| Title: | Most Probable Number and Other Microbial Enumeration Techniques |
|---|---|
| Description: | Calculates the Most Probable Number (MPN) to quantify the concentration (density) of microbes in serial dilutions of a laboratory sample (described in Jarvis, 2010 <doi:10.1111/j.1365-2672.2010.04792.x>). Also calculates the Aerobic Plate Count (APC) for similar microbial enumeration experiments. |
| Authors: | Martine Ferguson [aut] (original R code), John Ihrie [cre, aut] |
| Maintainer: | John Ihrie <[email protected]> |
| License: | Unlimited |
| Version: | 0.4.0 |
| Built: | 2025-02-20 04:14:39 UTC |
| Source: | https://github.com/cran/MPN |
apc calculates the Aerobic Plate Count (APC) point estimate
and confidence interval of colony forming units (CFU). Adjusts for
too-numerous-to-count (TNTC) plates using the maximum likelihood method of
Haas et al. (2014).
apc( count, amount_scor, amount_tntc = NULL, tntc_limit = 250, conf_level = 0.95, tol = 1e-06 )apc( count, amount_scor, amount_tntc = NULL, tntc_limit = 250, conf_level = 0.95, tol = 1e-06 )
count |
A vector of CFU counts in each scorable (countable) plate. |
amount_scor |
A vector of inoculum amounts (in ml) in each scorable plate. See Details section. |
amount_tntc |
A vector of inoculum amounts (in ml) in each TNTC plate. |
tntc_limit |
A vector (or scalar) of the limit above which the plate counts are considered too-numerous-to-count (often 100, 250, or 300). Each plate can potentially have a different value. Default is 250. |
conf_level |
A scalar value between zero and one for the confidence level. Typically 0.95 (i.e., a 95 percent confidence interval). |
tol |
A scalar value for tolerance to be passed to
|
As an example, assume we start with four plates and 1 ml of
undiluted 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 plates
were TNTC with limits of 300 and 250. The other plates had CFU counts of
28 and 20. We now have
count = c(28, 20),
amount_scor = 1 * c(.001, .001),
amount_tntc = 1 * c(.01, .01), and
tntc_limit = c(300, 250).
Currently, confidence intervals can only be calculated using the likelihood ratio (LR) approach described in Haas et al. (2014).
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.
The likelihood ratio confidence interval assumptions depend on asymptotic theory. Therefore, the confidence interval results will be better with larger experiments.
apc() will fail in certain cases where the TNTC results are
extremely unlikely to occur when taking the scorable (countable) plates
into consideration. In other words, if the countable plates suggest a low
concentration of microbes, then TNTC plates at higher dilution levels are
probably due to experimental error. Mathematically, the probability is so
small that the likelihood function is essentially zero.
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 risk assessment, Second Ed." John Wiley & Sons, Inc., ISBN 978-1-118-14529-6.
mpn for Most Probable Number
#------- "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 6.2 my_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 <- 100 apc(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 <- 80 apc(my_count, my_amount_scor, my_amount_tntc, my_tntc_limit) #Haas & Heller: APC = 13.28 CFU/ml#------- "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 6.2 my_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 <- 100 apc(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 <- 80 apc(my_count, my_amount_scor, my_amount_tntc, my_tntc_limit) #Haas & Heller: APC = 13.28 CFU/ml
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).
mpn( positive, tubes, amount, conf_level = 0.95, CI_method = c("Jarvis", "LR"), tol = 1e-06 )mpn( positive, tubes, amount, conf_level = 0.95, CI_method = c("Jarvis", "LR"), tol = 1e-06 )
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 |
tol |
A scalar value for tolerance to be passed to
|
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
instead of since these are one-sided intervals. The Ridout
(1994) LR approach uses the same technique (with ) 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.
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.
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.
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 a serial dilution test." Communications in Statistics: Theory and Methods, 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 model agree with its outcome?" Model Assisted Statistics and Applications, 5(3), 209-215.
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.
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.
Shiny app: https://pub-connect.foodsafetyrisk.org/microbial/mpncalc/
apc for Aerobic Plate Count
# 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))$MPN # Compare MPN and 95% LR CI to Ridout (1994) -------------------------------- # Table 1 mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(.1, .01, .001), CI_method = "LR") #Ridout: 0 (0, 9.0) mpn(positive = c(2, 2, 0), tubes = c(3, 3, 3), amount = c(.1, .01, .001), CI_method = "LR") #Ridout: 21.1 (6.2, 54.3) mpn(positive = c(3, 3, 3), tubes = c(3, 3, 3), amount = c(.1, .01, .001), CI_method = "LR") #Ridout: Inf (465.1, Inf)# Compare MPN, 95% CI, and RI to Jarvis ------------------------------------- # Table 1 mpn(positive = c(3, 1, 1), tubes = c(3, 3, 3), amount = c(1, .1, .01)) #Jarvis: 7.5 (1.9, 30) RI = .209 mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(1, .1, .01)) #Jarvis: 0 (0, 1.1) RI = 1 mpn(positive = c(0, 0, 0), tubes = c(3, 3, 3), amount = c(1, .1, .01), conf_level = .975)$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)