MPN: Most Probable Number for Serial Dilutions

Introduction

The MPN package computes the Most Probable Number (i.e. microbial density) and other microbial enumeration metrics derived from serial dilutions.

mpn()

MPN includes the mpn() function to estimate the Most Probable Number (MPN), its variance and confidence interval, and Blodgett’s (2, 3, 4) Rarity Index (RI).

The user inputs the number of dilutions, number of tubes, number of positive tubes, amount of inocula, confidence level, and confidence interval method.

Maximum Likelihood Estimation

As discussed in the references, MPN is estimated by maximizing likelihood. Combining the notaton of Blodgett (2) and Jarvis et al. (7), we write the likelihood function as:

$$L = L(\lambda; x_i, n_i, z_i, i = 1,...,k) = \prod_{i=1}^k \binom{n_i}{x_i} {(1-exp(-{\lambda}{z_i}))} ^ {x_i} {(exp(-{\lambda}{z_i}))} ^ {n_i-x_i}$$

where

• λ is the microbial density (concentration) to be estimated
• k is the number of dilution levels
• x_i is the number of positive tubes at the i^th dilution level
• n_i is the total number of tubes at the i^th dilution level
• z_i is the amount of inoculum per tube at the i^th dilution level

As an R function:

#likelihood
L <- function(lambda, positive, tubes, amount) {
  binom_coef <- choose(tubes, positive)
  exp_term   <- exp(-lambda * amount)
  prod(binom_coef * ((1 - exp_term) ^ positive) * exp_term ^ (tubes - positive))
}
L_vec <- Vectorize(L, "lambda")

As is typical of maximum likelihood approaches, mpn() uses the score function (derivative of the log-likelihood) to solve for λ̂, the maximum likelihood estimate (MLE) of λ (i.e., the point estimate of MPN). However, let’s demonstrate what is happening in terms of the likelihood function itself. Assume we have 10g of undiluted inoculum in each of 3 tubes. Now we use a 10-fold dilution twice (i.e., the relative dilution levels are 1, .1, .01). Also assume that exactly 1 of the 3 tubes is positive at each dilution level:

#MPN calculation
library(MPN)
my_positive <- c(1, 1, 1) #xi
my_tubes    <- c(3, 3, 3) #ni
my_amount   <- 10 * c(1, .1, .01)  #zi
(my_mpn <- mpn(my_positive, my_tubes, my_amount))
#> $MPN
#> [1] 0.1118255
#> 
#> $MPN_adj
#> [1] 0.08725827
#> 
#> $variance
#> [1] 0.00431064
#> 
#> $var_log
#> [1] 0.3447151
#> 
#> $conf_level
#> [1] 0.95
#> 
#> $CI_method
#> [1] "Jarvis"
#> 
#> $LB
#> [1] 0.03538176
#> 
#> $UB
#> [1] 0.3534288
#> 
#> $RI
#> [1] 0.005814327

If we plot the likelihood function, we see that λ̂ maximizes the likelihood:

my_mpn$MPN
#> [1] 0.1118255
lambda <- seq(0, 0.5, by = .001)
my_L   <- L_vec(lambda, my_positive, my_tubes, my_amount)
plot(lambda, my_L, type = "l", ylab = "Likelihood", main = "Maximum Likelihood")
abline(v = my_mpn$MPN, lty = 2, col = "red")

If none of the tubes are positive, the MLE is zero:

no_positive <- c(0, 0, 0) #xi
(mpn_no_pos <- mpn(no_positive, my_tubes, my_amount)$MPN)
#> [1] 0
L_no_pos <- L_vec(lambda, no_positive, my_tubes, my_amount)
plot(lambda, L_no_pos, type = "l", xlim = c(-0.02, 0.2), ylab = "Likelihood",
     main = "No Positives")
abline(v = mpn_no_pos, lty = 2, col = "red")

If all of the tubes are positive, then no finite MLE exists:

all_positive <- c(3, 3, 3) #xi
mpn(my_tubes, all_positive, my_amount)$MPN
#> [1] Inf
lambda <- seq(0, 200, by = .1)
L_all_pos <- L_vec(lambda, all_positive, my_tubes, my_amount)
plot(lambda, L_all_pos, type = "l", xlim = c(0, 100), ylim = c(0, 1.1),
     ylab = "Likelihood", main = "All Positives")
abline(h = 1, lty = 2)

From a practical perspective, if all the tubes are positive, then the scientist should probably further dilute the sample until some tubes are negative.

Bias Adjustment

mpn() also returns a bias-adjusted version (9, 5, 6) of the point estimate:

my_mpn$MPN
#> [1] 0.1118255
my_mpn$MPN_adj
#> [1] 0.08725827

Confidence Intervals

As discussed in the references, many different confidence intervals (CIs) can be calculated for the Most Probable Number. Currently, mpn() computes a CI using the approach of Jarvis et al. (7) or the likelihood ratio approach of Ridout (8). However, since these approaches rely on large-sample theory, the results are more reliable for larger experiments.

my_positive <- c(1, 1, 1)
my_tubes    <- c(3, 3, 3)
my_amount   <- 10 * c(1, .1, .01)
mpn(my_positive, my_tubes, my_amount)  #Jarvis approach
#> $MPN
#> [1] 0.1118255
#> 
#> $MPN_adj
#> [1] 0.08725827
#> 
#> $variance
#> [1] 0.00431064
#> 
#> $var_log
#> [1] 0.3447151
#> 
#> $conf_level
#> [1] 0.95
#> 
#> $CI_method
#> [1] "Jarvis"
#> 
#> $LB
#> [1] 0.03538176
#> 
#> $UB
#> [1] 0.3534288
#> 
#> $RI
#> [1] 0.005814327
mpn(my_positive, my_tubes, my_amount, CI_method = "LR")  #likelihood ratio
#> $MPN
#> [1] 0.1118255
#> 
#> $MPN_adj
#> [1] 0.08725827
#> 
#> $variance
#> [1] NA
#> 
#> $var_log
#> [1] NA
#> 
#> $conf_level
#> [1] 0.95
#> 
#> $CI_method
#> [1] "LR"
#> 
#> $LB
#> [1] 0.02745596
#> 
#> $UB
#> [1] 0.2975111
#> 
#> $RI
#> [1] 0.005814327

Rarity Index

As Jarvis (7) explains, Blodgett’s (2, 3, 4) Rarity Index is a ratio of two likelihoods. The likelihood in the numerator is for the actual results (i.e., evaluated at the MPN point estimate). The likelihood in the denominator is for the (hypothetical) results that would have given the largest possible likelihood. So RI is larger than 0 and at most 1. Values of RI that are very small are unlikely; therefore, the results should be regarded with suspicion.

Conclusion

The MPN package is more versatile than static Most Probable Number tables in that the number of tubes can vary across dilution levels, the user can choose any number (or levels) of dilutions, and the confidence level can be changed. Also, the Rarity Index, which quantifies the validity of the results, is included.