A series of basic statistics by Tom Lang

4. Understanding Measures of Risk


Safety and risk are among the most commonly discussed issues in public health: "What is my risk of measles?" "Does this chemical increase the risk of cancer?" "How much does this drug reduce the risk of stroke?" "How likely is the surgery to be successful?" In medicine, the goal of any therapy is to increase the probability of benefit and to reduce the probability of harm. These probabilities are often reported as risks.
The US National Academy of Science defines risk as "A combination of the probability of an event—usually an adverse event—and the nature and severity of the event." In other words, risk is the possibility of suffering harm or loss; the probability of an unfavorable event occurring during a given period."

Measures of Frequency

How often or how likely risk occurs can be indicated by "measures of frequency."
A proportion or fraction is a measure in which the numerator is included in the denominator: fetal deaths/all deaths. Proportions are often expressed as percentages.
A rate is a change in proportion over time, although sometimes the time period is assumed or not specified. For example, "the fetal survival rate was 90%" means that 90% of the infants alive at the beginning of the period were alive at the end of the period.
Finally, a ratio is a relationship between two independent quantities in which the numerator is not included in the denominator. For example, the fetal death ratio is expressed as "fetal deaths:live births."
Some specific proportions, rates, and ratios have defined numerators and denominators, such as the example given for fetal death rate. Most are also associated with a time period (eg, per year; per lifetime) and a unit of population (a unit multiplier, such as x 1000 people) that should be reported.

Measures of Risk

Morbidity Rates: Prevalence and Incidence
Morbid means "or of relating to disease," so a morbidity is a disease, disability, or any poor health condition from any cause. "Disease," "disorder," "morbidity," and "illness" are often used interchangeably. Morbidity is often expressed in terms of prevalence and incidence.
Prevalence is the proportion of people with a diagnosis at a given time:
The prevalence of prostate cancer in 1999 was 11 per 1,000 men, or 1.11%.
Two common types of prevalence (there are several) are point prevalence, which is the proportion of people with a diagnosis at a single point in time, and period prevalence, which is the proportion of people with a diagnosis during a given period of time, such as lifetime prevalence or lifetime risk, or the proportion of people who will have the diagnosis at some point in their lives.
The lifetime risk of prostate cancer is 14.3%; prostate cancer will be
diagnosed in about 14% of all men at some point in their lifetime.
The incidence rate is the rate at which new cases are diagnosed; that is, the number of new cases identified in a given period among all people in whom the disease can occur:
The incidence of prostate cancer in 2012 was 105 per 100,000 men, or 0.11%.
Two of the most common incidence rates are cumulative incidence (also called the incidence rate, incidence proportion, or cumulative proportion), which is a measure of disease frequency during a period of time.
The cumulative incidence of prostate cancer in black men up to age 69 years is 15.0%.
Many people confuse incidence with prevalence. Remember that Prevalence is a Proportion, whereas incidence is a rate.

Mortality Rate
The mortality rate is the proportion of people dying during a given period:
The annual death rate from prostate cancer in 2012 was 21 per 100,000 men, or 0.02%.
Case fatality rate is a type of mortality rate that is especially useful in epidemics. It is the proportion of people who die from the disease in a given period.
The 10-year case fatality rate from prostate cancer is about 1%.

Risk is the probability of an unfavorable event occurring during a given period of time. How risk is reported is important. A risk of 1 in 20 is seen as lower than a risk of 1 in 43 because 20 is lower than 43, when in fact it indicates a higher risk. Similarly, a risk of 1 in 20 is considered less likely than 10 in 200 although the risk is the same, and a probability of 6 in 100 is the same as 6% and 0.06, but all are often thought to be different.
Absolute risk, or simply risk: the probability that a specified condition will affect the health of an individual or a population
Risk of Prostate
No. men with
prostate cancer
No. men in whom prostate
cancer can develop
In epidemiology, absolute risk may also require a defined geographical area and period
Absolute risk of prostate cancer
for men living in Ohio in 2014

No. of men in Ohio with
prostate cancer in 1997
Estimated No. of men in Ohio as of
July 1, 2014, in whom prostate
cancer could developp
The absolute risk of death from prostate cancer with
watchful waiting (no treatment) is 13.6%.
The absolute of death from prostate cancer with prostate resection is 7.4%.
The absolute risk difference, attributable risk, or absolute risk reduction (ARR) is simply the difference between two absolute risks:
The absolute risk difference in mortality from prostate cancer treated with
resection as opposed to watchful waiting is 6.2%
(from the above data, 13.6% - 7.4% = 6.2%).
In other words, resection reduces the risk of death by 6.2%.
The relative risk reduction (relative risk, or RRR) is the absolute risk difference expressed as a percent of the risk of the control or untreated group:
The relative risk reduction in mortality from prostate cancer attributable
to prostate resection (using the above data) is 46% (6.2% ÷ 13.6% = 46%).
A risk ratio is simply a ratio of two risks. When the risk ratio is 1, the risk in the two groups is equal:。
The risk ratio of death from prostate cancer with watchful waiting is 1.8 (13.6%÷7.4% = 1.8);
men who choose watchful waiting over prostate resection are 1.8 times as likely to die
from the disease as those who choose resection.
Because the risk ratio is the risk of one group divided by another, it matters which group is in the numerator and which is in the denominator:
If the risk ratio is 2, the risk for one group is 2 times (200%) as likely as it is for the other.
If the risk ratio is 0.5, the risk for one group is half (50%) the risk of the other.
Here, both ratios indicate that the risk in one group is twice as great as the risk in the other. Thus, by convention, protective factors are described with ratios of less than 1 and harmful factors are described with ratios of greater than 1. So:
The risk ratio of death from prostate cancer with resection is 0.54 (7.4%÷13.6% = 0.54);
men who choose resection over watchful waiting are about half as likely to die
from the disease as those who choose resection.

