# Presentation: 346617

## Modeling the Frequency and Duration of Microbial Contamination Events: Considering Uncertainty and Variability in Censored Data

- Mark Powell, USDA, Wash., DC
- Greg Paoli, Decisionalysis, Ottawa, CA
- IAFP 2004
- 8-11 August, Phoenix, AZ

## Overview

- Frequency and duration of *Listeria* contamination event data for RTE products are “censored.”
- Statistical methods specifically designed for analyzing censored data are available.
- The analysis considers:  natural variability in the frequency and duration of contamination events; parameter uncertainty (within model); model uncertainty (between models).

## Duration of Listeria spp. contamination events

| Frequency | Duration of contamination event (d) | Censor
type |
| --- | --- | --- |
| 483 | 0-7 | left |
| 136 | 7-14 | interval |
| 36 | 14-21 | interval |
| 32 | 21-28 | interval |
| 44 | 35+ | right |

- Source: Tompkin (2002)

- Frequency of *Listeria spp*. Contamination Events

- Source: FSIS (2003)

## Exponential Variability Distribution

- Assumes random event occurs with a constant probability over time.
- ß = mean time until occurrence.

## Weibull Variability Distribution

- Probability of occurrence varies with time:
- prob. inc
- α > 1
- prob. dec.
- α < 1

## Maximum Likelihood Estimation for Censored Data

- where: DL = “detection limit”;  f = prob. density;    F = cum. prob.;  x = uncensored data

## Parameter Uncertainty – Large Sample Size

***Empirical bootstrap*** for large sample sizes - duration of contamination events
- Advantage: relax statistical assumptions about parameter distributions
- Repeatedly sample with replacement from the data and apply MLE procedure each time.
- Do this separately for the parameters of the exponential and Weibull distributions.

## Parameter Uncertainty –Small Sample Size 

***Parametric bootstrap*** for small samples - duration of contamination events.
- Bootstrap sample distributions represent parameter uncertainty for each time-to-event model.

## Model Uncertainty – Model Weights

- Bayesian Information Criterion (BIC)

- Bayes Factor (BF)

- Model Weight

## Results: Duration of Contamination

| DURATION | exponential | Weibull |
| --- | --- | --- |
| α | 1 | 0.6292
(0.5621-0.7042) |
| β | 8.3561
(7.7370-9.0248) | 5.9663
(5.1041-6.9743) |
| Improvement in Fit (Chi-square) | - | 86.6821
(<0.0001)  |
| BIC | -831.6952  | -791.6514 |
| Model Weight | 0 | 1 |

## Duration of Contamination Events: Weibull Model

- Large variability, little parameter or model uncertainty.

- Median = 4 d
- 95% CL -95th %ile = 40 d

## Results: Frequency of Contamination

| FREQUENCY | exponential | Weibull |
| --- | --- | --- |
| α | 1 | 1.0501
(0.7093-1.5543) |
| β | 19.6923
(11.4345- 33.9139) | 19.8464
(11.7719-33.4596) |
| Improvement in Fit (Chi-square) | - | 0.0577
(0.81)  |
| BIC | -24.2642 | -25.5178 |
| Model Weight | 0.78 | 0.22 |

## Frequency of Contamination Events

- Large variability and parameter uncertainty, some model uncertainty.

- Median = 14 d
- 5% CL, 5th  %ile = 0.7 d

## Cautionary Example of Large Model Uncertainty

- Source: Salin et al. 2002. Survival Analysis of U.S. Meat and Poultry Recalls, 1994-200, Fig. 10 (Time Until Class 1 Recall, Pre-HACCP)

## Next Steps

- Some data we talk about as censored, like microbial concentrations, really aren't.
- Zero counts for discrete microorganisms are not censored data below an analytical “detection limit,” as assumed by models for continuously distributed chemical concentrations.
- Zero microbial counts arise from variation in sampling and detection processes.
- Methods (e.g., Poisson regression) are available for microbial concentration data including zero counts, non-detects, and MPN, but we’ll save those for another day.