Statistical quality control and social processes: A drug testing application

~

Socio-Econ. Plann. Sci. Vol. 31, No. 1, pp. 69-82, 1997

Pergamon

P I I : S0038-0121(96)00026-2

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Statistical Quality Control and Social Processes: A Drug Testing Application L A N C E A. M A T H E S O N ~, P A M E L A K. L A T T I M O R E 2 a n d J O A N N A R. B A K E R 3 ~Department of Management Science, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0235, U.S.A. 2633 Indiana Ave., NW, National Institute of Justice, Washington, DC 20531, U.S.A. 3MIS/OM Department, Belk College of Business Administration, UNC-Charlotte, Charlotte, NC 28223, U.S.A. A~traet--Traditional acceptance sampling procedures have been used to monitor the quality of outgoing items from a production process within a manufacturing environment. Generally, however, this has represented a reactive, rather than a proactive, approach to quality control. Importantly, the manufacturing sector has refocused itself on more proactive techniques that attempt to improve item quality by repairing the process at the closest feasible point of intervention. This type of intervention is less possible, however, in most social service environments since: (1) the process may not be visible; and (2) the relationship between intervention and outcome is not well-understood and/or well-defined.Largely because of the complexity of social services, statistical quality control procedures have not generally been applied to improve process quality or to otherwise affect process outcomes. Because of these difficulties, the benefits of statistical quality control procedures and their ability to make processes more efficient and/or effective have largely been ignored in the literature on quality control. However, provided one can identify an objective outcome measure from a process (regardless of the complexity of that process), and make some assumptions about the prior distribution, acceptance sampling can be an appropriate and useful technique for monitoring process outcomes. Indeed, for many social processes, acceptance sampling, or testing "after the fact" may be the only approach available for monitoring shifts in process quality. We here demonstrate the utility of Bayesian acceptance sampling in the context of a timely social process, testing a population for the use of illegal drugs. The use of drugs, and the desire on the part of businessesand the criminal justice system to control or deter use has been a major focal point for policy and decision makers for more than a decade. Given that budgets to institute drug testing and/or screening programs are not unlimited, a technique that reduces the cost of a drug treatment program while maintaining deterrence and monitoring effects would indeed be useful to practitioners. We thus propose an application within the framework of an economic model of drug use, and show that adoption of the testing approach can reduce the expected cost of testing. © 1997 Elsevier Science Ltd. All rights reserved

INTRODUCTION The e n v i r o n m e n t a l , social, a n d i n s t i t u t i o n a l factors a n d p r o g r a m s to which individuals are exposed constitute processes that produce, or c o n t r i b u t e to, i n d i v i d u a l behaviors. A l t h o u g h these processes m a y n o t be as well defined as, for example, a m a n u f a c t u r i n g process, it is often the case that their " o u t p u t s " are sufficiently well-defined as to be measurable. Thus, for example, e d u c a t i o n a l a t t a i n m e n t is routinely m e a s u r e d as the o u t p u t o f a social process that includes the c o n t r i b u t i o n s o f schools as well as the influences o f parents, peers, a n d e n v i r o n m e n t . W i t h i n the e d u c a t i o n a l e n v i r o n m e n t , testing accomplishes at least two objectives. First, it allows individuals to be gauged with respect to their peers a n d establishes s t a n d a r d s o f performance. Second, it encourages students to c o m p l y with these s t a n d a r d s - - i . e , students study (at least partially) because they will be tested. Other testing p r o g r a m s established by society t h r o u g h business or g o v e r n m e n t a l institutions m a y be seen to address similar objectives. I n this paper, we develop a n d apply a statistical quality c o n t r o l model to a social process. We suggest that social processes are often sufficiently a n a l o g o u s to p r o d u c t i o n processes to allow a p p l i c a t i o n o f quality c o n t r o l techniques. F u r t h e r , we d e m o n s t r a t e that when the objective of a testing p r o g r a m is to o b t a i n c o m p l i a n c e with standards, significant cost savings m a y be realized by replacing 100% inspection with a r a n d o m s a m p l i n g program. I n particular, we apply Bayesian 69

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acceptance sampling techniques to the monitoring of illicit drug use in a population. In doing so, we identify cost-minimizing acceptance sampling plans based on an expected total cost model that includes a prior distribution over drug use, costs for "accepting the population" based on the results of testing a sample from the population, and expected costs for "rejecting the population" followed by 100% inspection. The next section provides background information on drug-testing programs. Subsequently, an economic model of drug testing is developed, followed by a description of our data and results. The final section presents our conclusions.

DRUG TESTING Control of drug use emerged as a priority public initiative in the 1980s. Recent estimates suggest that the abuse of illegal drugs cost U.S. society $58.3 billion in 1988 alone [16 pp. 23-24]. This estimate reflects both the direct cost of health care for drug abusers, as well as indirect costs due to reduced or lost productivity and crime. The Federal government has responded with legislation, including the Drug-Free Workplace Act of 1988, that requires Federal grantees and contractors to maintain drug-free workplaces. (See Ref [25] for a discussion of national drug policy.) State legislatures have responded with m a n y new laws, passing more than 450 new drug laws in 1990 alone [21]. Firms have responded with a variety of programs, including those for drug-testing. In 1988, for example, companies polled by The Conference Board were testing job applicants or employees [2]. More recently, the American Management Association reported that 85% of large companies are conducting workplace drug testing--an increase from the 74% that reported testing in 1992 [24]. Drug testing is also viewed as a deterrent and a monitoring tool by such agencies as the Department of Transportation, which has published regulations requiring the drug-testing o f over six million employees of America's transportation sector [25]. Finally, drug-testing programs have been established for criminal populations in all fifty States [8]. The environment within which drug-testing is conducted is generally more complex, and poorly controlled, than a typical manufacturing environment. The social process that generates drug use is not well defined or even well understood. In this regard, environmental factors such as the availability of illegal drugs and the acceptability of drug use within peer groups and family are likely factors in the decision to use drugs. Within this environment, individuals choose to use illegal drugs if the perceived benefit of doing so exceeds the perceived costs (see Ref [3]). Note that even the addicted user can be assumed to behave rationally if benefits and costs are properly defined--for example, including withdrawal as a "cost" of abstaining. Finally, if individual discount rates are large, or the likelihood of detection and penalty is perceived to be small, relatively small perceived benefits may generate drug use. Although the social process producing drug use is not well defined, the output of this process-illegal drug use--is well-defined and can be detected and measured. Currently, drug-testing methods involve the chemical analysis of a urine specimen, although hair and other bodily fluids (e.g. sweat) are currently being considered as alternative sources (e.g. see Refs [26]). The sensitivity and specificity of drug tests vary, as does the period over which drug metabolites can be detected (e.g. see Refs [9, 17, 21,23]). For example, Visher and McFadden [23] (also see [22]) report false-positive rates for the detection of cocaine of 1.9 to 4.1% and false-negative rates of 18 to 89% for four c o m m o n drug screening testst. Wish and Gropper [26] report that the duration of detectability is about 48 h for amphetamines, 3 days for cocaine metabolites and single use of marijuana, and 21 to 27 days for chronic, heavy use of marijuana. Just as educational testing programs have both individual monitoring and compliance objectives, drug-testing programs are applied to detect and monitor the use of illegal drugs by individuals as tFor the Visher and McFadden [23] study, the methods included standard thin-layer chromatography and three commonly used immunoassays--AbuscreentmRIA, EMIT'm,and TDx'm. Results were compared to results of gas chromatography/ mass spectrometry (GC/MS) analyses of the samples. Urine specimens were analyzed for opiates, marijuana, PCP, amphetamines, as well as cocaine. Fretthold [9] summarizes the results of government testing programs: "The false-positiverate approximates zero to such a degree that the tens of thousands of negative proficiencytest specimens were unable to produce such an incident. The false-negativerate appears to be approximately 3% to 5%." The analytic method was EMIT'm with GC/MS confirmation of positive tests.

