age-test markov chains for drug testing and detection

~ Pergamon Socio-Econ. Plann. S¢i. Vol. 30, No. 4, pp. 303--313, 1996 Published by Elsevier Science Ltd

Printed in Great Britain. All rights reserved S0038-0121(96)00022-5 0038-0121/96 $15.00 + 0.00

Age-test Markov Chains for Drug Testing and Detectiont

JAMES P. BOYLE ~ and T H E O D O R E J. THOMPSON2:~ ~Navy Personnel Research and Development Center, 53335 Ryne Road, San Diego, CA 92152-7250,

U.S.A. :Centers for Disease Control and Prevention, National Center for Chronic Disease Prevention and Health Promotion, Division of Diabetes Translation (K-10), 4770 Buford Highway, NE, Atlanta,

GA 30341-3724, U.S.A.

Abstract--Random urinalysis strategies stratified by time since the last test are characterized with a set of Markov chain models. The probability of a person being tested depends on the amount of time since the person's last test. The Nuclear Regulatory Commission (NRC) has proposed a two strata drug testing strategy based on time since last test. The proposal included a high testing rate for people not yet tested in a given time period and a low testing rate for people testing negative in a given time period. Southern California Edison has implemented a variation of the NRC proposal. These strategies can be modeled within a Markov chain framework. Time to detection is calculated as a function of testing probabilities and drug usage levels. Drug user gaming strategies are discussed with illustrations. These models are implemented as part of a U.S. Navy drug policy analysis system. Published by Elsevier Science Ltd

I N T R O D U C T I O N

The U.S. Navy's zero tolerance drug policy has been in effect since 1981. Since then, the Navy has pursued an aggressive urinalysis testing program. The objectives of this program have been to deter and detect drug abuse, as well as provide data on the prevalence of drug abuse. All officer and enlisted personnel are subject to random urinalysis testing on a continuing basis. Service members are processed for separation for the first drug abuse incident. The Navy's random urinalysis program has been considered successful. The proportion of service members sampled testing positive for drugs thus fell from 7 to 1% between 1983 and 1991. However, the ongoing political climate has put pressure on all military programs to be cost-effective. Navy managers thus seek a drug testing program that achieves the same levels of detection and deterrence but costs less than it might have in the past. As part of a continuing evaluation and improvement of the Navy's program, alternative testing strategies have been developed and analyzed. This paper presents age-test Markov models as a method to analyze urinalysis testing strategies stratified by time since last test.

In 1988 [14], the Nuclear Regulatory Commission (NRC) proposed a urinalysis testing strategy based on time since last test. That is, the probability of a person being tested depends on the amount of time since the person was last tested. Southern California Edison (SCE) has implemented a variation [13] of the NRC proposal at the San Onofre Nuclear Generating Station. Urinalysis testing strategies based on time since last test are defined by (1) a high testing rate for personnel not yet tested in a given time period, and (2) a low testing rate for previously tested personnel with negative results in a given time period. This strategy may provide balance among program objectives, including detection, deterrence, a high probability of testing some fixed numbers of times, a low probability of testing more than some fixed number of times, cost effectiveness, ease of administration, and avoidance of discrimination. The NRC's adopted rules and regulations [15]

tThe opinions expressed in this paper are those of the authors, are not official, and do not necessarily reflect the views of the Navy Department. This work was performed while Mr Thompson was a mathematical statistician at Navy Personnel R&D Center, San Diego.

~;Corresponding author.

303

304 James P. ,Boyle and Theodore J. Thompson

for urinalysis do not require a time since last test strategy. However, SCE continues to use its own variation of this strategy with NRC approval.

There is a limited body of literature that applies quantitative methods to the analysis of random drug testing. Evanovich [6], for example, uses Markov chains to estimate probability of detection of drug users. The states correspond to the number of days since the drug was last used. A simple random sampling urinalysis strategy is assumed. Stoloff [17] uses empirical data from the Marine Corps to model drug use deterrence as a function of testing rates by employing cross-sectional and time series methods. Baker, Lattimore and Matheson [2] examine drug testing within the criminal justice system. They apply acceptance sampling methods to the construction of optimal drug testing plans. Thompson and Boyle [19] develop probability models for current Navy practice, show its susceptibility to gaming, and propose an alternative system.

