Comparing RR, PR, OR and POR - SHS - PDF

Comparing RR, PR, OR and POR (1/2) Comments • The names “POR” and “OR” are used in cross-sectional and case-control studies, respectively. However, they are numerically identical and can be treated equally when modeling data. • In case-control studies, neither the RR nor the PR can be estimated, and...

Comparing RR, PR, OR and POR (1/2) Comments • The names “POR” and “OR” are used in cross-sectional and case-control studies, respectively. However, they are numerically identical and can be treated equally when modeling data. • In case-control studies, neither the RR nor the PR can be estimated, and we must use the OR. • In cross-sectional studies, both POR and PR can be estimated. However, PR is preferred because it is easier to interpret. • All relative measures of association, RR, PR and OR, consider the exposed group in the numerator. Hence: • If the relative measure > 1, then the exposure E is a potential risk factor for the disease D. • If the relative measure < 1, then the exposure E is a potential protective factor for the disease D. • If the relative measure = 1, then the exposure E and the disease D are independent. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 18 / 32 Comparing RR, PR, OR and POR (2/2) RR and PR: equal but different As seen before, if both a cohort study and a cross-sectional study provided identical 2 ⇥ 2 contingency tables at the end of the study, the values of RR for the former and PR for the latter would be identical. However, the interpretation would be different. • In a cohort study, the risk ratio (RR) compares the probability of developing the disease during the follow-up between the two groups of exposure. However, it could result in a biased estimation if the follow-up time is not similar for all individuals. • In cross-sectional studies, the prevalence ratio (PR) compares the probability of having the disease between the two groups of exposure, which is useless as an indicator of a causal relationship between the exposure and the disease because we cannot guarantee that the exposure preceded the disease. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 19 / 32 RR vs OR (1/3) RR and OR use different metrics RR and OR compare risks between exposed and non exposed but using different metrics. For instance, in the previous example, RR = 1.50 while OR = 1.71. 2.0 If we denote ⇡0 := P(D|Ē), prove that: OR 1 ⇡0 • = RR 1 ⇡0 RR • RR > 1 ! OR > RR • RR < 1 ! OR < RR • RR = 1 ! OR = 1 OR Exercise π0 = P(D |E) 1.0 0 0.001 0.01 0.02 0.05 0.1 0.25 0.5 0.5 1.0 2.0 RR Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 20 / 32 RR vs OR (2/3) RR vs OR: Error size when reporting OR as if it was RR Error(%) = OR RR RR ⇥ 100% = ⇡0 (OR 1) ⇥ 100% 20 15 Error (%) 10 5 0 π0 = P(D |E) −5 0.001 0.01 0.02 0.05 0.1 0.25 −10 −15 −20 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 OR Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 21 / 32 RR vs OR (3/3) Using (carefully) OR as an approximation for RR • As seen, the lower P(D|E) and P(D|Ē), the closer RR and OR. Hence, for rare diseases, the values of RR and OR are similar. • Usually, rare diseases are analyzed using case-control ( have previously seen, OR ⇡ RR. Why?). In those cases, as we • In the logistic regression model, which is used to model a binary outcome, the coefficients have a straightforward interpretation in terms of OR. However, such coefficients cannot be interpreted in terms of RR. Hence, despite RR is easier to interpret than OR, we are forced to use OR to interpret risks for a binary outcome when it is modeled with a logistic regression model, even in the case of cross-sectional data (i.e. modeling a prevalence). In R • RR and OR can be calculated with riskratio and oddsratio, respectively, in the epitools package. • The epi.2by2 function in the epiR provides a detailed analysis of the 2⇥2 contingency table, taking into account the study design. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 22 / 32 Confidence intervals for OR, RR and PR (1/3) Notation Disease Exposed Yes Yes No n++ n + ⇡+ := P(D|E), OR point estimate No n+ n d= OR n++ , n++ + n+ Jose Barrera (ISGlobal & UAB) ⇡ ˆ = n n + +n = n++ n n + n+ RR and PR point estimate ⇡ := P(D|Ē). Point estimates for probabilities ⇡ ˆ+ = ⇡ ˆ+ 1 ⇡ ˆ+ ⇡ ˆ 1 ⇡ ˆ . n++ (n + + n ) ˆ+ c = PR c = ⇡ RR = ⇡ ˆ n + (n++ + n+ ) + Statistics in Health Sciences, 2023/2024 23 / 32 Confidence intervals for OR, RR and PR (2/3) Confidence intervals for OR d is normally distributed, so that • Asymptotically (high sample size), log(OR) ! r ⇣ ⌘ d d d CI1 ↵ (OR) ⇡ OR exp ±z1 ↵/2 Var log(OR) , where d = n++ n OR n + n+ and ⇣ ⌘ d ⇡ n++1 + n 1 + n d log(OR) Var + 1 + +n 1 . • The exact calculation of confidence intervals for OR is based in the hypergeometric distribution. • In R, confidence intervals can be calculated using epitools::oddsratio or epiR::epi.2by2. