Nonparametric (distribution-free) tolerance limits can be used to statistically test whether a specified area or room in a building is contaminated with biological agents, chemicals or radionuclides at concentrations greater than their respective fixed action levels (ALs). The statistical meaning, use, and computation of tolerance limits are discussed in Hahn and Meeker (1991) and Helsel (2005, Chapter 6).
In this module VSP computes the number of measurements, \(n\), needed to determine the nonparametric one-sided upper tolerance limit to statistically test if the true \(P\)th percentile of the population exceeds a fixed AL. Then VSP conducts the test and reports the results. A discussion of this use of tolerance limits is in Millard and Neerchal (2001, page 339).
A nonparametric tolerance limit is valid regardless of the probability distribution of the population of measurements. That is, the data distribution need not be known. If the distribution is known with confidence to be a normal distribution, the tolerance limits for the normal distribution should be used.
A one-sided upper tolerance limit is identical to a one-sided confidence limit on a specified percentile \(P\) of the population of measurements. The \(P\)th percentile is the value above which \(100(1-P)\%\) of the population lies and below which \(100P\%\) of the population lies.
A one-sided upper tolerance limit on the \(P\)th percentile of a population is a value, denoted here by \(UTL_{P,\alpha}\), such that at least \(100P\%\) of the population of measurements is less than \(UTL_{P,\alpha}\) with \(100(1-\alpha)\%\) confidence. For example, if \(P\) = 0.90 and \(\alpha\) = 0.05, then at least 90% of the population is less than the computed value \(UTL_{0.90,0.05}\) with 95% confidence.
The following assumptions are needed when using nonparametric tolerance limits:
Representative measurements have been obtained from a defined target population (e.g., a wall, section of a wall, the floor in a hallway, the walls and floors of a selected set of rooms, etc.) using simple random sampling, systematic sampling on a grid, or some other suitable probability-based sampling design
The measurements are statistically independent, i.e., there is no spatial correlation (no spatial patterns of contaminant levels throughout the target population).
There are no outliers, i.e., there are no observations that are mistakes or that do not belong to the population being studied.
The assumption of statistical independence implies that tolerance limits may be most useful for building areas that are not expected to contain hot spots or other dominant patterns. Hence, they may be most useful after decontamination has occurred and the objective is to test if the building is ready to be re-occupied.
Nonparametric tolerance limits are determined in VSP as follows:
1. The VSP user specifies the desired values of \(P\) and \(\alpha\) and the action level, AL. VSP uses the following equation to compute the number of measurements that should be obtained (Hahn and Meeker, 1991, page 169): $$n = \frac{\text{ln}(\alpha)}{\text{ln}(P)}$$
3. The VSP user obtains the \(n\) representative measurements from the target population using a probability-based design, e.g., using simple random sampling or sampling on a square or triangular grid pattern.
4. Then \(UTL_{P,\alpha}\) = the largest of the \(n\) measurements obtained is the nonparametric upper \(100(1-\alpha)\%\) tolerance limit on the \(P\)th percentile.
5. If \(UTL_{P,\alpha} \ge\) AL, then the null hypothesis, \(H_0\) : \(P\)th percentile of the population \(\ge\) AL, is not rejected and the statement Conclude Site is Dirty is given by VSP.
6. If \(UTL_{P,\alpha} \lt\) AL, then the null hypothesis is rejected and the statement Conclude the Site is Clean is given by VSP.
The assumptions of representative measurements and no outliers are important because if one or more outliers occur, the maximum of the \(n\) measurements will not be a valid one-sided upper nonparametric tolerance limit.
When \(n\) random samples have been collected with none greater than or equal to the AL, and the required fraction of the population (\(P\)) to be less than the action level is specified, VSP has the option of calculating the actual confidence level achieved. This is achieved by rearranging the equation in step 2 above to be to solve for \(\alpha\): $$\alpha = \text{exp}(n*\text{ln}(P))$$
The confidence limit is then computed as \(100 \times (1-\alpha)\).
VSP also has the option of calculating the minimum \(P\) which achieves the desired confidence level. The equation rewritten to solve for \(P\): $$P = \text{exp}\Big(\frac{\text{ln}(\alpha)}{n}\Big)$$
Hahn, G.J. and W.Q. Meeker. 1991. Statistical Intervals. Wiley & Sons, Inc, New York, NY.
Helsel, D.R. 2005. Nondetects and Data Analysis, Statistics for Censored Environmental Data, Wiley & Sons, New York, NY.
Millard, S.P. and N.K. Neerchal. 2001. Environmental Statistics with S-Plus. CRC Press, New York, NY.
Required fraction of the population to be less than the action level