Sample size: two stage sample

Theory

The sample size required for detection of an event if it is present in a population from two-stages sampling is determined by calculating independently the number of herds from which the individuals will be sampled and the number of individuals per herd to include in the sample.

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_h The number of herds in the population.

The number of individuals to sample per herd is given by:

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

Only 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

This procedure is used to calculate the sample size needed to determine if an event is present in a population above a stated level of prevalence, when the selection of individuals is done by two-stage sampling. The two-stage method is advised when the total list of the individuals from the population is not available but there is a complete list of herds or farms. The procedure produces two results: (1) the number of herds from which the individuals are going to be chosen; and (2) the number of individuals per herd that have to be included in the sample.

Level of confidence The confidence that 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 null and this type of error would never occur. Unfortunately, the perfect method does not exit 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.

Minimum expected prevalence of positive herds State the lower proportion of affected herds if the event really existed in the region under study.
Acceptable values: ≥ 0 and ≤ 1.

Number of herds The total number of farms in the population under study.
Acceptable values: any positive integer.

Minimum expected prevalence of positive animals State the lower proportion of affected herds if the event really existed in the region under study.
Acceptable values: ≥ 0 and ≤ 1.

Number of animals per herd The mean number of animals per herd.
Acceptable values: any positive integer.

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.


Instituto Nacional de Tecnología Agropecuaria	EpiCentre, IVABS, Massey University

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_h	The number of herds in the population.

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_i	The number of individuals per herd.

Diagnostic strategy	Sensitivity	Specficity
Only 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 )

Level of confidence	The confidence that 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 null and this type of error would never occur. Unfortunately, the perfect method does not exit 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.
Minimum expected prevalence of positive herds	State the lower proportion of affected herds if the event really existed in the region under study. Acceptable values: ≥ 0 and ≤ 1.
Number of herds	The total number of farms in the population under study. Acceptable values: any positive integer.
Minimum expected prevalence of positive animals	State the lower proportion of affected herds if the event really existed in the region under study. Acceptable values: ≥ 0 and ≤ 1.
Number of animals per herd	The mean number of animals per herd. Acceptable values: any positive integer.
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.