Measures of association - Biostatistics
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The document covers biostatistics concepts related to measures of association, including risk difference, risk ratio, and odds ratio, along with their calculation and interpretation using 2×2 tables. It discusses the importance of controlling for confounding variables through methods like stratification and regression modeling and provides examples from randomized controlled trials. Additionally, it compares crude and adjusted measures of effect, highlighting the impact of confounding on statistical analysis.
Measures of association - Biostatistics
- 2. Lecture 9 Review – Measures of association Measures of association – – – Risk difference Risk ratio Odds ratio Calculation & interpretation interval for each measure of confidence of association
- 3. 2×2 table - Measures of association Outcome - binary Measure of Effect Formula Risk difference p1-p0 Risk ratio p1 / p0 Odds ratio (d1/h1) / (d0/h0)
- 4. Differences in measures of association • When there is no association between exposure and outcome, – – – risk difference = 0 risk ratio (RR) = 1 odds ratio (OR) = 1 • • Risk difference can be negative or positive RR & OR are always positive • For rare outcomes, OR ~ RR • OR is always further from 1 than corresponding RR – If RR > 1 then OR > RR – If RR < 1 the OR < RR
- 5. Interpretation of measures of association • RR & OR < 1, associated with a reduced risk / odds (may protective) be – RR = 0.8 (reduced risk of 20%) • RR – & OR > 1, associated with an increased risk / odds RR = 1.2 (increased risk of 20%) • RR & OR – further the risk is from 1, stronger the association between exposure and outcome (e.g. RR=2 versus RR=3).
- 6. Comparing the outcome measure of two exposure groups (groups 1 & 0) s.e.(log RR) = − + −e eR (log e OR ) d h d h Outcome variable – data type Population parameter Estimate of population parameter from sample Standard error of loge(parameter) 95% Confidence interval of loge(population parameter) Categorical Population risk ratio p1/p0 1 1 1 1 d1 n1 d0 n0 logeRR ±1.96× s.e.(log R ) Categorical Population odds ratio (d1/h1) / (d0/h0) s.e. = 1 + 1 + 1 + 1 1 1 0 0 logeOR ±1.96xs.e.(log eOR)
- 7. Calculation of p-values for comparing two groups 1 0 z = s.e.(lo g ( RR )) s.e.(log (OR)) Outcome variable – data type Population parameter Population parameter under null hypothesis Test statistic Categorical π1-π0 Population risk ratio Population odds ratio π1-π0=0 Population risk ratio=1 Population odds ratio=1 p − p s.e.( p 1 − p 0 ) z = loge (RR) e z = loge (OR) e
- 8. Comparing the outcome measure of two exposure groups (TBM trial: dexamethasone versus placebo) Outcome variable – data type Population parameter under null hypothesis Estimate of population parameter from sample 95% confidence interval for population parameter Two-sided p-value Categorical Population risk difference = 0 p1-p0 = -0.095 -0.175, -0.015 0.020 Categorical Population risk ratio = 1 p1/p0 = 0.77 0.62, 0.96 0.016 Categorical Population odds ratio = 1 (d1/h1) / (d0/h0) = 0.66 0.46, 0.93 0.021
- 9. 2×2 table – TBM trial example Odds ratio for death = (d1/h1) / (d0/h0) = 0.465 / 0.704 = 0.66 Odds ratio for exposure to dexamethasone = (d1/d0) / (h1/h0) = 0.777 / 1.176 = 0.66 Odds ratio for not dying = (h1/d1) / (h0/d0) = 2.149 / 1.420 = 1.51 = (1/0.66) Odds ratio for exposure to placebo = (d0/d1) / (h0/h1) = 1.287 / 0.850 = 1.51 = (1/0.66) Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (d1) 187 (h1) 274 (n1) Placebo (group 0) 112 (d0) 159 (h0) 271 (n0) Total 199 346 545
- 10. Measure of association Study Design Risk difference Risk Ratio Odds Ratio Randomised controlled trial √ √ √ Cohort Study √ √ √ Case-control Study × × √
- 11. Lecture 10 – Controlling for confounding: stratification and regression • A description of confounding • How to control for confounding analysis by – Stratification – Regression modelling in statistical • A brief description of the role of multiple linear or logistic regression in adjusting for confounding
