Presence/Absence Acceptance Sampling

Background Information

The objective of this design is to demonstrate, with high probability, that a high percentage of the decision area (or population) is acceptable, while allowing some of the observed samples to be unacceptable. The following discussion is presented in terms of sampling conducted within a decision area (such as room or collection of rooms within a building). However, this methodology is equally applicable to the sampling of any finite population of items, in which case the decision area is analogous to the population of items and the grid cell sampling locations are analogous to the individual items that will be sampled.

The hypergeometric model used in this design requires that each sample result can be categorized as a binary outcome, such as 1) the presence or absence of a particular quality, 2) a sample result being acceptable or unacceptable as defined by an action level threshold, 3) contamination being detected or not detected, etc. This statistical sampling approach employed here is known as Acceptance Sampling for Attributes (Schilling and Neubauer 2009, Chapter 5).

Definitions

Each of the required inputs in the VSP dialog is labeled in the image shown below the definition section.

$ Grid Cell $	A grid cell is a small unit area on a room surface that will be inspected or measured to determine if it exceeds the action level (AL) or some other criterion. For example, a single grid unit might be a 10cm by 10cm area that will be swiped and then measured for a biological, chemical or radiological agent.
$ N $	The total number of grid units in the sampling area (or target population). The value of $ N $ is supplied by the VSP user as indicated below.
$ n $	The number of grid units that are randomly selected to be measured or inspected. The value of $n $ is computed in VSP as described below.
$ X $	The number of unacceptable grid units observed in the sample. $ X $ may range from $ 0 $ to $ n $.
$ D $	The true (but unknown) number of defective grid cells.
$ 1- Po $	The desired fraction of the decision area that will be acceptable with $ (1- \alpha ) \times 100 \% $ confidence.
$H_0$	The null hypothesis that is being tested. $H_0$ is assumed to be true unless proved otherwise by the data. For this design, $ D_0 > P_0N $ (i.e. the decision area is unacceptable).
$ D_0 $	The largest whole number of unacceptable grid cells that may be in the population if $ H_0 $ is not true. Specifically, $ D_0 = P_0N $.
$ H_a$	The alternative hypothesis, typically the hypothesis we desire to prove. For this design, $ H_a \leq D_0 $ (i.e. decision area is acceptable). Ha can be accepted as being true with $(1- \alpha ) \times 100\% $ confidence when $H_0$ is rejected.
$ C $	The critical value that defines the decision rule for rejecting $ H_0$. Specifically, the null hypothesis is rejected if $ X \leq C $ . It is computed by VSP as described below.
$ \alpha $	The desired probability of rejecting $ H_0$ when $H_0$ is really true (Type I error). The value of $ 1 - \alpha $ , which is the confidence level, is provided by the VSP user as indicated below.
$ P_a $	A hypothetical proportion of unacceptable grid units (less than $P_0$) for which we desire a relatively low probability, $\beta$, of not rejecting $H_0$ (a Type II error). The value of $P_a$ is provided by the VSP user as indicated below.
$ D_a $	A hypothetical number of $ D_a = P_aN $ defective grid units (less than $D_0$) for which we desire a relatively low probability, $ \beta $ , of not rejecting $H_0$.
$ \beta $	The desired probability of not rejecting $H_0$ when $ D = D_a $ , where $ D_a < D_0 $. (a Type II error). The value of $ \beta $ is provided by the VSP user as indicated below.
$ Decision Rule $	If $ C$ or fewer of the $ n $ grid units are unacceptable ($X \leq C$), then reject $H_0$ and conclude with $(1- \alpha ) \times 100\% $ confidence that $ D \leq D_0 $ . In other words, conclude with $(1- \alpha ) \times 100\% $ confidence that $(1- P_0) \times 100\% $ of the grid cells are acceptable.

$image\Acceptance.gif$

Assumptions Underlying Acceptance Sampling for Attributes

The size of the grid unit has been determined to be appropriate for the measurement (inspection) method to be performed. For example, an appropriate grid unit size might be a 10cm by 10cm surface area.
The total number of grid units in the target population, $ N $, is known.
All $ N $ grid units are the same size.
$n$ of the $ N $ grid units are selected using simple random sampling.
The $ n $ grid units selected are representative of the total population of $ N $ grid units.
Each of the $n $ grid units is measured or inspected using an approved method.
Each sample is correctly classified as being acceptable or unacceptable (no false positives or false negatives).

Method Used in VSP to Compute the Number of Grid Units to be Measured, $ n $ , and the Acceptance Number, $C$

The method described below is similar to Desu and Raghavarao (1990, pages 66 and 67) and Bowen and Bennett (1988, pages 880-888). The minimum value of $ n $ and the value of the critical value $ C $ are determined such that the desired Type I and Type II error probabilities are achieved.

