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Causal Diagrams -- DAGs

Causal Diagrams -- DAGs. DAGs = directed acyclic graphs; aka chain graphs Consist of nodes (variables) and arrows “Directed”: all arrows have one-way direction and depict causal relationships “Acyclic”: there is never a complete circle (i.e. no factor can cause itself)

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Causal Diagrams -- DAGs

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  1. Causal Diagrams -- DAGs • DAGs = directed acyclic graphs; aka chain graphs • Consist of nodes (variables) and arrows • “Directed”: all arrows have one-way direction and depict causal relationships • “Acyclic”: there is never a complete circle (i.e. no factor can cause itself) • Better than the rough criteria for confounding when planning studies and analyses E Research Question: Does E cause D? C • Frontier of epidemiologic theory ? • Forces investigator to conceptualize system D • Identifies pitfalls of adjusting and not adjusting for certain variables

  2. Use of DAGs to Identify What is Not Confounding RQ: Does lack of folate intake cause birth defects? Folate Intake ? Stillbirths Birth Defects Stillbirths are a “common effect” of both the exposure and disease – not a common cause. Common effects are called “colliders” Adjusting for colliders OPENS paths. Will actually result in bias. It is harmful. Hernan AJE 2002

  3. DAGs point out special issue when estimating direct effects • RQ: Does aspirin prevent CHD in a pathway other than through platelet aggregation • Assumes no common cause of platelet agg. and D Would be correct to adjust But if • Assume common cause (e.g., genetic component) • Need other statistical methods to resolve Aspirin Platelet Aggregation ? Coronary Heart Disease Aspirin Would be incorrect to adjust OR not to adjust for platelet aggregation Platelet Aggregation ? Genetic factors (not measured) Coronary Heart Disease Cole and Hernan IJE 2002

  4. Confounding and Interaction: Part III • Methods to reduce confounding • during study design: • Randomization • Restriction • Matching • during study analysis: • Stratified analysis • Forming “Adjusted” Summary Estimates • Concept of weighted average • Woolf’s Method • Mantel-Haenszel Method • Handling more than one potential confounder • Role of an analysis plan • Another design technique: Instrumental variables • Quantitative assessment of unmeasured confounding • Limitations of stratification • motivation for multivariable regression • Limitations of conventional adjustment • motivation for other “causal” techniques

  5. Effect-Measure Modification Crude RR crude= 1.7 Heavy Caffeine Use No Caffeine Use Stratified RRcaffeine use = 0.7 RRnocaffeine use = 2.4 . cs delayed smoking, by(caffeine) caffeine | RR [95% Conf. Interval] M-H Weight -----------------+------------------------------------------------- no caffeine | 2.414614 1.42165 4.10112 5.486943 heavy caffeine | .70163 .3493615 1.409099 8.156069 -----------------+------------------------------------------------- Crude | 1.699096 1.114485 2.590369 M-H combined | 1.390557 .9246598 2.091201 -----------------+------------------------------------------------- Test of homogeneity (M-H) chi2(1) = 7.866 Pr>chi2 = 0.0050 Report interaction; confounding is not relevant

  6. Report vs Ignore Effect-Measure Modification?Some Guidelines Is an art form: requires consideration of clinical, statistical and practical considerations

  7. Does AZT after needlesticks prevent HIV? Crude ORcrude =0.61 Stratified Minor Severity Major Severity OR = 0.35 OR = 0.0 Report or ignore interaction?

  8. General Framework for Stratification • Design phase: Create a DAG • Decide which variables to control for • Implementation phase: measure the confounders (or other variables needed to block path) • Analysis phase: Report Effect-Measure Modification? (assess clinical, statistical, and practical considerations) Report stratum-specific estimates no yes Derive summary “adjusted” estimate Decide which variables to adjust for in final estimate none some Report crude estimate, 95% CI, p value Report adjusted estimate, 95% CI, p value

  9. Assuming Interaction is not Present, Form a Summary of the Unconfounded Stratum-Specific Estimates • Construct a weighted average • Assign weights to the individual strata • Summary Adjusted Estimate = Weighted Average of the stratum-specific estimates • a simple mean is a weighted average where the weights are equal to 1 • which weights to use depends on type of effect estimate desired (OR, RR, RD), characteristics of the data, and goal of research • e.g., • Woolf’s method • Mantel-Haenszel method • Standardization (see text)

  10. Forming a Summary Adjusted Estimate for Stratified Data Crude ORcrude = 0.61 Stratified Minor Severity Major Severity OR = 0.0 OR = 0.35 How would you weight these strata?

