## Technical Terms

It is not always easy to evaluate (health) risks. To facilitate risk comprehension we have summarized and explained technical terms and criteria in a glossary.

### All Terms:

**Absolute risk reduction**

A measure of the efficacy of a treatment in terms of the absolute number of people saved or lost. For instance, if a treatment reduces the number of people who die of a disease from 6 to 4 in 1,000, then the absolute risk reduction is 2 in 1,000, or 0.2 percent.

**Average**

A measure for the central tendency of a sample of observations. The term "average" is most often used for the arithmetic mean, but sometimes also for the median. For instance, suppose the yearly income of five brokers is $80,000, $90,000, $100,000, $130,000, and $600,000. The arithmetic mean is the sum of these values divided by their number, that is, $200,000. The median is obtained by ranking the values (as above) and taking the one in the middle, that is, $100,000. When the distribution is asymmetric, as it is often with income, the mean and the median are not the same, and it can be the case that most people earn less than the mean.

**Base rate**

The base rate of an attribute (or event) in a population is the proportion of individuals manifesting that attribute (at a certain point in time). A synonym for base rate is prevalence.

**Bayes's rule**

A procedure for updating the probability of some hypothesis in the light of new evidence. The origin of the rule is attributed to the Reverend Thomas Bayes. For the simple case of a binary hypothesis (H and not-H, such as cancer and not cancer) and data D (such as a positive test), the rule is:

p(H|D) = p(H)p(D|H)/[p(H)p(D|H) + p(not-H)p(D|not-H)]

where p(D|H) is the posterior probability, p(H) is the prior probability, p(D|H) is the probability of D given H, and p(D|not-H) is the probability of D given not-H.

Many professionals have problems understanding this rule. The interesting point is that the calculation of p(H|D) becomes more intuitive and much simpler when the input is in natural frequencies rather than probabilities. For natural frequencies, the rule is:

p(H|D) = a/(a + b)

where a is the number of D and H cases, and b is the number of D and not-H cases.

**Clouded thinking**

A form of innumeracy, in which a person knows about the risks but not how to draw conclusions or inferences from them. For instance, physicians often know the error rates of mammography and the base rate of breast cancer, but not how to infer from this the chances that a woman with a positive test actually has breast cancer. Mind tools for overcoming clouded thinking are representations that facilitate drawing conclusions, such as natural frequencies.

**Conditional probability**

The probability that an event A occurs given event B, usually written as p(A|B). An example of a conditional probability is the likelihood of a woman actually having breast cancer if a screening mammogram is positive; it is around 10 percent. The probability p(A), for instance, is not a conditional probability. Conditional probabilities are notoriously misunderstood, and that in two different ways. One is to confuse the probability of A given B with the probability of A and B; the other is to confuse the probability of A given B with the probability of B given A. One can reduce this confusion by replacing conditional probabilities with natural frequencies.

**Degrees of belief**

One of the three major interpretations of probability (besides relative frequencies and propensities). The probability of an event is the subjective degree of belief a person has in that event. Historically, degrees of warranted belief entered probability theory from applications in the courtroom, such as the credibility of witnesses. Degrees of belief are constrained by the laws of probability (e.g., probabilities need to add up to one), that is, beliefs need to follow these laws to qualify as subjective probabilities.

**Early detection**

Early detection of a disease is the goal of screening for it. Early detection can reduce mortality. Early detection, however, does not imply mortality reduction. For instance, if there is no effective therapy, then early detection, including treatment, will not reduce mortality.

**Error**

A test can result in one of two errors, a false positive or a false negative. These errors can result from various sources, including human error (e.g., the laboratory assistant confuses two samples, labels, or enters the wrong result into the computer) and medical conditions (e.g., a positive HIV test can result from rheumatological diseases and liver diseases that have nothing to do with HIV). Errors can be reduced but not completely eliminated, and they may even be indispensable to adaptation and survival, as the copying errors (mutations) in DNA illustrate.

**Evidence-based medicine**

To treat patients consistent with the best scientific evidence available, taking the patient's values into consideration.

