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16: Odds Ratios [from case-control studies]

16: Odds Ratios [from case-control studies]. Case-control studies get around several limitations of cohort studies. Cohort Studies (Prior Chapter). Use incidences to assess risk Exposed cohort  incidence 1 Non-exposed cohort  incidence 0 Compare incidences via risk ratio ().

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16: Odds Ratios [from case-control studies]

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  1. 16: Odds Ratios [from case-control studies] Case-control studies get around several limitations of cohort studies

  2. Cohort Studies (Prior Chapter) • Use incidences to assess risk • Exposed cohort  incidence1 • Non-exposed cohort  incidence0 • Compare incidences via risk ratio ()

  3. Hindrances in Cohort Studies • Long induction between exposure & disease may cause delays • Study of rare diseases require large sample sizes to accrue sufficient numbers • When studying many people  information by necessity can be limited in scope & accuracy • Case-control studies were developed to help overcome some of these limitations

  4. Levin et al. (1950) Historically important study (not in Reader) • Selection criteria • 236 lung cancer cases -- 156 (66%) smoked • 481 non-cancerous conditions (“controls”) -- 212 (44%) smoked • Although incidences of lung cancer cannot be determined from data, we see an association between smoking and lung cancer

  5. How do we quantify risk from case-control data? • Two article shed light on this question • Cornfield, 1951 Cornfield, J. (1951). A method of estimating comparative rates from clinical data. Application to cancer of the lung, breast, and cervix. Journal of the National Cancer Institute, 11, 1269-1275. • Miettinen, 1976 Miettinen, O. (1976). Estimability and estimation in case-referent studies. American Journal of Epidemiology, 103, 226-235.

  6. Cornfield, 1951 • Justified use of odds ratio as estimate of relative risk • Recognized potential bias in selection of cases and controls

  7. Miettinen, 1976 • Conceptualized case-control study as nested in a population • all population cases studied • sample of population non-cases studied

  8. Miettinen (1976) Density Sampling • Imagine 5people followed over time • At time t1 (shaded), D occurs in person 1 • You select at random a non-cases at this time Note: person #2 becomes a case later on but can still serve as a control at t1

  9. How incidence density sampling works The ratio of exposed to non-exposed time in the controls estimates the ratio of exposed to non-exposed controls in the population(see EKS for details)

  10. Data Analysis • Ascertain exposure status in cases and controls • Cross-tabulate counts to form 2-by-2 table • Notation same as prior chapter

  11. Calculate Odds Ratio (^) Cross-product ratio

  12. Illustrative Example (Breslow & Day, 1980) • Dataset = bd1.sav • Exposure variable (alc2) = Alcohol use dichotomized • Disease variable (case) = Esophageal cancer

  13. Interpretation of Odds Ratio • Odds ratios are relative risk estimates • Risk multiplier • e.g., odds ratio of 5.64 suggests 5.64× risk with exposure • Percent relative risk difference = (odds ratio – 1) × 100% • e.g., odds ratio of 5.64 • Percent relative risk difference = (5.64 – 1) × 100% = 464%

  14. 95% Confidence Interval • Calculations • Convert ψ^ to ln scale • selnψ^ = sqrt(A1-1 + A0-1 + B1-1 + B0-1) • 95% CI for lnψ = (lnψ^) ± (1.96)(se) • Exponentiate limits • Illustrative example • ln(ψ^) = ln(5.640) = 1.730 • selnψ^ = sqrt(96-1 + 104-1 + 109-1 + 666-1) = 0.1752 • 95% CI for lnψ = 1.730 ± (1.96)(0.1752) = (1.387, 2.073) • 95% CI for ψ = e(1.387, 2.073) = (4.00, 7.95)

  15. SPSS Output Odds ratio point estimate and confidence limits Ignore “For cohort” lines when data are case-control

  16. Interpretation of the 95% CI • Locates odds ratio parameter (ψ) with 95% confidence • Illustrative example: 95% confident odds ratio parameter is no less than 4.00 and no more than 7.95 • Confidence interval width provides information about precision

  17. Testing H0: ψ = 1 with the Confidence Interval • 95% CI corresponds to a = .05 • If 95% CI for odds ratio excludes 1 odds ratio is significant • e.g., (95% CI: 4.00, 7.95) is a significant positive association • e.g., (95% CI: 0.25, 0.65) is a significant negative association • If 95% CI includes 1 odds ratio NOT significant • e.g., (95% CI: 0.80, 1.15) is not significant (i.e., cannot rule out odds ratio parameter of 1 with 95% confidence

  18. p value • H0: ψ = 1 (“no association”) • Use chi-square test (Pearson’s or Yates’) or Fisher’s test, as covered in prior chapters Fisher’s exact test by computer

  19. Chi-Square, Pearson c2Pearson's =  (96 - 42.051)2 / 42.051    + (109 – 162.949)2 / 162.949   +              (104 - 157.949)2 / 157.949 + (666 – 612.051)2 / 612.051            =  69.213                +    17.861                  +              18.427                +   4.755           =  110.256 c = sqrt(110.256) = 10.50  off chart (way into tail)  p < .0001

  20. Chi-Square, Yates c2Pearson's =  (|96 - 42.051| - ½)2 / 42.051    + (|109 – 162.949| - ½)2 / 162.949   +              (|104 - 157.949| - ½)2 / 157.949 + (|666 – 612.051| - ½)2 / 612.051            =  67.935                +    17.532                  +              18.087                +   4.668           =  108.221 c = sqrt(108.22) = 10.40  p < .0001

  21. SPSS Output Pearson = uncorrected Yates = continuity corrected Fisher’s unnecessary here Linear-by-linear not covered

  22. Interpreting the p value • "If the null hypothesis were correct, the probability of observing the data is p“ • e.g., p = .000 suggests association is unlikely due to chance (we can be confident in rejecting H0)

  23. Validity! • Before you get too carried away with the odds ratio (or any other statistic), remember they assume validity • No info bias (exposure and disease accurately classified) • No selection bias (cases and controls are fair reflection of population analogues) • No confounding

  24. Matched-Pairs • Matching can be employed to help control for confounding • e.g., matching on age and sex • Each pair represents an observation • Classify each pair • Concordant pairs • case is exposed & control is exposed • case is non-exposed & control is non-exposed • Discordant pairs • case is exposed & control is non-exposed • case is non-exposed & control is exposed

  25. Tabulation & Notation Tabular display is optional Odds ratio for matched pair data:

  26. Example (Matched Pairs)

  27. Confidence Interval for Matched Pairs

  28. McNemar’s Test for Matched Pairs H0: ψ = 1 (“no association”) df = 1 for McNemar’s OK to convert to chi-statistic  chi-table 

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