statistics, hypothesis testing, and regression analysis. Environ Toxicol Chem 2018;37:643-656. © 2017 SETAC.

Quantification of Campylobacter jejuni contamination on chicken carcasses in France

Highly prevalent in poultry, Campylobacter is a foodborne pathogen which remains the primary cause of enteritis in humans. Several studies have determined prevalence and contamination level of this pathogen throughout the food chain. However it is generally performed in a deterministic way without considering heterogeneity of contamination level. The purpose of this study was to quantify, using probabilistic tools, the contamination level of Campylobacter spp. on chicken carcasses after air-chilling step in several slaughterhouses in France. From a dataset (530 data) containing censored data (concentration <10CFU/g), several factors were considered, including the month of sampling, the farming method (standard vs certified) and the sampling area (neck vs leg). All probabilistic analyses were performed in R using fitdistrplus, mc2d and nada packages. The uncertainty (i.e. error) generated by the presence of censored data was small (ca 1 log₁₀) in comparison to the variability (i.e. heterogeneity) of contamination level (3 log₁₀ or more), strengthening the probabilistic analysis and facilitating result interpretation. The sampling period and sampling area (neck/leg) had a significant effect on Campylobacter contamination level. More precisely, two "seasons" were distinguished: one from January to May, another one from June to December. During the June-to-December season, the mean Campylobacter concentration was estimated to 2.6 [2.4; 2.8] log₁₀ (CFU/g) and 1.8 [1.5; 2.0] log₁₀ (CFU/g) for neck and leg, respectively. The probability of having >1000CFU/g (higher limit of European microbial criterion) was estimated to 35.3% and 12.6%, for neck and leg, respectively. In contrast, during January-to-May season, the mean contamination level was estimated to 1.0 [0.6; 1.3] log₁₀ (CFU/g) and 0.6 [0.3; 0.9] log₁₀ (CFU/g) for neck and leg, respectively. The probability of having >1000CFU/g was estimated to 13.5% and 2.0% for neck and leg, respectively. An accurate quantification of contamination level enables industrials to better adapt their processing and hygiene practices. These results will also help in refining exposure assessment models.

Bacteriophage removal efficiency as a validation and operational monitoring tool for virus reduction in wastewater reclamation: Review

The multiple-barrier concept is widely employed in international and domestic guidelines for wastewater reclamation and reuse for microbiological risk management, in which a wastewater reclamation system is designed to achieve guideline values of the performance target of microbe reduction. Enteric viruses are one of the pathogens for which the target reduction values are stipulated in guidelines, but frequent monitoring to validate human virus removal efficacy is challenging in a daily operation due to the cumbersome procedures for virus quantification in wastewater. Bacteriophages have been the first choice surrogate for this task, because of the well-characterized nature of strains and the presence of established protocols for quantification. Here, we performed a meta-analysis to calculate the average log₁₀ reduction values (LRVs) of somatic coliphages, F-specific phages, MS2 coliphage and T4 phage by membrane bioreactor, activated sludge, constructed wetlands, pond systems, microfiltration and ultrafiltration. The calculated LRVs of bacteriophages were then compared with reported human enteric virus LRVs. MS2 coliphage LRVs in MBR processes were shown to be lower than those of norovirus GII and enterovirus, suggesting it as a possible validation and operational monitoring tool. The other bacteriophages provided higher LRVs compared to human viruses. The data sets on LRVs of human viruses and bacteriophages are scarce except for MBR and conventional activated sludge processes, which highlights the necessity of investigating LRVs of human viruses and bacteriophages in multiple treatment unit processes.

Campylobacter QMRA: A Bayesian Estimation of Prevalence and Concentration in Retail Foods Under Clustering and Heavy Censoring

A Bayesian statistical temporal-prevalence-concentration model (TPCM) was built to assess the prevalence and concentration of pathogenic campylobacter species in batches of fresh chicken and turkey meat at retail. The data set was collected from Finnish grocery stores in all the seasons of the year. Observations at low concentration levels are often censored due to the limit of determination of the microbiological methods. This model utilized the potential of Bayesian methods to borrow strength from related samples in order to perform under heavy censoring. In this extreme case the majority of the observed batch-specific concentrations was below the limit of determination. The hierarchical structure was included in the model in order to take into account the within-batch and between-batch variability, which may have a significant impact on the sample outcome depending on the sampling plan. Temporal changes in the prevalence of campylobacter were modeled using a Markovian time series. The proposed model is adaptable for other pathogens if the same type of data set is available. The computation of the model was performed using OpenBUGS software.

