MPN Profile Likelihood Bounds

Estimate MPN and profile-likelihood bounds from tube inoculum volumes and counts; caller supplies a likelihood-ratio critical value (about 3.84146 for asymptotic 95%).

Description

An MPN point estimate can be supplemented with likelihood-based concentration bounds.

An MPN point estimate can be supplemented with likelihood-based concentration bounds. The critical likelihood-ratio value determines the interval rule and must be supplied rather than assumed to make every interval exact 95% coverage.^{[1, 2]}

Estimate MPN and profile-likelihood bounds from tube inoculum volumes and counts; caller supplies a likelihood-ratio critical value (about 3.84146 for asymptotic 95%).

When to use MPN Profile Likelihood Bounds

Use this tool to estimate MPN and profile-likelihood bounds from tube inoculum volumes and counts; caller supplies a likelihood-ratio critical value (about 3.84146 for asymptotic 95%).

Prepare the sample, count, growth or assay measurements requested by this tool. Confirm units, sampling periods and reporting bases; supply independently established comparison criteria where requested.

How to use MPN Profile Likelihood Bounds and understand its results

Prepare the sample, count, growth or assay measurements requested by this tool. Confirm units, sampling periods and reporting bases; supply independently established comparison criteria where requested.
Enter volumesMl, positiveTubes, totalTubes, likelihoodRatioCritical using the field table and calculation rules, then run the tool.
Read mpnPerMl, lowerMpnPerMl, upperMpnPerMl. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
volumesMl	Volumes ml	list of number	Required
positiveTubes	Positive tubes	list of number	Required
totalTubes	Total tubes	list of number	Required
likelihoodRatioCritical	Likelihood ratio critical	number	Required

Enter the tube design supported by the MPN helper and a positive likelihood-ratio critical value. After the same finite mixed-outcome MPN fit, the tool bisects the lower bound between zero and the estimate. It expands the upper bracket then bisects it, using 150 iterations for each bound. Return estimate and lower/upper concentrations per mL; failure to bracket finite results is an error.

bound cutoff = logLikelihood(MPN)−likelihoodRatioCritical/2

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
mpnPerMl	Mpn per ml	number
lowerMpnPerMl	Lower mpn per ml	number
upperMpnPerMl	Upper mpn per ml	number

json

{
  "input": {
    "volumesMl": [
      1,
      1,
      1
    ],
    "positiveTubes": [
      1,
      1,
      1
    ],
    "totalTubes": [
      2,
      2,
      2
    ],
    "likelihoodRatioCritical": 1.7260924347106854
  },
  "output": {
    "mpnPerMl": 0.6931471805599452,
    "lowerMpnPerMl": 0.2876820724517808,
    "upperMpnPerMl": 1.3862943611198904
  }
}

Limitations and assumptions

The critical value and asymptotic approximation are supplied assumptions. Small-sample coverage is not guaranteed and these are not necessarily the FDA table's interval construction. Independent Poisson occupancy and correct original-sample volumes remain essential.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Tube Design MPN Estimate: Fit a Poisson tube-positivity MPN by maximum likelihood using actual inoculum volumes, replacing a fixed three-tube lookup table. For a complementary task, use Water Coliform Sample Compliance: Audit water-sample coliform observations against a supplied per-sample allowance, retaining tested volumes; nondetection is not guaranteed sterility.

Further background: Most probable number from serial dilutions.

References

MPN Profile Likelihood Bounds — tool.io
Most probable number from serial dilutions — U.S. Food and Drug Administration

Similar or alternative tools

Tube Design MPN Estimate
Fit a Poisson tube-positivity MPN by maximum likelihood using actual inoculum volumes, replacing a fixed three-tube lookup table.
Water Coliform Sample Compliance
Audit water-sample coliform observations against a supplied per-sample allowance, retaining tested volumes; nondetection is not guaranteed sterility.