Share with your friends


Analytics Magazine

The Five-Minute Analyst: Drug tests for needy families

July/August 2011


New Florida bill requires welfare applicants to take a drug test to continue receiving benefits. Will the policy save the state money?

Harrison Schramm, CAPBy Harrison Schramm

The state of Florida has signed into law a bill that requires welfare applicants to take a drug test to continue receiving benefits [1, 2]. A twist to this law is that applicants will pay for the drug test out of pocket. If they pass the drug test, their fee ($10-$25) [3] will be reimbursed, and they will continue to receive benefits under the Temporary Assistance for Needy Families program (TANF) for six months. The value of this benefit is $300 per month. Approximately 4,000 persons apply for the benefit each month.

Will this policy save money for Florida? Let’s examine the policy mathematically from four points of view:

A. The state’s point of view: perfect drug testing

Drug tests for needy families

The cost of the program with no drug screening is: NB=$7.2 million

The cost of the program with drug screening is:  Drug tests for needy families

Assuming 10 percent applicant drug use, the cost of the program with perfect drug testing is $6.5 million, with a net savings to the taxpayers of $700,000. Drug testing will be cost effective [4] if the proportion of drug users who apply for aid is greater than 1 percent.

Consider now an imperfect drug test — one that is susceptible to both false positives [5] and false negatives [6]. A ‚”decision tree‚” for an imperfect drug test is shown in Figure 1:

Figure 1: Imperfect drug test decision tree.
Figure 1: Imperfect drug test decision tree.

B. The drug user’s decision

A drug user is interested in the outcome ‚”drug user, tests negative,‚” and specifically in the rate of false positives that make the benefit of assistance (weighted by his probability of getting it) equals his potential loss paying for the test [7]. A drug user should take the test if it has a false-negative rate ? .5 percent. As a policy implication, drug users will want the state to choose the most inexpensive test available, as it makes the price of entering the ‚”lottery‚” cheaper. An inexpensive drug test may also have a higher false positive rate.

C. The non-drug user’s decision

An applicant who does not use drugs should not use an expected value model for their decision; the assistance program is not a lottery to them but rather a means to survive. Assuming the refund for the test is timely, non drug-users will want the state to have the test that will have the lowest false positive rate Imperfect drug test regardless of expense.

D. The state’s decision — revisited

The state may be interested in the probability that a person using drugs is receiving the benefit. We can solve this by reversing the conditioning and applying Bayes Theorem [8]. Reducing the population from ‚”everyone‚” to ‚”everyone who tests negative for drugs‚” may be thought of as: ‚”drug users who test negative‚” divided by ‚”everyone who tests negative‚” [9].

If the state were to use a drug test with draug test and 1 percent of the applicants use drugs, then the probability a drug user is receiving assistance is less than 1 in 10,000. If 10 percent of the applicants use drugs, then the probability a drug user is receiving assistance remains less than 1 in 1,000.

In conclusion, a drug test for welfare applicants is a mathematically defensible policy. This is not the end of the debate on this issue but rather the beginning.  Armed with statistical reasoning, we may now ask the real question: How do the savings under this policy compare with the significant ethical and legal issues it raises?

Harrison Schramm ( is a military instructor in the Operations Research Department at the Naval Postgraduate School in Monterey, Calif.


  4. We set drug tests and solve for drug tests
  5. Reporting the presence of drugs where there are none
  6. Reporting the absence of drugs when they are present
  7. drug test
  8. Wolcott and Smith have an excellent, accessible discussion of Bayes Theorem in ‚”The Cartoon Guide to Statistics,‚” p.46-50.
  9. The formula is: drug tests


Analytics Blog

Electoral College put to the math test

With the campaign two months behind us and the inauguration of Donald Trump two days away, isn’t it time to put the 2016 U.S. presidential election to bed and focus on issues that have yet to be decided? Of course not.



Study: Salaries for early career data scientists decrease for first time

Salaries for early career data scientists decreased year over year for the first time in four years as did the percentage of early career data scientists with a Ph.D. while demand for data scientists continued to increase, according to a recently released Burtch Works’ 2017 salary study of data scientists. Salaries for more experienced data scientists generally held steady or increased slightly depending on an individual’s focus area, responsibility and geographic base, according to the report. Read more →

Generous health insurance plans encourage overtreatment, but may not improve health

Offering comprehensive health insurance plans with low deductibles and co-pay in exchange for higher annual premiums seems like a good value for the risk averse, and a profitable product for insurance companies. But according to a forthcoming study in a leading scholarly marketing journal, the INFORMS journal Marketing Science, such plans can encourage individuals with chronic conditions to turn to needlessly expensive treatments that have little impact on their health outcomes. This in turn raises costs for the insurer and future prices for the insured. Read more →




2017 INFORMS Healthcare Conference
July 26-28, 2017, Rotterdam, the Netherlands


CAP® Exam computer-based testing sites are available in 700 locations worldwide. Take the exam close to home and on your schedule:

For more information, go to