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Measures of Risk

Epidemiologic data obtained by clinical studies allow researchers to determine the likelihood of developing a certain outcome of interest within a studied population. This likelihood, or risk, can be quantified through what are known as measures of risk, which are mathematical formulas derived from contingency tables. These measures of risk include absolute risk ( AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation), relative risk (RR), attributable risk, and odds ratios (OR), each of which offers different kinds of information according to the needs of the researchers.

Last updated: 19 May, 2022

Editorial responsibility: Stanley Oiseth, Lindsay Jones, Evelin Maza

Contingency Tables

Definition

A contingency table lists the frequency distributions of variables from a study and is a convenient way to look at any relationships between variables.

Table structure

  • A 2×2 grid comparing whether 2 different variables are associated with each other 
  • Each box in the table indicates the number of people in the study who have that specific combination of variables and is labeled with a letter A–D. 
  • Formulas to calculate different measures of risk use these letters.
  • In order for the formulas to work, the tables need to be set up the same way each time; for clinical trials:
    • The rows refer to whether or not a patient was exposed to the risk factor being tested (e.g., smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases)
    • The columns refer to whether or not a patient developed the outcome being studied (e.g., lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer)
    • The “Yes” answer comes 1st, the “No” answer comes 2nd.
  • Letter labels Labels Analogs of those substrates or compounds which bind naturally at the active sites of proteins, enzymes, antibodies, steroids, or physiological receptors. These analogs form a stable covalent bond at the binding site, thereby acting as inhibitors of the proteins or steroids. Immunoassays:
    • A = A patient was exposed to the risk factor and developed the outcome (e.g., a smoker develops lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer).
    • B = A patient was exposed to the risk factor but did not develop the outcome (e.g., a smoker does not develop lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer).
    • C = A patient was not exposed to the risk factor but developed the outcome anyway (e.g., nonsmoker develops lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer).
    • D = A patient was not exposed to the risk factor and did not develop the outcome (e.g., nonsmoker does not develop lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer).
  • Totals:
    • Each row and column has subtotals called marginal totals in the far right column and bottom row.
    • N is the total number of people in the population set; this grand total is reported in the bottom right box
  • Allows for the rapid calculation of several measures of association and risk

Example

Below is an example of a 2×2 contingency table. The cells show the frequencies of distribution (A, B, C, D) for different combinations of the two variables (outcome, exposure), for a population of size N.

Contingency table

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Absolute Risk

Definition

The AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation is the risk of developing a disease or condition after an exposure. 

  • The AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation is also the cumulative incidence rate Incidence rate (Number of new cases of disease occurring in the population during a specified period of time)/(Total person-time, or the sum of the time periods of observation of each person who was observed) X 1,000. Measures of Disease Frequency (I) of exposed (or unexposed) patients Patients Individuals participating in the health care system for the purpose of receiving therapeutic, diagnostic, or preventive procedures. Clinician–Patient Relationship.
  • Note: Both absolute risk and attributable risk (discussed below) are frequently abbreviated as AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation, which is why absolute risk is often written as incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency (I) instead.

Calculations of Absolute Risk

The AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation is calculated as the number of people who have a particular outcome divided by the total number of people with the same exposure (or the same nonexposure). This risk can be calculated for both exposed and unexposed populations.

Steps:

Start by setting up a contingency table:

Contingency table

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Using the contingency table, the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation in the exposed group is calculated as:

$$ Absolute\ risk\ of\ the\ exposed\ group = \frac{A}{A + B} $$

where A = a patient was exposed to the risk factor and developed the outcome and B = a patient was exposed to the risk factor but did not develop the outcome.

Using the contingency table, the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation in the unexposed group is calculated as:

$$ Absolute\ risk\ of\ the\ unexposed\ group = \frac{C}{C + D} $$

where C = a patient who was not exposed to the risk factor but developed the outcome anyway and D = a patient who was not exposed to the risk factor and did not develop the outcome.

Example of calculating AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation

Example 1: In a population of 100 smokers, 75 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 25 did not. What is the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation of developing lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer if you are a smoker?

