![• “We can never finally prove our scientific
theories, we can merely (provisionally)
confirm or (conclusively) refute them.”
– - Karl Popper
Sir Karl Raimund Popper CH FBA FRS
[4]
(28 July 1902 – 17 September 1994) was an Austrian-British[5]
philosopher and professor at the London School of Economics.[6]
He is generally regarded o regarded as
one of the greatest philosophers of science of the 20th century.[7][8]
Popper is known for his rejection of the
classical inductivist views on the scientific method, in favour of empirical falsification: regarded as one of the
greatest philosophers of science of the 20th century.[7][8]
(wikipedia.com)
2014 Page 5](https://image.slidesharecdn.com/4-160809065802/85/4-Threats-to-validity-from-confounding-bias-and-effect-modification-5-320.jpg)
![2
BMJ 2004;329:868-869 (16 October)
Why is confounding so important in
epidemiology?
● BMJ Editorial: “The scandal of poor epidemiological
research” [16 October 2004]
● “Confounding, the situation in which an apparent
effect of an exposure on risk is explained by its
association with other factors, is probably the
most important cause of spurious associations in
observational epidemiology.”
2014 Page 13](https://image.slidesharecdn.com/4-160809065802/85/4-Threats-to-validity-from-confounding-bias-and-effect-modification-13-320.jpg)
![Effect modification: test of homogeneity
• Null hypothesis: The individual stratified estimates of the effect do not
differ from some uniform estimate of effect (such as a Mantel Haenszel
estimator)
• Notation:
–
– N is the number of strata (N=2 in our smoking/matches example);
– ln^Ri
is the natural logarithm of the estimated (hence the “^”) effect
measure for each stratum (ORi
in our example);
– ln^R is the natural logarithm of the uniform effect estimate (e.g. ORMH
in
X
2
(N-1)
is chi-square with (N-1) degrees of freedom;
our example—the computer will use the maximum likelihood estimate)
• One formula to test homogeneity:
X
2
(N-1)
=
∑
[ln(^ Ri
) – ln(RMH
)]2
Var[ln(^
Ri
)]
N
i= 1
78
JC: Comment on choice of signifciance level for test of homogeneity2014 Page 91](https://image.slidesharecdn.com/4-160809065802/85/4-Threats-to-validity-from-confounding-bias-and-effect-modification-91-320.jpg)
4 Threats to validity from confounding bias and effect modification
- 1. Threats to Validity from Confounding and Effect Modification • Overview: Random vs. systematic error • Confounding • Effect Modification • Logistic regression (time permitting) • Special thanks for some of the materials in these lecture: – Professor Jen Ahern (UCB) – Professor Madhu Pai (McGilll—a former 250b GSI) 1 2014 Page 1
- 2. 1 The cardinal rule of epidemiology • Remember that all results based on epidemiology studies are likely to be … 2014 Page 2
- 3. The cardinal rule of epidemiology (continued) • WRONG… – unless proper care has been taken to eliminate all sources of error in the estimate (…and sometimes even then the results will be wrong because of unknown sources of error) 2 2014 Page 3
- 4. Example: Confounding • A colleague with outside funding believes that cigarette smoke is not a “cause” (in any sense) of lung cancer but that exposure to matches (yes, matches) is the cause. This colleague has conducted a large case control study to test the null hypothesis: Ho : “Matches are not associated with lung cancer”. • What’s the rationale (in the Popperian sense) for stating the null hypothesis rather than the alternative: HA : “Matches are associated with lung cancer”. • What does the colleague hope to do (in terms of hypothesis testing) • What do you think of the term “associated” –would it be better to write “a cause of”? 2014 Page 4
- 5. • “We can never finally prove our scientific theories, we can merely (provisionally) confirm or (conclusively) refute them.” – - Karl Popper Sir Karl Raimund Popper CH FBA FRS [4] (28 July 1902 – 17 September 1994) was an Austrian-British[5] philosopher and professor at the London School of Economics.[6] He is generally regarded o regarded as one of the greatest philosophers of science of the 20th century.[7][8] Popper is known for his rejection of the classical inductivist views on the scientific method, in favour of empirical falsification: regarded as one of the greatest philosophers of science of the 20th century.[7][8] (wikipedia.com) 2014 Page 5
- 6. Confounding: smoking, matches, 10 and lung cancer • Your colleague has located 1000 cases of lung cancer, of whom 820 carry matches. • Among 1000 reference patients (selected randomly from a population with recently taken normal chest x-rays), 340 carry matches. • Strengths of the reference selection process? Weaknesses? • Describe the relationship between matches and lung cancer in your colleague’s data. • Would you like to analyze the data in any other fashion? 2014 Page 6
- 7. Confounding: smoking, matches, and lung cancer • Odds ratio = (820 * 660) / (180 * 340) • OR = 8.8 • 95% CI (7.2, 10.9) Cancer No cancer Matches 820 340 No matches 180 660 2014 Page 7
- 8. Confounding: smoking, matches, and lung cancer • You decide to look at the relationship between matches and lung cancer in the smokers separately from the non- smokers. • You find that among the 1000 cases, 900 are smokers and 810 (of the 900) carry matches • Among the 1000 reference patients, 300 are smokers and 270 (of the 300) carry matches • Calculate the relevant measure(s) of effect. • What should your colleague do about future funding? 2014 Page 8
