Fitting a distribution to microbial counts: making sense of zeroes

The Most Probable Curve method - A robust approach to estimate kinetic models from low plate count data resulting in reduced uncertainty

A novel method is proposed for fitting microbial inactivation models to data on liquid media: the Most Probable Curve (MPC) method. It is a multilevel model that makes a separation between the "true" microbial concentration according to the model, the "actual" concentration in the media considering chance, and the actual counts on the plate. It is based on the assumptions that stress resistance is homogeneous within a microbial population, and that there is no aggregation of microbial cells. Under these assumptions, the number of colonies in/on a plate follows a Poisson distribution with expected value depending on the proposed kinetic model, the number of dilutions and the plated volume. The novel method is compared against (non)linear regression based on a normal likelihood distribution (traditional method), Poisson regression and gamma-Poisson regression using data on the inactivation of Listeria monocytogenes. The conclusion is that the traditional method has limitations when the data includes plates with low (or zero) cell counts, which can be mitigated using more complex (discrete) likelihoods. However, Poisson regression uses an unrealistic likelihood function, making it unsuitable for survivor curves with several log-reductions. Gamma-Poisson regression uses a more realistic likelihood function, even though it is based mostly on empirical hypotheses. We conclude that the MPC method can be used reliably, especially when the data includes plates with low or zero counts. Furthermore, it generates a more realistic description of uncertainty, integrating the contribution of the plating error and reducing the uncertainty of the primary model parameters. Consequently, although it increases modelling complexity, the MPC method can be of great interest in predictive microbiology, especially in studies focused on variability analysis.

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.

Informing water distribution line rehabilitation through quantitative microbial risk assessment

Poor urban water quality has been linked to diminished source water quality, poorly functioning water treatment systems and infiltration into distribution lines after treatment resulting in microbiological contamination. With limited funding to rehabilitate distribution lines, developing nations need tools to identify the areas of greatest concern to human health so as to target cost effective remediation approaches. Herein, a case study of Hyderabad, Pakistan was used to demonstrate the efficacy of combining quantitative microbial risk assessment (QMRA) for multiple pathogens with spatial distribution system modeling to identify areas for pipe rehabilitation. Abundance of Escherichia coli, Enterococcus (enterococci), Salmonella spp., Shigella spp., Giardia intestinalis, Vibrio cholera, norovirus GI and adenovirus 40/41, were determined in 85 locations including the source water, treatment plant effluent and the city distribution lines. Bayesian statistics and Monte Carlo simulations were used in the QMRA to account for left-censored microbial abundance distributions. Bacterial and viral abundances in the distribution system samples decreased as follows: 9400 ± 19,800 norovirus gene copies/100 mL (average ± standard deviation, 100% of samples positive); 340 ± 2200 enterococci CFU/100 mL (94%), 71 ± 97 Shigella sp. CFU/100 mL (97%), 60 ± 360 E. coli CFU/100 mL (89%), 35 ± 79 adenovirus gene copies/100 mL (100%), and 21 ± 46 Salmonella sp. CFU/100 mL (76%). The QMRA revealed unacceptable probabilities of illness (>1 in 10,000 illness level) from the four exposure routes considered (drinking water, or only showering, tooth brushing, and rinsing vegetables consumed raw). Disease severity indices based on the QMRA combined with mapping the distribution system revealed areas for targeted rehabilitation. The combined intensive sampling, risk assessment and mapping can be used in low- and middle-income countries to target distribution system rehabilitation efforts and improve health outcomes.

Multilevel modelling as a tool to include variability and uncertainty in quantitative microbiology and risk assessment. Thermal inactivation of Listeria monocytogenes as proof of concept

Variability is inherent in biology and also substantial for microbial populations. In the context of food safety risk assessment, it refers to differences in the response of different bacterial strains (between-strain variability) and different cells (within-strain variability) to the same condition (e.g. inactivation treatment). However, its quantification based on empirical observations and its incorporation in predictive models is a challenge for both experimental design and (statistical) analysis. In this article we propose the use of multilevel models to quantify (different levels of) variability and uncertainty and include them in the predictions. As proof of concept, we analyse the microbial inactivation of Listeria monocytogenes to thermal treatments including different levels of variability (between-strain and within-strain) and uncertainty. The relationship between the microbial count and time was expressed using a (non-linear) Weibullian model. Moreover, we defined stochastic hypotheses to describe the different types of variation at the level of the kinetic parameters, as well as in the observations (microbial counts). The model parameters (kinetic parameters and variances) are estimated using Bayesian statistics. The multilevel approach was compared against an analogous, single-level model. The multilevel methodology shrinks extreme parameter estimates towards the mean according to uncertainty, thus mitigating overfitting. In addition, this approach enables to easily incorporate different levels of variation (between-strain and/or within-strain variability and/or uncertainty) in the predictions. On the other hand, multilevel (Bayesian) models are more complex to define, implement, analyse and communicate than single-level models. Nevertheless, their ability to incorporate different sources of variability in predictions make them very suitable for Quantitative Microbial Risk Assessment.