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well as to deter drug use through the threat of sanctions for positive results (e.g. see Refs [2] and [26]). As noted by Wish and Gropper [26], "Monitoring programs may also deter persons not being tested from using drugs. This is the primary rationale for random testing in the workplace." Several researchers have recently found that urine testing reduces drug use in criminal justice populations [5, 6, 14]. More recently, Kennedy [13] has suggested that "urine collection alone may produce a sufficient perception of vulnerability to deter continued drug use." He tested this hypothesis in a short-term experiment and "found that testing a one-third random sample of collected specimens, rather than all collected specimens, did not lead to increased drug use or to a reduction in self-disclosure validity over a one-month period." Only 10% of drug-test outcomes were positive for both the experimental (one-in-three) and the control (all) groups. This finding, in itself, suggests urine testing deters drug use as all study subjects had been identified as serious drug abusers. Currently, most drug testing programs target entire populations (e.g. all employees or all probationers reporting to an office) or selected subsets (e.g. employees in sensitive positions, job applicants, or probationers identified with serious drug abuse problems). These programs generally require that tests are performed on all members of the identified population, although the frequency of testing may vary for individuals. In those few cases where a program chooses to test only a subset, ad hoc or naive approaches have been used to establish the sampling frame. For example, an intensive drug supervision probation program in a large metropolitan probation agency routinely tests only one in two collected specimens; Kennedy [13] chose to test one in three. To the extent that the primary purpose of a drug-testing program is to deter drug use in a population and not to identify the drug use of specific individuals, a statistical quality control approach to drug testing may be more cost-effective than either screening, ad hoc partial testing, or no testing. These latter approaches do not consider the underlying distribution of users in the population or the direct and indirect costs associated with the drug-testing program--i.e, factors routinely considered when sampling is applied in a manufacturing environment. In particular, the costs associated with failing to detect drug use and the costs of treating (or punishing) those who test positive are not generally considered in developing the drug-testing programt. Further, if, say, l0 individuals test positive out of a sample of 100, there is no consideration given to whether that is "too m a n y " positives. As noted earlier, the social process that produces drug use is not well defined. Thus, proactive approaches to quality control, such as "zero defects" and "continuous improvement," adapted by manufacturers and service providers over the past twenty years, are not entirely appropriate to our drug-testing problem. Rather, we rely on acceptance sampling--a more traditional and reactive approach to quality control that focuses more on the process output. While statistical quality control is more routinely used in manufacturing, a recent article by Sower et al. [18] discusses the relationship between acceptance sampling and statistical quality control, showing that acceptance sampling is appropriate in certain areas. Here, acceptance sampling allows us to explicitly consider all program costs in an economic model and provides a decision criterion--the acceptance n u m b e r - - f o r program management. The economic model includes not only a probability distribution of drug use in the population but, also, as we detail below, the direct and indirect costs of the testing program. We use a Bayesian approach so that we may treat the percentage of drug users in the population as a random variable (e.g. [4, 7, 10, 15]). The next section develops our proposed economic model of drug testing. A N E C O N O M I C M O D E L OF D R U G T E S T I N G To provide context for our drug-testing program, we assume it will be administered to a probation population. An economic or expected total cost model will be developed to identify a cost-minimizing acceptance sampling plan. Acceptance sampling entails the random selection of tThe U.S. Department of Health and Human Services[20]established proceduresfor workplacedrug testing. Health and Human Servicessuggests a cost/benefitanalysis that compares only the costs of testing (including confirmationtests) plus stafl~ng/trainingwith potential productivitysavings. The American Probation and Parole Association [1] specifies, for testing probation and parole populations. "Drug testing should be implemented only after the need for such a program has been established. Documentation of the nature and extent of illegal drug use within a jurisdiction will substantiate the illegal drug use problem."

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a sample of items from a "lot" (or population) of N finished items and inspection of the sampled items on the basis of the selected attribute(s). When an economic or expected total cost model is used to identify the acceptance sampling plan, the goal is to select the sample size n and the acceptance number a that jointly minimize expected total cost. These decision variables are referred to as the sampling plan (n, a). The lot is accepted if there are a or fewer "defective" items in the sample; otherwise, the lot is rejected. Rejected lots may be scrapped or subjected to 100% inspection. We make the following assumptions in developing our model: 1. A urine specimen is "defective" or non-conforming if it tests positive for one or more of a finite set of illicit drugs. 2. A "lot" consists of N urine specimens collected from a homogeneous population during a short period of time (t, t + dt). These assumptions simply ensure that the specimens can be considered a set of items produced by a like-process, i.e. by individuals with similar backgrounds, similar living environments, exposed to the same standards of conduct. For example, a "lot" might consist of specimens produced by our probation population during a one-week period. (For a business, the "lot" might consist of specimens produced by employees in the shipping department.) 3. The inspection procedure is 100% accurate. This simplifying assumption allows us to ignore the costs associated with failures to detect defective items (i.e. false negatives) and identifying non-defective items as defective (i.e. false positives). 4. The individuals in the population are "rational" consumers of illicit drugs. Specifically, we assume that an individual will use drugs if the subjective expected benefits of drug use exceed the subjective expected costs of drug use (see Ref. [3]). The expected benefits of drug use include mood changes such as euphoria (see Ref. [21]), for a summary of the effects and side effects of illicit drug use). The subjective expected costs of using illicit drugs are a function of the perceived probability of the drug use being detected, and of the subjective evaluation of the penalty associated with detection. The penalty would depend upon the circumstances of testing. Thus, for a job applicant, the penalty could be a failure to obtain employment. For a probationer, the penalty could entail a prison or jail sentence. 5. Rejected lots will be subjected to 100% inspection. 6. The costs of inspecting, accepting and rejecting the lot (see below) can be identified or estimated. 7. Inspection is for a single attribute "use of drugs." Thus, we ignore the potential multi-attribute problem of testing for multiple drugs. Instead, we assume that the distributions and costs are appropriate for this "aggregate" drug. We define the following terms for a single period: N = population size; D -- number of users in the population; 7t = population proportion of drug users, D/N; n -- number of people tested in the population; d = number of drug users in the sample; and a = the acceptance number. The objective is to derive a sampling plan (n, a) for a single period that minimizes the expected total cost of administering the plan. The plan is implemented by selecting n people at random. If more than a specimens test positive, the remaining ( N - - n ) people are also tested. Next, we discuss the development of a cost function for the single-period model. Costs of drug testing programs