- 9. Confounding: smoking, matches, and lung cancer • ORpooled = 8.84 (7.2, 10.9) • ORsmokers = 1.0 (0.6, 1.5) • ORnonsmokers = 1.0 (0.5, 2.0) Pooled Cancer No cancer Matches No Matches Smokers Matches 820 180 Cancer 810 340 660 No cancer 270 No Matches Non-smoker Matches No Matches 90 Cancer 10 90 30 No cancer 70 630 13 2014 Page 9
- 10. Confounding: smoking, matches, and lung cancer • To be complete, you also decide to examine the relationship between smoking and lung cancer. • What tables should you construct to do this? 14 2014 Page 10
- 11. Confounding: smoking, matches, and lung cancer ’ • ORpooled = 21.0 (16.3, 27.1) • ORmatches = 21.0 (10.5, 46.2) • ORno matches = 21.0 (12.9, 34.7) • Discuss your intuitions about the 95% CI s Pooled Cancer No cancer Smoking No Smoking Matches Smoking 900 100 Cancer 810 300 700 No cancer 270 No Smoking No matches Smoking No Smoking 10 Cancer 90 90 70 No cancer 30 630 16 2014 Page 11
- 12. Confounder? ? ? ? Unadjusted RR Exposure Disease ? Adjusted RR 19 2014 Page 12
- 13. 2 BMJ 2004;329:868-869 (16 October) Why is confounding so important in epidemiology? ● BMJ Editorial: “The scandal of poor epidemiological research” [16 October 2004] ● “Confounding, the situation in which an apparent effect of an exposure on risk is explained by its association with other factors, is probably the most important cause of spurious associations in observational epidemiology.” 2014 Page 13
- 14. Overview 3 ● Causality is the central concern of epidemiology ● Confounding is the central concern with establishing causality ● Confounding can be understood using multiple different approaches ● A strong understanding of various approaches to confounding and its control is essential for all those who engage in health research 2014 Page 14
- 15. 10 Adapted from: Maclure, M, Schneeweis S. Epidemiology 2001;12:114-122. Causal Effect Random Error Confounding Information bias (misclassification) Selection bias Bias in inference Reporting & publication bias Bias in knowledge use Confounding is one of the key biases in identifying causal effects RR causal “truth” RR association 2014 Page 15
- 16. 11 Confounding: 4 ways to understand it! 1. “Mixing of effects” 2. “Classical” approach based on a priori criteria 3. Collapsibility and data-based criteria 4. “Counterfactual” and non-comparability approaches 2014 Page 16
- 17. 12 Rothman KJ. Epidemiology. An introduction. Oxford: Oxford University Press, 2002 First approach: Confounding: mixing of effects ● “Confounding is confusion, or mixing, of effects; the effect of the exposure is mixed together with the effect of another variable, leading to bias” - Rothman, 2002 Latin: “confundere” is to mix together 2014 Page 17
- 18. Example Association between birth order and Down syndrome 13 Data from Stark and Mantel (1966) Source: Rothman 2002 2014 Page 18
- 19. Association between maternal age and Down syndrome 14 Data from Stark and Mantel (1966) Source: Rothman 2002 2014 Page 19
- 20. Association between maternal age and Down syndrome, stratified by birth order 15 Data from Stark and Mantel (1966) Source: Rothman 2002 2014 Page 20
- 21. Mixing of Effects: the water pipes analogy Exposure 16 Adapted from Jewell NP. Statistics for Epidemiology. Chapman & Hall, 2003 Outcome Confounder Mixing of effects – cannot separate the effect of exposure from that of confounder Exposure and disease share a common cause (‘parent’) 2014 Page 21
- 22. Mixing of Effects: “control” of the confounder Exposure 17 Adapted from: Jewell NP. Statistics for Epidemiology. Chapman & Hall, 2003 Outcome Confounder Successful “control” of confounding (adjustment) If the common cause (‘parent’) is blocked, then the exposure – disease association becomes clearer 2014 Page 22
- 23. Second approach: “Classical” approach based on a priori criteria 18 “Bias of the estimated effect of an exposure on an outcome due to the presence of a common cause of the exposure and the outcome” – Porta 2008 ● A factor is a confounder if 3 criteria are met: ● a) a confounder must be causally or noncausally associated with the exposure in the source population (study base) being studied; ● b) a confounder must be a causal risk factor (or a surrogate measure of a cause) for the disease in the unexposed cohort; and ● c) a confounder must not be an intermediate cause (in other words, a confounder must not be an intermediate step in the causal pathway between the exposure and the disease) 2014 Page 23
- 24. 19 Exposure E Disease (outcome) D Confounder C Confounding Schematic Szklo M, Nieto JF. Epidemiology: Beyond the basics. Aspen Publishers, Inc., 2000. Gordis L. Epidemiology. Philadelphia: WB Saunders, 4th Edition. 2014 Page 24
- 26. Exposure E Confounder C General idea: a confounder could be a ‘parent’ of the exposure, but should not be be a ‘daughter’ of the exposure Disease D 21 2014 Page 26
- 27. Example of schematic (from Gordis) 22 2014 Page 27
- 28. Birth Order E 23 Down Syndrome D Confounding factor: Maternal Age C Confounding Schematic 2014 Page 28
- 29. HRT use Heart disease Association between HRT and heart disease Confounding factor: SES 24 Are confounding criteria met? 2014 Page 29
- 30. BRCA1 gene Breast cancer Confounding factor: Age x 25 Are confounding criteria met? Should we adjust for age, when evaluating the association between a genetic factor and risk of breast cancer? No! 2014 Page 30
- 31. Sex with multiple partners Cervical cancer Confounding factor: HPV Are confounding criteria met? 26 2014 Page 31
- 32. Sex with multiple partners HPV Cervical cancer 27 What if this was the underlying causal mechanism? 2014 Page 32
- 33. Obesity Mortality Are confounding criteria met? Confounding factor: Hypertension 28 2014 Page 33