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 24 / 32 Confidence intervals for OR, RR and PR (3/3) Confidence intervals for RR and PR c is normally distributed, so that • Asymptotically (high sample size), log(RR) ! r ⇣ ⌘ c d c CI1 ↵ (RR) ⇡ RR exp ±z1 ↵/2 Var log(RR) , where c = n++ (n + + n ) RR n + (n++ + n+ ) ⇣ ⌘ d log(RR) c Var ⇡ and n+ + n++ (n++ + n+ ) n n + (n + +n ) . • The exact calculation of confidence intervals for RR is based in the multinomial distribution. • In R, confidence intervals can be calculated using epitools::oddsratio or epiR::epi.2by2. • Formulas above also apply to PR. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 25 / 32 Measures of impact: PAR Population attributable risk (PAR) • While PR, RR and OR are measures of associations, the population attributable risk is a measure of impact. • The population attributable risk (PAR), or attributable fraction among the population (AFp ), is the proportion of cases in the population which are attributable to the exposure: PAR = AFp = Risk of D Risk of D among non exposed Risk of D = P(D) P(D|Ē) =1 P(D) P(D|Ē) . P(D) • Previous expression can be written as PAR = AFp = Pe (RR 1) , 1 + Pe (RR 1) where Pe is the exposed proportion of the population (i.e. the exposure prevalence). Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 26 / 32 Measures of impact: PAR Population attributable risk (PAR): comments • P(D|Ē) = 0 =) PAR = 1. • P(D|Ē) = P(D) =) PAR = 0. • PAR can be interpreted as the fraction of cases that could be avoided if the exposure would have been removed from the population. However, such interpretation assumes that: • • • • The exposure status among the individuals is time invariant. There is a casual relationship between E and D. Removing E does no modify other potential effects on D due to other variables. The PAR estimation has been obtained after adjusting for potential confounding. Exercise Compute and interpret the PAR for the table in slide 5, assuming that, in that example, the sample is representative of the population and that the prevalence of the exposure in the whole population is 5%. Answer: 2.44%. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 27 / 32 Measures of impact: EAR Exposure attributable risk (EAR) • The exposure attributable risk, EAR, or attributable fraction among the exposed (population), AFe , is the proportion of cases among exposed (population) that are attributable to the exposure: EAR = AFe = Risk of D among exposed Risk of D among non exposed Risk of D among exposed = P(D|E) P(D|Ē) P(D|Ē) =1 . P(D|E) P(D|E) • Previous expression can be written as EAR = AFe = 1 Jose Barrera (ISGlobal & UAB) 1 . RR Statistics in Health Sciences, 2023/2024 28 / 32 Measures of impact: EAR Exposure attributable risk (EAR): comments • P(D|Ē) = 0 =) EAR = 1. • P(D|E) = P(D|Ē) =) EAR = 0. • The numerator of EAR is known as “excess risk”: ER = P(D|E) P(D|Ē). Exercise Compute and interpret the EAR for the table in slide 5, assuming that, in that example, the sample is representative of the population. Answer: 33.33%. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 29 / 32 Measures of impact: attributable fractions vs attributable numbers Attributable fractions vs attributable numbers • In epidemiology, attributable fractions are usually reported as a percentage to describe the impact of the exposure. • Absolute impacts are also usually reported, which are known as attributable numbers (AN). • ANp = AFp · np , where np is the number of cases among the whole population, is the number of cases among the whole population that would not have occurred in the absence of exposure (under assumptions described previously). • ANe = AFe · np is the number of cases among the exposed population that would not have occurred in the absence of exposure (under assumptions described previously). • (Optional) further reading: Steenland and Armstrong [1] . Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 30 / 32 Measures of impact: exercises Exercises 1 2 3 According to the work by Basagaña et al [2] , how the concept “heat wave” is defined? What was the overall impact estimate of heat waves on mortality? Is it PAR or EAR? Why? Interpret the result. According to the work by Khomenko et al [3] , what is the overall impact estimate of not compliance with WHO air pollution guidelines on mortality among the analyzed European cities? Is it PAR or EAR? Why? Interpret the result. Solve exercises in the document Exercises_SHS-CDA_v0_5.pdf, section “Measures of the disease”. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 31 / 32 References [1] K. Steenland and B. Armstrong. An overview of methods for calculating the burden of disease due to specific risk factors. Epidemiology, 17(5):512–519, 2006. URL https://doi.org/10.1097/01.ede.0000229155.05644.43. [2] X. Basagaña, C. Sartini, J. Barrera-Gómez, P. Dadvand, J. Cunillera, B. Ostro, J. Sunyer, and M. Medina-Ramón. Heat waves and causespecific mortality at all ages. Epidemiology, 22(6):765–772, 2011. URL https://doi.org/10.1097/EDE.0b013e31823031c5. [3] S. Khomenko, M. Cirach, E. Pereira-Barboza, N. Mueller, J. Barrera-Gómez, D. Rojas-Rueda, K. de Hoogh, G. Hoek, and M. Nieuwenhuijsen. Premature mortality due to air pollution in european cities: a health impact assessment. The Lancet Planetary Health, 5(3): e121–e134, 2021. URL https://doi.org/10.1016/S2542-5196(20)30272-2. Jose Barrera (ISGlobal & UAB) Statistics in Health Sciences, 2023/2024 32 / 32