Community hub
0 subscribers8 pages, 0 posts
Recent from talks
All channels
Be the first to start a discussion here.
Be the first to start a discussion here.
Be the first to start a discussion here.
Be the first to start a discussion here.
Contribute something
Welcome to the community hub built to collect knowledge and have discussions related to Risk difference.
Nothing was collected or created yet.
Risk difference
View on Wikipediafrom Wikipedia
The risk difference (RD), excess risk, or attributable risk[1] is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as , where is the incidence in the exposed group, and is the incidence in the unexposed group. If the risk of an outcome is increased by the exposure, the term absolute risk increase (ARI) is used, and computed as . Equivalently, if the risk of an outcome is decreased by the exposure, the term absolute risk reduction (ARR) is used, and computed as .[2][3]
The inverse of the absolute risk reduction is the number needed to treat, and the inverse of the absolute risk increase is the number needed to harm.[2]
Usage in reporting
[edit]It is recommended to use absolute measurements, such as risk difference, alongside the relative measurements, when presenting the results of randomized controlled trials.[4] Their utility can be illustrated by the following example of a hypothetical drug which reduces the risk of colon cancer from 1 case in 5000 to 1 case in 10,000 over one year. The relative risk reduction is 0.5 (50%), while the absolute risk reduction is 0.0001 (0.01%). The absolute risk reduction reflects the low probability of getting colon cancer in the first place, while reporting only relative risk reduction, would run into risk of readers exaggerating the effectiveness of the drug.[5]
Authors such as Ben Goldacre believe that the risk difference is best presented as a natural number - drug reduces 2 cases of colon cancer to 1 case if you treat 10,000 people. Natural numbers, which are used in the number needed to treat approach, are easily understood by non-experts.[6]
Inference
[edit]Risk difference can be estimated from a 2x2 contingency table:
| Group | ||
|---|---|---|
| Experimental (E) | Control (C) | |
| Events (E) | EE | CE |
| Non-events (N) | EN | CN |
The point estimate of the risk difference is
The sampling distribution of RD is approximately normal, with standard error
The confidence interval for the RD is then
where is the standard score for the chosen level of significance[3]
Bayesian interpretation
[edit]We could assume a disease noted by , and no disease noted by , exposure noted by , and no exposure noted by . The risk difference can be written as
Numerical examples
[edit]Risk reduction
[edit]| Quantity | Experimental group (E) | Control group (C) | Total |
|---|---|---|---|
| Events (E) | EE = 15 | CE = 100 | 115 |
| Non-events (N) | EN = 135 | CN = 150 | 285 |
| Total subjects (S) | ES = EE + EN = 150 | CS = CE + CN = 250 | 400 |
| Event rate (ER) | EER = EE / ES = 0.1, or 10% | CER = CE / CS = 0.4, or 40% | — |
| Variable | Abbr. | Formula | Value |
|---|---|---|---|
| Absolute risk reduction | ARR | CER − EER | 0.3, or 30% |
| Number needed to treat | NNT | 1 / (CER − EER) | 3.33 |
| Relative risk (risk ratio) | RR | EER / CER | 0.25 |
| Relative risk reduction | RRR | (CER − EER) / CER, or 1 − RR | 0.75, or 75% |
| Preventable fraction among the unexposed | PFu | (CER − EER) / CER | 0.75 |
| Odds ratio | OR | (EE / EN) / (CE / CN) | 0.167 |
Risk increase
[edit]| Quantity | Experimental group (E) | Control group (C) | Total |
|---|---|---|---|
| Events (E) | EE = 75 | CE = 100 | 175 |
| Non-events (N) | EN = 75 | CN = 150 | 225 |
| Total subjects (S) | ES = EE + EN = 150 | CS = CE + CN = 250 | 400 |
| Event rate (ER) | EER = EE / ES = 0.5, or 50% | CER = CE / CS = 0.4, or 40% | — |
| Variable | Abbr. | Formula | Value |
|---|---|---|---|
| Absolute risk increase | ARI | EER − CER | 0.1, or 10% |
| Number needed to harm | NNH | 1 / (EER − CER) | 10 |
| Relative risk (risk ratio) | RR | EER / CER | 1.25 |
| Relative risk increase | RRI | (EER − CER) / CER, or RR − 1 | 0.25, or 25% |
| Attributable fraction among the exposed | AFe | (EER − CER) / EER | 0.2 |
| Odds ratio | OR | (EE / EN) / (CE / CN) | 1.5 |
See also
[edit]References
[edit]- ^ Porta M, ed. (2014). Dictionary of Epidemiology (6th ed.). Oxford University Press. p. 14. doi:10.1093/acref/9780199976720.001.0001. ISBN 978-0-19-939006-9.