- 34. Obesity Hypertension Mortality 29 What if this was the underlying causal mechanism? 2014 Page 34
- 35. Direct vs indirect effects Obesity Hypertension Mortality Obesity Indirect effect Hypertension Mortality Direct effect Direct effect is portion of the total effect that does not act via an intermediate cause 30 Indirect effect 2014 Page 35
- 36. Hernan MA, et al. Causal knowledge as a prerequisite for confounding evaluation: an appl3 ic3 ation to birth defects epidemiology. Am J Epidemiol 2002;155(2):176-84. Simple causal graphs E DC Maternal age (C) can confound the association between multivitamin use (E) and the risk of certain birth defects (D) 2014 Page 36
- 37. 34 Complex causal graphs Hernan MA, et al. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. Am J Epidemiol 2002;155(2):176-84. E DC U History of birth defects (C) may increase the chance of periconceptional vitamin intake (E). A genetic factor (U) could have been the cause of previous birth defects in the family, and could again cause birth defects in the current pregnancy 2014 Page 37
- 38. 35 Smoking A E Calcium D Bone fractures C BMI supplementation U Physical Activity B Source: Hertz-Picciotto More complicated causal graphs! 2014 Page 38
- 39. The ultimate complex causal graph! 36 A PowerPoint diagram meant to portray the complexity of American strategy in Afghanistan! 2014 Page 39
- 40. 38 Third approach: Collapsibility and data- based approaches ● According to this definition, a factor is a confounding variable if ● a) the effect measure is homogeneous across the strata defined by the confounder and ● b) the crude and common stratum-specific (adjusted) effect measures are unequal (this is called “lack of collapsibility”) ● Usually evaluated using 2x2 tables, and simple stratified analyses to compare crude effects with adjusted effects “Collapsibility is equality of stratum-specific measures of effect with the crude (collapsed), unstratified measure” Porta, 2008, Dictionary 2014 Page 40
- 41. 39 Crude vs. Adjusted Effects ● Crude: does not take into account the effect of the confounding variable ● Adjusted: accounts for the confounding variable(s) (what we get by pooling stratum-specific effect estimates) ● Generating using methods such as Mantel-Haenszel estimator ● Also generated using multivariate analyses (e.g. logistic regression) ● Confounding is likely when: ● ● RR crude =/= RR adjusted OR crude =/= OR adjusted 2014 Page 41
- 42. 42 Crude 2 x 2 table Calculate Crude OR (or RR) Stratify by Confounder Calculate OR’s for each stratum If stratum-specific OR’s are similar, calculate adjusted RR (e. g. MH) Crude Stratum 1 Stratum 2 If Crude OR =/= Adjusted OR, confounding is likely If Crude OR = Adjusted OR, confounding is unlikely OR Crude OR1 OR2 Stratified Analysis JC: introduce “test of homogeneity” 2014 Page 42
- 43. Examples: crude vs adjusted RR Study Crude RR Stratum1 Stratum2 Adjusted Confound RR RR RR ing? 1 6.00 3.20 3.50 3.30 2 2.00 1.02 1.10 1.08 3 1.10 2.00 2.00 2.00 4 0.56 0.50 0.60 0.54 5 4.20 4.00 4.10 4.04 6 1.70 0.03 3.50 48 2014 Page 43
- 44. 49 Maldonado & Greenland, Int J Epi 2002;31:422-29 Fourth approach: Causality: counterfactual model ● Ideal “causal contrast” between exposed and unexposed groups: ● “A causal contrast compares disease frequency under two exposure distributions, but in one target population during one etiologic time period” ● If the ideal causal contrast is met, the observed effect is the “causal effect” 2014 Page 44
- 45. 52 What happens actually? RR assoc = I exp / I substitute RR causal = I exp / I unexp IDEAL ACTUAL 2014 Page 45
- 46. 50 I exp Iunexp Maldonado & Greenland, Int J Epi 2002;31:422-29 Counterfactual, unexposed cohort RR causal = I exp / I unexp “A causal contrast compares disease frequency under two exposure distributions, but in one Exposed cohort Ideal counterfactual comparison to determine causal effects target population during one etiologic time period” “Initial conditions” are identical in the exposed and unexposed groups – because they are the same population! 2014 Page 46
- 47. 51 I exp Iunexp Counterfactual, unexposed cohort Exposed cohort Substitute, unexposed cohort Isubstitute What happens actually? counterfactual state is not observed A substitute will usually be a population other than the target population during the etiologic time period - INITIAL CONDITIONS MAY BE DIFFERENT 2014 Page 47
- 48. 53 Maldonado & Greenland, Int J Epi 2002;31:422-29 Counterfactual definition of confounding ● “Confounding is present if the substitute population imperfectly represents what the target would have been like under the counterfactual condition” ● “An association measure is confounded (or biased due to confounding) for a causal contrast if it does not equal that causal contrast because of such an imperfect substitution” RR causal =/= RR assoc 2014 Page 48
- 49. Residual confounding • Confounding can persist, even after adjustment • Why? – All confounders were not adjusted for (unmeasured confounding) – Some variables were actually not confounders! – Confounders were measured with error (misclassification of confounders) – Categories of the confounding variable are improperly defined (e.g. age categories were too broad) 51 2014 Page 49
- 50. 55 Simulating the counter-factual comparison: Experimental Studies: RCT Randomization helps to make the groups “comparable” (i.e. similar initial conditions) with respect to known and unknown confounders Therefore confounding is unlikely at randomization - time t0 Eligible patients Treatment Randomization Placebo Outcomes Outcomes 2014 Page 50