- 12. Outcome and exposure variables (RECAP) Outcomes are variables of interest (population health relevance) whose patterns and determinants we wish to learn about from data • • Exposures are the variables we think might explain observed variation in the outcomes • Statistical analysis can be used to quantify the association between outcomes and exposures
- 13. What is confounding? A confounding variable 1) 2) 3) is associated with the outcome variable; is associated with the exposure variable; does not lie on the causal pathway. Outcome variableExposure variable Confounding variable Failing to control for confounding may result in a biased estimate of the magnitude of the association between exposure and outcome
- 14. Example of confounding Exposure variable Outcome variable Alcohol intake Heart disease Confounding variables Cigarette smoking
- 15. Control of confounding Design of Study • Randomisation (randomised controlled trial: e.g. TBM trial) • Restriction (only include those with one value of confounder) • Matching
- 16. Control of confounding Statistical analysis • Stratification • Regression modelling
- 17. Hypothetical example of a case-control study Association between energy intake and heart disease Odds Odds of heart disease in high energy intake group = 730/600 = 1.22 of heart disease in low energy intake group = 700/540 = 1.30 Odds ratio = 1.22 / 1.30 = 0.94 95% confidence interval: 0.80 up to 1.10 Heart disease Energy intake Yes No Total High (group 1) 730 (d1) 600 (h1) 1330 (n1) Low (group 0) 700 (d0) 540 (h0) 1240 (n0) Total 1430 1140 2570
- 18. Is this association confounded by physical activity? Exposure variable Outcome variable Energy intake Heart disease Confounding variables Physical activity
- 19. Stratify by physical activity….. Calculate the stratum specific odds ratios… Energy intake High physical activity Low physical activity Heart disease Heart disease Yes No Yes No High (group 1) 500 510 230 90 Low (group 0) 100 150 600 390
- 20. Stratify by physical activity….. For high physical activity group: OR (95% CI) = 1.47 (1.11, 1.95) For low physical activity group: OR (95% CI) = 1.66 (1.26, 2.19) Energy intake High physical activity Low physical activity Heart disease Heart disease Yes No Yes No High (group 1) 500 510 230 90 Low (group 0) 100 150 600 390
- 21. Is this association confounded by physical ??? activity? Exposure variable Energy intake Outcome variable Heart disease ?????? Confounding variables Physical activity
- 22. Confounding – condition 1 Association between physical activity and heart disease ** Look particularly in those who are not exposed to the factor of interest** For low energy intake group: OR (95% CI) = 0.43 (0.33, 0.58) For high energy intake group: OR (95% CI) = 0.38 (0.29, 0.50) Physical activity High energy intake Low energy intake Heart disease Heart disease Yes No Yes No High (group 1) 500 510 100 150 Low (group 0) 230 90 600 390
- 23. Confounding – condition 2 Association between energy intake and physical activity • In a case-control study: examine the association in the controls • In a cohort study: use the whole cohort
- 24. Confounding – condition 2 Association between energy intake and physical activity for those without heart disease (n=1140) Proportion in high energy intake group who report high physical activity = 510/600 = 0.85 (85%) Proportion in low energy intake group who report high physical activity = 150/540 = 0.28 (28%) Odds Ratio = (510/90) / (150/390) = 14.7; 95% CI: 11.0 up to 19.7 Physical activity Energy intake High Low Total High (group 1) 510 90 600 Low (group 0) 150 390 540
- 25. Is this association confounded by physical ??? activity? Exposure variable Energy intake Outcome variable Heart disease High energy intake: OR = 0.38 (95% CI: 0.29, 0.50) Low energy intake: OR = 0.43 (95% CI: 0.33, 0.58) High energy intake associated with high physical activity Confounding variables Physical activity
