MPN (version 0.3.0)

mpn: Calculate most probable number (MPN)

Description

mpn calculates the Most Probable Number (MPN) point estimate and confidence interval for microbial concentrations. Also calculates Blodgett's (2002, 2005, 2010) Rarity Index (RI).

Usage

mpn(positive, tubes, amount, conf_level = 0.95, CI_method = c("Jarvis",
  "LR"))

Arguments

positive

A vector of number of positive tubes at each dilution level.

tubes

A vector of total number of tubes at each dilution level.

amount

A vector of the amount of inoculum per tube at each dilution level. See Details section.

conf_level

A scalar value between zero and one for the confidence level. Typically 0.95 (i.e., a 95 percent confidence interval).

CI_method

The method used for calculating the confidence interval. Choices are "Jarvis" or "LR" (likelihood ratio). See Details section.

Value

A list containing:

• MPN: The most probable number point estimate for microbial density (concentration).

• MPN_adj: The bias-adjusted point estimate for MPN.

• variance: The estimated variance (see Jarvis et al.) of the MPN estimate if CI_method = "Jarvis". If all tubes are positive or all negative, variance will be NA. If CI_method is not "Jarvis", variance will be NA.

• var_log: The estimated variance of the natural log of the MPN estimate (see Jarvis et al.) using the Delta Method. If all tubes are positive or all negative, var_log will be NA. If CI_method is not "Jarvis", var_log will be NA.

• conf_level: The confidence level used.

• CI_method: The confidence interval method used.

• LB: The lower bound of the confidence interval.

• UB: The upper bound of the confidence interval.

• RI: The rarity index.

Warnings

The Jarvis confidence interval assumptions of approximate normality (Delta Method and asymptotic normality of maximum likelihood estimators) depend on large-sample theory. The likelihood ratio assumptions also depend on large-sample theory. Therefore, the Jarvis and LR confidence interval approaches work best with larger experiments.

Details

As an example, assume we start with 3g of undiluted inoculum per tube, then use a 10-fold dilution for 2 dilutions. We now have amount = 3 * c(1, .1, .01).

When all tubes are negative, the point estimate of MPN is zero (same approach as Jarvis et al.). The point estimate for the BAM tables "is listed as less than the lowest MPN for an outcome with at least one positive tube" (App.2).

When all tubes are positive, the point estimate for MPN is Inf (same approach as Jarvis et al.) since no finite maximum likelihood estimate (MLE) exists. The BAM tables "list the MPN for this outcome as greater than the highest MPN for an outcome with at least one negative tube" (App.2). Here, the difference is probably trivial since the sample should be further diluted if all tubes test positive.

The bias adjustment for the point estimate uses the method of Salama et al. (1978). Also see Haas (1989).

Currently, confidence intervals can only be calculated using the Jarvis (2010) or likelihood ratio (LR) approach (Ridout, 1994). The BAM tables use an alternate approach. We slightly modified Jarvis' approach when all tubes are positive or all are negative; we use \(\alpha\) instead of \(\alpha / 2\) since these are one-sided intervals. The Ridout (1994) LR approach uses the same technique (with \(\alpha\)) for these two extreme cases.

If the Rarity Index is less than 1e-04, the experimental results are highly improbable. The researcher may consider running the experiment again and/or changing the dilution levels.

References

Bacteriological Analytical Manual, 8th Edition, Appendix 2, https://www.fda.gov/Food/FoodScienceResearch/LaboratoryMethods/ucm109656.htm

Blodgett RJ (2002). "Measuring improbability of outcomes from a serial dilution test." Communications in Statistics: Theory and Methods, 31(12), 2209-2223. https://www.tandfonline.com/doi/abs/10.1081/STA-120017222

Blodgett RJ (2005). "Serial dilution with a confirmation step." Food Microbiology, 22(6), 547-552. https://doi.org/10.1016/j.fm.2004.11.017

Blodgett RJ (2010). "Does a serial dilution experiment's model agree with its outcome?" Model Assisted Statistics and Applications, 5(3), 209-215. https://doi.org/10.3233/MAS-2010-0157

Haas CN (1989). "Estimation of microbial densities from dilution count experiments" Applied and Environmental Microbiology 55(8), 1934-1942.