- 51. Confounding: Methods to control or reduce confounding • Methods used in study design to reduce confounding – Randomization – Restriction – Matching • Methods used in study analysis to reduce confounding – Stratified analysis – Multivariate analysis 31 2014 Page 51
- 52. Confounding:The use of randomization to “ ” reduce confounding • Randomization – Useful only for intervention studies – Definition: random assignment of study subjects to exposure categories – The special strength of randomization is its ability to control/reduce the effect of confounding variables about which the investigator is unaware – If there is maldistribution of potentially confounding variables after randomization (the reason for the classic “Table I: Baseline characteristics” in the randomized trial) then other confounding control options (see below) are 32applied 2014 Page 52
- 53. Substitute, unexposed cohort 54 Maldonado & Greenland, Int J Epi 2002;31:422-29 Counterfactual, unexposed cohort Exposed cohort “Confounding is present if the substitute population imperfectly represents what the target would have been like under the counterfactual condition” 2014 Page 53
- 54. Confounding: The use of restriction to reduce confounding • Confounding cannot occur if the distribution of the potential confounding factors do not vary across exposure or disease categories – Implication of this is that an investigator may restrict study subjects to only those falling with specific level (s) of a confounding variable • Extreme example: an investigator only selects subjects of exactly the same age. • Advantages of restriction – straightforward, convenient, inexpensive 33 2014 Page 54
- 55. Confounding: The use of restriction to reduce confounding (cont.) • Disadvantages – May limit number of eligible subjects – Residual confounding may persist if restriction categories not sufficiently narrow (e.g. “decade of age” might be too broad) – Not possible to evaluate the relationship of interest at different levels of the confounder • Question: How does restriction differ from matching? 34 2014 Page 55
- 56. Confounding:The use of matching to reduce confounding • Subjects with all levels of a potential confounder are admitted into the study BUT the control/reference subjects (either with respect to exposure in a cohort or disease in a case-reference study) are chosen to have the same distribution of the potential confounder • The use of matching (may) also require special analysis techniques (matched analyses and conditional logistic regression) 35 2014 Page 56
- 57. • Disadvantages of matching – Finding appropriate control/reference subjects may be difficult and expensive and limit sample size – Matching is most often used in case-reference (i.e. case- control studies because in a large cohort study the cost of matching may be prohibitive) • Thus, in cohort studies it’s often cheaper to just enroll available controls and use analytic methods (below) to control confounding)—this doesn’t apply to computerized “free” data 36 2014 Page 57
- 58. Confounding: The use of matching to reduce confounding (cont.) • Disadvantages of matching (cont.) – Confounding factor used to match subjects cannot be itself evaluated with respect to the outcome/disease – Obviously, matching does not control for confounding by factors other than that used to match – The use of matching makes the use of stratified analysis (for the control of other potential but non-matched factors) very difficult • One way around this problem is the use of conditional logistic regression but there is a large reduction in “effective” sample size because only discordant pairs are used. 37 2014 Page 58
- 59. • Advantages of matching – Matching may be the only way to obtain sufficient numbers of control/reference subjects with relevant levels of the confounding factor(s) – Example: controlling for “neighborhood” (and all that it implies) by any approach other than matching is very difficult 38 2014 Page 59
- 60. • Advantages of matching (cont.) – Useful in very small studies in which chance differences in confounding factors are likely to exist between the study groups and other forms of control for the confounders (such as stratification or multivariate adjustment) are not possible (because of the limited sample size) – The full benefit of matching (in terms of the reduction of confounding) is obtained only if the proper form of matched analysis is used (to be reviewed later in the course) 39 2014 Page 60
- 61. • Basic goal of stratification is to evaluate the relationship between the predictor (“cause”) and outcome (“effect”) variable in strata homogenous with respect to potentially confounding variables 40 2014 Page 61
- 62. Confounding:The use of stratification to reduce confounding • For example, to examine the relationship between smoking and lung cancer while controlling for the potentially confounding effect of gender: – Create a 2x2 table (smoking vs. lung cancer) for men and women separately – To control for multiple confounders simultaneously, stratify by pairs (or triplets or higher) of confounding factors. For example, to control for gender and race/ethnicity determine the OR for smoking vs. lung cancer in multiple strata: white women, black women, Hispanic women, white men, black men, Hispanic men,etc. 41 2014 Page 62