A relevant issue here is drug detection time in urine, which varies by drug. For example, marijuana is detectable for approximately 2-4 days, cocaine for approximately 1-2 days, and methamphetamine for approximately 2-3 days. Detection times also vary by dose, analytical test method used, individuals' physical condition, fluid intake, and method and frequency of ingestion. Discussions of urinary excretion kinetics of cocaine, marijuana, methamphetamines, and amphetamines appear in Ambre et al. [1], Beckett and Rowland [3], Hamilton et al. [7], Johansson et al. [10], Johansson and Halldin [11], and Cook et al. [5]. The models developed here assume that if an individual were tested every day in a month he or she would test positive on a fixed number of days. These proportions are model parameters specified by the model user.

The remainder of the paper is organized as follows. The next section discusses theory and applications related to nonusers. The probability distribution of random urinalysis tests is modeled under a general class of age-test urinalysis strategies. Age-test is a particular Markov chain with the probability of being tested defined as a function of time since last test. The NRC proposal and the SCE program can be modeled as age-test Markov chains with 13 states. In this case, states one through 12 correspond to one through 12 months since last test and state 13 is defined more than 12 months since last test. The probability of being tested, given the current state, defines an age-test urinalysis strategy. Both age-test strategies are analyzed. (Hoel, Port and Stone [8, Chapters 1 and 2] provide an introduction to the theory of finite state recurrent Markov chains used in this section.)

The following section reports on work to quantify the extent to which age-test urinalysis strategies can be gamed by drug users. This section extends the age-test model to include an absorbing state for detection of drug users and reanalyzes both age-test urinalysis strategies. We use results taken from Taylor and Karlin [18] and Hoel, Port and Stone [8] concerning the behavior of absorbing Markov chains.

The final section contains a discussion and provides several conclusions.

AGE-TEST MODELS: NON-USERS

Model

We here define a class of age-test Markov chains where the states are defined by time since last test. The transition matrix is given by eqn (1):

pl ql 0

p2 0 q2

p3 0 0

p =

0

0

q3

P~ 0 0 0 " qd

p~+l 0 0 0 " " qd+l

(1)

A similar model, called age-replacement, where p~+ 1 ~ 1, is presented in [18]. However, we will be interested primarily in the case where d = 12 and testing is conducted monthly. Note that the

Age-test Markov chains for drug testing 305

value for d could be set to 52 or 365 for weekly or daily testing and all our theoretical results would still hold. In general, an individual is in state i if last tested i time periods ago. An individual resides in state d + 1 if last tested d + 1 or more time periods ago. For each state, an individual is either tested with probability pi, in which case the individual moves to state one, or the individual ages one time period with probability qj = 1 -p~. Hence, the chain is named age-test.

When the restrictions 0 < pl ~ p2 ~< • • • ~< p~ < 1 and pa ~ pal+ I ~< 1 are placed on the transition matrix in (1), all states lead to all other states with positive probabilities. The chain constitutes an irreducible recurrent class with a unique stationary distribution. A closed form solution for the stationary distribution, H, is given in the Appendix. Since pl > 0, H is steady state with n~ equal to the reciprocal of the mean return time to state i.

We calculate the occupation time probabilities for state 1 using the age-test model. The occupation time probabilities for 12 time periods provide the distribution of the annual number of tests. A visit to state 1 is equivalent to being tested. Observe that from any initial state there are 212 possible paths over 12 periods. The probability of each path is calculated and the results are aggregated in order to generate the conditional probabilities. (An efficient algorithm for these calculations is provided by [9].) The unconditional distribution of the number of tests in 12 time periods, (N12(1)), with initial distribution set equal to steady state (H = rrl . . . . . ;rl3) is determined by

13

Pr(Nlffl) = m) = ~ ;tiP,(N,ffl) = m). i ~ 1

We also observe the average annual number of tests is E(N~2(1)= 12~rl since H is stationary. Finally, note that the probability of not being tested within a year is

Pr(Nlffl) = 0) = rr13.

Specifically, the event Nl2(1) = 0 is equivalent to the chain being in state 13 at time 12. Because the initial distribution is stationary, all distributions are stationary and Pr(Xl2 = 13) = zr13. See [16] for further details.

Applications

The age-test Markov chain was used here to analyze two different urinalysis testing strategies described below: the NRC proposal and the SCE process.

This age-test Markov chain has 13 states. The process is observed monthly and states 1 through 12 are defined by the number of months since an individual was last tested. State 13 includes individuals who were tested 13 or more months ago. The optimization problems were solved using the Microsoft Excel Solver [12] on an IBM compatible personal computer. The specific examples follow.