- ^ a b Porta, Miquel, ed. (2014). "Dictionary of Epidemiology - Oxford Reference". Oxford University Press. doi:10.1093/acref/9780199976720.001.0001. ISBN 9780199976720. Retrieved 2018-05-09.
- ^ a b J., Rothman, Kenneth (2012). Epidemiology : an introduction (2nd ed.). New York, NY: Oxford University Press. pp. 66, 160, 167. ISBN 9780199754557. OCLC 750986180.
{{cite book}}: CS1 maint: multiple names: authors list (link) - ^ Moher D, Hopewell S, Schulz KF, Montori V, Gøtzsche PC, Devereaux PJ, Elbourne D, Egger M, Altman DG (March 2010). "CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials". BMJ. 340: c869. doi:10.1136/bmj.c869. PMC 2844943. PMID 20332511.
- ^ Stegenga, Jacob (2015). "Measuring Effectiveness". Studies in History and Philosophy of Biological and Biomedical Sciences. 54: 62–71. doi:10.1016/j.shpsc.2015.06.003. PMID 26199055.
- ^ Ben Goldacre (2008). Bad Science. New York: Fourth Estate. pp. 239–260. ISBN 978-0-00-724019-7.
Risk difference
View on Grokipediafrom Grokipedia
Fundamentals
Definition
The risk difference (RD), also known as the attributable risk, is a fundamental measure in epidemiology that quantifies the absolute difference in the probability of an adverse event occurring between two groups, typically an exposed group (e.g., those subjected to a risk factor) and an unexposed group (e.g., a control or reference group).[1] This metric captures the excess risk directly attributable to the exposure, expressed on an absolute scale rather than a proportional one.[4] In formal terms, the risk difference is calculated as the arithmetic difference in event probabilities: where denotes the risk, or cumulative incidence, of the event in the specified group.[1] Here, cumulative incidence refers to the proportion of individuals in a defined population who experience the event (such as disease onset or death) over a fixed time period, assuming the population is initially free of the event.[5] This definition presupposes basic probability concepts, where risks are estimated from cohort or cross-sectional data as proportions between 0 and 1.[4] Unlike relative measures of association, such as the risk ratio (which compares risks multiplicatively) or the odds ratio (which compares odds of the event), the RD emphasizes the practical magnitude of the effect by highlighting the actual change in event probability.[1] For instance, a RD of 0.10 indicates that the exposure increases the event risk by 10 percentage points, regardless of baseline risk levels, whereas relative measures might overstate or understate impact depending on the unexposed risk.[1] This absolute focus makes the RD particularly valuable for public health decision-making, as it directly informs the scale of harm or benefit associated with an exposure.[4] The concept of risk difference originated in epidemiology during the mid-20th century, amid growing interest in quantifying exposure-disease associations in cohort studies.[6] Early applications appeared in investigations of major public health issues, notably the pioneering work by Richard Doll and Austin Bradford Hill in the 1950s, who examined smoking as a risk factor for lung cancer using British physician cohorts; their analyses helped establish attributable risk concepts in modern epidemiological practice.[6] [7] The RD is inversely related to the number needed to treat (NNT), a clinical metric where, for beneficial effects, NNT equals 1 divided by the RD (or absolute risk reduction).[8]Calculation
The risk difference (RD) is computed from a 2×2 contingency table that categorizes study participants by exposure status and outcome occurrence.[9] The table is structured as follows, where rows represent exposure groups and columns represent outcome status:| Outcome Present | Outcome Absent | Total | |
|---|---|---|---|
| Exposed | |||
| Unexposed | |||
| Total |
Interpretation
Absolute Risk Measures
The risk difference (RD), also known as absolute risk reduction or excess risk, quantifies the absolute change in the probability of an event occurring between two groups, such as exposed and unexposed populations, providing a direct measure of effect magnitude in additive terms.[13] Unlike relative risk, which expresses the effect on a multiplicative scale as a ratio of probabilities, RD captures the actual difference in event occurrence, such as a shift from 5% to 10% risk representing an RD of 5%.[15] This absolute perspective is particularly valuable for understanding the tangible impact of an exposure or intervention without distortion from baseline risk levels.[4] The direction of the RD indicates the nature of the association: a positive value signifies an increased risk in the exposed group compared to the unexposed, while a negative value denotes a reduction in risk, often interpreted as a protective effect.