- 63. • (From the earlier example): Goal: create a summary or “adjusted” estimate for the relationship between matches and lung cancer while adjusting for the two levels of smoking (the potential confounder) • This process is analgous to the standardization of rates earlier in the course—in those examples the purpose of adjustment was to remove the confounding effect of age on the relationship between populations (A vs. B etc.) and rates of disease or death. • In the present example the goal is to remove the confounding effect of smoking on the relationship between matches and lung cancer. 42 2014 Page 63
- 64. Confounding:Types of summary estimators to determine uniform effect over strata • Mantel-Haenszel – We will use this estimator in the present course – Resistant to the effects of small strata or cells with a value of “0” – Computationally a piece of cake • Directly pooled estimators (e.g. Woolf) – Sensitive to small strata and cells with value “0” – Computationally messy but doable • Maximum likelihood – The most “appropriate” estimator – Resistant to the effects of small strata or cells with a value of “0” – Computationally challenging 43 2014 Page 64
- 65. Confounding: smoking, matches, and lung cancer • ORpooled = 8.84 (7.2, 10.9) • ORsmokers = 1.0 (0.6, 1.5) • ORnonsmokers = 1.0 (0.5, 2.0) Pooled Cancer No cancer Matches No Matches Smokers Matches 820 180 Cancer 810 340 660 No cancer 270 No Matches Non-smoker Matches No Matches 90 Cancer 10 90 30 No cancer 70 630 44 2014 Page 65
- 66. An aside: Terminology • Pooled = combined = collapsed = unadjusted • Adjusted = summary = weighted, etc. – All of these reflect some adjustment process such as Mantel-Haenszel or Woolf or maximum likelihood estimation to weight the strata and develop confidence intervals about the estimate. 45 2014 Page 66
- 67. Confounding:Notation used in Mantel- Haenszel estimators of relative risk • Notation for case-control or cohort studies with count data Case-control: RR = OR = ad / bc Cohort: RR = Ie I0 46 = a / (a + b) c/ (c + d) Cases Controls Total Exposed Nonexposed a c b d a + b c + d Total a + c b + d a + b + c + d = T 2014 Page 67
- 68. Confounding:Notation used in Mantel- Haenszel estimators of relative risk (cont.) • Notation for cohort studies with person-time data RR = Ie I0 = a / PY1 47 c / PY0 Cases Controls Exposed Nonexposed a c --- --- PY1 PY0 Total a + c T 2014 Page 68
- 69. Confounding:Mantel-Haenszel estimators of relative risk for stratified data Case-Control Study: RRMH = ∑(ad / T)i ∑(bc / T)i Cohort Study with Count Denominators: RRMH = ∑{a(c + d) / T}i ∑{b(a + b) / T}I Cohort Study with Person-years Denominators: RRMH = ∑{a(PY0 ) / T}i ∑{b(PY1 ) / T}i 48 2014 Page 69
- 70. Confounding: smoking, matches, and lung cancer • ORpooled = 8.84 (7.2, 10.9) • ORsmokers = 1.0 (0.6, 1.5) • ORnonsmokers = 1.0 (0.5, 2.0) No Matches 90 630 51 Pooled Cancer No cancer Matches 820 340 No Matches 180 660 Smokers Cancer No cancer Matches 810 270 No Matches 90 30 Non-smoker Cancer No cancer Matches 10 70 2014 Page 70
- 71. Confounding:Mantel-Haenszel estimators of relative risk for stratified data (smoking, matches, lung cancer RRMH = ∑(ad / T)i / ∑(bc / T)i Numerator of MH estimator: • For smokers: (ad/T)=(810*30)/1200=20.25; • For nonsmokers: (ad/T)=(10*630)/800=7.88; • Add these together: 20.25 + 7.88=28.13 (numerator) Denominator of MH estimator: • For smokers: (bc/T)=(270*90)/1200=20.25; • For nonsmokers: (bc/T)=(90*70)/800=7.88; • Add these together: 20.25 + 7.88=28.13 •ORMH = 28.13 / 28.13 = 1.0 (as expected since both stratified OR’s were = 1.0) •Be sure to try this on stratified data in which the two strata are not exactly equal to each other (but also not so different as to suggest that effect modification is present 52 2014 Page 71
- 72. Confounding:Interpretation of ORMH • If ORMH (=1.0 in this example) “differs meaningfully” from ORunadjusted (=8.8 in this example) then confounding is present • What does “differs meaningfully” mean – This is a matter of judgment based on biologic/clinical sense rather than on a statistical test – Even if they “differ” only slightly, generally the ORMH rather than the ORcombined is reported as the summary effect estimate • But what is one disadvantage of reporting ORMH ? – Although there do exist statistical tests of confounding they are not widely recommended (these tests evaluate53 Ho: OR MH = OR unadjusted 2014 Page 72
- 73. 67 JC: test of homogeneity 2014 Page 73
- 74. Hennekens, 1987, p305 54 2014 Page 74
- 75. 55 2014 Page 75
- 76. 56 2014 Page 76
- 77. Review what the X^2 means in this context. 58 2014 Page 77
- 78. 59 2014 Page 78
- 79. • Confounding “pulls” the observed association away from the true association – It can either exaggerate/over-estimate the true association (positive confounding) • Example – RRcausal = 1.0 – RR observed = 3.0 or – It can hide/under-estimate the true association (negative confounding) • Example – RRcausal = 3.0 – RR observed = 1.0 Direction of Confounding Bias 40 2014 Page 79
- 80. Confounding:Summary of steps to evaluate confounding Table 12-10. Steps for the control of confounding and the evaluation of effect modification through stratified analysis 1. Stratify by levels of the potential confounding factor. 2. Compute stratum-specific unconfounded relative risk estimates. 3. Evaluate similarity of the stratum-specific estimates by either eyeballing or performing test of statistical significance. (More on this step later) 4. If the effect is thought to be uniform, calculate a pooled unconfounded summary estimate using RRMH . If effect is not uniform (i.e. effect modification is present, skip to step 6) 5. Perform hypothesis testing on the unconfounded estimate, using Mantel-Haenszel chi-square and compute confidence interval. 6. If effect is not thought to be uniform (i.e., if effect modification is present): a. Report stratum-specific estimates, results of hypothesis testing, and confidence intervals for each estimate b.If desired, calculate a summary unconfounded estimate using a standar6 d6 ized formula 2014 Page 80