Haas CN, Rose JB, Gerba CP (2014). "Quantitative microbial risk assessment, Second Ed." John Wiley & Sons, Inc., ISBN 978-1-118-14529-6.

Jarvis B, Wilrich C, Wilrich P-T (2010). "Reconsideration of the derivation of Most Probable Numbers, their standard deviations, confidence bounds and rarity values." Journal of Applied Microbiology, 109, 1660-1667. https://doi.org/10.1111/j.1365-2672.2010.04792.x

Ridout MS (1994). "A comparison of confidence interval methods for dilution series experiments." Biometrics, 50(1), 289-296.

Salama IA, Koch GG, Tolley DH. (1978) "On the estimation of the most probable number in a serial dilution technique." Communications in Statistics - Theory and Methods, 7(13), 1267-1281.

See Also

Shiny app: https://mpncalc.galaxytrakr.org/

apc for Aerobic Plate Count

Examples

# NOT RUN {
# Compare MPN, 95% CI, and RI to Jarvis -------------------------------------

# Table 1
mpn(positive = c(3, 1, 1), tubes = c(3, 3, 3), amount = c(1, .1, .01))
  #Jarvis: 7.5 (1.9, 30) RI = .209

mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(1, .1, .01))
  #Jarvis: 0 (0, 1.1) RI = 1
mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(1, .1, .01),
    conf_level = .975)$UB  #alpha / 2

mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(1, .1, .01))
  #Jarvis: Inf (36, Inf) RI = 1
mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(1, .1, .01),
    conf_level = .975)$LB  #alpha / 2

# Table 2
mpn(positive = c(20, 14, 3), tubes = c(20, 20, 20), amount = c(1, .1, .01))
  #Jarvis: 13 (7.6, 21) RI = 0.794

mpn(positive = c(50, 35, 7), tubes = c(50, 50, 50),
    amount = 2 * c(1, .1, .01))
  #Jarvis: 6.3 (4.5, 8.7) RI = .806

mpn(positive = c(1, 5, 3, 1, 1), tubes = c(1, 5, 5, 5, 5),
    amount = c(5, 1, .5, .1, .05))
  #Jarvis: 2.7 (1.3, 5.5) RI = .512

# Compare MPN and 95% CI to BAM tables --------------------------------------

# Table 1
mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(.1, .01, .001))
  #BAM: <3.0 (-, 9.5)

mpn(positive = c(0, 0, 1), tubes = c(3, 3, 3), amount = c(.1, .01, .001))
  #BAM: 3.0 (0.15, 9.6)

mpn(positive = c(2, 2, 0), tubes = c(3, 3, 3), amount = c(.1, .01, .001))
  #BAM: 21 (4.5, 42)

mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(.1, .01, .001))
  #BAM: >1100 (420, -)
mpn(positive = c(3, 3, 2), tubes = c(3, 3, 3), amount = c(.1, .01, .001))$MPN


# Table 2
mpn(positive = c(0, 0, 0), tubes = c(5, 5, 5), amount = c(.1, .01, .001))
  #BAM: <1.8 (-, 6.8)
mpn(positive = c(0, 0, 1), tubes = c(5, 5, 5), amount = c(.1, .01, .001))$MPN

mpn(positive = c(4, 0, 2), tubes = c(5, 5, 5), amount = c(.1, .01, .001))
  #BAM: 21 (6.8, 40)

mpn(positive = c(5, 5, 5), tubes = c(5, 5, 5), amount = c(.1, .01, .001))
  #BAM: >1600 (700, -)
mpn(positive = c(5, 5, 4), tubes = c(5, 5, 5), amount = c(.1, .01, .001))$MPN

# Compare MPN and 95% LR CI to Ridout (1994) --------------------------------

# Table 1
mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(.1, .01, .001),
    CI_method = "LR")
  #Ridout: 0 (0, 9.0)
mpn(positive = c(2, 2, 0), tubes = c(3, 3, 3), amount = c(.1, .01, .001),
    CI_method = "LR")
  #Ridout: 21.1 (6.2, 54.3)
mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(.1, .01, .001),
    CI_method = "LR")
  #Ridout: Inf (465.1, Inf)

# }