The urinalysis testing alternative proposed by the Nuclear Regulatory Commission [14] is analyzed first. The proposal requires that at least 90% of the individuals be tested each year and that testing rates for individuals already tested with negative results be at least 30% per year, or 30/12 = 2.5% per month. The age-test model with

pi >10.3/12 = 0.025, i = 1 . . . . . 12

zrl3 ~< 0.1 (2)

satisfies these requirements. Minimizing rrl, the average number of tests per month per person, subject to (2), yields a solution with pl =p2 . . . . . pl2 = 0.025 and p,3 = 0.6338. Results are summarized in Table 1. The last column in the table was calculated by enumerating all possible transitions over 12 months. The advantages of this alternative include: (1) the average number of tests is 1.03, (2) 90% of the people are tested at least once per year, and (3) 12% of the people are tested more than once. The major disadvantage is that once tested, people have a 74% chance of not being tested for the next 12 months. By comparison, simple random sampling at the 103% annual rate gives a 66% chance of at least one test, a 28% chance of more than one test, but a 34% chance, once tested, of not being tested for the next 12 months.

Two additional models of the NRC proposal were developed: a weekly model and a daily model.

306 James P. Boyle and Theodore J. Thompson

The weekly model has 53 states, pi i> 0.3/52, its3 ~< 0.1. The least cost solution has 1.093 tests per person per year; 90% of the people are tested at least once per year, while 12.6% are tested more than once. The daily model has 366 states, pt/> 0.3/365, Zt366 ~< 0.1. The least cost solution has 1.042 tests per person per year, while 90% of the people are tested at least once per year, and 12.9% are tested more than once. These models show that as the number of states is increased with a fixed restriction on the p/s, the annual cost increases and the distribution of the number of tests within a year shifts slightly to the right.

The NRC proposal suggests the following more general optimization problem:

Minimize 7r~

subject to pi>tfl , f o r i = 1,2 . . . . . 12

Pr(Ni2 = 0) = rcl3 ~< 8

where 0 < 8, fl < 1. Here, we have set N,2 = NI2(1). The cost-minimizing solution to this problem is developed in [20]. The optimization problem has multiple solutions of the form pl = p2 . . . . . p,, = fl, p,3 = (1 - pt2)/y(8, fl), fl ~ pl2 ~< p,3, where y(8, fl) = (8/1 - 8)[1 - (1 - fl)~2]/fl(1 - fl)" and 0 < 6 ~< (1 - fl),2. All of these solutions yield the same distribution for Nt2, with the unique solution pi =p2 . . . . . p~l = f l , p l 2 = p l 3 = 7 = 1/(1 + y ( 8 , fl)) minimizing the probability of not being tested for the next 12 months once tested. The minimum value of rt~ is f l ( 1 - ~ ) / [ 1 - ( 1 - fl),2]. Table 2 lists several examples of these cost- minimizing age-test models with comparisons to simple random sampling. Values of fl ranging from 0.01 to 0.50 have been chosen with 8's at (1 - fl)t2/2 and 0.10. Note that the solution in Table 1 could be replaced with the solution in Table 2 where fl = 0.025 and 6 = 0.10. This would reduce the 74% chance of not being tested for the next 12 months, once tested, to a 46% chance. The last entry in Table 2 relates to the SCE process discussed later in this section. A number of properties of these models are revealed below:

1. For a fixed minimum monthly testing rate (fl), as the probability of not being tested in a given year (6) increases, the mean of the number of tests per year (N,2) decreases while the variance of the number of tests per year increases. Notice that when 8 = (1 - fl)~2 in the above formula, the cost-minimizing solution reduces to simple random sampling with all the testing rates (pi, i = 1 . . . . . 13) equal to ft.

2. The variance of the number of tests per year (N i2 ) is smaller for age-test sampling than for simple random sampling with the same mean.

3. For age-test sampling, Pr(N12 = 0) is smaller and Pr(N~2 = l) is larger than for simple random sampling with the same mean.

4. As fl increases, the age-test distribution of N~2 converges to the distribution of N~2 under simple random sampling.

A referee has pointed out that these results are intuitive. That is, if you wish to minimize the annual testing rate subject to a minimum monthly rate and a limit on how many people go the

Table I. Probabilities from the age-test model of the NRC proposal

Probability

l 0.025 0.086 0.738 2 0.025 0.084 0.277 3 0.025 0.082 0.104 4 0.025 0.080 0.039 5 0.025 0.078 0.015 6 0.025 0.076 0.006 7 0.025 0.074 0.002 8 0.025 0.072 0.001 9 0.025 0.070 0.000

10 0.025 0.068 0.000 11 0.025 0.067 0.000 12 0.025 0.065 0.000 13 0.634 0.100 0.000

