2.3: Sampling to Detect Disease
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- 96636
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As part of many disease control or eradication programs, entire herds or flocks arc tested to ascertain if the specified disease is present or, conversely, to ensure that the disease is absent. However, testing entire herds or flocks is expensive, and the veterinarian may have to accept the results of testing only a portion of the animals
When sampling is used for this purpose, a frequently asked question is, What sample size is required so that the veterinarian can be 950/o or 9911/o confident that the herd or flock is disease-free if no animals or birds in the sample give a positive test result? To actually prove (i.e., be 100% certain) that a disease is absent from a population requires testing almost every individual. For example, to prove that atrophic rhinitis was not present in a 5000 pig feeder operation would require the examination of the snout of virtually every pig.
Despite these limitations, sampling can provide valid insight into the health status of the population, because it is rare for only one animal in a herd to have the disease of interest. Infectious diseases tend to spread, and even infrequent noninfectious diseases would be expected to cluster somewhat within a herd, assuming environmental determinants of the disease are present. Thus for many diseases, if the disease is present at all, the herd will be likely to contain more than one diseased individual. This knowledge may be utilized when sampling to detect disease. The sampling strategy is designed to detect disease if more than a specified number or percentage ( > 0) of animals have the disease. The actual number or percentage of diseased animals to specify when making the sample size calculations should be based on knowledge of the biology of the disease. Often, the results of previous testing campaigns will supply useful information. For example, available data might indicate that the percentage of cattle with bovine tuberculosis in infected herds averages between 5 and lOo/o. These could be used as starting points to determine the possible range of sample sizes required to detect bovine tuberculosis when it is present.
Table 2.3 contains the sample size required to be 95% or 990/o certain that at least one animal in the sample would be diseased if the disease were present at or above the specified level. The minimum number of diseased animals assumed to be present in a herd is one, and for populations of greater than 100 individuals, the number of diseased animals is based on assumed prevalences ranging from 1-500/o. Note that a formal random sampling method, with individuals as the sampling units, is required if the desired confidence level shown is to be attained. If no formal random selection is used, the confidence one can have in the result is unknown, at least quantitatively. This circumstance may arise when animals are examined at slaughter for the presence of disease (e.g., in slaughter checks of pigs for respiratory disease). The pigs examined may not be representative of the source population; for example, the disease of interest may have a high case fatality rate and hence only disease-free animals survive to market age and weight. Although sample size requirements may be calculated to assist in evaluating the potential workload, one should be cautious and assign only a judgmental level of confidence if no diseased animals are observed in an informal sample such as this. Sometimes it may be assumed with a high degree of certainty that the level of disease in culled animals is much higher than in the source population; these diseases influencing the withdrawal of the animal in the first instance. If a sufficient number of these animals are examined and are found to be disease-free, the source herd or flock may be deemed disease-free, although no formal sampling was used in selecting the culled animals to be examined. (In fact, if a high percentage of culled animals are tested at slaughter, the tested animals essentially are a census of all culled animals. The problem in this case is not so much concerned with sampling, but with the amount of information about the population of interest provided by testing the culled animals.)
Assume that in a population of 1000 (N) swine, there will be at lea'it lO (D) pigs with atrophic rhinitis, if it is present at alL The sample size required to be 95% (a = 0.95) sure of detecting at least one pig with rhinitis is:
\[n=\left[1-(1-0.95)^{0.1}\right][1000-(9 / 2)]=0.259 \times 995.5=258\]
To be 99% certain of detecting at least one pig with rhinitis under the conditions in this example, the required sample size is:
\[n=0.369 \times 995.5=367\]
The previous formula may be solved for \(D\), rather than \(n\), and the following formula results:
\[D=\left[1-(1-a)^{1 / n}\right) \quad(N-[(n-1) / 2])\]
This formula is useful to provide the maximum number of disceased animals (\(D\)) expected in a population with confidence \(a\), when \(n\) individuals are examined and found to be free of disease.
If 20 randomly selected layer hens from a flock of 5000 are examined and found to be free of pullorum disease, the maximum expected number of infected birds in that. flock would be:
\[\begin{aligned}
D & =\left[1-(1-0.95)^{0.05}\right][5000-(19 / 2)] \\
& =0.139 \times 4990.5=694
\end{aligned}\]
giving a maximum percentage with pullorum disease of 13.9%. If 200 randomly selected hens were all negative, the maximum expected number infected in the flock would be 73, or a maximum prevalence of 1.5%.
As noted, Table 2.3 can be used to obtain the maximum number diseased by changing the column header DIN to n!N where nlN represents the percentage of the population examined and found disease-free. The body of the table will provide the maximum number of diseased individuals expected in a population of size N.