The relationship between purely stochastic sampling error and the number of technical replicates used to estimate concentration at an extreme dilution

Authors: Irwin, Peter; Nguyen, Ly Huong; Chen, Chinyi

Submitted to: Analytical and Bioanalytical Chemistry
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 6/22/2010
Publication Date: 7/1/2010
Citation: Irwin, P.L., Nguyen, L.T., Chen, C. 2010. The relationship between purely stochastic sampling error and the number of technical replicates used to estimate concentration at an extreme dilution. Analytical and Bioanalytical Chemistry. DOI:10.1007/s00216-010-3967-2

Interpretive Summary: The contamination of foods with pathogenic bacteria (e.g., Salmonella or E. coli O157:H7) may lead to substantial food poisoning epidemics. In order to determine if food poisoning bacteria are present in foods it is sometimes necessary to detect as few as 1 entity (i.e., bacterial DNA molecule or cell or clump of cells) per sampled volume or mass because the level of pathogenic bacteria in foods is very low. In this report we quantitatively model the detection of bacteria at such low levels in order to determine the minimum sample size necessary to detect the pathogens which occasionally populate various food products. This information is useful to food safety microbiologists in the detection of pathogenic bacteria from foods using sensitive biosensor methods such as the quantitative DNA-based (polymerase chain reaction or PCR) techniques.

Technical Abstract: For any analytical system the population mean (mu) number of entities (e.g., cells or molecules) per tested volume, surface area, or mass also defines the population standard deviation (sigma = square root of mu ). For a preponderance of analytical methods, sigma is very small relative to mu due to their large limit of detection (> 100 per volume). However, in theory at least, DNA-based detection methods (real-time, quantitative or qPCR) can detect about 1 DNA molecule per tested volume (i.e., mu ~ 1) whereupon errors of random sampling can cause sample means to substantially deviate from mu if the number of samplings (n), or “technical replicates”, per observation are too few. In this work the behavior of two measures of sampling error (each replicated five-fold) are examined under the influence of n. For all data (mu = 1.25, 2.5, 5, 7.5, 10, and 20) a large sample of individual analytical counts (x) were created and randomly assigned into N integral-valued sub-samples each containing between 2 and 50 repeats (n) whereupon N × n = 322 to 361. From these data the average mu-normalized deviation of sigma from each sub-sample’s standard deviation estimate was calculated (Delta). Alternatively, the average mu-normalized deviation of mu from each sub-sample’s mean estimate was also evaluated (Delta’). It was found that both of these empirical measures of sampling error were proportional to the inverse square root of {n mu}. Derivative (with respect to n) analyses of our results indicate that a large number of samplings (n ~ 33 +/- 3.1) are requisite to achieve a nominal sampling error for samples with a u ~ 1. This result argues that pathogen detection is most economically performed, even using highly sensitive techniques such as qPCR, when some form of organism cultural enrichment is utilized and which results in a binomial response. Thus, using a specific gene PCR-based (+ or –) MPN assay one could detect anywhere from 0.2 to 100,000 CFU/mL using 18 to 48 reactions (i.e., 8 dilutions × 6 replicates per dilution) depending on the initial concentration of the pathogen.