The accuracy of any model depends on the effective estimation of parameters. As previously noted, our acceptance sampling model includes the direct and indirect costs of testing, both of which can be easily underestimated. For example, the costs may be estimated as those of the testing procedure and, perhaps, the administration needed to run the program (see footnote on p. 71). This approach ignores such costs as those attributable to failing to detect a user or providing a

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confirmation test and treatment and/or sanction to those who test positive. The presence of indirect costs, including costs incurred if a defective item is not detected, contingencies for inspecting the entire lot if the sample is rejected, or allowance for various dispositions of the lot, have appeared in several papers in the context of Bayesian acceptance sampling in a production setting. For example of earlier work, see Caplen [4] and Heeremans [10]. In a more recent study, Moskowitz et al. [15] (see also Ref. [19]) developed a cost function for a multiple-stage, multiple-attribute Bayesian acceptance sampling plan for a serial assembly process that allows for alternative disposition strategies for rejected lots. The expected total cost model developed by Moskowitz et al. [15] is an appropriate model for our drug-testing application with the following modifications. First, the cost of inspection in the Moskowitz et al. [15] model is expanded to include the costs of both collecting samples and performing tests. Second, subjects who are not detected but who are using drugs are presumed to impose a cost on society, e.g. through criminal activity related to their drug use or reduced productivityt. Third, the cost of rejecting the lot if the number of positives exceeds the acceptance number is included. The Moskowitz et al. model allows for a variety of lot dispositions including, for example, repair or scrapping of non-conforming items. Repair/replacement of all nonconforming items is called rectification. Obviously, human beings cannot be scrapped or thrown out, nor can they be "rectified". Instead, we assume that drug use can be reduced by two avenues--sanctions, which serve to deter future use, and treatment, which can be viewed as "repairing" the person. In a production environment, it is easy to imagine a single process that "repairs" a defective item with certainty. Within the parameters of a drug-testing program, neither the assumption of a "single repair process" nor the assumption of "certainty of repair" seems appropriate. Instead, there is a range of possible responses that will vary with respect to cost, likelihood, and effectiveness. Specifically, sanctions may range from a relatively inexpensive "slap on the wrist" to house arrest to a very expensive alternative, incarceration. The likelihood that an individual will receive a particular sanction will be affected by the individual's number of previous positives, the availability of resources to impose a sanction, and the external influences of the judicial system itself. Effectiveness is likely to vary with the degree of punitiveness, with more punitive (and, generally more expensive) sanctions being more likely to deter future drug use. Similar distributions exist over treatment. The costs of treatment options will vary from, for example, requirements to attend Narcotics Anonymous, which is relatively inexpensive, to long-term residential treatment, which may cost thousands of dollars. As in the case of sanctions, there is a probability distribution over the likelihood that treatment will be successful and an individual will remain drug-free. Note that because neither sanctions nor treatment can be assumed to repair the non-conforming individual with certainty, that this dramatically increases the costs associated with each defective item. For example, if a $1,000 program has a probability of success of 0.1, then the expected cost of the program, defined as the cost of producing one successful treatment, is $10,000. Having discussed some of the major differences between costs incurred in a drug testing environment as compared to a production setting, we next derive a model of expected total cost. This model consists of three components: (1) the cost of inspection, (2) the cost of accepting a lot, and (3) the cost of rejecting a lot. Each of these is presented in turn. Inspection costs

Inspection costs consist of two c o m p o n e n t s - - t h e cost of collecting specimens from the sample and the cost of conducting the tests. We differentiate between these since collection costs, while negligible in most manufacturing environments, are not negligible for our problem as the population to be tested is at large in the community. We identify the costs of collection and inspection as Cc and C,, respectively, where Cc < C,. Specimens can be collected either from all N individuals or from only n individuals. tA multiple-stage serial production process may pass defectives "downstream", in which case the penalty for a defective item not being detected "upstream" is manifest in the cost of repairing or scrapping a more expensive, downstream item. The analogy to drug-testing is appropriate, but the penalty cost is far more difficult to estimate in drug-testing situations.

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Under the first assumed scenario, 100% of the population is required to submit urine specimens and a random sample, n is chosen for testing, yielding inspection costs of:

ICx = Cc.N + Ci.n.

(1)

Alternatively, if only n, specimens are collected and tested, the inspection costs would be:

IC2 = (Cc + C,).n.

(2)

Under this second scenario, the (N - n) unsampled people would have to be recalled to test the entire population. A recall cost would be incurred, Cr'(N -- n), where Cf is the per-person recall cost. Because of the delay between the time the sample of n specimens was collected and rejected, and the time it takes to recall the (N - n) individuals for testing, the composition of the lot may have changed. Thus, we assume 100% collection.

Acceptance costs The population will be assumed to be at an acceptable level of usage if no more than a people test positive, where a is the non-negative acceptance number. The probability of accepting a lot, P .... is: Pacc =

E h(dlD).g(D), D=O d=O

(3)

where h(dlD) is the hypergeometric probability of having d positive urine specimens in a sample of n people when there are D users in the population, and g(D) is the probability of having D users in the population. If the lot is accepted, then d users are identified and subjected to treatment and/or sanction a t an average cost to society of Cr per person, where CT > Ct, and a probability that the treatment will be successful ofpf. As a lower bound, CT will include the cost of confirmatory testingt. There are (D -- d) additional drug users still at large in the population who will, it is assumed, continue their habits. Each user costs society a penalty cost, CA, which could include, for example, the cost of crimes associated with drug use or the cost of crimes committed to support drug use. The expected cost of accepting the lot is:

EC.~c=D=oLd=0~I cA "(D -

d) + (-~r ) d ] "h (dlD ) "g(D )"

(4)

Rejection costs I f there are more than a positive tests in the sample, the lot is rejected and an additional (N -- n) specimens must be tested. As noted earlier, we assume that specimens from all N members of the population were collected at one time (the IC~ case). Assuming perfect testing, and having collected all N specimens (i.e. no recall was necessary), we identify D users in the population who will undergo treatment and/or sanction. The expected cost of rejection, ECroj, is:

Expected test costs The expected total cost is simply the sum of the inspection costs and the expected acceptance and rejection costs. Thus, the expected total cost (ETC) is:

E T C = IC, + EC~cc + ECr~j.