[1] RD is closely related to attributable risk, which represents the excess risk directly due to the exposure, and can be extended to population-level metrics. For instance, the population attributable risk (PAR), which estimates the excess risk in the population attributable to the exposure, is calculated as the product of the RD and the prevalence of exposure in the population: where is the proportion exposed.[16] Compared to relative measures, RD offers advantages in interpretability, especially for rare events or when baseline risks vary substantially, as relative risks can exaggerate effects and mislead clinical or public health decisions by overemphasizing proportional changes over actual differences.[13] Critiques from the 1990s highlighted this overemphasis on relative risk, advocating for absolute measures like RD to better reflect public health impact and avoid undue alarm from inflated ratios in low-incidence scenarios.[17] For beneficial effects, RD also relates inversely to the number needed to treat (NNT), computed as 1 divided by the absolute value of RD.[13]Clinical Relevance
In clinical practice, the risk difference (RD) plays a key role in patient decision aids by providing a clear, absolute measure of treatment benefits or harms, facilitating informed counseling. For instance, clinicians can explain to patients that a particular intervention reduces the absolute risk of an adverse event from 10% to 7%, corresponding to an RD of 3 percentage points, which helps patients weigh options based on their personal circumstances rather than relative percentages that may exaggerate effects.[18] This approach enhances shared decision-making, as evidenced by systematic reviews showing that decision aids incorporating absolute risks improve patient knowledge and reduce decisional conflict without increasing anxiety.[19] Such tools, often presented via visual aids like icon arrays or simple statements, promote autonomy by grounding discussions in tangible probabilities tailored to individual baseline risks.[20] In public health policy, RD informs the evaluation and prioritization of interventions by quantifying the population-level impact of preventive measures, as seen in guidelines assessing disease burden and intervention efficacy. Organizations such as the World Health Organization employ metrics based on attributable risks, including population attributable fractions derived from risk differences, in reports on environmental and lifestyle interventions to estimate excess disease burden and guide resource allocation for broad-scale programs.[21] A notable example is in vaccine efficacy trials post-2000, where RD highlights absolute benefits in diverse populations; for COVID-19 vaccines, phase 3 trials reported RDs of approximately 0.7-1.2 percentage points for preventing symptomatic infection, underscoring modest but critical public health gains during low-prevalence periods.[22] This metric supports policy decisions, such as vaccination mandates, by emphasizing real-world reductions in incidence rather than solely relative efficacy. Despite its clarity, RD communication can lead to misinterpretation if presented without context on baseline risks, potentially understating or overstating benefits in varying populations. The CONSORT 2010 guidelines, updated in the 2010s, recommend reporting RD alongside relative measures (e.g., risk ratios) for binary outcomes in trial reports to provide comprehensive effect estimates and avoid misleading inferences about clinical importance.[23] For example, an RD of 5% might seem substantial in high-baseline-risk groups but trivial in low-risk settings, necessitating explicit baseline details to prevent cognitive biases in interpretation.[24] Ethically, using RD in low-risk populations helps mitigate overstatement of benefits or harms, aligning with principles of non-maleficence and respect for autonomy by ensuring realistic expectations. In scenarios like cancer chemoprevention trials, where baseline event risks are often below 1%, emphasizing RD prevents undue alarm or false reassurance from relative measures, which could otherwise inflate perceived intervention value and erode trust.[25] This transparent approach, as discussed in ethical analyses of screening and preventive care, supports equitable decision-making by highlighting that small RDs may not justify risks for individuals with minimal baseline probability.[26] For intuitive understanding, RD can be reframed as the number needed to treat (NNT = 1/RD), offering a patient-friendly metric for weighing personal trade-offs.[27]Statistical Analysis
Frequentist Inference
