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Animal and Plant Health Inspection Service
U.S. Department of Agriculture

# Probability of Failure to Detect Diseased Animals Calculator

A fundamental objective in controlling/eradicating any disease outbreak is identifying all infected animals in a target population as quickly as possible. However, because a list (sampling frame) of each animal (sampling unit) in this population will likely be difficult to construct, animals are typically aggregated into herds/flocks as the initial sampling unit. To this end, the minimum number of samples taken in each herd/flock should reflect the degree of confidence you have set (e.g. 95%) that, based on the limitations of the diagnostic test being used to identify truly infected animals (its Se), you will find one or more of these animals at the time of testing if infection IS PRESENT in the herd/flock at or above the level set by you (design prevalence).

Resource limitations (e.g. money, personnel) or other factors, however, may sometimes dictate a need to reduce the number of samples taken from a herd/flock from the minimum quantity identified in the Animal Sample Size Calculator. Use this Probability of Failure to Detect Diseased Animals calculator to evaluate how significant any deviations from your estimated minimum sample size compromise your ability to detect one or more diseased animals if they are present in the herd/flock from which the sample was taken.

It is noteworthy to point out that the this probability estimate, in turn, is used to establish the herd sensitivity of detection (HSe) for the disease in question:

HSe = 1- probability of failing to detect at least one diseased animal from those sampled

The HSe is the probability that an infected herd/flock will yield a positive result,  i.e. one or more samples are positive to your screening test (taking into account its specific diagnostic Se), given that the herd/flock is infected at a design prevalence equal to or greater than the one set by you.

### How to use this Matrix Tip: Since this calculator must use information from the Animal Sample Size Calculator in order to provide meaningful results, make sure to review its input values before proceeding.

1. Open the Probability of Failure to Detect Diseased Animals calculator
2. Supply the input data for diagnostic test sensitivity (Se) and herd/flock size in the spreadsheet where prompted.
3. View the matrix and the column that corresponds to the design prevalence of disease (0.01%-90%) for your herd/flock population of animals to be sampled.
4. Select the row that is equal to (or nearest to) the number (1-100,000) of samples to be tested.
5. Read the percentage listed at the intersection of the selected row and column

This number corresponds to what the estimated probability is of failing to detect any positives (using your chosen test per its Se) from this particular sample size if repeatedly obtained randomly from this herd/flock population of animals with the specified prevalence of disease.

Herd Sensitivity of detection (HSe) is 1 minus this probability.

The values that populate the cells of the matrix are derived using the  formula:

Probability of failure to detect disease (β)  =  [1 - (n · Se)/N]p · N

Where,

• p = design prevalence of infection in the population of animals (herd/flock) to be sampled.
• n = number of animals to be sampled among all animals (N) in the herd/flock
• Se = diagnostic sensitivity of the test
• N = herd/flock size

Herd sensitivity (HSe) of detection of disease is equal to 1 – β.

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