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MPN: Most Probable Number and Other Microbial Enumeration Techniques

Description

MPN is a package for calculating the Most Probable Number (MPN) toquantify the concentration of microbes in serial dilutions of laboratorysamples. Also calculates the Aerobic Plate Count (APC) for similarexperiments.

Functions

The functionmpn calculates the Most Probable Number (MPN)point estimate and confidence interval for microbial concentrations. Alsocalculates Blodgett's (2002, 2005, 2010) Rarity Index (RI). The MPNcalculation is described in the Bacteriological Analytical Manual (BAM, 8thed., Appendix 2) and Jarvis et al. (2010).

apc calculates the Aerobic Plate Count (APC) point estimateand confidence interval of colony forming units (CFU). Adjusts fortoo-numerous-to-count (TNTC) plates using the maximum likelihood method ofHaas et al. (2014).

Author(s)

Maintainer: John IhrieJohn.Ihrie@fda.hhs.gov

Authors:

• Martine FergusonMartine.Ferguson@fda.hhs.gov (original R code)

References

Bacteriological Analytical Manual, 8th Edition, Appendix 2,https://www.fda.gov/food/laboratory-methods-food/bam-appendix-2-most-probable-number-serial-dilutions

Bacteriological Analytical Manual, 8th Edition, Chapter 3,https://www.fda.gov/food/laboratory-methods-food/bam-chapter-3-aerobic-plate-count

Blodgett RJ (2002). "Measuring improbability of outcomes from aserial dilution test."Communications in Statistics: Theory andMethods, 31(12), 2209-2223.

Blodgett RJ (2005). "Serial dilution with a confirmation step."Food Microbiology, 22(6), 547-552.

Blodgett RJ (2010). "Does a serial dilution experiment's modelagree with its outcome?"Model Assisted Statistics and Applications,5(3), 209-215.

Haas CN, Heller B (1988). "Averaging of TNTC counts."Applied and Environmental Microbiology, 54(8), 2069-2072.

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

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

Jarvis B, Wilrich C, Wilrich P-T (2010). "Reconsideration of thederivation of Most Probable Numbers, their standard deviations, confidencebounds and rarity values."Journal of Applied Microbiology, 109,1660-1667.

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

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

See Also

Useful links:

• https://pub-connect.foodsafetyrisk.org/microbial/mpncalc/

Calculate aerobic plate count (APC)

Description

apc calculates the Aerobic Plate Count (APC) point estimateand confidence interval of colony forming units (CFU). Adjusts fortoo-numerous-to-count (TNTC) plates using the maximum likelihood method ofHaas et al. (2014).

Usage

apc(  count,  amount_scor,  amount_tntc = NULL,  tntc_limit = 250,  conf_level = 0.95,  tol = 1e-06)

Arguments

Details

As an example, assume we start with four plates and 1 ml ofundiluted inoculum. For the first two plates we use a 100-fold dilution;for the other two plates we use a 1,000-fold dilution. The first two plateswere TNTC with limits of 300 and 250. The other plates had CFU counts of28 and 20. We now havecount = c(28, 20),amount_scor = 1 * c(.001, .001),amount_tntc = 1 * c(.01, .01), andtntc_limit = c(300, 250).

Currently, confidence intervals can only be calculated using thelikelihood ratio (LR) approach described in Haas et al. (2014).

Value

A list containing:

• APC: The aerobic plate count point estimate in CFU/ml.

• conf_level: The confidence level used.

• LB: The lower bound of the confidence interval.

• UB: The upper bound of the confidence interval.

Warnings

The likelihood ratio confidence interval assumptions depend on asymptotictheory. Therefore, the confidence interval results will be better withlarger experiments.

apc() will fail in certain cases where the TNTC results areextremely unlikely to occur when taking the scorable (countable) platesinto consideration. In other words, if the countable plates suggest a lowconcentration of microbes, then TNTC plates at higher dilution levels areprobably due to experimental error. Mathematically, the probability is sosmall that the likelihood function is essentially zero.

References

Bacteriological Analytical Manual, 8th Edition, Chapter 3,https://www.fda.gov/food/laboratory-methods-food/bam-chapter-3-aerobic-plate-count

Haas CN, Heller B (1988). "Averaging of TNTC counts."Applied and Environmental Microbiology, 54(8), 2069-2072.