Tested Steady Not tested State This month State next 12 months

Age-test Markov chains for drug testing

Table 2. Several examples of age-test models with comparisons to simple random sampling

fl ~, Minimum (pt - p . ) 6 (ptr-p~3) E(NI2) Vat(N,2) Pr(N~: = 0) Pr(N~ = 1) Pr(NI2 >t 2) Pr(Ni2 = 0 IT)*

307

0.01 0.443** 0.090 0.588 0.307 0.443 0.527 0.030 0.815 n, = 0.049*** (0.559) (0.547) (0.338) (0.115) (0.547)

0.01 0.10 0.415 0.951 0.152 0. I 0 0.851 0.049 0.524 n, = 0.079 (0.875) (0.371) (0.383) (0.246) (0.371)

0.025 0.369** 0.110 0.723 0.399 0.369 0.547 0.084 0.674 ~, = 0.060 (0.679) (0.475) (0.365) (0.160) (0.475)

0.025 0.10 0.394 1.031 0.252 0.10 0.780 0.120 0.459 nt = 0.086 (0.942) (0.340) (0.384) (0.276) (0.340)

0.05 0.270** 0.143 0.953 0.569 0.270 0.542 0.188 0.487 n~ = 0.079 (0.877) (0.371) (0.384) (0.245) (0.371)

0.05 0.10 0.358 1.175 0.441 0.10 0.668 0.232 0.365 ~, = 0.098 (1.060) (0.290) (0.378) (0.332) (0.290)

0.10 0.141"* 0.210 1.436 0.953 0.141 0.451 0.408 0.248 ~t, = 0.120 (1.264) (0.217) (0.353) (0.430) (0.217)

0.10 0.10 0.282 1.505 0.895 0.10 0.472 0.428 0.225 ~, = 0.125 (1.316) (0.200) (0.345) (0.455) (0.200)

0.20 0.034** 0.341 2.489 1.770 0.034 0.214 0.752 0.057 ~t~ = 0.207 (1.972) (0.061) (0.193) (0.746) (0.061)

0.50 0.00012** 0.667 6.001 2.996 0.00012 0.003 0.997 0.00016 ~, = 0.500 (3.000) (0.00024) (0.003) (0.997) (0.00024)

0.0596 0.0518 0.515 1,300 0.462 0.052 0.661 0.287 0.246 ~tz = 0,108 (1.159) (0.253) (0.368) (0,379) (0.253)

*Pr(N~2 = 01 T) is the probability of not being tested in the next 12 months given having just been tested. **~ = (1 - #)~2/2.

***E(Nu) = 12 x nt. Notes: Entries in parentheses are the corresponding quantities for the binomial distribution with the same mean. Ni2 will follow a binomial

distribution in the case of simple random sampling when all p : s are equal.

entire year without being tested, simply test as little as possible until just before the year has elapsed. Of course, the Markov structure is needed to calculate the higher testing rate at the end of the year, to calculate steady state distributions, and to calculate the distribution of the number of tests within a specified time frame.

Southern California Edison has implemented a composite random sampling [13] approach to urinalysis. Their approach is based on a scheme that is part sampling with replacement and part sampling without replacement. The entire population is sampled at a specified rate with replacement. People who have not been sampled within the past year are sampled at another specified rate without replacement. SCE states a 5% annual chance of not being tested and a 130% average annual testing rate. The process in use at SCE can be modeled as an age-test Markov chain. Such a model with

7r~ ~< 1.30/12 months = 0.1083

7113 ~ 0 . 0 5

pl = p2 . . . . . pl: (3)

meets then specifications. There is, however, no feasible solution to eqn (3). Changing the model to a weekly or daily model allows the SCE results to be duplicated exactly. These changes resulted in no substantive changes to our results and are thus not included here.

The following two optimizations were performed to find an age-test strategy as close to meeting (3) as possible without a 13 state model. Recall that rr~, the steady state probability for state 1, represents the average monthly testing rate, while 7r,3, the steady state probability for state 13, represents the average annual not tested rate. Minimizing ~rt subject to the restrictions on p, and nr3 in eqn (3) yields p~ = p: . . . . = p~2 = 0.0645, p~3 = 1.0 and 7rt = 0.1113. Minimizing rt23 subject to the restrictions on pi and rrl in (3) yields p~ = p2 . . . . . p~2 = 0.0596, p~3 = 1.0 and n~3 = 0.0518. The solutions from both of these optimizatons define drug testing strategies that give results close to those reported by SCE [13]. Results from the latter optimization are summarized in Table 3 and Fig. 1. A problem with this process is that every month almost half (0.052/0.108 = 0.48) the tests are given to people who are in state 13 and, therefore, know they are being tested that month. This can be ameliorated by replacing the process by the one defined in the last row of Table 2. Note that the distribution of N~2 remains the same, but being in state 13 no longer guarantees a test. The probability of not being tested in the next 12 months, once tested, is reduced from 0.478 to 0.246. However, the requirement of equal p~ for i = 1 . . . . . 12 in eqn (3) is not met.