(6)

Given cost estimates, and some assumptions concerning the distribution of D, eqn (6) can be "['Kaufmann [12] stresses "screening tests should never be used without automatic confirmatory testing of positive results by a more specific method" (italics his).

Statistical quality control and social processes Table 1. Prior distributions of drug use p (n) SYMM. a*b (c) a*c 0.0 0.0013 0.0 0.003235 0.0031 0.000155 0.00736 0.0067 0.00067 0.011655 0.0132 0.00198 0.01572 0.0237 0.00474 0.01925 0.0389 0.009725 0.02211 0.0583 0.01749 0.02422 0.0800 0.0280 0.02556 0.1001 0.04004 0.02619 0.1146 0.05157 0.0262 0.1199 0.05995 0.02563 0.1146 0.06303 0.0246 0.1001 0.06006 0.023205 0.0800 0.0520 0.02163 0.0583 0.04081 0.019875 0.0389 0.029175 0.01800 0.0237 0.01896 0.01615 0.0132 0.01122 0.01431 0.0067 0.00603 0.01254 0.0031 0.002945 0.0109 0.0013 0.0013 E (n) = 0.3638 E (n) = 0.4998 n = Proportion of users in population. P (n) = Probability that the proportion of users is n. E (n) = Expected proportion of users in the population.

Proportion of users (n) (a) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

p (x) LEFT (b) 0.0485 0.0647 0.0736 0.0777 0.0786 0.0770 0.0737 0.0692 0.0639 0.0582 0.0524 0.0466 0.0410 0.0357 0.0309 0.0265 0.0225 0.0190 0.0159 0.0132 0.0109

p (n) RIGHT (d) 0.0109 0.0132 0.0159 0.0190 0.0225 0.0265 0.0309 0.0357 0.0401 0.0466 0.0524 0.0582 0.0639 0.0692 0.0737 0.0770 0.0786 0.0777 0.0736 0.0647 0.0485

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a*d 0.0 0.00066 0.00159 0.00285 0.0045 0.006625 0.00927 0.012495 0.0164 0.02097 0.0262 0.03201 0.03834 0.04498 0.05159 0.05775 0.06288 0.066045 0.06624 0.061465 0.0485 E (n) = 0.6314

m i n i m i z e d with respect to n a n d a to p r o v i d e an o p t i m a l s a m p l i n g plan (n*, a*). W e d e m o n s t r a t e this a p p r o a c h in the next section.

DATA AND RESULTS This section addresses the q u e s t i o n o f h o w m a n y specimens to test in a p o p u l a t i o n given that a single test is c o n d u c t e d to detect d r u g use. W i t h respect to drug use in the p o p u l a t i o n , the p a r a m e t e r s o f the p r i o r d i s t r i b u t i o n are u n k n o w n . Clearly, d e p e n d i n g u p o n the type o f d r u g and the n a t u r e o f the p o p u l a t i o n , the p e r c e n t a g e o f users co u l d vary f r o m 0 to 100%. F o r example, a p o p u l a t i o n o f law e n f o r c e m e n t officers is likely to have a low p e r c e n t a g e o f users, d e m o n s t r a t e d by a d i s t r i b u t i o n that is highly skewed to the left. A p o p u l a t i o n o f p r o b a t i o n e r s with a history o f d r u g use is m o r e likely to be c h a r a c te r iz e d by a sy m m et r i cal or right-skewed distribution o f use. In o r d e r to d e m o n s t r a t e the m o d e l , we allow the p r o p o r t i o n o f users in the p o p u l a t i o n to vary f r o m 0.0 to 1.0 a n d thus e x a m i n e results for left-skewed, symmetrical, an d right-skewed distributions. Because d r u g use is p r e s u m e d to be p r ev al en t in o u r h y p o t h e t i c a l p o p u l a t i o n o f p r o b a t i o n e r s , we will focus on the right-skewed p r i o r distribution, recognizing that the expected val u e for this d i s t r i b u t i o n is higher t h a n o n e w o u l d expect for a n o n - p r o b a t i o n e r p o p u l a t i o n . W e shall d e m o n s t r a t e , h o w e v e r , that the results g en er at ed are m o r e d e p e n d e n t on the relative m a g n i t u d e o f the a c c e p t a n c e a n d rejection costs t h a n they are on the shape o f the p r i o r distribution. W e a p p r o x i m a t e a c o n t i n u o u s d i s t r i b u t i o n with a p r o b a b i l i t y mass f u n c t i o n o v e r ~ as s h o w n in T a b l e 1, while a s s u m i n g a p o p u l a t i o n size, N, o f 100. T h e m o d e l presented in e q n (6) requires that we identify cost coefficients. C u r r e n t estimates suggest that the cost o f a d r u g test is a b o u t $50t. F o r o u r model, we consider cost estimates on a relative scale, n o r m a l i z i n g the costs against the inspection or testing cost (Ct). W e set the testing cost (6",) at $1 a n d the cost o f collecting a specimen (Cc) at 5 0 % o f this, or $0.50. W e c o n s i d e r a range o f values for the p e n a l ty for failing to identify a d r u g user d u r i n g any one testing period, CA. T h e s e p e n a lt y costs include health care costs associated with the d r u g use, crime c o m m i t t e d to support, or as a result of, d r u g use, an d lost p r o d u c t i v i t y (e.g. see Refs [16] an d [1 I]). W e allow the value o f CA to range f r o m $10, 10 times the cost o f a test, to $300, 300 times the cost o f a test. T h e value o f CA w o u l d d e p e n d , o f course, on the characteristics o f the p o p u l a t i o n . tThe American Management Association survey outlined in The Wall Street Journal [24] reported an average cost of testing of $43.