- 26. So physical activity is a potential confounder Control for confounding - Stratified analyses 1) Start with stratum specific estimates differences, rate ratios of odds ratios, risk ratios, risk 2) Calculate a weighted average of the ‘pooled’ estimate stratum-specific estimates Usual method is Mantel-Haenszel method – Weights assigned according to amount of information in each stratum
- 27. Calculate a pooled OR (600×90)/1310) = 41.2 For low physical activity: OR = 1.66 w= (d0×h1)/n = For high physical activity: OR = 1.47 w= (d0×h1)/n = (100×510)/1260) = 40.5 Energy intake High physical activity (n=1260) Low physical activity (n=1310) Heart disease Heart disease Yes No Yes No High (group 1) 500 (d1) 510 (h1) 230 (d1) 90 (h1) Low (group 0) 100 (d0) 150 (h0) 600 (d0) 390 (h0)
- 28. Calculate a pooled OR (600×90)/1310) = 41.2 Mantel-Haenszel estimate of pooled odds ratio: ∑(wi × ORi ) OR =MH ∑wi Stratum ‘i’ For low physical activity: OR = 1.66 w= (d0×h1)/n = For high physical activity: OR = 1.47 w= (d0×h1)/n = (100×510)/1260) = 40.5
- 29. Calculate a pooled OR (600×90)/1310) = 41.2 Mantel-Haenszel estimate of pooled odds ratio: (40.5×1.47) + (41.2×1.66) OR =1.57=MH (40.5+ 41.2) 95% CI: 1.29 up to 1.91 Recall that the crude OR was 0.94 (95% CI 0.80-1.10) Is there a difference between crude and adjusted measures of effect? For low physical activity: OR = 1.66 w= (d0×h1)/n = For high physical activity: OR = 1.47 w= (d0×h1)/n = (100×510)/1260) = 40.5
- 30. Association between energy intake & heart disease adjusting for physical activity ORMH = 1.57 95% CI: 1.29, 1.91 Exposure variable Energy intake Outcome variable Heart disease High energy intake: OR = 0.38 (95% CI: 0.29, 0.50) Low energy intake: OR = 0.43 (95% CI: 0.33, 0.58) High energy intake associated with high physical activity Confounding variables Physical activity
- 31. Multiple logistic regression Outcome variable (y-variable) – binary e.g. dead or alive; treatment failure or success; disease or no disease.. Measure of association – Odds ratio Multiple logistic regression model – loge(odds of outcome) = β0 + β1X1 + β2X2 + β3X3 +…. + βkXk β1,…βk – loge(odds ratios) X1, …..Xk – k different exposure variables (do not need to be binary but can be categorical with more than 2 categories or numerical) Useful when there are many confounding variables…
- 32. Logistic regression Example – Association between energy intake and heart disease Outcome variable (y-variable) – heart disease (coded as yes-1 & no-0) Logistic regression model – loge(odds of outcome) = β0 + β1X1 β1 – loge(odds ratios) X1 – energy intake (high versus low) Exposure Odds Ratio (expβi) 95% Confidence Interval Energy intake (high vs low) 0.94 0.80, 1.10
- 33. Multiple logistic regression Example – Association between energy intake and heart disease Outcome variable (y-variable) – heart disease (coded as yes-1 & no-0) Multiple logistic regression model – loge(odds of outcome) = β0 + β1X1 + β2X2 β1, β2 – loge(odds ratios) X1 – energy intake (high versus low) X2 – physical activity (high versus low) Exposure Odds Ratio (expβi) 95% Confidence Interval Energy intake (high vs low) 1.57 1.29, 1.91 Physical activity (high vs low) 0.41 0.33, 0.49
- 34. Multiple linear regression Outcome variable (y-variable) – numerical e.g. blood pressure, forced expiratory volume in 1 sec (FEV1) Linear regression model – y = β0 + β1X1 + β2X2 + β3X3 +…. + βkXk y – numerical outcome variable, β1,…βk – increase in y for every unit increase in x X1, …..Xk – k different exposure variables (can be numerical or categorical with 2+ categories) Useful when there are many confounding variables…
- 35. Lecture 10 - Objectives • Understand confounding • Calculate the Mantel-Haenszel estimate of pooled odds ratio the • Understand the difference between linear and logistic regression