Probabilistic model for estimating Listeria monocytogenes concentration in cooked meat products from presence/absence data

A quantitative probabilistic model was developed to estimate the concentration of Listeria monocytogenes in cooked meat products based on presence/absence data and an assumed zero-inflated distribution, i.e. zero-inflated Poisson (ZIP) or zero-inflated Poisson lognormal (ZIPL) distribution. The performance of these two distributions was compared in two data sets (data set A and B), which represented L. monocytogenes prevalence and concentrations in cooked meat products. In this study, L. monocytogenes contamination data consisted of 4.23% (8/189) and 4.17% (5/120) non-zero counts for data set A and B, respectively. The contamination level of L. monocytogenes, determined by the most probable number (MPN) technique, ranged from 3 to 93 MPN/g among 13 positive samples. The goodness-of-fit test indicated that the ZIPL distribution was better than the simpler ZIP distribution, when L. monocytogenes contamination levels on positive cooked meat samples illustrated large heterogeneity. Results obtained from ZIPL distribution showed that the logarithmic mean value of L. monocytogenes positive samples was 1.5 log MPN/g (log σ = 0.4) for data set A and B. This study provides an alternative probabilistic method when only qualitative data is available in Quantitative microbial risk assessment (QMRA), in particular if pathogen concentrations consist of large numbers of zero counts and represent high variability.

Learning Something From Nothing: The Critical Importance of Rethinking Microbial Non-detects

Accurate estimation of microbial concentrations is necessary to inform many important environmental science and public health decisions and regulations. Critically, widespread misconceptions about laboratory-reported microbial non-detects have led to their erroneous description and handling as "censored" values. This ultimately compromises their interpretation and undermines efforts to describe and model microbial concentrations accurately. Herein, these misconceptions are dispelled by (1) discussing the critical differences between discrete microbial observations and continuous data acquired using analytical chemistry methodologies and (2) demonstrating the bias introduced by statistical approaches tailored for chemistry data and misapplied to discrete microbial data. Notably, these approaches especially preclude the accurate representation of low concentrations and those estimated using microbial methods with low or variable analytical recovery, which can be expected to result in non-detects. Techniques that account for the probabilistic relationship between observed data and underlying microbial concentrations have been widely demonstrated, and their necessity for handling non-detects (in a way which is consistent with the handling of positive observations) is underscored herein. Habitual reporting of raw microbial observations and sample sizes is proposed to facilitate accurate estimation and analysis of microbial concentrations.

Antibiotic Resistances in Livestock: A Comparative Approach to Identify an Appropriate Regression Model for Count Data

Antimicrobial resistance in livestock is a matter of general concern. To develop hygiene measures and methods for resistance prevention and control, epidemiological studies on a population level are needed to detect factors associated with antimicrobial resistance in livestock holdings. In general, regression models are used to describe these relationships between environmental factors and resistance outcome. Besides the study design, the correlation structures of the different outcomes of antibiotic resistance and structural zero measurements on the resistance outcome as well as on the exposure side are challenges for the epidemiological model building process. The use of appropriate regression models that acknowledge these complexities is essential to assure valid epidemiological interpretations. The aims of this paper are (i) to explain the model building process comparing several competing models for count data (negative binomial model, quasi-Poisson model, zero-inflated model, and hurdle model) and (ii) to compare these models using data from a cross-sectional study on antibiotic resistance in animal husbandry. These goals are essential to evaluate which model is most suitable to identify potential prevention measures. The dataset used as an example in our analyses was generated initially to study the prevalence and associated factors for the appearance of cefotaxime-resistant Escherichia coli in 48 German fattening pig farms. For each farm, the outcome was the count of samples with resistant bacteria. There was almost no overdispersion and only moderate evidence of excess zeros in the data. Our analyses show that it is essential to evaluate regression models in studies analyzing the relationship between environmental factors and antibiotic resistances in livestock. After model comparison based on evaluation of model predictions, Akaike information criterion, and Pearson residuals, here the hurdle model was judged to be the most appropriate model.