Microbiological Sampling Plans

Microbiological sampling plans: a tool to explore ICMSF recommendations (some explanation)

Two-class sampling plans (2-class enrichments and 2-class counts)

The sheet has several components containing graphs, input/output fields, and calculation results. For the normal user only the graphs and input/output fields shown in lines 1 to 37 and columns A to U should be of interest.

Input fields are shown in yellow. In the following text information that can be entered in such fields is written in italics.

Graphs for two-class sampling plans (from left to right):

Plot 1 – Common type of operating characteristic (OC) curve relating probability of accepting a lot to the proportion defective it contains.

Acceptance probabilities are calculated for given number of sampling units that are examined, n, given microbiological limit specified by the sampling plan, m, and given maximum number of sampling units that are allowed to exceed the limit, c.

Plot 2 – Normal frequency distribution assumed for Log-transformed colony count numbers per gram to be found in sampling units drawn randomly from a lot characterized by given mean Log count per gram and given standard deviation sigma.

The pink vertical line indicates the microbiological limit, m, specified in the sampling plan. Right to mthe area under the curve corresponds to the proportion defective the lot contains.

Plot 3 – Using these proportions defective and the given sampling plan specifications for n and ccorresponding probabilities of lot acceptance are calculated. Results are plotted (black) to show the OC curve in relation to Log arithmetic mean counts per gram.

Input fields for two-class sampling plans:

Yellow fields in the center

  • Lot characteristics: mean Log count per gram and standard deviation sigma, shown in Plot 2.
  • Sampling plan specifications: the microbiological limit, m, (and for enrichments the sample amount), the number of sampling units, n, and the number of sampling units that are allowed to exceed the limit, c.

To change any of these values go to that field, type in a new value, and press Enter. Calculation results, like P(accept), and graphs will be changed accordingly.

  • The desired acceptance probability P(accept).

The desired P(accept) can be changed for example to 5%, and by clicking the upper grey box the mean Log count per gram accepted with this new probability P(accept) is calculated and graphs are changed accordingly.

Yellow field on the left

To calculate the acceptance probability P(accept) for a specific proportion defective type in this proportion as Pd and press Enter. The result will be shown in the next field to the right.

For this calculation only the given values for n and c are needed (see Plot 1).

Yellow fields on the right

Only sampling plan specifications (n and c) can be manipulated here, lot characteristics remain unchanged. When new values for n and/or c are entered the corresponding acceptance probability P(accept) can be calculated using the microbiological limit m as given. When the grey box on the right side is clicked a new value for m is derived that yields the same combination of mean Log count per gram and lot acceptance probability P(accept) that is achieved by the sampling plan described in the center.

Furthermore the upper percentile of the concentration distribution can be determined (implied acceptance level).

Three-class sampling plans (3class)

Graphs for three-class sampling plans (from left to right):

Plot 1 – Operating characteristic (OC) surface showing probabilities of accepting a lot depending on two proportions: the proportion defective in the lot exceeding the microbiological limit M, named Pd, and the proportion marginally defective between the two microbiological limits m and M, named Pm.

Acceptance probabilities are calculated for given number of sampling units that are examined, n, given microbiological limits m and M, and given maximum number of sampling units that are allowed to be marginally defective, c, i.e. that are allowed to exceed the limit m but not M. The number of sampling units that are allowed to exceed M is assumed to be zero.

Plot 2 – Normal frequency distribution assumed for Log-transformed colony count numbers per gram to be found in sampling units drawn randomly from a lot characterized by given mean Log count per gram and given standard deviation sigma.

The red vertical line indicates the microbiological limit M specified in the sampling plan. Right to Mthe area under the curve corresponds to the proportion defective the lot contains, Pd. The pink vertical line indicates the microbiological limit m specified in the sampling plan. The area under the curve between m and M corresponds to the proportion marginally defective in the lot, Pm.

Plot 3 – Using these proportions defective, these proportions marginally defective, and the given sampling plan specifications for n and c corresponding probabilities of lot acceptance are calculated. Results are plotted (black) to show the OC curve in relation to Log arithmetic mean counts per gram.

Input fields for two-class sampling plans:

Yellow fields in the center

  • Lot characteristics: mean Log count per gram and standard deviation sigma shown in Plot 2.
  • Sampling plan specifications: the microbiological limits, m and M, the number of sampling units, n, and the number of sampling units that are allowed to be marginally defective, c, i.e. that are allowed to exceed the limit m but not to exceed M.

To change any of these values go to that field, type in a new value, and press Enter. Calculation results, like P(accept), and graphs will be changed accordingly.

  • The desired acceptance probability P(accept)

The desired P(accept) can be changed to for example 5% and by clicking the upper grey box the mean Log count per gram accepted with this new probability P(accept) is calculated and graphs are changed accordingly.

Yellow fields on the left

To calculate the acceptance probability P(accept) for a specific combination of proportion defective, i.e. exceeding M, and proportion marginally defective, i.e. between m and M, type in these proportion as Pd and Pm, respectively, and press Enter. The result will be shown in the next field to the right.

For this calculation only the given values for n and c are needed (see Plot 1).

Yellow fields on the right

Only sampling plan specifications (n and m) can be manipulated here, lot characteristics remain unchanged. When new values for n and/or c are entered the corresponding acceptance probability P(accept) can be calculated using the microbiological limits m and M as given. When the grey box on the right side is clicked a new value for m is derived that yields the same combination of meanLog count per gram and lot acceptance probability P(accept) that is achieved by the sampling plan described in the center. The microbiological limit M is left unchanged (as long as M>m).

Furthermore the upper percentile of the concentration distribution can be determined (implied acceptance level).