Variability and uncertainty for C. botulinum spore loads in food materials.

A representation for C. botulinum spore contamination in food materials begins with the spore concentration, s (per kilogram). In foods, C. botulinum spore loads are very small (typically an s of ∼1 kg⁻¹) so that concentration is well defined (i.e., it is a continuous variable) only in relatively large volumes (i.e., V ≫ 1 kg). In this case, a (batch) processing volume, typically 10³ kg, can be identified as the primary element for consideration. A batch is often identified uniquely for product tracing purposes. In this representation, each batch is considered homogenous so that the contamination of smaller volumes can be considered in terms of independent random samples, but the contamination of batches from a single food category is variable. This contamination structure, expressing within-batch variability as small compared to between-batch variability, is consistent with the aggregation property of the food material categorization scheme and with the stochastic nature of detection that generally applies to small numbers of spores of C. botulinum. In relation to the contamination of food by pathogenic microorganisms, the role of between-batch and within-batch variability has been considered in detail by Commeau et al. (23).

A practical representation is based on log-normal variability for the batch concentration of C. botulinum

where the parameters μ and σ are the means and the standard deviations of ln(s). These parameters are related to <s> and σ_s, the mean and standard deviation of s [i.e., <s> = e^μ+σ2/2 and σ_s/<s> = eσ2−1]. The variability of the spore concentration accounts for different loads in distinct batches of materials. The log-normal form of the variability corresponds to many multiplicative effects on contamination, but in realistic situations where the origin of contamination is simply unknown, it is also a convenient method of expressing the variation of a numerical quantity over several orders of magnitude. We choose to parameterize the variability of the spore concentration by the mean value (location), <s> (per kilogram), and by the coefficient of variation (spread), σ_s/<s>. This parameterization is particularly convenient for considering spore loads in food materials for two reasons: first, when the load is small, the mean value of concentration is more easily interpreted than μ; second, the coefficient of variation is directly related to the standard deviation of the logarithm of the concentration, σ, and is independent of the actual value.

In turn, the parameters of the variability are considered to be uncertain, i.e., each parameter has a single value, but we do not have sufficient information to make a precise estimate. We choose to represent prior belief about the mean spore concentration with a BetaPert uncertainty distribution

where, for each food category, <s>_min, <s>_mod, and <s>_max (per kilogram) are the minimum, modal, and maximum values for the mean concentration, respectively. The BetaPert distribution is a unimodal form (a constrained beta distribution) that is traditionally used for representing numeric values within a finite range, but other unimodal distributions would be equally valid. Prior beliefs concerning the coefficient of variation of the spore concentration are represented by a uniform uncertainty distribution

where (σ_s/<s>)_min and (σ_s/<s>)_max are the limiting values. In the majority of situations, the coefficient of variation for batch concentration is largely unquantified and the uniform distribution represents uninformed beliefs; other featureless distributions are equally valid. A uniform distribution of the coefficient of variation emphasizes the larger values in the corresponding distribution of σ.

We assume that the parameter values for the uncertainty distribution of the mean spore load can be established from the database, and in turn the uncertainty distributions can be considered informative priors in a Bayesian framework. In practice, the database records contain an inseparable mixture of variability and uncertainty. Although there are many ways to evaluate the records, we have used the following consistent scheme for estimating the parameters for the uncertainty distribution of the mean batch concentration. (i) Records were extracted from the database according to material category. (ii) For each record, an MPN measure and an associated standard error for the spore concentration were established. (iii) For each material category, the estimate for <s>_mod was set as the uniformly weighted average of the MPN values (spores per kilogram). (iv) For each material category, the estimate for <s>_max was set as the uniformly weighted average of the values for the upper 95% confidence limit for the MPN (spores per kilogram). (v) We fixed <s>_min as a spore load of zero.

Table 2 indicates parameter values for uncertainty distributions of the mean spore concentration that have been extracted from the collected data set. Each database record includes a number of distinct tests. For example, the 81 shellfish records include 3,874 distinct tests, and the 13 mushroom and fungus records represent 1,410 individual tests.