  • This question is asking about the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation in the exposed group
  • Set up a contingency table with the exposure ( smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases) on the vertical axis Axis The second cervical vertebra. Vertebral Column: Anatomy and the outcome ( lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer) on the horizontal axis Axis The second cervical vertebra. Vertebral Column: Anatomy (see below)
  • Answer: AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation = A / (A + B) = 75 / (75 + 25) = 75 / 100 = 0.75

Example 2: In a population of 100 nonsmokers, 10 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 90 did not. What is the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation of developing lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer if you are not a smoker?

  • This question is asking about the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation in the unexposed group
  • Set up a contingency table with the exposure ( smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases) on the vertical axis Axis The second cervical vertebra. Vertebral Column: Anatomy and the outcome ( lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer) on the horizontal axis Axis The second cervical vertebra. Vertebral Column: Anatomy (see below)
  • Answer: AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation = C / (C + D) = 10 / (10 + 90) = 10 / 100 = 0.1
Contingency table lung cancer

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Absolute risk reduction (ARR)/absolute risk increase (ARI)

ARR or ARI is a measure of the reduction or increase in risk of developing a disease or condition as the result of an exposure.

Other ways to conceptualize ARR:

  • The difference in AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation between exposed and unexposed groups 
  • The difference in incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency rates
  • The difference between the percentage of subjects who have a condition in both exposed and unexposed groups.

ARR can be interpreted as the health “gained” or ”lost” as a result of the exposure. For example, if you don’t smoke, by what percent can you reduce your risk of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer?

The ARR between the exposed and unexposed groups can be calculated as:

$$ ARR = I_{Exposed} – I_{Unexposed} $$

where I = incidence rate Incidence rate (Number of new cases of disease occurring in the population during a specified period of time)/(Total person-time, or the sum of the time periods of observation of each person who was observed) X 1,000. Measures of Disease Frequency. Because I is the same as AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation, this formula can be calculated from a contingency table:

$$ ARR = \frac{A}{A + B} – \frac{C}{C + D} $$

Example of calculating ARR

In a population of 100 smokers, 75 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 25 did not. In a population of 100 nonsmokers, 10 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 90 did not. By how much does not smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases reduce the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation of developing lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer?

  • This question is asking about the ARR achieved by avoidance of the exposure ( smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases).
  • Set up a contingency table with the exposure ( smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases) on the vertical axis Axis The second cervical vertebra. Vertebral Column: Anatomy and the outcome ( lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer) on the horizontal axis Axis The second cervical vertebra. Vertebral Column: Anatomy (see below).
  • Answer: 
    • AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic RegurgitationExposed = A / (A + B) = 75 / (75 + 25) = 75 / 100 = 0.75
    • AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic RegurgitationUnexposed = C / (C + D) = 10 / (10 + 90) = 10 / 100 = 0.1
    • ARR = IExposed ‒ IUnexposed = AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic RegurgitationExposed AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic RegurgitationUnexposed = 0.75 ‒ 0.1 = 0.65
  • Interpretation: Based on this data set, not smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases reduces a person’s absolute risk of developing lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer by 65%.
Contingency table lung cancer

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Number needed to treat Number needed to treat The number needed to treat (NNT) is the number of patients that are needed to treat to prevent 1 additional adverse outcome (e.g., stroke, death). For example, if a drug has an NNT of 10, it means 10 people must be treated with the drug to prevent 1 additional adverse outcome. Number Needed to Treat/harm

These are numbers typically reported when testing new therapeutic options. In these cases:

  • The “exposure” is the new medication/procedure.
  • The “outcome” is either the benefit of the procedure or a potential adverse effect.