  11. Summary Estimators: Woolf’s Method • aka Directly pooled or precision estimator • Woolf’s estimate for adjusted odds ratio • where wi • wi is the inverse of the variance of the stratum-specific log(odds ratio)

  12. Calculating a Summary Effect Using the Woolf Estimator • e.g., AZT use, severity of needlestick, and HIV Crude ORcrude =0.61 Stratified Minor Severity Major Severity OR = 0.0 OR = 0.35 Problem: cannot take log of 0; cannot divide by zero

  13. Summary Adjusted Estimator: Woolf’s Method • Conceptually straightforward • Best when: • number of strata is small • sample size within each strata is large • Cannot be calculated when any cell in any stratum is zero because log(0) is undefined • 1/2 cell corrections have been suggested but are subject to bias • Formulae for Woolf’s summary estimates for other measures (e.g., risk ratio, RD) available in texts and software documentation

  14. Summary Adjusted Estimators: Mantel-Haenszel • Mantel-Haenszel estimate for odds ratios • ORMH = • wi = • wi is inverse of the variance of the stratum-specific odds ratio under the null hypothesis (OR =1)

  15. Summary Adjusted Estimator: Mantel-Haenszel • Relatively resistant to the effects of large numbers of strata with few observations • Resistant to cells with a value of “0” • Computationally easy • Most commonly used in commercial software

  16. Calculating a Summary Adjusted Effect Using the Mantel-Haenszel Estimator • ORMH = • ORMH = Crude ORcrude =0.61 Stratified Minor Severity Major Severity OR = 0.0 OR = 0.35

  17. Calculating a Summary Effect in Stata • To stratify by a third variable: • cs varcase varexposed, by(varthird variable) • cc varcase varexposed, by(varthird variable) • Default summary estimator is Mantel-Haenszel • “ , pool” will also produce Woolf’s method epitab command - Tables for epidemiologists

  18. Calculating a Summary Effect Using the Mantel-Haenszel Estimator • e.g. AZT use, severity of needlestick, and HIV • . cc HIV AZTuse,by(severity) pool • severity | OR [95% Conf. Interval] M-H Weight • -----------------+------------------------------------------------- • minor | 0 0 2.302373 1.070588 • major | .35 .1344565 .9144599 6.956522 • -----------------+------------------------------------------------- • Crude | .6074729 .2638181 1.401432 • Pooled (direct) | . . . • M-H combined | .30332 .1158571 .7941072 • -----------------+------------------------------------------------- • Test of homogeneity (B-D) chi2(1) = 0.60 Pr>chi2 = 0.4400 • Test that combined OR = 1: • Mantel-Haenszel chi2(1) = 6.06 • Pr>chi2 = 0.0138 Crude ORcrude =0.61 Stratified Minor Severity Major Severity OR = 0.0 OR = 0.35

  19. Calculating a Summary Effect Using the Mantel-Haenszel Estimator • In addition to the odds ratio, Mantel-Haenszel estimators are also available in Stata for: • risk ratio • “cs varcase varexposed, by(varthird variable)” • rate ratio • “ir varcase varexposed vartime, by(varthird variable)”

  20. After Confounding is Managed: Confidence Interval Estimation and Hypothesis Testing for the Mantel-Haenszel Estimator • e.g. AZT use, severity of needlestick, and HIV • . cc HIV AZTuse,by(severity) pool • severity | OR [95% Conf. Interval] M-H Weight • -----------------+------------------------------------------------- • minor | 0 0 2.302373 1.070588 • major | .35 .1344565 .9144599 6.956522 • -----------------+------------------------------------------------- • Crude | .6074729 .2638181 1.401432 • Pooled (direct) | . . . M-H combined | .30332 .1158571 .7941072 • -----------------+------------------------------------------------- • Test of homogeneity (B-D) chi2(1) = 0.60 Pr>chi2 = 0.4400 • Test that combined OR = 1: • Mantel-Haenszel chi2(1) = 6.06 • Pr>chi2 = 0.0138 • What does the p value = 0.0138 mean?

  21. Mantel-Haenszel Confidence Interval and Hypothesis Testing

  22. Mantel-Haenszel Techniques • Mantel-Haenszel estimators • Mantel-Haenszel chi-square statistic • Mantel’s test for trend (dose-response)

  23. Spermicides, maternal age & Down Syndrome Crude OR = 3.5 Age < 35 Age > 35 Stratified OR = 3.4 OR = 5.7 Which answer should you report as “final”? What undesired feature has stratification caused?