- 81. 67 JC: test of homogeneity 2014 Page 81
- 82. 68 Effect modification (Interaction) • Goals of stratification of data – Evaluate and reduce/remove confounding – Evaluate and describe effect modification • Description of effect modification – A change in the magnitude of an effect measure (between exposure and disease) according to the level of some third variable – What two “classes” of effect measures have we used so far in the course? 2014 Page 82
- 83. Effect modification: example #1 • Disease incidence by exposure and age – Does the relationship between exposure and disease change over the value of the potential confounder (age)? How? 69 2014 Page 83
- 84. Effect modification: example #2 • Disease incidence by exposure and age • Does the relationship between exposure and disease change over the value of the potential confounder (age)? How? Rothman ’86 (p 178) 70 2014 Page 84
- 85. Effect modification: contrast with confounding • Confounding – A bias that an investigator hopes to remove – A nuisance that may or may not be present in a given study design • Properties of a confounding variable: (Rothman, p123): – a) be a risk factor for disease among the non-exposed; – b) be associated with the exposure variable; and – c) not be an intermediate step in the “causal pathway” 71 2014 Page 85
- 86. Effect modification: contrast with confounding • Effect modification – A more detailed description of the “true” relationship between the exposure and the outcome – Effect modification is a finding to be reported (even celebrated), not a bias to be eliminated – Effect modification is a “natural phenomenon” that exists independently of the study design – The presence and interpretation of effect modification depends upon the choice of effect measure (ratio vs. difference) 72 2014 Page 86
- 87. 73 Some lingo • Covariate – Confounder, potential confounder – Effect modification, interaction – Intermediate variable 2014 Page 87
- 88. Effect modification: contrast with confounding • Note that for any association under study, a given factor may be: – Both a confounder and an effect modifier or – A confounder but not an effect modifier or An effect modifier but not a confounder or – neither 74 2014 Page 88
- 89. Examples of confounding/effect modification 76 Level 1 Level 2 Crude/ collapsed/ Combined “unadjusted” Uniform estimate (ORMH ) / “adjusted” Confounding present Interaction present 4.0 4.0 4.0 4.0 NO NO 4.0 0.25 1.0 1.0 NO YES 1.0 1.0 8.4 1.0 YES NO 4.0 0.25 1.0 2.0 YES (?relevance) YES 2014 Page 89
- 90. 77 2014 Page 90
- 91. Effect modification: test of homogeneity • Null hypothesis: The individual stratified estimates of the effect do not differ from some uniform estimate of effect (such as a Mantel Haenszel estimator) • Notation: – – N is the number of strata (N=2 in our smoking/matches example); – ln^Ri is the natural logarithm of the estimated (hence the “^”) effect measure for each stratum (ORi in our example); – ln^R is the natural logarithm of the uniform effect estimate (e.g. ORMH in X 2 (N-1) is chi-square with (N-1) degrees of freedom; our example—the computer will use the maximum likelihood estimate) • One formula to test homogeneity: X 2 (N-1) = ∑ [ln(^ Ri ) – ln(RMH )]2 Var[ln(^ Ri )] N i= 1 78 JC: Comment on choice of signifciance level for test of homogeneity2014 Page 91
- 92. Paradox • If effect modification is present, a uniform estimator of effect (such as ORMH ) cannot (or at least should not) be reported. • However, in order to determine if effect modification is present, it is necessary to calculate the value of a uniform estimator of effect (such as ORMH ) because it is needed in the calculation of the test of homogeneity. 79 2014 Page 92
- 93. Effect modification: test of homogeneity (or is heterogeneity?) • Comments – If the test of homogeneity is “significant” (=“reject homogeneity”) this is evidence that there is heterogeneity (i.e. no homogeneity) and that effect modification may be present. • (Null hypothesis: The individual stratified estimates of the effect do not differ from some uniform estimate of effect) – The choice of a significance level (e.g. p < 0.05) is somewhat open to interpretation. • One “conservative” approach, because of inherent limitations in the power of the test of homogeneity, is to treat the data as if interaction is present for p < 0.20). • In other words, one would rather err on the side of assuming that interaction is present (and reporting the stratified estimates of effect) than on reporting a uniform estimate that may not be true across strata. 80 2014 Page 93
- 94. UC Berkeley 34 2014 Page 94
- 95. 81 2014 Page 95
- 96. Additive versus multiplicative scale effect modification ● Notation: RXZ ● No additive interaction if (R11 – R01) = (R10 – R00) ○ Rewrite as: (R11-R01)-(R10-R00)=0 ● In words: Difference in risk for (X=1 vs. X=0) when Z=1 is equal to difference in risk for (X=1 vs. X=0) when Z=0 ● Note: the values R11, R10, etc. are risks (not counts) 2014 Page 96
- 97. Additive versus multiplicative scale effect modification ● Notation: RXZ ● No multiplicative interaction if (R11/R01)=(R10/R00) Rewrite as: (R11/R01)/(R10/R00)=1 ● In words: Ratio of risks/rates when X=1 vs. X=0 when Z=1 is equal to ratio of risks/rates when X=1 vs. X=0 when Z=0 2014 Page 97