Number needed to treat Number needed to treat The number needed to treat (NNT) is the number of patients that are needed to treat to prevent 1 additional adverse outcome (e.g., stroke, death). For example, if a drug has an NNT of 10, it means 10 people must be treated with the drug to prevent 1 additional adverse outcome. Number Needed to Treat ( NNT NNT The number needed to treat (NNT) is the number of patients that are needed to treat to prevent 1 additional adverse outcome (e.g., stroke, death). For example, if a drug has an nnt of 10, it means 10 people must be treated with the drug to prevent 1 additional adverse outcome. Number Needed to Treat):

  • Represents the number of patients Patients Individuals participating in the health care system for the purpose of receiving therapeutic, diagnostic, or preventive procedures. Clinician–Patient Relationship who would need to be treated to achieve 1 additional case of the outcome 
  • NNT NNT The number needed to treat (NNT) is the number of patients that are needed to treat to prevent 1 additional adverse outcome (e.g., stroke, death). For example, if a drug has an nnt of 10, it means 10 people must be treated with the drug to prevent 1 additional adverse outcome. Number Needed to Treat = 1 / ARR (i.e., the inverse of the ARR)

Number needed to harm Number needed to harm The NNH is the additional number of individuals who need to be exposed to risk (harmful exposure or treatment) to have 1 extra person develop the disease compared to that in the unexposed group. Number Needed to Treat ( NNH NNH The NNH is the additional number of individuals who need to be exposed to risk (harmful exposure or treatment) to have 1 extra person develop the disease compared to that in the unexposed group. Number Needed to Treat):

  • Typically used when reporting on experimental treatments with potential adverse effects
  • Represents the number of patients Patients Individuals participating in the health care system for the purpose of receiving therapeutic, diagnostic, or preventive procedures. Clinician–Patient Relationship who would need to be treated to achieve the outcome (the harmful effect) 
  • NNH NNH The NNH is the additional number of individuals who need to be exposed to risk (harmful exposure or treatment) to have 1 extra person develop the disease compared to that in the unexposed group. Number Needed to Treat = 1 / ARI (i.e., the inverse of the ARI)

Relative Risk

Definition

Relative risk (RR) is the risk of a disease or condition occurring in a group or population with a particular exposure relative to a control (unexposed) group.  

  • Can be equivalently stated as the risk of developing the disease after an exposure compared to the risk of developing the disease without exposure.
  • RR shows how strongly exposure is associated with the risk of the disease.

Calculations of RR

RR is typically among the most important numbers calculated. Cohort studies Cohort studies Studies in which subsets of a defined population are identified. These groups may or may not be exposed to factors hypothesized to influence the probability of the occurrence of a particular disease or other outcome. Cohorts are defined populations which, as a whole, are followed in an attempt to determine distinguishing subgroup characteristics. Epidemiological Studies are the only type of observational study that can determine the RR.

The relative risk is calculated as the frequency of a disease or condition in the exposed group (IE) divided by the frequency in an unexposed control group (IO), which is represented by the formula:

$$ RR = \frac{I_{E}}{I_{O}} $$

Again, since the incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency rates are the same as the AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation, the RR can be calculated from a contingency table by using the following expanded formula:

$$ RR = \frac{\frac{A}{A + B}}{\frac{C}{C + D}} $$

Interpretation of RR

  • RR = 1: The risk of the outcome for the exposed group and the unexposed group is the same.
  • RR > 1: The risk of the exposed group is greater than the risk of the unexposed group; evidence of positive association/possible causal factor.
  • RR < 1: The risk of the exposed group is lower than the risk of the unexposed group; evidence of negative association/possible protective factor.

Example of calculating RR

In a population of 100 smokers, 75 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 25 did not. In a population of 100 nonsmokers, 10 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 90 did not. What is the risk of getting lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer if you smoke compared to the risk of getting lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer if you do not smoke?

  • This question is asking about the RR of getting lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer
  • Set up a contingency table (see below)
  • Answer: 
    • RR = IE ÷ IO = AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic RegurgitationExposed ÷ AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic RegurgitationUnexposed = [ A / (A + B) ] ÷ [ C / (C + D) ]
    • RR = 0.75 ÷ 0.1 = 7.5
  • Interpretation: Based on this data sample, smokers are 7.5 times more likely to develop lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer than nonsmokers.
Contingency table lung cancer

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Relative risk reduction (RRR)/relative risk increase (RRI)

Definition:

  • The RRR is defined as the reduction (or increase) in risk of a particular outcome, in a group with a known exposure, relative to the unexposed control group.
  • Put another way: RRR is the proportion of baseline risk that is reduced through nonexposure.
  • Example: If people don’t smoke, how much less lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer can you expect?