  24. Effect of Adjustment on Precision (Variance) • Adjustment can increase or decrease standard errors (and CI’s) depending upon: • Nature of outcome (interval scale vs. binary) • Measure of association desired • Method of adjustment (Woolf vs M-H vs MLE) • Strength of association between potential confounding factor and exposure/disease • Complex and difficult to memorize • Good news: adjustment for strong confounders removes bias and often improves precision • Bad news: adjustment for less-than-strong confounders can often (but not always) worsen precision

  25. Effect of Adjustment on Precision Crude ORcrude = 21.0 (95% CI: 16.4 - 26.9) Stratified Matches Present Matches Absent ORmatches = 21.0 OR nomatches = 21.0 ORadj= 21.0 (95% CI: 14.2 - 31.1)

  26. Whether or not to accept the “adjusted” summary estimate instead of the crude? • Methodologic literature is inconsistent on this • Bias-variance tradeoff • Scientifically most rigorous approach is to: • Create the DAG and identify potential confounders • Prior to adjustment, create two lists of potential confounders • “A” List: Those factors for which you will accept the adjusted result no matter how small the difference from the crude. • Factors strongly believed to be confounders • “B” List: Those factors for which you will accept the adjusted result only if it meaningfully differs from the crude (with some pre-specified difference, e.g., 5 to 10%). • “Change-in-estimate” approach • Factors you are less sure about • For some analyses, may have no factors on A list. For other analyses, no factors on B list. • Always putting all factors on A list may seem conservative, but not necessarily the right thing to do in light of penalty of statistical imprecision Bias control paramount Need for tradeoffs

  27. Choosing the crude or adjusted estimate? • Assume all factors are on B list and a 10% change-in-estimate rule is in place

  28. No Role for Statistical Testing for Confounding • Testing for statistically significant differences between crude and adjusted measures is inappropriate • e.g., examining an association for which a factor is a known confounder (say age in the association between hypertension and CAD) • if the study has a small sample size, even large differences between crude and adjusted measures may not be statistically different • yet, we know confounding is present • therefore, the difference between crude and adjusted measures cannot be ignored as merely chance. • bias must be prevented and hence adjusted estimate is preferred • we must live with whatever effects we see after adjustment for a factor for which there is a strong a priori belief about confounding • the issue of confounding is one of bias, not of sampling error. • Other than in RCTs, we’re not concerned that sampling error is causing confounding and therefore we don’t have to worry about testing for role of chance

  29. Spermicides, maternal age & Down Syndrome Crude OR = 3.5 Age < 35 Age > 35 Stratified OR = 3.4 OR = 5.7 Which answer should you report as “final”?

  30. Stratifying by Multiple Potential Confounders Crude Stratified <40 smokers 40-60 smokers >60 smokers <40 non-smokers 40-60 non-smokers >60 non-smokers

  31. The Need for Evaluation of Joint Confounding • Variables that evaluated alone show no confounding may show confounding when evaluated jointly Crude Stratified by Factor 1 alone by Factor 2 alone by Factor 1 & 2

  32. WHO Causal Model of Coronary Heart Disease Murray et al. Population Health Metrics 2003

  33. Approaches for When More than One Potential Confounder is Present • Backward vs forward variable selection strategies • relevant both for stratification and multivariable regression modeling (“model selection”) • Backwards Strategy • initially evaluate all potential confounders together (i.e., look for joint confounding) • preferred because in nature variables act together • Procedure: • with all potential confounders considered, form adjusted estimate. This is the “gold standard” • Of variables on the B list, one variable can then be dropped and the adjusted estimate is re-calculated (adjusted for remaining variables) • if the dropping of the first variable results in a non-meaningful (eg < 5 or 10%) change compared to the gold standard, it can be eliminated • continue until no more variables can be dropped (i.e. all remaining variables are relevant) • Problem: • With many potential confounders and multiple stratified analyses, p values (too small) & confidence intervals (too narrow) lose their nominal interpretation • Active area of methodologic interpretation • With many potential confounders, cells become very sparse and many strata provide no information

  34. Approaches for When More than One Potential Confounder is Present • Forward Strategy • start with the variable that has the biggest “change-in-estimate” impact when evaluated individually • then add the variable with the second biggest impact • keep this variable if its presence meaningfully changes the adjusted estimate • procedure continues until no other added variable has an important impact • Advantage: • avoids the initial sparse cell problem of backwards approach • Problem: • does not evaluate joint confounding effects of many variables • Multiple analyses again lead to problems in interpreting p values and CI’s