A Comparison of the β-Substitution Method and a Bayesian Method for Analyzing Left-Censored Data

Classical statistical methods for analyzing exposure data with values below the detection limits are well described in the occupational hygiene literature, but an evaluation of a Bayesian approach for handling such data is currently lacking. Here, we first describe a Bayesian framework for analyzing censored data. We then present the results of a simulation study conducted to compare the β-substitution method with a Bayesian method for exposure datasets drawn from lognormal distributions and mixed lognormal distributions with varying sample sizes, geometric standard deviations (GSDs), and censoring for single and multiple limits of detection. For each set of factors, estimates for the arithmetic mean (AM), geometric mean, GSD, and the 95th percentile (X0.95) of the exposure distribution were obtained. We evaluated the performance of each method using relative bias, the root mean squared error (rMSE), and coverage (the proportion of the computed 95% uncertainty intervals containing the true value). The Bayesian method using non-informative priors and the β-substitution method were generally comparable in bias and rMSE when estimating the AM and GM. For the GSD and the 95th percentile, the Bayesian method with non-informative priors was more biased and had a higher rMSE than the β-substitution method, but use of more informative priors generally improved the Bayesian method's performance, making both the bias and the rMSE more comparable to the β-substitution method. An advantage of the Bayesian method is that it provided estimates of uncertainty for these parameters of interest and good coverage, whereas the β-substitution method only provided estimates of uncertainty for the AM, and coverage was not as consistent. Selection of one or the other method depends on the needs of the practitioner, the availability of prior information, and the distribution characteristics of the measurement data. We suggest the use of Bayesian methods if the practitioner has the computational resources and prior information, as the method would generally provide accurate estimates and also provides the distributions of all of the parameters, which could be useful for making decisions in some applications.

Fitting a distribution to microbial counts: making sense of zeroes

The accurate estimation of true prevalence and concentration of microorganisms in foods is an important element of quantitative microbiological risk assessment (QMRA). This estimation is often based on microbial detection and enumeration data. Among such data are artificial zero counts, that originated by chance from contaminated food products. When these products are not differentiated from uncontaminated products that originate true zero counts, the estimates of true prevalence and concentration may be inaccurate. This inaccuracy is especially relevant in situations where highly pathogenic bacteria are involved and where growth can occur along the food pathway. Our aim was to develop a method that provides accurate estimates of concentration parameters and differentiates between artificial and true zeroes, thus also accurately estimating true prevalence. We first show the disadvantages of using a limit of quantification (LOQ) threshold for the analysis of microbial enumeration data. We show that, depending on the original distribution of concentrations and the LOQ value, it may be incorrect to treat artificial zeroes as censored below a quantification threshold. Next, a method is developed that estimates the true prevalence of contamination within a food lot and the parameters characterizing the within-lot distribution of concentrations, without assuming a LOQ, and using raw plate count data as an input. Counts resulting both from contaminated and uncontaminated sample units are analysed together. This procedure allows the estimation of the proportion of artificial zeroes among the total of zero counts, and therefore the estimation of true prevalence from enumeration results. We observe that this method yields best estimates of mean, standard deviation and prevalence at low true prevalence levels and low expected standard deviation. Furthermore, we conclude that the estimation of prevalence and the estimation of the distribution of concentrations are interrelated and therefore should be estimated simultaneously. We also conclude that one of the keys to an accurate characterization of the overall microbial contamination is the correct identification and separation of true and artificial zeroes. Our method for the analysis of quantitative microbial data shows a good performance in the estimation of true prevalence and the parameters of the distribution of concentrations, which indicates that it is a useful data analysis tool in the field of QMRA.

Impact of microbial count distributions on human health risk estimates

Quantitative microbiological risk assessment (QMRA) is influenced by the choice of the probability distribution used to describe pathogen concentrations, as this may eventually have a large effect on the distribution of doses at exposure. When fitting a probability distribution to microbial enumeration data, several factors may have an impact on the accuracy of that fit. Analysis of the best statistical fits of different distributions alone does not provide a clear indication of the impact in terms of risk estimates. Thus, in this study we focus on the impact of fitting microbial distributions on risk estimates, at two different concentration scenarios and at a range of prevalence levels. By using five different parametric distributions, we investigate whether different characteristics of a good fit are crucial for an accurate risk estimate. Among the factors studied are the importance of accounting for the Poisson randomness in counts, the difference between treating "true" zeroes as such or as censored below a limit of quantification (LOQ) and the importance of making the correct assumption about the underlying distribution of concentrations. By running a simulation experiment with zero-inflated Poisson-lognormal distributed data and an existing QMRA model from retail to consumer level, it was possible to assess the difference between expected risk and the risk estimated with using a lognormal, a zero-inflated lognormal, a Poisson-gamma, a zero-inflated Poisson-gamma and a zero-inflated Poisson-lognormal distribution. We show that the impact of the choice of different probability distributions to describe concentrations at retail on risk estimates is dependent both on concentration and prevalence levels. We also show that the use of an LOQ should be done consciously, especially when zero-inflation is not used. In general, zero-inflation does not necessarily improve the absolute risk estimation, but performance of zero-inflated distributions in QMRA tends to be more robust to changes in prevalence and concentration levels, and to the use of an LOQ to interpret zero values, compared to that of their non-zero-inflated counterparts.