In frequentist inference for the risk difference (RD), point estimation begins with the unbiased estimator , where and represent the sample proportions of events in the exposed and unexposed groups, respectively, with denoting the number of events and the sample size in group .[28] The variance of this estimator arises from the binomial nature of the proportions, leading to the standard error , which quantifies the sampling variability under the assumption of independent binomial outcomes. This standard error forms the basis for subsequent inferential procedures, enabling assessment of precision in the estimate. Confidence intervals provide a range of plausible values for the true RD based on the sampling distribution. The standard 95% Wald confidence interval is constructed as , relying on the asymptotic normality of . However, this interval can exhibit poor coverage probabilities in small samples or when proportions are near 0 or 1, prompting the use of alternatives such as Newcombe's method, which employs score intervals for individual proportions adjusted for their correlation to achieve better performance. Hypothesis testing typically evaluates the null hypothesis against a two-sided alternative, using the test statistic , which follows a standard normal distribution under for large samples. The p-value is then obtained from the cumulative distribution function of the standard normal, with rejection of at if . For study planning, power analysis determines the sample size required to detect a meaningful RD of size with power . Assuming equal group sizes and known planning values and , the formula per group is where and are the critical values from the standard normal distribution. These methods assume independent observations across groups and rely on large-sample approximations for the central limit theorem to hold, ensuring the normality of the sampling distribution. When data involve clustering or stratification (e.g., due to confounding factors), adjustments such as the Mantel-Haenszel estimator pool stratum-specific RDs via inverse-variance weighting, providing a summary estimate that accounts for the layered structure while maintaining asymptotic efficiency.[29] Unlike Bayesian approaches that incorporate prior information to update beliefs, frequentist inference focuses solely on long-run frequency properties of the estimator under repeated sampling.Bayesian Inference
In the Bayesian framework for the risk difference (RD), the probabilities of events in the two groups, denoted and , are modeled as parameters governing independent binomial likelihoods for the observed counts of successes out of sample sizes and . Conjugate beta priors are specified for and , enabling closed-form posterior updates; a common non-informative choice is the uniform prior, which assumes no prior preference across the [0,1] interval for probabilities.[30] The posteriors for and follow updated beta distributions: and similarly for , where and are the observed successes. For the RD = , the posterior lacks a simple closed form due to the difference of dependent betas, so inference typically relies on simulation methods like Markov chain Monte Carlo (MCMC) to draw samples from the joint posterior, from which the posterior mean or median of RD and credible intervals (e.g., 95% equal-tailed) are derived directly from the sample quantiles.[30] Prior elicitation plays a central role, with skeptical priors such as (the Jeffreys prior) providing weak information equivalent to half an observation while shrinking extreme estimates, or informative priors derived from meta-analyses of prior studies to incorporate external evidence on expected risks. These priors are updated via the conjugate rule to yield the posteriors, allowing flexible incorporation of domain knowledge. Bayesian methods for RD offer advantages in handling small or sparse samples by leveraging priors to regularize estimates and avoid unstable frequentist approaches, particularly for rare events where zero counts are common. They enable direct probability statements, such as , quantifying the posterior probability of a positive effect. Implementation is facilitated by post-2010 software developments, including the R package brms, which fits binomial models with beta priors via Stan's MCMC engine and supports derived RD inference through posterior sampling.[30]Examples and Applications
Numerical Illustrations
To illustrate the risk difference (RD), consider a hypothetical cohort study with 100 exposed individuals, among whom 20 events occur, and 100 unexposed individuals, among whom 10 events occur. The risk in the exposed group is 20/100 = 0.20, and the risk in the unexposed group is 10/100 = 0.10, yielding an RD of 0.20 - 0.10 = 0.10, or 10%. This indicates that exposure increases the risk of the event by 10 percentage points.[1]| Group | Events | No Events | Total | Risk |
|---|---|---|---|---|
| Exposed | 20 | 80 | 100 | 0.20 |
| Unexposed | 10 | 90 | 100 | 0.10 |