The bound for the Type I Error rate of this test is given by

\begin{equation} \large \alpha \geq P( X \leq C | D > D_0 ) \geq P( X \leq C | D = D_0 + 1) \end{equation}

Which is equivalent to

\begin{equation} \large P( X > C | D = D_0 +1) = \displaystyle\sum_{X=C+1}^{min( D_0 + 1, n)} \frac{ \dbinom{D_0 + 1}{x} \dbinom{N - D_0 - 1}{n-x}}{ \dbinom{N}{n}} \geq 1 - \alpha \end{equation}

The bound for the Type II error rate is given by

\begin{equation} \large P( X > C | D = D_a) = \displaystyle\sum_{X=C+1}^{min( D_a, n)} \frac{ \dbinom{D_a}{x} \dbinom{N - D_a}{n-x}}{ \dbinom{N}{n}} \leq \beta \end{equation}

The optimal values of $ C $ and $ n $ are determined using an iterative method. As illustrated in Bowen and Bennett (1988, page 885-886), a solution is obtained by choosing successively larger values of $ C $ beginning with $ C = 0 $, and then determining the minimum value of n for which both (2) and (3) are satisfied.

The achieved confidence, and the observed Type II error rate for the hypothetical number of defectives, $ D_a $, may be calculated directly using the right hand sides of equations (2) and (3), respectively.

References:

Bowen, W.M. and C.A. Bennett. 1988. Statistical Methods for Nuclear Material Management. NUREG/CR-4604. U.S. Nuclear Regulatory Commission, Washington, DC.

Schilling, E.G. and D.V. Neubauer. 2009. Acceptance Sampling in Quality Control, 2nd ed. CRC Press, Taylor & Francis Group, NY.

Desu, M.M. and D. Raghavarao. 1990. Sample Size Methodology. Academic Press, NY.

The Acceptance Sampling dialog contains the following controls:

Total Number of Grid Cells in Decision Area

Confidence (1-$\alpha$)

Minimum Percentage of Acceptable Grid Cells (1-$ P_0$)

Hypothetical Percentage of Unacceptable Grid Cells ($P_a$)

False Acceptance Rate (Beta)

Number of Samples

The number of grid cells that may be unacceptable (C)

Sample Placement page

Cost page

Data Analysis page

Data Entry sub-page

Summary Statistics sub-page

Tests sub-page

Plots sub-page

Analyte page

\( Grid Cell \)	A grid cell is a small unit area on a room surface that will be inspected or measured to determine if it exceeds the action level (AL) or some other criterion. For example, a single grid unit might be a 10cm by 10cm area that will be swiped and then measured for a biological, chemical or radiological agent.
\( N \)	The total number of grid units in the sampling area (or target population). The value of \( N \) is supplied by the VSP user as indicated below.
\( n \)	The number of grid units that are randomly selected to be measured or inspected. The value of \(n \) is computed in VSP as described below.
\( X \)	The number of unacceptable grid units observed in the sample. \( X \) may range from \( 0 \) to \( n \).
\( D \)	The true (but unknown) number of defective grid cells.
\( 1- Po \)	The desired fraction of the decision area that will be acceptable with \( (1- \alpha ) \times 100 \% \) confidence.
\(H_0\)	The null hypothesis that is being tested. \(H_0\) is assumed to be true unless proved otherwise by the data. For this design, \( D_0 > P_0N \) (i.e. the decision area is unacceptable).
\( D_0 \)	The largest whole number of unacceptable grid cells that may be in the population if \( H_0 \) is not true. Specifically, \( D_0 = P_0N \).
\( H_a\)	The alternative hypothesis, typically the hypothesis we desire to prove. For this design, \( H_a \leq D_0 \) (i.e. decision area is acceptable). Ha can be accepted as being true with \((1- \alpha ) \times 100\% \) confidence when \(H_0\) is rejected.
\( C \)	The critical value that defines the decision rule for rejecting \( H_0\). Specifically, the null hypothesis is rejected if \( X \leq C \) . It is computed by VSP as described below.
\( \alpha \)	The desired probability of rejecting \( H_0\) when \(H_0\) is really true (Type I error). The value of \( 1 - \alpha \) , which is the confidence level, is provided by the VSP user as indicated below.
\( P_a \)	A hypothetical proportion of unacceptable grid units (less than \(P_0\)) for which we desire a relatively low probability, \(\beta\), of not rejecting \(H_0\) (a Type II error). The value of \(P_a\) is provided by the VSP user as indicated below.
\( D_a \)	A hypothetical number of \( D_a = P_aN \) defective grid units (less than \(D_0\)) for which we desire a relatively low probability, \( \beta \) , of not rejecting \(H_0\).
\( \beta \)	The desired probability of not rejecting \(H_0\) when \( D = D_a \) , where \( D_a < D_0 \). (a Type II error). The value of \( \beta \) is provided by the VSP user as indicated below.
\( Decision Rule \)	If \( C\) or fewer of the \( n \) grid units are unacceptable (\(X \leq C\)), then reject \(H_0\) and conclude with \((1- \alpha ) \times 100\% \) confidence that \( D \leq D_0 \) . In other words, conclude with \((1- \alpha ) \times 100\% \) confidence that \((1- P_0) \times 100\% \) of the grid cells are acceptable.