Table 3. Probabilities from age-test model of SCE's process

Probability

Tested Steady Not tested State this month state next 12 months

1 0.06 0.108 0.478 2 0.06 0.102 0 3 0.06 0.096 0 4 O.O6 O.O9O 0 5 0.06 0.085 0 6 0.06 0.080 0 7 0.06 0.075 0 8 0.06 0.070 0 9 0.06 0.066 0

10 0.06 0.062 0 11 0.06 0.059 0 12 0.06 0.055 0 13 1 0.052 0

The U.S. Navy has decided not to use age-test strategies in their random drug urinalysis program at this time. However, the methods would be easy to implement, as follows. All new recruits already are drug tested at basic training prior to official entry into the Navy. This pre-employment testing is not considered part of the random drug testing program. After basic training, recruits would be assigned to state i with probability rci. This guarantees the process to be in steady state. All previously stated results would then hold exactly. For example, if the 13 state NRC model were used, the distribution of the number of tests within the first year of enlistment would be: 0 tests 10%, 1 test 78%, 2 tests l l%, 3 or more tests 1%. The same methods [9] could be used to calculate the number of tests within a typical enlistment of three years: 0 tests 0%, l test 0.1%, 2 tests 19.1%, 3 tests 56.7%, 4 tests 20.2%, 5 tests 3.5%, 6 or more tests 0.4%. Precise estimates of the model's behavior on a daily basis could be obtained using the 366 state model previously described.

Mean = 1 .30 Var iance = 0 . 4 6 2

0.8

• 0.661

.~ 0.6

2 0- 0 4

o.o52_ o .o49_ o.oo7 o.oo 0

0 1 2 3 4 5 +

Number of Tests

Fig. !. S t e a d y s ta te d i s t r i bu t i on o f n u m b e r o f tests wi th in 12 m o n t h s for age- tes t m o d e l o f S C E ' s process .


AGE-TEST MODELS: D R U G USERS

Model

We again consider finite state Markov chains, but with the following special structure. There are r non-recurrent or transient states labeled 1, 2 . . . . . r and one absorbing state, a. A state a is absorbing if, once in a, the chain stays in a for all time. Starting at any transient state, the chain must eventually be trapped by the absorbing state. The transition matrix for such a Markov chain can be partitioned as

P = 0 . ' . 0 '

where Q is r x r, and R is r x 1, and the last row corresponds to the absorbing state. The matrix W = [W~k] = (I -- Q)-l is often called the fundamental matrix. We define the random variable Wk as the number of visits to the transient state k for k = 1, 2 , . . . , r and let the random variable T = W1 + " - + Wr denote the time to absorption.

We now define a class of age-test models extending the transition matrix is

(1--Cel)pl ql 0 0 " " 0

(1--~2)p2 0 q2 0 " '" 0

(1--~3)p3 0 0 q3 " '" 0

p =

models previously presented. The

0qpl

~2p2

~t3p3

(1--o~a)pa 0 0 0 " " qa ~dpa

(1 -- ~d+l)pa+l 0 0 0 " " " q d + l O~d+lpa+l

0 0 0 0 ' ' ' 0 1

(4)

In eqn (4), an individual is in state i (l ~ i ~< d) if the individual tested negative i time periods ago, and in state d + 1 if he/she tested negative d + 1 or more time periods ago. The p,'s are the probabilities of being tested given state i, while the ~i's are the conditional probabilities of a positive result given the individual is tested. Stage d + 2 is the single absorbing state defined in eqn (4). A person is absorbed or caught if tested positive. To summarize, an individual residing in any of the states 1 to d + 1 must, in the next period, test negative, moving to state 1, or age, or be absorbed by testing positive.

As before, we impose the restrictions 0 < p1 <~ pz ~ " " " ~ pd < 1 andp~ ~< p~+ 1 ~< 1. We allow the ~t~'s to be unrestricted probabilities, i.e., 0 ~< 0tl . . . . . ctu+l ~< 1, except that we insist at least one of the 0Cs be positive. All of this guarantees that states 1 through d + 1 must lead to the absorbing state and these states are transient.