Fitting a distribution to censored contamination data using Markov Chain Monte Carlo methods and samples selected with unequal probabilities

The fitting of statistical distributions to chemical and microbial contamination data is a common application in risk assessment. These distributions are used to make inferences regarding even the most pedestrian of statistics, such as the population mean. The reason for the heavy reliance on a fitted distribution is the presence of left-, right-, and interval-censored observations in the data sets, with censored observations being the result of nondetects in an assay, the use of screening tests, and other practical limitations. Considerable effort has been expended to develop statistical distributions and fitting techniques for a wide variety of applications. Of the various fitting methods, Markov Chain Monte Carlo methods are common. An underlying assumption for many of the proposed Markov Chain Monte Carlo methods is that the data represent independent and identically distributed (iid) observations from an assumed distribution. This condition is satisfied when samples are collected using a simple random sampling design. Unfortunately, samples of food commodities are generally not collected in accordance with a strict probability design. Nevertheless, pseudosystematic sampling efforts (e.g., collection of a sample hourly or weekly) from a single location in the farm-to-table continuum are reasonable approximations of a simple random sample. The assumption that the data represent an iid sample from a single distribution is more difficult to defend if samples are collected at multiple locations in the farm-to-table continuum or risk-based sampling methods are employed to preferentially select samples that are more likely to be contaminated. This paper develops a weighted bootstrap estimation framework that is appropriate for fitting a distribution to microbiological samples that are collected with unequal probabilities of selection. An example based on microbial data, derived by the Most Probable Number technique, demonstrates the method and highlights the magnitude of biases in an estimator that ignores the effects of an unequal probability sample design.

Development of a time-to-detect growth model for heat-treated Bacillus cereus spores

The microbiological safety and quality of Refrigerated Processed Foods of Extended Durability (REPFEDs) relies on a combination of mild heat treatment and refrigeration, sometimes in combination with other inhibitory agents that are ineffective when used alone. In this context, a predictive model describing the time-to-detect growth (measured by turbidimetry) of psychrotrophic Bacillus cereus spores submitted to various combinations of pH, water activity (aw), heat treatment and storage temperature was developed. As the inoculum was high, the time-to-detect growth was the sum of two times: for a large part of the spore lag time (time before germination and outgrowth) and to a lesser extent of the time to have subsequent vegetative cells growing up to a detectable level. A dataset of 434 combinations (of pH, aw, heat treatment, storage temperature and B. cereus strain), originally collected at Ghent University to build a growth/no-growth model for two Bacillus cereus strains, was re-interpreted as time-to-detect growth values. In the growth area (223 combinations) the time-to-detect growth was set as the longest time where none, or only one, of the 8 replicated wells showed growth. In the no-growth area (211 combinations) the time-to-detect growth was set as longer than the time where the experiment was stopped (60days or more) and analysed as a censored response. The factors of variation were heat-treatment intensity (85°C, 87°C and 90°C in a time range of 1 to 38min), storage temperature (8-30°C), pH (5.2-6.4) and aw (0.973-0.995). Two different strains were analysed. The model had a Gamma multiplicative structure; it was solved by Bayesian inference with informative prior distributions. To be implemented in a decision tool, for instance to calculate the process and formulation conditions required to achieve a given detection time, each Gamma term had some constraints: they had to be monotonous, continuous and algebraically simple mathematical functions (i.e. having analytical solution). Overall, the cumulative effect of various stressful conditions (pasteurisation process, low temperature, and low pH) enables to extend the time-to-detect growth up to 60days or more, whereas the heat-treatment on its own did not have a similar effect. For example, with the most heat resistant strain (strain 1, FF140), for a product at aw0.99, stored at 10°C, heat-treated at 90°C for 10min, a time-to-detect growth of 2days was expected when the pH equalled 6.5. Under the same conditions, if the pH was reduced to 5.8, the time-to-detect growth was predicted to be 11days (and 33days at pH5.5). After a pasteurisation at 90°C for 10min, for a product kept at 10°C, combinations of pH and aw such as pH6.0-aw0.97, pH5.7-aw0.98 or pH5.5-aw0.99 were predicted to extend the time-to-detect growth up to 30days. The developed model is a useful tool for REPFED producers to guarantee the safety of their products towards psychrotrophic B. cereus.