**Expert witness**

A person identified by the court as an expert allowed to testify before the court to facts, but also to draw conclusions by bringing together the available data and to testify to matters not accessible to lay knowledge, such as insanity, testament capacity, and standards of care.

**Fact boxes**

Fact boxes are designed to convey bottom-line information about the effectiveness and side effects of drugs and the utility of screening tests in a quick and transparent way. Based on representative studies, health features are compared with and without the use of the particular drug. For screening tests such as those used for early detection of cancer, overall and cancer specific mortality rates are compared for groups that participate in screening and those that do not. It is also specified which number of screening participants will receive a positive test result even though they do not have cancer (called a false positive result), and how many healthy people will be treated unnecessarily due to these false positive results.

**Frequencies**

A number of observations in a class of events. Frequencies can be expressed as relative frequencies, absolute frequencies, or natural frequencies.

**Fact witness**

A person who testifies in court to phenomena legally called "facts" that are usually limited to direct observation, such as what he or she has seen, heard, or done. Unlike experts, fact witnesses are not permitted to offer opinions or draw conclusions from their observations.

**False negative**

When a test is negative (e.g., a pregnancy test finds no sign of pregnancy) but the event is actually there (the woman is pregnant); this diagnosis is called a false negative or miss.

**False positive**

When a test is positive (e.g., a positive pregnancy test) but the event is not extant (the woman in not pregnant); this diagnosis is called a false positive or false alarm.

**False negative rate**

The proportion of negative tests among people with the disease is called the false negative rate. It is typically expressed as a conditional probability or a percentage. For instance, mammography screening has a false negative rate of 5 percent to 20 percent depending on age, that is, 5 percent to 20 percent of women with breast cancer receive a negative test result. The false negative rate and the sensitivity (hit rate) of a test add up to 100 percent.

**False positive rate**

The proportion of positive tests among people without the disease is called the false positive rate. It is typically expressed as a conditional probability or a percentage. For instance, mammography screening has a false positive rate of 5 percent to 10 percent depending on age, that is, 5 percent to 10 percent of women without breast cancer nevertheless receive a positive test result. The false positive rate and the specificity (power) of a test add up to 100 percent. The rates of the two errors are dependent: Decreasing the false positive rate of a test increases the false negative rate, and vice versa.

**Franklin's law**

"Nothing is certain but death and taxes." A reminder that in all human conduct, uncertainty is prevalent due to human and technical errors, limited knowledge, unpredictability, deception, or other causes.

**Guilt probability**

The probability p(guilt|match) that a person is guilty given the evidence, such as a DNA match.

**Ignorance of risks**

An elementary form of innumeracy in which a person does not know, not even roughly, how great a relevant risk is. It differs from the illusion of certainty (e.g., "smoking cigarettes does not cause lung cancer") in that the person is aware that there are uncertainties, but does not know how great they are.

**Illusion of certainty.**

The belief that an event is absolutely certain although it may not be. For instance, people tend to believe that the results of modern technologies, such as HIV testing, DNA fingerprinting, ordinary fingerprinting techniques, medical tests, or even the mechanical vote-counting machines used in elections are certain, that is, is error-free. The illusion can have benefits, such as reassurance, but also costs, such as suicide after a false positive HIV test. With respect to morals, religion, and political values, the illusion of certainty may be a requirement for being accepted by a social group, fostering social control.

*Incidence rate. *

Unlike prevalence (base rate), which refers to the proportion of individuals in a population manifesting an attribute (or event) at a certain point in time, the incidence rate is the proportion of individuals in a given population developing this attribute within a specified time interval. For instance, the proportion of men with prostate cancer at age 50 is a prevalence; the proportion of men who will develop prostate cancer between 50 and 60 is an incidence rate.

*Independence*

Two events are independent if knowing the outcome of one does not inform us about the outcome of the other. Formally, two events A and B are independent if the probability p(A&B) that A and B occur together is the product of p(A) times p(B). The concept of independence is crucial, for instance, to evaluating a match between a defendant's DNA and that found on the victim. Assume only 1 out of 1 million men show such a match. If the DNA of all a country's citizens are in a data bank, and one randomly selects one, then the probability of a match is about 1 in a million. If the defendant, however, has an identical twin, the probability that the twin also shows a match is 1 (except for procedural errors), not 1 in 1 million. Similarly, if the defendant has brothers, the probability that they match is considerably higher than for the general population. The DNA of relatives is not independent; knowing that one matches increases the chances that the relative also matches.