- 98. Effect modification is scale-dependent • Evidence for effect modification/statistical interaction if the RR or the AR differs between two groups • However, effect modification/statistical interaction is scale-dependent – If you do not have interaction on the additive scale (AR is homogenous) then you will have interaction on the multiplicative scale (RR must be heterogeneous) – If you do not have interaction on the multiplicative scale (RR is homogenous) then you will have interaction on the additive scale (AR must be heterogeneous) – Note: It is common to have evidence of interaction on both scales. 2014 Page 98
- 99. Example ● No additive scale interaction if (R11-R01)-(R10-R00)=0 ● No relative scale interaction if (R11/R01)/(R10/R00)=1 ● Additive scale: (60-20) - (50-10) = 0 ○ Interaction not present on the additive scale ● Relative scale: (60/20) / (50/10)=0.6 ○ Interaction present on the relative scale Z=1 Z=0 X=1 60 50 X=0 20 10 2014 Page 99
- 100. Example ● No additive scale interaction if (R11-R01)-(R10-R00)=0 ● No relative scale interaction if (R11/R01)/(R10/R00)=1 ● Additive scale: (60-20) - (30-10) = 20 ○ Interaction present on the additive scale ● Relative scale: (60/20) / (30/10)=1 ○ Interaction not present on the relative scale Z=1 Z=0 X=1 60 30 X=0 20 10 2014 Page 100
- 101. Logistic Regression (time permitting) 2014 Page 101
- 102. Confounding: smoking, matches, and lung cancer ’ • ORpooled = 21.0 (16.3, 27.1) • ORmatches = 21.0 (10.5, 46.2) • ORno matches = 21.0 (12.9, 34.7) • Discuss your intuitions about the 95% CI s Pooled Cancer No cancer Smoking No Smoking Matches Smoking 900 100 Cancer 810 300 700 No cancer 270 No Smoking No matches Smoking No Smoking 10 Cancer 90 90 70 No cancer 30 630 84 2014 Page 102
- 103. A brief introduction to logistic regression Let X1 = smoking (1=yes; 0=no) Let X2 = matches (1=yes; 0=no) Let Cancer = cancer (1=yes; 0=no) Recall earlier tables: OR=21.0 OR=21.0 OR=21.0 Conclusions: No confounding by matches of the relationship between smoking and lung cancer; no effect modification by matches of the relationship between smoking and lung cancer 85 Collapsed Cancer =1 Cancer=0 X1 =1 900 300 X1 =0 100 700 X2 =1 Cancer=1 No Cancer=0 X2 =0 Cancer=1 No Cancer=0 X1 =1 810 270 X1 =1 90 30 X1 =0 10 70 X1 =0 90 630 2014 Page 103
- 104. Data structure for computer analysis • Most computer programs would want to see the data for the individual subjects in the study in the following form: H 0 0 0 86 Subject ID X1 X2 Cancer How many? A 1 1 1 B 1 1 0 C 0 1 1 D 0 1 0 E 1 0 1 F 1 0 0 G 0 0 1 2014 Page 104
- 105. Data structure for computer analysis • Most computer programs would want to see the data for the individual subjects in the study in the following form: 87 Subject ID X1 X2 Cancer How many? A 1 1 1 810 of these B 1 1 0 270 of these C 0 1 1 10 of these D 0 1 0 70 of these E 1 0 1 90 of these F 1 0 0 30 of these G 0 0 1 90 of these H 0 0 0 630 of these 2014 Page 105
- 106. 88 The basic logistic equation for this problem • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • ln (odds of disease) = a + b1 (smoking) + b2 (matches) + b3 (smoking)(matches) 2014 Page 106
- 107. Solving a logistic equation • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 0 and X2 = 0, solve for “a” • ln (odds) = a = ln ( ) = • a = • So now: ln (odds) = 89 2014 Page 107
- 108. OR=21.0 OR=21.0 90 X2 =1 Cancer=1 No Cancer=0 X2 =0 Cancer=1 No Cancer=0 X1 =1 810 270 X1 =1 90 30 X1 =0 10 70 X1 =0 90 630 2014 Page 108
- 109. Solving a logistic equation • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 0 and X2 = 0, solve for “a” • ln (odds) = a = ln (90/630) = -1.946 • a = -1.946 • So now: ln (odds) = -1.946 + b1 X1 + b2 X2 + b3 X1 X2 91 2014 Page 109
- 110. 92 Solving a logistic equation (cont.) • When X1 = 1 and X2 = 0, solve for b1 • ln (odds) = • b1 = • So now: ln (odds) = 2014 Page 110
- 111. 93 OR=21.0 OR=21.0 X2 =1 Cancer=1 No Cancer=0 X2 =0 Cancer=1 No Cancer=0 X1 =1 810 270 X1 =1 90 30 X1 =0 10 70 X1 =0 90 630 2014 Page 111
- 112. 94 Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 1 and X2 = 0, solve for b1 • ln (odds) = ln (90/30) = 1.099 = -1.946 + b1 • b1 = 3.045 • So now: ln (odds) = -1.946 + 3.045X1 + b2 X2 + b3 X1 X2 2014 Page 112
- 113. 95 Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 0 and X2 = 1, solve for b2 : • ln (odds) = ln ( ) = • b2 = • So now: ln (odds) = 2014 Page 113
- 114. 96 X2 =1 Cancer=1 No Cancer=0 X2 =0 Cancer=1 No Cancer=0 X1 =1 810 270 X1 =1 90 30 X1 =0 10 70 X1 =0 90 630 OR=21.0 OR=21.0 2014 Page 114
- 115. 97 Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 0 and X2 = 1, solve for b2 : • ln (odds) = ln (10/70) = -1.946 + 0 + b2 X2 + 0 • b2 = 0 • So now: ln (odds) = -1.946 + 3.045X1 + 0 + b3 X1 X2 2014 Page 115
- 116. Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 1 and X2 = 1 then: • ln (odds) = • ln (odds) = • Solve for b3 • ln (odds) = • b3 = • So now: ln (odds) = 98 2014 Page 116
- 117. 99 X2 =1 Cancer=1 No Cancer=0 X2 =0 Cancer=1 No Cancer=0 X1 =1 810 270 X1 =1 90 30 X1 =0 10 70 X1 =0 90 630 OR=21.0 OR=21.0 2014 Page 117
- 118. Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • When X1 = 1 and X2 = 1 then: • ln (odds) = -1.946 + b1 + b2 + b3 • ln (odds) = -1.946 + 3.045 + 0 + b3 • Solve for b3 • ln (odds) = ln (810/270) = 1.099 = -1.946 + 3.045 + b3 • b3 = 0 • So now: ln (odds) = -1.946 + 3.045X1 + 0 + 0 100 2014 Page 118