Calculation:

The RRR is calculated as the difference between the incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency of a disease in an exposed (IE) and an unexposed (IO ) group divided by the incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency in the unexposed group, which is calculated with the following formula:

$$ RRR = \frac{ARR}{I_{O}} = (\frac{I_{E} – I_{O}}{I_{O}}) = \frac{\frac{A}{A + B} – \frac{C}{C + D}}{\frac{C}{C + D}} $$

Example of calculating RRR

In a population of 100 smokers, 75 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 25 did not after 10 years in a cohort study. In a population of 100 nonsmokers, 10 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 90 did not. If people didn’t smoke, how much less lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer can be expected in the population?

  • This question is asking about the RRR achieved by not smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases.
  • Set up a contingency table (see below).
  • Answer: RRR = ARR ÷ IO = 0.65 ÷ 0.1 = 6.5
  • Interpretation: Based on this data set, avoiding smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases reduces the RR of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer by 650%.
Contingency table lung cancer

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Attributable Risk

Definition

The attributable risk is a measure of the risk of developing an outcome associated with a particular exposure.  

  • That is, how much of the outcome can we contribute to the behavior?
  • Attributable risk is sometimes called the excess risk because it represents how much increase in risk a particular behavior will add to the baseline risk of developing a particular outcome. 
  • The formula for attributable risk is similar to that for ARR, but attributable risk is used in epidemiologic studies.
  • 2 kinds of attributable risk:
    • Attributable risk in the exposed group 
    • Population attributable risk (PAR)
  • Note: Both absolute risk and attributable risk are frequently abbreviated as AR AR Aortic regurgitation (AR) is a cardiac condition characterized by the backflow of blood from the aorta to the left ventricle during diastole. Aortic regurgitation is associated with an abnormal aortic valve and/or aortic root stemming from multiple causes, commonly rheumatic heart disease as well as congenital and degenerative valvular disorders. Aortic Regurgitation, which is why absolute risk is often described as incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency (I) instead.

Attributable risk in the exposed group

The attributable risk in the exposed group is the difference in the rate of a disease between the exposed and the unexposed groups. For example, what percentage of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer cases are likely due to smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases?
The attributable risk is calculated by subtracting the incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency in the unexposed group (IO) from the incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency of the exposed group (IE) and dividing by the incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency in the exposed group, which is expressed as:

$$ Attributable\ risk\ in\ the\ exposed\ group = \frac{(I_{E} – I_{O})}{I_{E}} = \frac{\frac{A}{A + B} – \frac{C}{C + D}}{\frac{A}{A + B}} = (\frac{(RR – 1)}{RR}) $$

Example: In a population of 100 smokers, 75 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 25 did not after 10 years in a cohort study. In a population of 100 nonsmokers, 10 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 90 did not. What percentage of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer cases are likely due to smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases?

  • This question is asking about the attributable risk in the exposed group.
  • Answer: (0.75 – 0.1 ) ÷ 0.75 = 0.65 ÷ 0.75 = 0.867
  • Interpretation: 86.7% of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer cases are due to smoking Smoking Willful or deliberate act of inhaling and exhaling smoke from burning substances or agents held by hand. Interstitial Lung Diseases.
Contingency table lung cancer

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Population attributable risk (PAR)

The PAR is the attributable risk for an entire population. It represents the fraction of cases that would not occur in a population if the exposure was eliminated.

For example, what percentage of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer cases could be prevented if nobody smoked?