An odds is the probability of an event happening divided by the probability of it not happening. Odds is not the same as risk:
The risk (probability) of drawing a heart from a deck of 52 cards is 13/52 = 1/4 = 25%.
The odds of drawing a heart is the probability of drawing a heart divided by the probability of not drawing a heart: 13 ÷ 39 = 1 ÷ 3 = 33%.
For uncommon outcomes, the odds and risk are similar. For example, the risk of drawing a face card from a deck is 12/52, or about 28.5%, whereas the odds are 12/40, or about 30%, not that much different from 28.5%. For common outcomes, the odds will be higher than the risk: the risk of drawing a card with an even number (not counting face cards) is 40/52, or about 77%, but the odds are 40/12, or 33%, which is nowhere near the 77%.
In a clinical trial, the odds of death with watchful waiting was 16.2 and with resection, 7.5.

Odds ratios
An odds ratio is the odds for one group divided by that for another. When the odds ratio is 1, both groups have the same odds.
Odds ratios are difficult to understand, but they are important because they are the output of logistic regression analyses and are used in retrospective studies where the true incidence of disease is unknown. For example, in a (retrospective) case-control study, the sample is selected after the event of interest has occurred. We don't know how many people might have been smoking 20 years ago, but we do know how many smokers are in our sample and how many died of heart attack. So, we can compute the odds of heart attack from smoking as the number of smokers with a heart attack divided by the number without a heart attack:
The odds of a heart attack for smokers is 14 ÷ 22, or 0.636.
To determine the odds ratio of heart attack of smoking to not smoking, we compute the odds of heart attack in nonsmokers:
The odds of a heart attack for nonsmokers is 5 ÷ 33, or 0.152.
We then compute the odds ratio:
The odds ratio for smoking and heart attack is 0.636 ÷ 0.152 = 4.2; that is,the odds of
smokers having a heart attack are 4.2 times as high as they are in nonsmokers.
To continue with the example of prostate cancer:
The odds ratio of dying with watchful waiting is 2.2 (16.2 ÷ 7.5).

Hazards Rate
A hazards rate is the probability that if an event has not occurred in one period, it will occur in the next. Hazards rates are seen in time-to-event studies with binary (only two) outcomes, often alive or dead. They are the output of Cox proportional hazards regression analyses, which can be also used to identify which factors are associated with living or dying.
Hazards rates and hazards ratios are interpreted the same as risk rates and risk ratios, except that they apply over a specified time period.

Effort-to-Yield Measures
An effort-to-yield measure indicates a relationship between a clinical input and a clinical outcome. The most common is probably the number needed to treat (NNT), which is the number of patients who are expected to be treated to achieve one more positive outcome. The NNT is calculated by dividing the absolute risk reduction into 1:
The number of men with prostate cancer who would have to
undergo radical prostatectomy to prevent a single death from
cancer is 16.1 (1 ÷ [0.136 – 0.074] = 16.1)
The second most common effort-to-yield measure is probably the complement to the NNT, the number needed to harm (NNH), which is the number of patients who are expected to be treated before one more negative outcome occurs.
The number of prostatectomies performed
for each additional case of erectile dysfunction is 3.

Natural Frequencies
A natural frequency is simply the number of people affected per unit of population. For example, the absolute risk of death with the two treatments for prostate cancer can also be expressed as natural frequencies:
Of 100 men with prostate cancer treated with resection, 7 will die.
Of 100 treated with watchful waiting, 11 will die.
This measure of risk seems to be the easiest to understand, as illustrated with data from the Women's Health Study (Table 1).
Table1 Table 1. The Health Effects of Hormone Replacement Therapy (HRT) in
Postmenopausal Women after 5 Years Reported as Natural Frequencies.
Natural frequencies seem to be the easiest measure of risk for most people to understand.

The Importance of How Risk is Reported
Reporting measures of risk is one thing; reporting measures that communicate risk to readers is another. All of the measures described here are common and acceptable, but each can give a different impression of how serious the risk is (Table 2). My research–h indicates that you should always report at least the absolute risk (and the size of the groups and the proportion experiencing the outcome of interest) because all other measures can be calculated from these data.
Table 2 Measures of the Risk of Death in Men with Prostate Cancer Treated with Watchful Waiting or Prostate Resection. Each measure is calculated from the same data and is a standard way to report risk, but each also gives a different impression of the risk of choosing either treatment.