In general, W~K equals the mean number of visits to state k prior to absorption, starting at state i. For the age-test model with an absorbing state, we will be interested in calculating the mean time to detection and the mean number of tests prior to absorption. The mean time to detection starting at state i is estimated by EK w~k, and the mean time to detection starting at steady state is estimated by Y,~ (m Xk W~k). Since testing negative is equivalent to visiting state 1, estimating the mean number of tests prior to absorption amounts to selecting the ith element from the first column of the fundamental matrix W.

Drug user gaming

An interesting problem arises in considering the process in eqn (4). Given a fixed set of p,'s, a drug user might be interested in choosing ~,'s in such a way that their usage rate is at a desired level and the mean time to absorption is a maximum. Here we define the usage rate as

~ l W i l "~- " " " "~- ~ d + l W i d + l y~ = W i l all- " " " "Jr- W i d + l


where, again, w~ equals the mean number of visits to state k prior to absorption, starting at state i. This represents the average time that the user would test positive prior to absorption divided by the average time to absorption.

Formally, the drug user, knowing the pj's, wishes to solve the problem

Maximize E ( T I X o = i) = w~ + . . . + w,a+l

subject to y~ =

0~<ct~< 1 Vi (5)

where ~ is some desired usage rate. The next section contains a number of examples illustrating eqn (5).

We first state a result proved elsewhere [4]. If the pi's are equal in eqn (4), or the ~ti's are equal to ~ in eqn (5), the mean time to absorption for the drug user starting at state 1 is given by 1/Tzt~. Here, n~ represents the expected number of tests for a non-user starting at steady state. This result is very important for two reasons. One, urinalysis strategies with equal pi's cannot be gamed. Two, age-test urinalysis strategies do not change mean time to detection for non-gaming drug users when the process starts in state 1.

As stated in the previous section, in order to implement these models, incoming employees are assigned to state i with probability n~. In this case, age-test has an advantage over simple random sampling. Mean time to detection for non-gaming drug users will be less under age-test. The mean time to detection for the non-gaming drug user is greatest when starting in state 1 since the pi's are non-decreasing. Therefore, a drug user randomly assigned to steady state will be caught sooner. Examples are shown in the next section.

A p p l i c a t i o n s

The two age-test strategies presented earlier are revisited here. Recall that these are the NRC proposal and an age-test model of the SCE process.

The time to detection, and the number of tests prior to detection for both non-gaming and gaming drug users are presented for both age-test strategies. Drug use (~) for the following examples is set to 0.10, 0.20, and 0.40. For example, when a = 0.20 drugs are detectable in a user's system an average of 20% of the time or about six days per month. This is roughly equivalent to using drugs twice a month, if one assumes a three-day detection time and use on two days separated by at least three days. Alternatively, this may be the result of four straight days of use, again assuming a three-day detection period.

From the previous section, the age-test strategy {p~ = p2 . . . . . p~2 = 0.025 and pl3 = 0.6338} is a least cost solution to the NRC proposal. This is only one of several least cost solutions that meets the criteria of the NRC proposal. The form of additional solutions was shown in the previous section. Analyses of other strategies meeting the NRC proposal are not presented here but can be carried out in a similar manner.

Time-to-detection results are presented in Table 4. We examine outcomes when average drug use is 10, 20, and 40% for both the non-gaming and gaming drug user. The gaming strategy employed is the obvious one. Drug usage in state 13 is set to zero. To keep average usage at the 10, 20 and 40% levels, drug usage in states 1 through 12 is increased to 11.1, 22.3 and 44.4%, respectively. For the 20% example, average time to detection increases significantly from 53.5 months to 201 months. Also, the average number of tests prior to detection increases from 4 tests to 16.7 tests. The same trends are observed when drug usage is 10 or 40%. Drug users at all levels can successfully game this age-test strategy. If detection of drug users and prevention of gaming is important, this age-test strategy should not be used.

Table 4 also contains time-to-detection estimates under simple random sampling with the annual rate the same as the NRC rate (103%). Recall that gaming makes no difference with this testing strategy. The times-to-detection are slightly longer under simple random sampling as compared to non-gaming drug users. For a 20% user, the mean time is 58 months vs 53.5 months.