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If the population were composed, for example, of serious offenders with a lengthy history of drug use/abuse, one might postulate that this cost would be very high. Alternatively, if the population consists of first-time offenders who are causal drug users, then there is no reason to expect the cost would be particularly high. Cost notwithstanding, in the current climate of zero tolerance for drug use, a very high penalty cost is given to undetected drug use. A distribution of values was also considered for CT, the cost of treating/sanctioning a detected user. The cost varied from $10 to $100. The value of CT depends on the policy of the agency with respect to positive test results. At a minimum, CT includes the cost of confirmatory testing. Low cost treatment options right include reprimanding an offender or referring them to Narcotics Anonymous. More costly treatments include private counseling or residential treatment programs. Punitive measures might include revoking probation and incarceration. For these analyses, we assume that the probability of a treatment being successful (pt) is 1.0. Later, we shall relax this assumption. Results were obtained for 140 combinations of CA and CT for each of the three prior distributions--left-skewed, symmetrical, and right-skewed. Sampling plans were generated using a PC-based F O R T R A N program. Results for the left-skewed prior distribution with an expected proportion, E(r0, of 0.3638, are shown in Table 2. Symmetrical prior distribution results with E(~) = 0.4998 are shown in Table 3; while right-skewed prior results with E(Tt) = 0.6314 are in Table 4. Results are presented for three cost scenarios: acceptance costs greater than (effective) treatment costs (CA > CT); acceptance costs equal to treatment costs (CA = CT); and acceptance costs less than treatment costs (CA < C~). Consider the results for the left-skewed distribution shown in Table 2. For the case where CA<< CT, e.g. CA = 10 and CT = 20, 30 . . . . . 100, the best solution is not to test. These results are reasonable, since, under these conditions, the cost of treating or penalizing an individual for a Table 2. Sampling plans for left-skewed distribution

c~ _CA 10

I0 20 I 30 { 40 I 50 { 60 I 70 { 80 I 90 i 1,00 8,4 No Sampling 335.52 368.34 20 72,22 15,6 5,3 No Samplmg 418.13 614.212 719.72 736.68 30 77,19 71,24 20,8 8,4 3,2 No Samplmg 434.27 682.82 885.38 1019.17 1095.35 1105.02 40 80,17 75,21 70,24 24,9 11,5 7,4 3,2 No Sampl~g 444.14 704.31 946.88 1154.32 1303.58 1406.96 1470.51 1473.36 50 82,15 77,20 74,22 70,24 27,10 14,6 8,4 5,3 No Samplmg 451.52 719.39 971.67 1210.79 1422.04 1581.21 1702.82 1788.20 1841.70 60 83,14 79,18 76,21 73,23 70,24 32,12 17,7 10,5 7,4 3,2 455.87 730.83 989.64 1237.30 1474.71 1689.02 1855.35 1989.10 2091.48 2168.18 70 82,14 80,17 77,20 74,22 72,24 69,24 35,13 19,8 13,6 8,4 460.94 740.33 1004.39 1258.18 ,!502.30 1738.54 1955.47 2127.36 2269.73 2,386.47 80 87,10 80,17 78,19 76,21 74,22 72,24 70,25, 35,13, 20,8 16,7 464.76 748.34 1016.88 1274.72 1525.05 1766.65 2002.30 2221.48 2397.65 2547.96 90 92,5 82,15 79,18 77,20 75,21 73,23, 71,24 70,25 38,14 23,9 478.65 755.46 1026.89 1289.38 1543.76 1791.00 2030.98 2266.02 2487.16 2667.29 fo0 69,5 83,14 79,18 78,19 76,21 74,22 73,23 71.24 70,25 41,15 483.73 759.99 1036.43 1302.45 1559.81 1811.56 2056.39 2295.07 2529.74 2752.59 150 49,0 87,10 83,14 80,17 78,18 78,19 77,20 76,21 75,22 73,23 495.32 781.70 1068.40 1348.73 1619.02 1884.99 2144.37 2399.24 2651.21 2898.40 200 55,0 46,0 87,10 83,14 80,17 80,17 79.18 78,19 77,20 76,21 495.70 843.11 1095.63 1376.80 1660.94 1933.10 2201.61 2467.52 2727.84 2985 99 250 59,0 53,0 92,5 83,14 83,14 82,15 80,17 79.18 79,18 78,19 495.91 843.91 1138.15 1398.23 1685.21 1971.22 2245.31 2516.74 2784.20 3050.06 300 61, 0 57, 0 51, 0 87, 10 83, 14 83, 14 82, 15 80, 17 80, 17 79, 18 496.06 844.18 1192.10 1417.07 1706.64 1993.62 2280.46 2557.52 2829.69 3099.33 CA = Societal penalty cost for undetected user. CT ~ Treatment cost for one detected user.

Statistical quality control and social processes

77

Table 3. Sampling plans for symmetricaldistribution

c¢ CA 10 20 30

10 1,1 498.12 71,27 606.23 74,24 623.91 77, 21 631.58 79,19 635.56 80,17 637.79

20

16,9 959.13 68,30 1043.83 72, 26 1073A8 73,25 1091.08 75,23 1103.95

I

30

I

40

25,13 1401.23 67, 30 1477.70 70,28 1516.79 72,26 1539.99

1,1 1496.38 33, 17 1838.78 67,31 1910.45 69,29 1955.04

81,17 77,21 73,25 639.44 1 1 1 1 . 2 3 1557.91 80 83,14 78,20 74,24 640.44 1 1 1 6 . 2 9 1572.60 90 83,14 78,20 75,23 641.05 1 1 2 0 . 2 2 1583.56 100 83,14 80,18 77,21 641.67 1 1 2 2 . 9 6 1590.88 150 92.5 83,14 80,18 644.52 1 1 3 0 . 9 2 1610.26 200 58,0 87,10 81,17 645.71 1 1 3 5 . 3 2 1619.29 250 62,0 92,5 83,14 645.72 1 1 3 9 . 0 3 1623.24 300 65,0 92, 5 87, 10 645.72 1 1 3 9 . 1 0 1628.09 CA= Societal penalty cost for undetected user. Cr = Treatment cost for one detected user.

71,27 1985.23 72,26 2006.51 73,25 2024.74 73,25 2040.18 77,21 2080.98 80,18 2097.55 81,17 2106.81 83, 14 2112.49

40 50 60

70

I

50

I 60 I 70 I 80 I 90 1100 No Sampling 499.85 No Sampling 999.70 No Samplmg 1499.55 1, 1 1, 1 No Sampling 1992.96 1998.07 1999A0 38,19 6,4 1,1 No Sampling 2273.96 2480.30 2494.65 2499.25 67,31 44,22 1~ 6 1,1 1, I No 2342.77 2708.10 2942.88 2991.23 2996.33 Sampling 2999.10 68,30 67,31 48,24 14,8 1,1 1,1 2390.66 2775.09 3141.59 3396A5 3487.80 3492.91 70,28 68,30 66,31 53,26 16,9 1,1 2427.09 2824.71 3207.06 3574.49 3844.92 3984.38 71.27 69,29 68,30 66,31 57,28 20,11 2452.80 2865.64 3258.75 3638.98 4006.82 4289.96 72,26, 71,27 69,29 67,30 66,31 59,29 2473.02 2896.66 3302.47 3692.25 4070.90 4438.89 76,22 73,25 73,26 72,26 71,27 69,29 2542.60 2989.28 3424.56 3853.50 4275.66 4686.85 78,20 77,21 75,23 74,24 73,25 72,26 2569.67 3035.05 3492.54 3938.17 4374.32 4805.59 80.18 78,20 77,21 76.22 75,23 74,24 2584.85 3058.05 3525.16 3987.70 4442.32 4886.86 81,17 80,18 78, 20 78, 20 77, 21 76, 22 2594.33 3072.14 3546.44 4015.19 4479.23 4939.36

positive test is expensive relative to society's penalty cost, CA, of failing to detect a user. When the minimum cost solution is not to test, i.e. when n* = 0, the expected total cost is: ETC..= o = CA'E(D),