**Informed consent**

The ideal that the patient should be informed about the pros and cons of a treatment and its alternatives, and on this basis should decide whether he or she wants to enter treatment. Today's medical practice has not yet generally reached this ideal, partly because patients want to be taken care of rather than be informed, and partly because physicians prefer to decide what treatment to apply.

*Innumeracy*

The inability to think with numbers. Statistical innumeracy is the inability to think with numbers that represent uncertainties. Ignorance of risk, miscommunication of risk, and clouded thinking are forms of innumeracy. Like illiteracy, innumeracy is curable. Innumeracy is not simply a mental defect "inside" an unfortunate mind but is in part produced by inadequate "outside" representations of numbers. Innumeracy can hence be cured from the outside.

**Life expectancy**

The expected number of years remaining to be lived by persons of a particular age.

*Mind tools*

Means to overcome the illusion of certainty and innumeracy, such as Franklin's law and proper representations of risks.

**Miscommunication of risks**

A form of innumeracy in which a person knows the risks of some event or action but does not know how to communicate these so that others understand them. Mind tools for overcoming miscommunication are representations that facilitate understanding.

**Mortality reduction**

A measure of the benefit of a treatment in terms of lives saved. The mortality reduction can be represented in many ways, including relative risk reduction, absolute risk reduction, and increased life expectancy.

**Natural frequencies**

Frequencies that correspond to the way humans encountered information before the invention of books and probability theory. Unlike probabilities and relative frequencies, they are "raw" observations that have not been normalized with respect to the base rates of the event in question. For instance, a physician has observed 100 persons, 10 of whom show a new disease and the others do not. Of these 10, 8 show a symptom, whereas 4 of the 90 without disease also show the symptom. Breaking these 100 cases down into four numbers (disease and symptom: 8; disease and no symptom: 2; no disease and symptom: 4; no disease and no symptom: 86) results in four natural frequencies 8, 2, 4, and 86. Natural frequencies facilitate Bayesian inferences. For instance, if observing a new person with the symptom, the physician can easily see that the chances that this patient also has the disease is 8/(8 + 4), that is, two-thirds. If the physician's observations, however, are transformed into conditional probabilities or relative frequencies (e.g., by dividing the natural frequency 4 by the base rate 90, resulting in .044 or 4.4 percent), then the computation of this probability becomes more difficult and requires Bayes's rule for probabilities. Natural frequencies help people to make sound conclusions, whereas conditional probabilities tend to cloud minds.

**Negative test result**

Typically good news. That is, no sign of a disease has been found.

*Number needed to treat (NNT)*

A measure of the efficacy of a treatment. For instance, if biannual mammogram screening eventually saves the life of 1 in 1,000 participating women, the NNT (to save one life) is 1,000. In other words, 999 women do not benefit in terms of mortality reduction. NNT is also used to measure the harm of a treatment, such as when about 1 in 7,000 women who take oral contraceptives get thromboembolism, the NNT (with oral contraceptives to cause one case of thromboembolism) is 7,000. In other words, 6,999 do not show this side effect.

**Number of people who match a characteristic**

A transparent way to express the meaning of an observed match between the characteristics of a defendant and the evidence. An example is the statement "1 out of 10,000 men in this population shows a match." In contrast, the following single-event statement (a random match probability) is mathematically equivalent but can easily lead to misunderstanding in court: "The probability that this match occurred by chance is 1 in 10,000 or .01 percent."

**Number of days/years gained or lost**

A measure of the efficacy of a treatment in terms of increase or decrease in life expectancy. For instance, 30 years of smoking one or two packs of cigarettes a day results in an average loss of 2,250 days or about 6 years of life.