Table 4. Time and number of tests to detection for various usage rates for age-test models of the NRC proposal and SCE process

NRC proposal Non-gaming user gaming user User (s.r.s.)*

10% 20% 40% 10% 20% 40% 10% 20% 40%

Mean Detection Time**: 111.7 53.5 24.4 401.1 200.9 100.8 116.4 58.2 29.1 1st Quartile**: 33 16 8 115 58 29 34 17 9 2nd Quartile**: 78 37 17 278 139 70 81 40 20 3rd Quartile**: 154 74 33 555 278 139 161 80 40 Mean No. of Tests to Detection: 9.0 4.0 1.5 33.9 16.7 8.1 9.0 4.0 1.5

SCE process Non-gaming user gaming user User (s.r.s.)*

10% 20% 40% 10% 20% 40% 10% 20% 40%

Mean Detection Time**: 89.2 43.1 20.0 168.9 84.2 42.3 92.3 46.2 23.1 1st Quartile**: 26 13 7 49 25 12 27 14 7 2nd Quartile**: 62 30 14 117 58 29 74 32 16 3rd Quartile**: 123 59 27 234 116 58 128 64 32 Mean No. of Tests to Detection: 9.0 4.0 1.5 17.6 8.5 3.9 9.0 4.0 1.5

*s.r.s. = simple random sampling. **All quantities in months.

Southern California Edison has implemented a composite random sampling [13] approach to urinalysis. The age-test strategy {p~ = p2 . . . . . pl2 = 0.0596, pl~ = 1.0} yields similar results. The distribution of number of tests within 12 months is shown in Fig. 1. Simple random sampling at the 130% annual rate gives a 25% chance of no tests and a 13% chance of 3 or more tests.

Time-to-detection results are presented in Table 4. The gaming strategy is, again, the obvious one. Drug usage in state 13 is set to zero. To keep average usage at 10, 20 and 40%, drug usage in states 1 through 12 is increased to 10.5, 21.1 and 42.2%, respectively. For fixed average drug usage, the average time-to-detection and the average number of tests prior to detection increase significantly. Again, if detection of drug users and prevention of gaming is important, this age-test strategy should not be used.

Table 4 also contains time-to-detection estimates under simple random sampling with an annual rate = 130%. The times-to-detection are slightly longer under simple random sampling as compared to non-gaming drug users. For a 20% user, the mean time is 46 months vs 43 months.

CONCLUSIONS AND DISCUSSION

Markov chains provide a framework for the systematic analysis of drug testing strategies stratified by time last tested. For non-users, the steady state distribution provides estimates of the number of tests per month. It also provides the number of people who have not been tested in the past year. In addition, the distribution of the number of tests in a fixed time period (e.g., year), given any initial state, can be calculated. Furthermore, given testing cost estimates, the relative merits of different testing strategies can be easily determined. For drug users, the distribution of time-to-detection and the expected number of tests prior to detection can be calculated given any initial state.

Key strengths and weaknesses of age-test urinalysis strategies can be summarized as follows.

1. Age-test urinalysis strategies trade off predictability of an individual being tested for reduced probability of not being tested and reduced probability of excessive tests. This reduced tail area of the distribution is sometimes perceived as more equitable,

2. Age-test can reduce detection time for non-gaming drug users. As long as new employees are randomly assigned to steady state and the annual testing rate is fixed, average time-to-detection is less using age-test strategies. Therefore, if gaming drug users are less a concern than perceived equity, age-test may be beneficial.

3. Age-test allows gaming drug users to significantly increase time-to-detection. The more extreme the age-test strategy, the more gaming is possible. Extreme strategies have large changes in probability tested over time (since last tested).


4. Constant testing strategies are resistant to gaming. Constant strategies are defined by equal probabilities of testing by state. Since the past and present provide no information about the future, gaming as defined here, is impossible.

An important issue, which is beyond the scope of this paper, is the relationship between drug testing strategies, including testing rates, and deterrence. All the analyses presented here assumed, in a sense, no deterrent effect. Average drug use remained constant regardless of the testing strategy. However, forcing a drug user to "game" could be considered deterrence. If a drug user's gaming strategy lead to reduced average usage, the drug testing strategy could be considered successful. More sophisticated models of drug user behavior are needed to address these issues. Many other issues, including severity of punishment, drug availability, and peer pressure, also need to be considered when modelling deterrence.

The Navy has decided not to implement any age-test urinalysis strategy. The primary reason is their susceptibility to gaming as demonstrated here. The Bureau of Naval Personnel has developed a microcomputer-based drug policy analysis system, with the Markov models developed here included in this system. Indeed, all of the calculations and optimizations presented can be performed interactively within this system. The system has provided Navy managers with the ability to analyze age-test urinalysis strategies for any given set of model parameters. Other models, including the probability models developed in [19], have also been implemented in the Navy's drug policy analysis system.

Acknowledgments--We thank Doug Hentschel for many helpful discussions and for programming the optimizations presented here. We also thank an anonymous referee for observations that lead to a better understanding of these age-test models.

REFERENCES

1. J. Ambre, T. I. Ruo, J. Nelson and S. Belknap. Urinary excretion of cocaine, benzoylecgonine, and ecgonine methyl ester in humans. J. Anal. Toxicol. 12, 301-306 (1988).