(7)

where E ( D ) is the expected number of drug users in the population. Note, conversely, that when drug testing is not conducted, it suggests an implicit decision on the part of the decision maker or manager that it is cheaper not to test than to test. For the symmetrical and the right-skewed prior distributions (see Tables 3 and 4, respectively), the results are similar, although the expected total costs are higher due to the increased expected number of users in the population. The optimal sampling plans derived when CA = CT or CA > CT show that n* increases and a* decreases as CA increases for a given value of CT. This reflects an increasing likelihood that a lot will be rejected as the penalty associated with failing to detect a user increases. When CA values are greater than ten times the value of CT, the solution approaches screening of the entire lot. However, for the parameter set selected here, the optimal solution is still to use acceptance sampling even when CA is large (e.g. $300). This suggests that the penalty cost must be extremely high relative to the cost of treatment for screening to be the best alternative. In order to demonstrate how much more expensive screening would be, as compared to acceptance sampling, the expected cost of the acceptance sampling plan and the cost of 100% testing (n = N = 100) were obtained for CT = $10, 20 . . . . . 100. For these acceptance sampling solutions, CA was set at $300, the most expensive case. These results are shown in the last row of data presented in Table 2, while results for the symmetrical prior distribution are shown in Table 3. Results for this case show that 100% sampling increases the expected cost of sampling from 5 to 24%. Thus, the savings from acceptance sampling are substantial. When an agency uses 100%

78

Lance A. Matheson et al. Table 4. Samplingplans for right-skeweddistribution

c¢

C^

10

20

I

30

I

8, 6 610.78 20 63, 29 18, 12 702.96 1150.58 30 69, 25 60, 31 23, 15 723.15 1237.19 1682.54 40 71,23 64,29 59,32 734.03 1269.95 1767.53 50 75, 20 68, 26 63, 30 740A8 1290.96 1808.05 60 76, 19 69, 25 65, 29 744.72 1304.49 1836.71 70 78, 17 71, 24 67, 27 747.80 1315.45 1857.99 80 78, 17 73, 22 69, 25 750.22 1324.00 1873.15 90 79, 16 74, 21 70, 24 752.38 1330.15 1885.77 100 81, 14 75, 20 70, 24 753.70 1335.04 1896.50 150 91, 5 78, 17 75. 20 763.79 1350.74 1929.59 200 48,0 81, 14 78, 17 770.19 1359.15 1946.43 250 54, 0 82, 13 79, 16 770.27 1364.96 1957.96 300 58, 0 91, 5 81, 14 770.32 1377.91 1964.60 CA= Societal penaltycost for undetecteduser. CT= Treatmentcost for one detected user.

40

10

1, 1 1892AI 25,16 2212.37 58, 32 2296.48 61, 31 2342.69 63, 30 2376.43 65, 28 2403.26 67, 27 2424.72 68, 26 2441.10 72, 23 2496.14 75,20 2524.14 77, 18 2541.70 78, 17 2554.20

I

501

60

I

70

I

8o

I

90

11oo

No Sampling 631.36 No Sampling 1262.72 No Sampling 1894.08 No Sampling 5,4 2525.44 2496.65 No Sampling 30, 19 8, 6 1, 1 3156.80 2741.09 3056.35 3155.10 1, 1 No Sampling 58, 33 33, 21 14, 10 3788.16 2824.59 3269.20 3602.54 3783.32 14, 10 6, 5 1, 1 60, 31 57, 33 35, 22 2875.28 3352.09 3796.81 4143A5 4379.27 4417.80 38,24 16,11 9,7 63, 30 60, 32 57, 33 2913.09 3406.00 3879.24 4324.11 4679.96 4945.74 57, 33 43, 27 19, 13 64, 29 62, 31 59, 32 2943.49 3447.85 3935.89 4406.40 4851.14 5215.30 59, 32 56, 33 43, 27 65, 28 63.30 61, 31 2969.66 3481.47 3980.88 4465.38 4933.16 5377.91 64, 29 63, 30 62, 31 70. 24 68, 26 66, 28 3047.83 3591.25 4124.41 4644.12 5154.89 5657.84 68, 26 67, 27 65, 28 73,22 71,24 69,25 3093.00 3649.97 4198.53 4741.40 5276.53 5801.65 70. 24 69, 25 68, 26 75, 20 74, 21 72, 23 3118.70 3688.50 4249.66 4801.64 5348.53 5891.55 73, 22 71, 24 70, 24 77, 18 75, 20 74, 21 3136.66 3713.25 4283.72 4848.07 5403.43 5952.97

sampling, they are implicitly m a k i n g the a s s u m p t i o n that the cost o f failing to detect a user is m u c h higher t h a n the cost o f treating that user. W h e n all m e m b e r s o f a lot are tested, i.e., acceptance s a m p l i n g is n o t performed, the expected cost of screening is: ETC..=N = (Cc + C~).N + CT.E(D).

(8)

The results o b t a i n e d when CA = CT are interesting in that they provide a n acceptance sampling s o l u t i o n when a user (or an agency) is u n c e r t a i n as to the relative m a g n i t u d e o f penalty versus treatment. This m a y actually be reasonable when the effectiveness of t r e a t m e n t is uncertain. The fact that we do drug testing at all within a p o p u l a t i o n suggests that we view the use o f drugs as inherently bad. I f the majority of the p o p u l a t i o n did n o t hold this view, then it would be illogical to c o n d u c t testing at all. This implies that agencies assume a large penalty associated with drug use a n d a high penalty cost associated with failure to detect its use. Thus, the c u r r e n t climate s u r r o u n d i n g drug testing suggests that CA>>CT. However, a l t h o u g h it is recognized that there is a p r o b a b i l i t y d i s t r i b u t i o n over the effectiveness of t r e a t m e n t (and the deterrent effect of most sanctions), that fact is generally n o t considered in the d e v e l o p m e n t of cost models. (Note: The results s h o w n in Tables 2 - 4 assume that t r e a t m e n t effectiveness is a certainty, i.e. pt = 1.0.) As a n example, consider a t r e a t m e n t of $1000. If the p r o b a b i l i t y o f the t r e a t m e n t being effective (i.e., the i n d i v i d u a l a b s t a i n s from taking drugs) is 0. l, then the expected cost of treatment, m e a s u r e d as the cost of p r o d u c i n g a drug-free individual, is $10,000. Thus, c o n s i d e r a t i o n of a p r o b a b i l i t y d i s t r i b u t i o n over effectiveness d r a m a t i c a l l y alters the drug testing e q u a t i o n a n d m a y (given some rather realistic a s s u m p t i o n s a b o u t effectiveness of treatment) produce expected t r e a t m e n t costs that are o n the same order o f m a g n i t u d e as p e n a l t y costs. As such, viewing these two costs as being a p p r o x i m a t e l y equal m a y provide a good, a n d realistic, starting p o i n t for testing programs. One could argue that in the long run, CA a n d CT would move t o w a r d equilibrium, and, hence, these

