The Probability of Penalizing the Innocent Due to Bad Test Results

In modern society two-outcome tests are everywhere. They include drug tests, sobriety tests, disease tests, genetic tests, etc.. The outcome of these tests are either positive or negative, yes or no. We like to think these tests are at least 99% accurate, and yet, horror stories of spurious results seem to abound. Take company-wide drug testing, opponents may claim that at least a third of those identified as drug users will actually be innocent. If we assume the test is 99% accurate, this claim sounds ridiculous. But is it?

To analyze the claim we will "grow" a decision tree. Decision trees are a wonderful little device for analyzing anything with two possible outcomes. Every time we reach the end of a branch and have  two possibilities we simply create a set of two new branches. For our analysis, we will assume that 2% of all employees actually use drugs. This is lower than the general population but keep in mind that a lot of drug users are unemployed. Also, a company with a clearly stated anti-drug policy will probably have a low proportion of users. The tree's trunk represents the population of all employees. The first set of branches (see figure 1) represent the two possible conditions: drug user, not drug user. The expression Pd = .02 indicates that there is a 2% probability of a person being a drug user. Pc = .98 indicates a 98% probability that a person is clean or drug free.

Next, we add two sets of branches representing the drug test* as shown in figure 2 . One set of branches is attached to each of the original two branches. Pw = .01 indicates that there is a 1% chance of getting a wrong or incorrect result from the test. Pr = .99 indicates that there is a 99% chance of getting a right or correct result from the test. Note that the probabilities associated with each set of branches must add up to 100%.

Finally we add the tree's leaves (see figure 3). Each leaf represents a possible final outcome of the entire process. Note that there are four possibilities. Two of the four possibilities are correct: drug users and drug free individuals are both correctly identified. However, two of the four possibilities are spurious: drug users and drug free individuals are not correctly identified. We are unlikely to hear complaints from a drug user who is incorrectly identified as being drug free. The drug free person identified as a user is another matter. This would be a very upsetting situation.

To find the probability of each final outcome as represented by the four leaves simply multiply the probabilities of each branch one must "climb" on the way to reaching the leaf. For example, the probability of a drug user being rightly identified is represented as Pdr and is calculated as follows:

Pdr = Pd * Pr

= 0.02 * 0.99

= 0.0198 or 1.98%

Note that all the leaf probabilities have to add up to 100%.

The population of people identified as drug users consists of individuals who actually are drug users (1.98% of the employees tested) and incorrectly identified individuals who actually are not drug users (0.98% of the employees tested). The percentage of people identified as drug users who are actually innocent can be calculated as follows:

Pinnocent = Pcw /( Pcw +Pdr )

= (0.98%) /(0.98% + 1.98%)

= .331 or 33.1%

The wild eyed claim that a third of all people accused of drug use will be innocent is not so ridiculous after all.

Figure 4 shows that the proportion of spurious results among people identified as drug users is surprisingly sensitive to test accuracy. An accuracy of 99% is marginal at best. However the biggest surprise is the fact that the proportion of spurious results among people failing drug tests approaches 100% as the proportion of drug users in the general population approaches zero. Drug testing in a drug free population amounts to a witch hunt.

 figure 1. First set of Branches
 figure 2. Second set of Branches
 figure 3. Complete Decision Tree

Drug testing can be administered in a manner which minimizes the proportion of innocent people among those dismissed for failing drug tests. First, test only when evidence for drug use exists, such as erratic behavior. A much higher percent of people showing drug abuse symptoms will actually be drug users and this reduces the the percentage of innocent people who are falsely accused (see figure 4).  Second dismiss an employee only after failing more than one test. Using multiple types of evidence including accurate drug testing is the key to preventing the firing of an innocent employee for drug abuse.

Police generally follow a similar procedure of using multiple types of evidence when making drunk driving arrests. First, before making an investigation they must establish a reasonable suspicion, such as the odor of alcohol on the breath or the sight of alcoholic beverage containers in the car. Second, they make a standardized field sobriety test. Third, they test the breath, blood, or urine for alcohol. Forth, they are required to inform suspects of their chemical test rights. These include the right to obtain a blood or urine sample for retesting (see Blood Alcohol Testing in Drunk Driving Cases).

No one should accept a conclusion based on only a single test result if acceptance carries severe or far reaching consequences. In the case of diagnostic tests for medical conditions it can lead to dangerous or unnecessary procedures. In the case of a drunk driving arrest it can lead to unjustified imprisonment. A low SAT score can lead to denial of admission to a well qualified candidate. Corroborating observations or data and multiple independent tests are the best defenses against spurious test results.

* Note: The analysis assumes that the probability of a false positive result is identical to the probability of a false negative test. In reality this is usually not the case. One will be at least slightly higher than the other. However, the decision tree analysis still works and still gives similar results. Testing in a compliant group without cause is still likely to be a witch hunt.