  35. An Analysis Plan • Available methods often arbitrary and invite fishing for desired answers • Solution: Analysis plan • Written before the data are analyzed • Content • Detailed description of the techniques to be used to analyze data, step by step • Forms the basis of “Statistical Analysis” section in manuscripts • Parameters/rules/logic to guide key decisions: • which variables will be assessed for interaction and for adjustment? • what p value will be used to guide reporting of interaction? • what is a meaningful change-in-estimate threshold between two estimates (e.g., 10%) to determine model selection? • Utility: A plan helps to keep the analysis: • Focused • Transparent • Reproducible • Honest (avoids p value shopping)

  36. Hour of birth Length of stay Unmeasured C Prenatal complications ? Neonatal outcomes Instrumental Variables to Manage Confounding Instrumental variable (IV) IV must be related to E but nothing else E Unmeasured C C1 C2 ? D Assess association between IV and D to estimate E-D relationship RQ: Does length of stay determine neonatal outcomes? Malkin et al. Heath Serv. Res., 2000

  37. Residual Confounding Four Mechanisms • Categorization of confounder too broad • e.g., Association between natural menopause and prevalent CHD Szklo and Nieto, 2007 • Misclassification of confounders • Can be differential or non-differential with respect to exposure and disease • If non-differential, will lead to adjusted estimates somewhere in between crude and true adjusted • If differential, can lead to a variety of unpredictable directions of bias

  38. Periodontal disease E Inflammatory Predisposition Unmeasured C Age CRP level ? ? CAD D Residual Confounding Mechanisms – cont’d • Variable used for adjustment is imperfect surrogate for true confounder • Unmeasured confounders

  39. Quantitative Analysis of Unmeasured Confounding • Can back calculate to determine how a confounder would need to act in order to spuriously cause any apparent odds ratio. Example: OR= 2.0 Prevalence of “high” level of unmeasured confounder Association between unmeasured confounder and exposure (prevalence ratio) Association between unmeasured confounder and disease (risk ratio) A (low prevalence scenario) = 7 B (high prevalence scenario) = 3.4 Winkelstein et al., AJE 1984

  40. Stratification to Manage Confounding • Advantages • straightforward to implement and comprehend • easy way to evaluate interaction • Limitations • Requires continuous variables to be discretized • loses information; possibly results in “residual confounding” • Discretizing often brings less precision • Deteriorates with multiple confounders • e.g., suppose 4 confounders with 3 levels • 3x3x3x3=81 strata needed • unless huge sample, many cells have “0”’s and strata have undefined effect measures • Solution: • Mathematical modeling (multivariable regression) • e.g. • linear regression • logistic regression • proportional hazards regression

  41. HAART CD4 count ? AIDS Severity of HIV (unmeasured) Limitation of Conventional Stratification (and Regression) • RQ: Does coffee use cause CAD? Behavioral factors (unmeasured) Coffee Cholesterol level ? CAD • RQ: Does HAART prevent AIDS/Death? Simultaneous desire to control for cholesterol/CD4 to manage confounding and NOT to control because they are intermediary variables

  42. When factors are simultaneously confounders and intermediaries, conventional techniques fail and “causal methods” are needed Causal methods: g-estimation, structural nested models, marginal structural models Cole et al, AJE 2003

  43. Regression is ahead but don’t forget about the simple techniques ….. • “Because of the increased ease and availability of computer software, the last few years have seen a flourishing of the use of multivariate analysis in the biomedical literature. These highly sophisticated mathematic models, however, rarely eliminate the need to examine carefully the raw data by means of scatter diagrams, simple n x k table, and stratified analyses.” Szklo and Nieto 2007 • “The widespread availability and user-friendly nature of computer software make the method accessible to some data analysts who may not have had adequate instruction in its appropriate applications. When they are misapplied, multivariate techniques have the potential to contribute to incorrect model development, misleading results, and inappropriate interpretation of the effect of hypothesized confounders.” Friis and Sellers, 2009 • “Statistical software is like raising the gas pedal in a car for a 4 year old.” Peter Bachetti (UCSF), date unknown

  44. Next Tuesday (12/9/08) • 8:45 to 10:15: Journal Club • 1:30 to 3:00 pm: Mitch Katz • “Conceptual approach to multivariable regression” • Note chapters in his textbook • 3:15 to 4:45: Last Small Group Section • Web-based course evaluation • Bring laptop • Distribute Final Exam (on line) • Exam due 12/16 in hands of Olivia by 4 pm by email (olivia@epi.ucsf.edu) or China Basin 5700

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