Statistical quality control and social processes

79

2700 2600

~ " 2500 0

2400

2300 IC. 2200 W 2100 2000

I 0

0 r

I

I

0 (kl

0 Or)

I 0 ~

I

I

0 LC)

0

i

0 ~

i

0 ~

i

0 O~

i

0 0

Sam pie Size

Fig. 1. E ( T o t a l cost) vs S a m p l e size. CA = Societal p e n a l t y cost for u n d e t e c t e d user; Cr = T r e a t m e n t cost for o n e detected user; CA = C r = $50.

solutions represent either a good starting point or the ultimate cost relationship that results from market pressures, ceteris paribus. Acceptance sampling plans along the diagonal of Table 2 show that the ratio of the acceptance number to the sample size (a*/n*) is the same (approximately) as the E(D) for the particular prior. As CA and CT increase, both the sample size (n*) and the acceptance number (a*) increase, but the ratio of a*/n* remains relatively constant. This pattern also holds for the symmetrical and right-skewed prior distribution results reported in Tables 3 and 4, respectively. Results for both the symmetrical and right-skewed prior distribution demonstrate that the optimal solution is dependent on the E(D) in the population. However, compared to the left-skewed distribution results, when the number of users increases (1) the expected total cost increases, (2) the pattern of n* and a* increases as costs increase along the diagonal, and (3) the magnitude of a*/n* relative to E(D) remains the same. At the optimum, the expected acceptance and rejection costs are approximately equal, while the cost curves reveal that the total cost curve is relatively flat. Figure 1 shows a cost curve for the optimal sampling plan when n varies from 1 to 100 with CA = CT = 50. This demonstrates an important point for practitioners, in that we can move away from the optimal solution by a considerable amount and the expected total cost does not increase dramatically. This provides the decision maker with a great deal of flexibility in knowing that movement away from the optimal acceptance sampling plan will not completely compromise the value of using acceptance sampling over screening or no testing. Thus, those in charge of collecting and sampling urines may be somewhat comforted by knowing that loss of a specimen or failure of one or more individuals to drop a specimen may not destroy the integrity of the acceptance sampling approach, provided they are willing to accept a "near-optimal", rather than the optimal, solution (see also Table 5). Table 5. Comparison of expected costs--acceptance sampling vs screening CT

Acceptance Sampling ETC* (CA = $300)

Screening ETC

% Change

I0 20 30 40 50 60 70 80 90 100

496.06 844.18 1192.10 1417.07 1706.64 1993.62 2280.46 2557.52 2829.69 3099.33

518.30 886.60 1254.90 1623.20 1991.50 2359.80 2728.10 3096.40 3464.70 3833.00

4.48 5.02 5.26 14.5 16.7 18.4 19.6 21.1 22.4 23.7

CA = Societal penalty cost for undetected user. CT = Treatment cost for one detected user.

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Lance A. Matheson et al.

Acceptance sampling provides a minimum expected total cost solution--compared with 100% inspection and no testing--under two cost scenarios: (1) when CA ~ CT, and (2) CA is less than one order of magnitude of CT. The results also suggest situations in which no sampling is optimal--CT is twice the level of CA--and that 100% sampling is never optimal, given the parameter values used in the analyses. This final result is important since it appears that the ad hoc procedure which is currently in use, i.e. test everyone, is more costly than acceptance sampling.

DISCUSSION Drug testing is routinely performed by testing all members of a population. Testing 100% of the population does not, by design, provide a decision rule by which the manager (or in our application, the probation officer) can determine, in an objective manner, whether there are too many positive tests in the population. In this scenario, it is unreasonable, without daily testing and the real presence of deterrents (i.e. incarceration or monitored treatment), to expect that a member of this population will abstain from usage for the length of their probation. It is more reasonable to expect that the best one can do is attempt to affect the prior in a way that does not increase the expected number of users. Certainly, if we could test "all the people, all of the time" a goal of zero non-conformers might be achieved for illegal drugs. However, budgets would not permit such a strategy, and evidence suggests that individuals would simply substitute other types of substance abuse to evade the testing procedure. Given that we cannot test everyone, acceptance sampling thus provides us with a minimum cost solution, coupled with a decision rule for partial testing of drug use in a population. In addition, the model is premised on the very realistic assumption that one cannot reduce the mean of the prior distribution of drug use to 0%. Rather, it allows the probation officer to recognize that a more realistic purpose of drug-testing is to monitor and control for an increase in use through a systematic, economically-based decision criterion. The proposed model also demonstrates how current drug testing protocols implicitly weight the indirect costs associated with drug use, drug treatment, and the failure to detect drug use. Given that budgets are not unlimited, and we, at the very least, do not wish to see drug use increase while probationers are in our custody, there are two possible objectives for drug testing: (1) we could attempt to reduce drug use (and maximize deterrence as measured by recidivism), subject to a budget constraint; or (2) we could minimize the total cost of testing subject to not affecting deterrence. It is not clear that testing all individuals in a population as often as an agency can afford to do so will allow either scenario to be achieved. In particular, suppose if (1) we consider the cost function for drug-testing to include the cost of testing but not the cost of sanction, and/or (2) individuals who test positive are not sanctioned, and/or (3) the cost to society of an uncaught user is ignored. We cannot then make any assumption about either scenario as we have not provided a framework in which drug use, and the cost of drug use, are combined in a meaningful fashion. Similarly, if we ignore the impact of the prior distribution in our cost model, we are then providing but a rather naive first approximation of the impact, on program costs, of drug use patterns. The Bayesian acceptance sampling model allows the manager of a drug testing program to more clearly evaluate the implications of current policy. For example, what is the implication of testing 100% of the population? Other than a perceived, but never proven, deterrent effect, testing everyone implies that the cost of a drug user going undetected is orders of magnitude higher than testing or treatment/sanction costs. This may be consistent with current public opinion, which holds that any and all drug use is bad and should be eradicated at any cost; but is it realistic? Doubtless, those who would argue in favor of 100% drug-testing either have not fully considered the cost tradeoff it implies, nor have they sufficiently considered the cost of treatment or any distribution over treatment effectiveness. I f we include a distribution over availability of treatment and sanctions, and a probability distribution over receipt of treatment or sanctions, the cost of this component increases dramatically. In reality, one could construct a not-too unrealistic scenario based on empirical probabilities that would drastically increase the expected cost of rejecting a lot. Rationally, this would yield a solution in which no sampling would be less expensive than either

Statistical quality control and social processes

81

acceptance sampling or 100% testing. Although we are not suggesting that no testing is an option, this is the lowest cost alternative, given treatment/sanction costs. Within the specific area of drug-testing, the proposed model includes a realistic cost function within the framework of a process that can be monitored (if not controlled). We would suggest that within the more general framework of social processes, many of which are subject to similar sets of indirect costs and (complex) behavioral interactions, this model is also applicable. With respect to such problems, one of the most useful outcomes is the ability of the planner or decision maker to rationally evaluate the implications of his/her decision(s), as well as allowing for a more efficient use of limited resources.