Prior belief concerning parameter values for uncertainty distribution of mean batch concentrations of nonproteolytic C. botulinum spores^a

For three material categories, we have compared the beta prior with an alternative construction that uses the average of many log-normal distributions, each constructed from an individual database record (again using derived MPN values). The comparison indicates that the parameterization scheme provides a suitable representation of database information. The informative beta prior consistently gives a smoother distribution over a range similar to that of the composite prior. Although the parameterization scheme truncates the uncertainty distribution of the mean value of the batch concentration at large values, it is important to appreciate that, by including log-normal variability, all nonzero spore concentrations are assigned a finite prior probability.

It is practical to assume prior beliefs about the coefficient of variation for the spore concentration that are independent of the material category. To represent prior belief about variability, we have used single values of (σ_s/<s>)_min = 0.5 and (σ_s/<s>)_max = 4. The maximum value for the coefficient of variation corresponds to a σ₁₀ of ∼0.73, so that it is clear that prior belief for the spore concentration includes the possibility of log-normal variability distributions for which 95% of the probability mass extends over three orders of magnitude [σ₁₀ = ln(10)σ is the standard deviation of log₁₀(s)]. The standard deviation, σ₁₀, varies at a very low rate (logarithmically) with the coefficient of variation, so that prior beliefs are relatively insensitive to the choice for (σ_s/<s>)_max. Limpert et al. (24) lists the coefficient of variation for numerous natural populations that occupy this range.

Parameter uncertainty can be combined with log-normal population variability to give a representation for belief concerning the concentration of nonproteolytic C. botulinum in food materials. A schematic illustration of the structure for the joint probability of the parameters and the variables is illustrated on the left side of Fig. 1. Marginal (prior) beliefs about the logarithm of the concentration of nonproteolytic C. botulinum spores in food materials are illustrated by broken dark lines in Fig. 2. The separation of uncertainty and variability imposed by this quantification scheme is not unique, but it is robust, reproducible, and flexible.

Network representation for a statistical model of spore concentration in food materials. The labeled objects (nodes) represent uncertain quantities, and arrows indicate dependency. In the first panel, <s> and σ_s are the means and the standard deviations of the spore concentration, s (kg⁻¹). In the second panel, W_Xi and S_Xi are the size and the load of the i^th sample and X_i is the outcome of the i^th test (positive or negative). In the third panel, S* (number of spores) is the limit of detection, and <S_i> and S_i are the expectation and the actual load in the i^th control experiment, respectively. The second and third panels extend to include the complete set of samples and controls.

Beliefs concerning the logarithm of the concentration for nonproteolytic C. botulinum spores in nine categories of food materials (fish [FI], meat [ME], dairy liquids [DL], dairy nonliquids [DN], cereals [CE], plant materials [PL], shellfish [SH], mushroom and fungi [MF], and herbs and spices [HS]). Broken lines represent prior beliefs, and full lines represent posterior beliefs. Additional dotted lines indicate the posterior belief given perfect detection of spores, and vertical arrows indicate an approximate 95% confidence interval for the maximum likelihood estimate of the limit of detection. For shellfish, the gray line indicates posterior beliefs following a hypothetical positive result of an additional test. For mushroom and fungi, the gray line indicates posterior belief following a hypothetical case in which two positive test results are absent. In the meat panel, two gray lines indicate spore concentration in 80:20 and 20:80 mixtures of meat and plant material. In the cereal panel, the gray line indicates posterior belief for the spore loads when a conflict, arising from a rare case in the evidence from positive controls, is removed. For fish, two gray lines indicate posterior beliefs for alternative forms of prior information: first when the prior probability for the coefficient of variation of the batch spore load is uniformly distributed in the range of 0.5 to 8, and second when the prior probability for the mean value of the batch spore load is uniformly distributed in the range of 250 to 260 spores kg⁻¹.