Sample size: simple random sample

Theory

Determination of the simple size required for detecting the presence of an event in a population (n) is based on the following formula:

Where:

CL	The level of confidence.
e	The number of detectable individuals with the event in the population. This value is the product of population size (N) by detectable prevalence. Detectable prevalence is the result of the product of expected prevalence (p) by sensitivity (Se) of the diagnostic method e = N × p × Se
N	The number of individuals in the population.

The sample size calculated this way takes into account the sensitivity (Se) of the diagnostic method (the lower the Se the larger the sample size). Test specificity is not considered in this calculation. The lack of Sp of a diagnostic test produces false positive results and increases the probability of a Type II error (that is, considering a population as affected by an event when it is actually free of it). ProMESA calculates, based on the maximum accepted probability of making a Type II error stated by the user, how many false positive results are expected to be obtained from a sample of size n (calculated by ProMESA) randomly taken from a population with a prevalence equal to the minimum expected prevalence stated by the user.

There are three scenarios:

• No positive result is obtained: the population under study can be declared, with low risk of error, as free of the event under study. Just A Type I error could be made (that is, declaring the population as free of the event when it is not - this is related to the lack of sensitivity of the diagnostic method). A Type II error will not be made because there are no positive results.
• The number of positive results is greater than the expected number of false positive results: in this case it can be said that the event is present in the population with low risk of error. A Type II error could be made (declaring that the event is present in a population that is actually free). This would happen if the specificity of the diagnostic method was lower than the one stated for the sample size calculation.
• The number of positive results is less than the expected number of false positive results: this is the most difficult situation to interpret. Both a Type I and Type II error could be made. The decision should take into account the effect that each type of error could have. In certain circumstances further examination of the test positive samples should be considered.

The sensitivity (Se) and specificity (Sp) of the diagnostic method are calculated as follows:

Diagnostic strategy	Sensitivity	Specficity
One test (Ts 1):	= Se Ts 1	= Sp Ts 1
Two tests in parallel:	= 1 - ( 1 - Se Ts 1 ) × ( 1 - Se Ts 2 )	= Sp Ts 1 × Sp Ts 2
Two tests in series:	= Se Ts 1 × Se Ts 2	= 1 - ( 1 - Sp Ts 1 ) × ( 1 - Sp Ts 2 )

Practice

Minimum expected prevalence	This procedure is used to calculate the sample size required to detect an event if it is present above a stated level of prevalence. Acceptable values: ≥ 0 and ≤ 1.
Population size	The number of individuals in the population under study. Acceptable values: any integer.
Level of confidence	The confidence the user wants to have in the results. Acceptable values: 90%, 95% or 99%.
Probability of making a Type II error	Also known as beta. This is the probability of concluding that the population under study is affected by the event when it is actually free of it. This type of error is due to lack of specificity of the diagnostic method. If the specificity of a test was perfect the probability of false positive results would be zero and this type of error would never occur. Unfortunately, perfect tests do not exist and it is important to have an estimation about how many misclassifications are to be expected using the available diagnostic tests. Acceptable values: ≥ 0 and ≤ 1.
Sensitivity	The probability that an individual having the event under study will be identified as positive by the diagnostic test. Acceptable values: ≥ 0 and ≤ 1.
Specificity	The probability that an individual not having the event under study will be identified as negative by the diagnostic test. Acceptable values: ≥ 0 and ≤ 1.

Combination must be entered when two diagnostic tests will be used.

Parallel interpretation: an individual is considered to be positive if one or both tests produce a positive result. This method increases the sensitivity but decreases the specificity.
Series interpretation: an individual is considered to be positive if both tests produce a positive result. This method decreases the sensitivity but increases the specificity.