- 119. Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • This simplifies (earlier calculations) to: – ln (odds) = -1.946 + 3.045X1 + 0 + 0 • One can now use the logistic equation to efficiently describe relationships in the table • Calculate the ln(odds) for a smoker who uses matches: ln (odds)= • Calculate the ln(odds) for a smoker who doesn’t use matches: ln(odds) = • Now calculate the odds ratio for (smokers vs. non-smokers// matches+) • At home, calculate the odds ratio for (smokers vs. non- smokers// matches-) 101 2014 Page 119
- 120. Solving a logistic equation (cont.) • ln (odds of disease) = a + b1 X1 + b2 X2 + b3 X1 X2 • This simplifies (earlier calculations) to: -1.946 + 3.045(X1 ) + 0(X2 ) + 0(X1 X2 ) • One can now use the logistic equation to efficiently describe relationships in the table • Calculate the ln(odds) for a smoker who uses matches (X1 = 1 and X2 = 1): ln (odds)= -1.946 + 3.045 = 1.099 • Calculate the ln(odds) for a smoker who doesn’t use matches (X1 = 1 and X2 = 0): ln(odds) = -1.946 + 3.045 = 1.099 • Now calculate the odds ratio for (smokers vs. non-smokers// matches) • At home, calculate the odds ratio for (smokers vs. non-smokers// 102 no matches) 2014 Page 120
- 121. Logistic Regression Using the logistic model model developed in class for the matches- smoking-lung cancer data (stratified by matches), evaluate the risk of lung cancer for: 1. (in-class) A smoker who uses matches vs. a non-smoker who uses matches. 2. (at home) A smoker who uses matches vs. a non-smoker who does not use matches SEPARATE ASSIGNMENT Develop a logistic model for the matches-smoking-lung cancer data (stratified by smoking status). Use this model to evaluate the risk of lung cancer for: 1. (at home) A user of matches who smokes vs. a non-user of matches who smokes. 2. (at home) A smoker who uses matches vs. a non-smoker who uses matches. Is this result consistent with that you arrived at in the 103 class example above? 2014 Page 121
- 122. Find OR for smokers (who use matches) vs. non-smokers (who use matches) For Smokers who use matches X1 = 1 X2 = 1 For non-smokers who use matches X1 = 0 X2 = 1 From prior slides we determined that: ln (odds) = -1.946 + 3.045 (X1 ) 105 2014 Page 122
- 123. For smokers who use matches (X1 = 1; X2 = 1) ln (odds) = -1.946 + 3.045 (1) = 1.0990 For non-smokers who use matches (X1 = 0; X2 = 1) ln (odds) = -1.946 + 0 + 0 + 0 = -1.946 We want to solve: ln OR = 1.0990 – (-1.946) = 3.045 eln OR = OR = e3.045 = 21.0 Therefore, the odds ratio (determined using logistic regression) comparing smokers using matches to non-smokers using matches is 21.0. This agrees with the stratified data presented earlier. 106 2014 Page 123
- 124. Confounding: smoking, matches, and lung cancer • ORpooled = 21.0 (16.3, 27.1) • ORmatches = 21.0 (10.5, 46.2) • ORno matches = 21.0 (12.9, 34.7) • Discuss your intuitions about the 95% CI s’ No Smoking 90 630 107 Pooled Cancer No cancer Smoking 900 300 No Smoking 100 700 Matches Cancer No cancer Smoking 810 270 No Smoking 10 70 No matches Cancer No cancer Smoking 90 30 2014 Page 124
- 125. Some concluding comments on logistic regression • Interpretations of the final logistic equation for these data: ln (odds of disease) = a + b1 (smoking) + b2 (matches) + b3 (smoking)(matches) ln(odds) = -1.946 + 3.045(smoking) + 0(matches) + 0(matches)(smoking) • This equation describes the data whether stratified either by matches or by smoking. • The relationship of multiple variables may be simultaneously adjusted for by the the logistic equations • The estimates of the coefficients for the equation are derived through maximum likelihood techniques • This technique is very widely used in epidemiologic (and other) applications when the outcome variable of interest is dichotomous. 108 2014 Page 125
- 126. Some concluding comments on logistic regression • Comments – Having multiple strata (how this technique makes possible) – Test of homogeneity (b3 ) Maximum likelihood estimation for coefficient estimation • Modifications of logistic regression exist for coping with – Outcome variables with multiple levels = polytomous logistic regression – Studies in which matching was used = Conditional logistic regression 109 2014 Page 126
- 127. . use http://www.stata-press.com/data/r8/lbw storage display value variable name type format variable label -------------------------------------------------------------- ----------------- 110 id low int byte %8.0g %8.0g identification code birth weight<2500g age lwt race smoke ptl ht ui ftv byte int byte byte byte byte byte byte %8.0g %8.0g %8.0g %8.0g %8.0g %8.0g %8.0g %8.0g age of weight race smoked mother at last menstrual period during pregnancy premature labor history (count) has history of hypertension presence, uterine irritability number of visits to physician during 1st trimester birth weight (grams)bwt int %8.0g 2014 Page 127
- 128. Special (and very useful) STATA command “xi” (=“interaction expansion”) • xi: logistic low age lowwt i.race smoke pt1 ht ui • In this example, a variable named “race” has three levels (e.g. white/hispanic/black) that might be coded as “0=white”; “1=hispanic”; “2=black” • The combined use of xi and i.race directs STATA to analyze all levels of race (and compare them to level 1)— this can be a HUGE time-saver (avoids the user having to manually recode such variables)! 111 2014 Page 128
- 129. Assignments • Write the logistic model describing these data (next slide). • What is the risk of low birth weight (LBW) for a smoker, adjusted for all other variables? • How can the 95% CI be determined? • What is the risk of LBW for an Hispanic baby (compared to a white baby)? • What is the risk of LBW for a black baby (compared to an Hispanic baby)? 112 2014 Page 129
- 130. 113 2014 Page 130
- 131. 114 Discuss intercept 2014 Page 131
- 132. 115 2014 Page 132