**Percentages**

There are three kinds of percentages. One is single-event probabilities multiplied by 100 (such as Washkansky has an 80 percent chance of survival). With this type of statement one can produce the same misunderstanding as with single-event probabilities. A second kind is conditional probabilities multiplied by 100. With this form of communication, one can produce the same confusion as with conditional probabilities. A third kind of percentages is (unconditional) relative frequencies multiplied by 100. An example is the finding of the 1962 Gallup poll that only 38 percent of adult Americans knew that cigarettes caused lung cancer. Such a percentage is easy to comprehend as long as the reference class is clear.

**Placebo effect**

A placebo operates through the mind instead of the body. For instance, when a physician gives a patient a sugar pill or injection that contains no ingredients known to influence the patient's cold symptoms or rash, but the patient nevertheless experiences relief, that is the placebo effect. Placebo is Latin for "I shall please." Placebos do not work all the time or for all diseases; their effect seems to depend on how strongly the patient believes that the treatment would actually be effective. Placebos are a challenge to the ideal of informed consent.

**Positive test result**

Typically not good news. That is, a possible sign of a disease has been found.

**Positive mammogram**

The result of a mammography is typically classified as positive (suspicious) or negative. Positive results are usually distinguished into three levels of suspicion, such as "additional evaluation needed," "suspicion of malignancy," and "malignant by radiological criteria." The vast majority of positive screening mammograms (over 90 percent) fall into the lowest level of suspicion.

*Posterior probability*

The probability of an event after a diagnostic result, that is, the updated prior probability, is called the posterior probability. It can be calculated from the prior probability using Bayes's rule.

**Prevalence**

See base rate.

**Principle of indifference**

When the prior probabilities are not known, the principle of indifference can be invoked. In the simplest case with two alternatives, the principle of indifference would assign each alternative a prior probability of one-half; for three alternatives the prior probability would be one-third each; and so on.

*Prior probability*

The probability of an event prior to new evidence is called the prior probability. Bayes' rule specifies how prior probabilities are updated in the light of new evidence.

**Probability**

A measure that quantifies the uncertainty associated with an event. If an event A cannot happen, the probability p(A) is zero; if an event happens with certainty, p(A) is one; otherwise the values of p(A) are between zero and one. For a set of events, A and B, which are mutually exclusive and exhaustive, the probabilities of the individual events sum up to 1.

**Proficiency tests**

A method to estimate the accuracy of a diagnostic testing procedure, such as DNA fingerprinting. For instance, a number of samples (e.g., DNA fingerprints) are sent to a large number of laboratories, which then independently analyze the evidence and determine whether some of the samples match. Results can be used to estimate the false negative and false positive rates, as well as the quality of the individual laboratories. Proficiency tests can be blind (that is, the laboratories and technicians do not know that they are being tested) or not blind (they do know), and internal or external (the samples are only analyzed in-house or by external laboratories).

**Prosecutor's fallacy**

Confusing the probability p(match) that the defendant matches the characteristics of the evidence available with the probability p(not guilty|match) that the defendant is not guilty given that he or she shows a match. Because p(match) is typically very small, such as when the evidence is a DNA trace, the confusion "serves" the prosecution because it makes the probability that the defendant is innocent appear equally small.

**Propensities**

One of the three major interpretations of probability (besides relative frequencies and degrees of belief). The probability of an event is defined by physical design. Historically, propensities entered probability theory from gambling, such as from the physical design of dice and roulette wheels. Propensities are limited to events whose underlying design or causal mechanism is known.

**Randomized trial**

A method for estimating the benefits of a treatment that uses randomization as a method of control. Participants in a randomized trial are randomly assigned to either a treatment (e.g., prostate cancer screening) or to a control group (no prostate cancer screening). After a certain number of years, the two groups are compared on criteria, such as mortality, to determine whether the treatment has been effective. Randomization allows the control of variables - such as age, education, and health - that could be alternative explanations (besides the treatment) for an observed difference in mortality. The randomized clinical trial uses the same idea of control as in experiments with random assignment to groups.

**Random match probability**

The relative frequency of a trait, or combination of traits, in a population. That is, the random match probability is the probability that a match occurs between a trait (e.g., DNA pattern) found on the scene of crime and a person randomly drawn from a population.