The PAR is calculated by subtracting the incidence rate Incidence rate (Number of new cases of disease occurring in the population during a specified period of time)/(Total person-time, or the sum of the time periods of observation of each person who was observed) X 1,000. Measures of Disease Frequency in the unexposed population from the incidence rate Incidence rate (Number of new cases of disease occurring in the population during a specified period of time)/(Total person-time, or the sum of the time periods of observation of each person who was observed) X 1,000. Measures of Disease Frequency in the entire population:

$$ PAR = \frac{(I_{T} – I_{O})}{I_{T}} = \frac{(\frac{A + C}{N} – \frac{C}{C + D})}{\frac{A + C}{N}} $$

Example: In a population of 100 smokers, 75 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 25 did not after 10 years in a cohort study. In a population of 100 nonsmokers, 10 developed lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer and 90 did not. What percentage of lung cancer Lung cancer Lung cancer is the malignant transformation of lung tissue and the leading cause of cancer-related deaths. The majority of cases are associated with long-term smoking. The disease is generally classified histologically as either small cell lung cancer or non-small cell lung cancer. Symptoms include cough, dyspnea, weight loss, and chest discomfort. Lung Cancer cases could be prevented if nobody smoked?

  • This question is asking about the population attributable risk.
  • Answer: 76.5% of lung cancers could be prevented if no one smoked
$$ PAR = \frac{(\frac{85}{200} – \frac{10}{100})}{\frac{85}{200}} = \frac{(0.425 – 0.1)}{0.425} = 0.765 $$
Contingency table lung cancer n = 200

Contingency table
N = total number of people in set population

Image by Lecturio. License: CC BY-NC-SA 4.0

Odds Ratio

Definition

An odds ratio (OR) is a statistic that quantifies the strength of association between 2 variables or events. 

  • OR calculates the odds of one variable Variable Variables represent information about something that can change. The design of the measurement scales, or of the methods for obtaining information, will determine the data gathered and the characteristics of that data. As a result, a variable can be qualitative or quantitative, and may be further classified into subgroups. Types of Variables (A) in the presence or absence of another variable Variable Variables represent information about something that can change. The design of the measurement scales, or of the methods for obtaining information, will determine the data gathered and the characteristics of that data. As a result, a variable can be qualitative or quantitative, and may be further classified into subgroups. Types of Variables (B). 
    • Odds are the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a thing happening divided by the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of that thing not happening.
    • For example, the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of “heads” on a coin toss is 50%; the odds are 1 (50% ÷ 50%).
  • In clinical studies, OR measures the association between an exposure and an outcome.
  • Does not imply causation
  • OR can be used to estimate the RR when incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency rates cannot be calculated, such as in case–control studies.
  • Used when diseases are rare (typically when the prevalence Prevalence The total number of cases of a given disease in a specified population at a designated time. It is differentiated from incidence, which refers to the number of new cases in the population at a given time. Measures of Disease Frequency is ≤ about 10%)
  • Also used in more complicated statistical analyses:
    • Can be thought of as a type of RR
    • Used to determine risk factors in studies

Calculations of the OR

An OR is used as an estimation of RR in case–control studies. OR is calculated by determining the odds of exposure among the diseased divided by the odds of exposure among the undiseased. This is represented as:

$$ OR = \frac{(Odds\ of\ exposure\ among\ diseased)}{(Odds\ of\ exposure\ among\ undiseased)} = \frac{A \div C}{B \div D} $$

where (A ÷ C) represents the number of exposed cases divided by the number of unexposed cases among those with the disease and (B ÷ D) is the number of exposed undiseased divided by the number of unexposed undiseased. 

Rearranging the formula gives the simplified equation:

$$ OR = ( AD AD The term advance directive (AD) refers to treatment preferences and/or the designation of a surrogate decision-maker in the event that a person becomes unable to make medical decisions on their own behalf. Advance directives represent the ethical principle of autonomy and may take the form of a living will, health care proxy, durable power of attorney for health care, and/or a physician’s order for life-sustaining treatment. Advance Directives) \div (BC) $$

Interpreting the OR

OR is interpreted in the same way as RR:

  • OR = 1: 
    • The risk of the outcome for the exposed group and the unexposed group is the same.
    • No association between the exposure and outcome
  • OR > 1: 
    • The risk in the exposed group is greater than the risk of the unexposed group. 
    • Evidence of a positive association/possible causal factor
  • OR < 1:
    • The risk in the exposed group is lower than the risk in the unexposed group. 
    • Evidence of a negative association/possible protective factor

Example

In this example, 6 people developed Creutzfeldt-Jakob disease Creutzfeldt-Jakob Disease A rare transmissible encephalopathy most prevalent between the ages of 50 and 70 years. Affected individuals may present with sleep disturbances, personality changes, ataxia; aphasia, visual loss, weakness, muscle atrophy, myoclonus, progressive dementia, and death within one year of disease onset. A familial form exhibiting autosomal dominant inheritance and a new variant cjd (potentially associated with bovine spongiform encephalopathy) have been described. Pathological features include prominent cerebellar and cerebral cortical spongiform degeneration and the presence of prions. Transmissible Spongiform Encephalopathies (CJD); 3 of them ate a significant amount of beef and 3 of them did not. These patients Patients Individuals participating in the health care system for the purpose of receiving therapeutic, diagnostic, or preventive procedures. Clinician–Patient Relationship were compared to a control population in a case–control study; in the control population of 10 people, 4 of them ate a significant amount of beef and 6 of them did not. What are the odds of developing CJD after eating a significant amount of beef?

  • Answer: OR = ( AD AD The term advance directive (AD) refers to treatment preferences and/or the designation of a surrogate decision-maker in the event that a person becomes unable to make medical decisions on their own behalf. Advance directives represent the ethical principle of autonomy and may take the form of a living will, health care proxy, durable power of attorney for health care, and/or a physician’s order for life-sustaining treatment. Advance Directives) ÷ (BC) = (3 x 6) ÷ (3 x 4) = 18 ÷ 12 = 1.5  
  • Interpretation: The odds of developing CJD are 50% higher in the group that ate beef, supporting the idea that eating beef is a risk factor for the development of CJD.
Contingency table for eating beef and creutzfeldt-jakob

Contingency table
N = total number of people in set population
Note: These are hypothetical data; the true incidence Incidence The number of new cases of a given disease during a given period in a specified population. It also is used for the rate at which new events occur in a defined population. It is differentiated from prevalence, which refers to all cases in the population at a given time. Measures of Disease Frequency rates of Creutzfeldt-Jakob disease Creutzfeldt-Jakob Disease A rare transmissible encephalopathy most prevalent between the ages of 50 and 70 years. Affected individuals may present with sleep disturbances, personality changes, ataxia; aphasia, visual loss, weakness, muscle atrophy, myoclonus, progressive dementia, and death within one year of disease onset. A familial form exhibiting autosomal dominant inheritance and a new variant cjd (potentially associated with bovine spongiform encephalopathy) have been described. Pathological features include prominent cerebellar and cerebral cortical spongiform degeneration and the presence of prions. Transmissible Spongiform Encephalopathies are much lower than these data suggest.

Image by Lecturio. License: CC BY-NC-SA 4.0

References

  1. Celentano, D., Szklo, M. (2019). Estimating risk: is there an association? Gordis Epidemiology (pp. 240–258).
  2. Szumilas M. (2010). Explaining odds ratios. Journal of the Canadian Academy of Child and Adolescent Psychiatry 19:227–229.
  3. Beaglehole, R. (2006). Basic epidemiology. Geneva: WHO.
  4. Bhopal, R. S. (2016). Concepts of Epidemiology. Oxford University Press. https://global.oup.com/academic/product/concepts-of-epidemiology-9780198739685?cc=de&lang=en&
  5. Edwards, A. W. F. (1963). The measure of association in a 2×2 table. Journal of the Royal Statistical Society A (General) 126:109–114.
  6. Morris, A., Gardner, M. (1988). Calculating confidence intervals for relative risks (odds ratios) and standardised ratios and rates. British Medical Journal Clinical Research 296: 1313–1316.
  7. Zhang, J., Yu, K. (1998). What’s the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. 280:1690–1691.

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