Causal Analytics for Applied Risk Analysis Louis Anthony Cox, Jr



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Excess Risk Factor Distribution

Our model assumes that T. gondii cases in humans are proportional to the overall prevalence of T. gondii in pigs. In this section we derive a factor to quantify the excess risk associated with shifting a given fraction of pigs from total confinement to open/free range operations, thereby boosting the overall average prevalence level. We will later use the excess risk factor derived below as a multiplier in determining the increase in adverse health outcomes.

Let ΔC represent pigs that are shifted from total confinement operations to open/free range operations expressed as a fraction of all pigs. Then the increase in overall net prevalence of T. gondii in pigs, ΔP, is distributed as:
ΔP ~ ΔC*(PO - Pc) (7.1)
where PC and PO are random variables drawn from the beta distributions for confined and open/free range operations respectively. Let:
Pb = baseline fraction of pigs with T. gondii antibodies = 0.027.
Then the relative risk factor, RR, associated with an increase, ΔC, in the fraction of pigs not in total confinement is distributed as:
Relative risk = RR ~ (Pb + ΔP)/Pb
And the excess (incremental) risk as a function of ΔC, denoted as r(ΔC), is RR -1:
r(ΔC) ~ (Pb + ΔP)/Pb - 1 = ΔC*(PO - Pc)/Pb (7.2)


Attribution to Pork

The fraction of human toxoplasmosis cases attributed to pork consumption is the product of the fraction attributed to all food as a source, and the fraction of all food cases attributed to pork. Food sources have been estimated to cause half of all cases in the U.S. (Scallan et al., 2011). The article notes that this estimate was based both on earlier estimates (Mead et al., 1999) and on a European six-city survey of pregnant women, which used survey data and logistic regression to estimate a food-attributable fraction between 0.30 to 0.63 Other mean estimates appearing in the literature include 0.32 (Cressey & Lake, 2005), 0.50 (Vaillant et al., 2005), and 0.56 (Havelaar et al., 2008), based on data and perceptions in New Zealand, France, and the Netherlands respectively. An assumption that the overall food attribution fraction varies uniformly over the interval 0.30 to 0.70 with a mean of 0.50 seems consistent with the data above (it covers the range of previous estimates), is symmetric, and better accounts for a high degree of uncertainty than would a peaked distribution.

In the available literature, fractions of cases attributable to specific foods within the foodborne category have been based on expert judgment. Structured expert elicitations have yielded a mean attribution for pork of 0.50, with a 5th and 95th percentile of 0.21 and 0.99 respectively (Havelaar et al., 2008); and a mean attribution of 0.41 for pork in relation to other possible U.S. food sources, with a standard error of 0.059 (Batz et al., 2012). Both studies relied upon panels of scientific experts, a panel of 16 in Havelaar et al., (11 microbiologists, 4 epidemiologists, and 1 food safety scientist) and forty five “nationally recognized food safety science experts” in Batz et al. Both the foodborne and pork-specific attribution values exhibit significant uncertainty and variability. Actual values will vary depending on food consumption patterns, climate, cultural practices, and other geographic specific dynamics. For the selection of pork within the food category we account for variability by using the mean and standard deviation parameters in the Batz et al. data to define a Normal (0.41, 0.059) probability distribution. Note that the resulting 95% confidence interval [0.29, 0.53] contains the Havelaar et al. mean estimate without introducing the statistical anomalies implied by its corresponding wide and asymmetric estimation interval. The food and pork attribution distributions are summarized in the top rows of Table 7.1.

Pork Attributable Human Case Rates – Adult Hospitalizations and Death

Estimates of human Toxoplasmosis case rates relied largely upon a single large CDC sponsored study of foodborne illness in the U.S (Scallan et al., 2011). That paper derived estimates from large national medical database systems, published research, and expert judgment. As with our study, uncertainty was captured by simulating probability distributions characterizing each component. A similar study, also using simulation modeling, was performed more recently to estimate foodborne illness rates in Canada (Thomas et al, 2013). The Canadian model utilized many estimates from Scallan et al., most notably, for toxoplasmosis, the foodborne attribution rate of 0.50. Similar to our approach, a separate model was used to estimate congenital cases. Given that the Canadian model used many of the same parameters, but was applied to a different population from that in Scallan et al., we concluded that it would have limited applicability to our situation. We used a slightly modified version of the Scallan et al. model as a submodel to ours to generate distributions of human health outcomes for a baseline prevalence level of human Toxoplasmosis. We did not do a deep re-evaluation of each individual distribution and parameter set in their model, nor did we try to combine their original parameter estimates with others. We felt that due to the extensive data, the expertise of the authors, the reasonableness of their assumptions, and the uniqueness of many of their results, their existing model would be quite suitable for the purposes of our study. An exception was the probability distribution for the proportion of cases that were foodborne, as discussed above. Also, since we are only interested in the pork attributable fraction of foodborne cases, the related distribution, also discussed above, was brought into the model. We used a slightly different form of the PERT-Beta distribution due to known mathematical shortcomings with theirs (Davis, 2008). We applied the Scallan et al. study parameters only to the non-infant population of the U.S in 2013 (estimated to be 310.98 M). A separate analysis, described later in this chapter, was used to derive congenital cases. The distributions and parameters derived from their study are summarized in the center portion of Table 7.1.

Following the model of Scallan et al., with modifications as discussed above, the total pork attributable case rate distribution can be stated as the following product:
Total Pork Attributable Cases = Population x Proportion Foodborne x Proportion from Pork x Incidence x Seroconversion Rate (7.3)
Distributions for hospitalizations and deaths were then computed as:
Hospitalizations = Total Pork Attributable Cases x Proportion hospitalized x Under-diagnosis factor (7.4)
Deaths = Total Pork Attributable Cases x Proportion who died x Under-diagnosis factor (7.5)
Using the probability model described by equations 7.3 through 7.5 (but without pork attribution), Scallan et al. estimated 86,686 (90% credible interval 64,861-111,912) domestically acquired foodborne cases of toxoplasmosis in the US, based on the 2006 population. The study also estimated 4,428 hospitalizations (90% range 2,634-6,674) and 327 deaths (90% range 200-482) attributable to foodborne infections. Multiplying 310.98M (estimated US non-infant population in Jan 2013) by the mean or mode of the appropriate distributions yields approximately 36,242 total cases, 1,885 hospitalizations, and 138 deaths attributable to T. gondii in pork. These are just slightly above the mean pork attributable outcomes given in Batz et al.(2012), Table 6, who also relied upon the model of Scallan et al. More detailed estimates are presented and discussed in the Results section.



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