**Reference class**

A class of events of objects to which a probability or frequency refers. In the frequency interpretation of the concept of probability, there is no probability without a specified reference class. This view excludes single-event probabilities which, by definition, specify no reference class.

**Relative frequencies**

One of the three major interpretations of probability (the others are degrees of belief and propensities). The probability of an event is defined as its relative frequency in a reference class. Historically, frequencies entered probability theory through mortality tables that provided the basis for calculating life insurance rates. Relative frequencies are constrained to repeated events that can be observed in large numbers.

**Relative risk reduction**

A measure of the efficacy of a treatment in terms of the relative number of people saved or lost. For instance, if a treatment reduces the number of people who die from 6 to 4 in 1,000, then the relative risk reduction is 33.3 percent. Reporting relative risks is popular because the numbers look larger than the absolute risk reduction (which would be 2 in 1,000 or 0.2 percent). Relative risks do not convey how large the risk is in absolute terms, and as a consequence are often misunderstood. For instance, if a treatment reduces the number of people who die from 6 to 4 in 10,000, the relative risk reduction is the same as for 1,000 (33.3 percent), although the absolute risk reduction has decreased to 0.02 percent.

**Reliability**

The extent to which a test produces the same results under different conditions (such as repeated measurements). High reliability is necessary but does not guarantee high validity.

**Risk**

If the uncertainty associated with an event can be quantified on the basis of empirical observations or causal knowledge, the uncertainty is called risk. Frequencies and probabilities are ways to express risks. Different from the everyday use of the term, a risk need not be associated with a harm; it can refer to a positive, neutral, or negative event.

**Screening**

The testing of a symptomless population in order to detect cases of a disease at an early stage.

**Sensitivity**

The sensitivity of a test is the percentage of individuals with a disease who test positive in the test, that is, who are correctly classified as having the disease. Formally, the sensitivity is the conditional probability p(positive|disease) of a positive test result given the disease. The sensitivity and the false negative rate add up to 100 percent. The sensitivity is also called the hit rate.

Sensitivity of mammography. The sensitivity is the proportion of women who test positive among those who have breast cancer. It ranges between about 80 percent and 95 percent, with the lower values in younger women. The sensitivity of mammography depends primarily on the ability of the radiologist to identify breast cancers and on the rate at which breast cancers double in size between screening examinations.

*Specificity*

The specificity of a test is the percentage of individuals without a disease who test negative in the test, that is, who are correctly classified as not having the disease. Formally, the specificity is the conditional probability p(negative| no disease) of a negative test result give no disease. The specificity and the false positive rate add up to 100 percent. The specificity is also called the power of a test.

Specificity of mammography. The specificity is the proportion of women who test negative among those who do not have breast cancer. It ranges between about 90 percent and 95 percent, with the lower values for younger women.

**Source probability**

The probability p(source|match) that a person is the source of a trace given a match. (An example for a trace is blood found on the scene of a crime.)

**Single-event probabilities**

A probability associated with a singular event for which no reference class is known or specified. For instance, the statement "there is a 30 percent chance that it will rain tomorrow" is a probability statement about a singular event - it either rains or does not rain tomorrow. In contrast, the statement that it will rain on 10 days in May is a frequency statement. The latter can be true or false; a single-event probability by itself can never be proven wrong (unless the probability is zero or one). Single-event probabilities can lead to misunderstandings because people tend to fill in different reference classes. For instance, people may understand that the statement "there is a 30 percent chance of rain tomorrow" means that it will rain 30 percent of the time, in 30 percent of the area, or on 30 percent of the days that are like tomorrow. Such miscommunication can be avoided by using frequencies instead of single-event probabilities, because frequencies spell out a reference class.

**Uncertainty**

An event or outcome that may or may not happen is called uncertain. When the uncertainty is quantified on the basis of empirical observations, it is called "risk."

**Validity**

The extent to which a test measures what it was intended to measure. High reliability is necessary but does not guarantee high validity.

Harding Center for Risk Literacy · Max Planck Institute for Human Development

Lentzeallee 94 · 14195 Berlin · +49 30 82406-723 · hardingcenter [at] mpib-berlin.mpg [dot] de