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

See Also

mpn for Most Probable Number

Examples

#------- "Quantitative Microbial Risk Assessment (Haas et al., 2014) --------# Table 6.1 (Sample A)my_count <- c(1, 2, 1, 0, 0, 1, 1, 3, 6, 8, 4)my_amount_scor <- c(1, 1, 1, 1, 1, 2.5, 2.5, 2.5, 2.5, 5, 5)apc(my_count, my_amount_scor)  #1.08# Table 6.1 (Sample B)my_count <- c(1, 0, 5, 1, 0, 5, 0, 1, 5, 1, 8)my_amount_scor <- c(1, 1, 1, 1, 1, 2.5, 2.5, 2.5, 2.5, 5, 5)apc(my_count, my_amount_scor)  #1.08# Table 6.2my_count <- c(12, 8, 15, 40, 58)my_amount_scor <- c(1, 1, 1, 10, 10)my_amount_tntc <- c(10, 100, 100, 100)my_tntc_limit <- 100apc(my_count, my_amount_scor, my_amount_tntc, my_tntc_limit) #~7 (6.03, 7.96)#----------- "Averaging of TNTC Counts" (Haas & Heller, 1988) ---------------# Note:#  Results are slightly different due mostly to differences in how the TNTC#  portion of the likelihood function is formulated (i.e., incomplete gamma#  function vs. infinite Poisson sum--see Haas et al. (2014) for details of#  this mathematical relationship).my_count <- c(10, 12, 23, 48, 63)my_amount_scor <- c(1, 1, 1, 5, 5)my_amount_tntc <- c(5, 10, 10)my_tntc_limit <- 80apc(my_count, my_amount_scor, my_amount_tntc, my_tntc_limit)#Haas & Heller: APC = 13.28 CFU/ml

Calculate most probable number (MPN)

Description

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

Usage

mpn(  positive,  tubes,  amount,  conf_level = 0.95,  CI_method = c("Jarvis", "LR"),  tol = 1e-06)

Arguments

Details

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

When all tubes are negative, the point estimate ofMPN iszero (same approach as Jarvis et al.). The point estimate for the BAMtables "is listed as less than the lowest MPN for an outcome with at leastone positive tube" (App.2).

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

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

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

If the Rarity Index is less than1e-04, the experimentalresults are highly improbable. The researcher may consider running theexperiment again and/or changing the dilution levels.

Value

A list containing:

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

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

• variance: The estimated variance (see Jarvis et al.) oftheMPN estimate ifCI_method = "Jarvis". If all tubesare positive or all negative,variance will beNA. IfCI_method is not"Jarvis",variance will beNA.

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

• 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 (DeltaMethod and asymptotic normality of maximum likelihood estimators) depend onlarge-sample theory. The likelihood ratio assumptions also depend onlarge-sample theory. Therefore, the Jarvis and LR confidence intervalapproaches work best with larger experiments.

References

Bacteriological Analytical Manual, 8th Edition, Appendix 2,https://www.fda.gov/food/laboratory-methods-food/bam-appendix-2-most-probable-number-serial-dilutions

Blodgett RJ (2002). "Measuring improbability of outcomes from aserial dilution test."Communications in Statistics: Theory andMethods, 31(12), 2209-2223.

Blodgett RJ (2005). "Serial dilution with a confirmation step."Food Microbiology, 22(6), 547-552.

Blodgett RJ (2010). "Does a serial dilution experiment's modelagree with its outcome?"Model Assisted Statistics and Applications,5(3), 209-215.

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

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

Jarvis B, Wilrich C, Wilrich P-T (2010). "Reconsideration of thederivation of Most Probable Numbers, their standard deviations, confidencebounds and rarity values."Journal of Applied Microbiology, 109,1660-1667.

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

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

See Also

Shiny app:https://pub-connect.foodsafetyrisk.org/microbial/mpncalc/

apc for Aerobic Plate Count

Examples

# Compare MPN, 95% CI, and RI to Jarvis -------------------------------------# Table 1mpn(positive = c(3, 1, 1), tubes = c(3, 3, 3), amount = c(1, .1, .01))  #Jarvis: 7.5 (1.9, 30) RI = .209mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(1, .1, .01))  #Jarvis: 0 (0, 1.1) RI = 1mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(1, .1, .01),    conf_level = .975)$UB  #alpha / 2mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(1, .1, .01))  #Jarvis: Inf (36, Inf) RI = 1mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(1, .1, .01),    conf_level = .975)$LB  #alpha / 2# Table 2mpn(positive = c(20, 14, 3), tubes = c(20, 20, 20), amount = c(1, .1, .01))  #Jarvis: 13 (7.6, 21) RI = 0.794mpn(positive = c(50, 35, 7), tubes = c(50, 50, 50),    amount = 2 * c(1, .1, .01))  #Jarvis: 6.3 (4.5, 8.7) RI = .806mpn(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 1mpn(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))$MPN# Table 2mpn(positive = c(0, 0, 0), tubes = c(5, 5, 5), amount = c(.1, .01, .001))  #BAM: <1.8 (-, 6.8)mpn(positive = c(0, 0, 1), tubes = c(5, 5, 5), amount = c(.1, .01, .001))$MPNmpn(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 1mpn(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)