2. 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).

3. A. H. Beckett and M. Rowland. Urinary excretion kinetics of amphetamine in man. J. Pharm. Pharmacol. 17, 628--639 (1965).

4. J. P. Boyle, D. J. Hentschel and T. J. Thompson. Markov Chains for Random Urinalysis II: Age-Test Model with Absorbing State. NPRDC TN-93-6. Navy Personnel Research and Development Center, San Diego, California (1993).

5. C. E. Cook, A. R. Jeffcoat, B. M. Sadler, J. M. Hill, R. D. Voyksner, D. E. Pugh, W. R. White and M. Perez-Reyes. Pharmacokinetics of oral methamphetamine and effects of repeated davy dosing in humans. Drug Metabolism and Disposition 20(6), 856-862 (1992).

6. P. Evanovich. A Model for Drug Testing (CRM 85-33). Center for Naval Analyses, Alexandria, Virginia (1985). 7. H. E. Hamilton, J. E. Wallace, E. L. Shimek, P. Land, S. C. Harris and J. G. Christenson. Cocaine and benzoylecgonine

excretion in humans. J. Forensic Sci. 22(4), 697-707 (1977). 8. P.G. Hoel, S. C. Port and C. J. Stone. Introduction to Stochastic Processes. Houghton Mifflin Company, Boston (1972). 9. K. James and B. James. Comments on JSAG Problem 92T, "Occupation Time Probabilities for a Markov Chain".

University of California, Berkeley, California (1992). 10. E. Johansson, H. K. Gillespie and M. M. Halldin. Human urinary excretion profile after smoking and oral

administration of[~4C]Al-tetrahydrocannabinol. J. Anal. Toxicol. 14, 176-180 (1990). 11. E. Johansson and M. M. Halldin. Urinary excretion half-life of A~-Tetrahydrocannabinol-7-oic acid in heavy marijuana

users after smoking. J. Anal. Toxicol. 13, 218-223 (1989). 12. Microsoft Corporation. Microsoft Excel Solver User's Guide. Microsoft Corporation, Redmond, Washington

(1991). 13. L. C. Murray and A. E. Talley. Composite Random Sampling. Southern California Edison, Rosemead, California

(1988). 14. Nuclear Regulatory Commission. Fitness-for-Duty Program: Proposed Rules. Federal Register (Vol. 53, No. 184)

(1988). 15. Nuclear Regulatory Commission. Fitness-for-Duty Program: Rules and Regulations. Federal Register (Vol. 54, No.

108) (1989). 16. T. M. Sellke. Memo on JSAG Problem 92T, "Occupation Time Probabilities for a Markov Chain. Stanford University

Department of Statistics, Stanford, California (1992). 17. P. H. Stoloff. The Effectiveness of Urinalysis as a Deterrent to Drug Use (CNR 111). Center for Naval Analyses,

Alexandria, Virginia (1985). 18. H. M. Taylor and S. Karlin. An Introduction to Stochastic Modeling. Academic Press, Inc., Orlando, Florida (1984). 19. T. J. Thompson and J. P. Boyle. Models for detection of drug use in the U.S. Navy. Socio-Economic Planning Sciences

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TN-93-5. Navy Personnel Research and Development Center, San Diego, California (1993).


A P P E N D I X

Le t I-I = (rtl, rr2 . . . . . na+ l). S o l v i n g I I = l i P w i t h X rck = 1 will g ive the s t a t i o n a r y d i s t r i bu t i on

17. W r i t i n g o u t the e q u a t i o n s , we h a v e the fo l lowing :

xl = plx~ + p2x2 + • • • + pdXd + pa+ IT'd+ 1

rt2 = qlrtl

7t3 = q2x2

x a + l = q a n a + q a + l X a + l

1 = ~1 + ~ 2 "['- " " " "Jr- ~ d "{- ~ d + 1.

So lv ing in t e rms o f nl, we ob ta in :

~1 = ~ 1

~ 2 = q l X l

n 3 = q 2 n 2 = q 2 q l n l

g k = q k - l X k - I = q k - l q k - 2 " ' " q1~1

na = q a - ~ n a - 1 = q a - l q d - 2 " " " q ln t

ha+, = (qa/pd+ 1)rid = (qaqa- l " " " q~/p,~+ 1)re1.

Since Y rck = 1, t hen

rCl = 1/[1 + q, + qlq2 + qtq2q3 + " " + (q lq2" " " q d - l ) + (q lq2" " " qa/pd+t)] .

age-test markov chains for drug testing and detection

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