SUMMARY AND CONCLUSIONS While the application of statistical quality control procedures, such as acceptance sampling in a manufacturing environment, may be viewed as reactive, within a social setting its use can provide an objective measure of quality measurement and monitoring that has yet to be explored. Thus, in social settings, where process may not be well-defined or directly or easily controlled, testing of an output, which may be well-defined, can provide a means for constructively affecting the process. The application of Bayesian acceptance sampling techniques to drug-testing appears to improve on the ad hoc approaches employed in the past. First, acceptance sampling allows the decision maker to enforce some type of decision rule when the number of individuals who test positive within a sample of the population is excessive. Thus, instead of applying a subjective measure to "too m a n y " positives, a sampling plan provides a clear delineation between too few and too many positives. Second, many drug testing programs are myopic in that they fail to consider all of the subjective and objective costs associated with drug testing and drug use. A Bayesian acceptance sampling plan gives the decision maker more complete information from which to plan a drug testing program. Thus, a manager may be able to use the dollars budgeted to test more effectively and efficiently, an essential outcome in the face of possible declining public funding. Drug testing was selected here as a social process that could benefit from the application of acceptance sampling. However, as previously noted, it is by no means the only type of process that might benefit from application of this technique. Provided that the process has an outcome or outcomes that avail themselves to objective measure, the technique is applicable independent of process complexity.

REFERENCES 1. American Probation and Parole Association. American Probation and Parole Association's Drug Testing Guidelines for Adult Probation and Parole Agencies. U.S. Department of Justice, Bureau of Justice Statistics, Washington DC (1991). 2. H. Axel. Corporate experience with drug testing programs. Conference Board Report No. 941, The Conference Board, Inc., New York, NY (1990). 3. J. R. Baker, P. K. Lattimore and L. A. Matheson. Constructing optimal drug-testing plans using a Bayesian acceptance sampling model. Mathematical and Computer Modeling 17(2), 77 88 (1993). 4. R. H. Caplen. A contribution to the problem of choosing a sampling inspection plan. The Quality Eng. 26, 103-107 (1962). 5. J. A. Carver. Drugs and Crime: Controlling Use and Reducing Risk through Testing. National Institute of Justice, Washington, DC (1986). 6. J. J. Collins. Policy choices in urine testing of probationers and parolees. Presentation to the American Society of Criminology, Reno, NV (1989). 7. G. W. Evans and S. M. Alexander. Multiobjective decision analysis for acceptance sampling plans, liE Trans. 19(3), 308-316 (1987). 8. The Executive Office of the President, Office of National Drug Control Policy. Office of National Drug Control Policy, Bulletin No. 3, August 1991. Washington, DC, Office of National Drug Control Policy (1991). 9. D. W. Frenhold. Drug-testing methods and reliability. J. Psychoactive Drugs 22(4), 419-428 (1990). 10. J. H. Heeremans. Determination of optimal in-process inspection plans. Industrial Quality Control 19, 22-37 (1962). 11. R. L. Hubbard, M. E. Marsden, J. V. Rachal, H. J. Harwood, E. R. Cavanaugh and H. M. Ginzburg. Drug Abuse Treatment: A National Study of Effectiveness. The University of North Carolina Press, Chapel Hill, NC (1989). 12. E. A. Kaufman. Testing for drugs of abuse: Methods and reliability. J. Law and Health 2(1), 1-13 (1986-87). 13. E. Kennedy. The Effects of Partial Drug Testing on Drug-Use Behavior and Self-Disclosure Validity. Illinois Criminal Justice Information Authority, Chicago, IL (1993).

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14. E. J. Latessa. The effects of random drug testing on probationers. Presentation to the Academy of Criminal Justice Sciences, Nashville, TN (1991). 15. H. Moskowitz, R. Plante and K. Tang. Multistage multiattribute acceptance sampling in serial production systems. lEE Trans. 16, 130-137 0986). 16. D. P. Rice, S. Kelman, L. S. Miller and S. Dunmeyer. The Economic Costs of Alcohol and Drug Abuse and Mental Illness: 1985. Report submitted to the Office of Financing and Coverage Policy of the Alcohol, Drug Abuse, and Mental Health Administration, U.S. Department of Health and Human Services. Institute for Health and Aging, University of California, San Francisco, CA (1990). 17. J. G. Schwartz, P. R. Zollars, A. O. Okorodudu, J. J. Carnahan, J. E. Wallace and J. E. Briggs. Accuracy of common drug screen tests. Amer. J. Emergency Medicine 9(2), 166-170 (1991). 18. V. E. Sower, J. Motwani and M. J. Savoie. Are acceptance sampling and SPC complementary or incompatible? Quality Prog. 26(9), 85-89 (1993). 19. K. Tang, R. Plante and H. Moskowitz. Multiattribute Bayesian acceptance sampling plans under nondestructive sampling. Management Sci. 32, 739 750 (1986). 20. U.S. Department of Health and Human Services, Public Health Service. Comprehensive Procedures for Drug Testing in the Workplace: A Process Model of Planning, Implementation, and Action. U.S. Department of Health and Human Services, Washington, DC (1991). 21. U.S. Department of Justice, Bureau of Justice Statistics. A National Report: Drugs, Crime and the Justice System. US Governmental Printing Office, Washington, DC (1992). 22. C. A. Visher. A comparison of urinalysis technologies for drug testing in criminal justice. National Institute of Justice, Washington, DC (Research Report, November 1991). 23. C. A. Visher and K. McFadden. A comparison of urinalysis technologies for drug testing in criminal justice. National Institute of Justice, Washington, DC (Research in Action, June 1991). 24. The Wall Street Journal. April 2, 1992. More U.S. Companies Test Employees for Drug Use. 25. The White House. National Drug Control Strategy: A Nation Responds to Drug Use. US Government Printing Office, Washington, DC (1992). 26. E. D. Wish and B. A. Gropper. Drug testing by the criminal justice system: Methods, research, and applications. In: Crime and Justice, Volume 13, Drugs and Crime, pp. 321-391. The University of Chicago, Chicago, IL (1990).