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 riskUse incidences to assess risk Exposed cohort  incidence 1Exposed cohort  incidence 1 Non-exposed cohort  incidence 0Non-exposed cohort  incidence 0 Compare incidences via risk ratio (  )Compare incidences via risk ratio (  )

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

Levin et al. (1950) Historically important study (not in Reader) Selection criteriaSelection criteria 236 lung cancer cases (66%) smoked236 lung cancer cases (66%) smoked 481 non-cancerous conditions (“controls”) (44%) smoked481 non-cancerous conditions (“controls”) (44%) smoked Although incidences of lung cancer cannot be determined from data, we see an association between smoking and lung cancerAlthough incidences of lung cancer cannot be determined from data, we see an association between smoking and lung cancer

How do we quantify risk from case-control data? Two article shed light on this questionTwo article shed light on this question Cornfield, 1951Cornfield, 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, Miettinen, 1976Miettinen, 1976 Miettinen, O. (1976). Estimability and estimation in case-referent studies. American Journal of Epidemiology, 103,

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

Miettinen, 1976 Conceptualized case-control study as nested in a populationConceptualized case-control study as nested in a population all population cases studiedall population cases studied sample of population non-cases studiedsample of population non-cases studied

Miettinen (1976) Density Sampling Imagine 5 people followed over timeImagine 5 people followed over time At time t 1 (shaded), D occurs in person 1At time t 1 (shaded), D occurs in person 1 You select at random a non-cases at this timeYou 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 t 1

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)

Data Analysis Ascertain exposure status in cases and controlsAscertain exposure status in cases and controls Cross-tabulate counts to form 2-by-2 tableCross-tabulate counts to form 2-by-2 table Notation same as prior chapterNotation same as prior chapter Disease + Disease - Total Exposed + A1A1A1A1 B1B1B1B1 N1N1N1N1 Exposed - A0A0A0A0 B0B0B0B0 N0N0N0N0 Total M1M1M1M1 M0M0M0M0N

Calculate Odds Ratio () Calculate Odds Ratio (  ^ ) Disease + Disease - Total Exposed + A1A1A1A1 B1B1B1B1 N1N1N1N1 Exposed - A0A0A0A0 B0B0B0B0 N0N0N0N0 Total M1M1M1M1 M0M0M0M0N Cross-product ratio

Illustrative Example (Breslow & Day, 1980) Dataset = bd1.savDataset = bd1.sav Exposure variable ( alc2 ) = Alcohol use dichotomizedExposure variable ( alc2 ) = Alcohol use dichotomized Disease variable ( case ) = Esophageal cancerDisease variable ( case ) = Esophageal cancer Alcohol CaseControlTotal  80 g/day < 80 g/day Total

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

95% Confidence Interval CalculationsCalculations Convert ψ ^ to ln scaleConvert ψ ^ to ln scale se ln ψ ^ = sqrt( A A B B 0 -1 ) se ln ψ ^ = sqrt( A A B B 0 -1 ) 95% CI for ln ψ = (ln ψ ^) ± (1.96)(se)95% CI for ln ψ = (ln ψ ^) ± (1.96)(se) Exponentiate limitsExponentiate limits Illustrative exampleIllustrative example ln( ψ ^) = ln(5.640) = 1.730ln( ψ ^) = ln(5.640) = se ln ψ ^ = sqrt( ) = se ln ψ ^ = sqrt( ) = % CI for ln ψ = ± (1.96)(0.1752) = (1.387, 2.073)95% CI for ln ψ = ± (1.96)(0.1752) = (1.387, 2.073) 95% CI for ψ = e (1.387, 2.073) = (4.00, 7.95)95% CI for ψ = e (1.387, 2.073) = (4.00, 7.95)

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

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

Testing H 0 : ψ = 1 with the Confidence Interval 95% CI corresponds to  =.0595% CI corresponds to  =.05 If 95% CI for odds ratio excludes 1  odds ratio is significantIf 95% CI for odds ratio excludes 1  odds ratio is significant e.g., (95% CI: 4.00, 7.95) is a significant positive associatione.g., (95% CI: 4.00, 7.95) is a significant positive association e.g., (95% CI: 0.25, 0.65) is a significant negative associatione.g., (95% CI: 0.25, 0.65) is a significant negative association If 95% CI includes 1  odds ratio NOT significantIf 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% confidencee.g., (95% CI: 0.80, 1.15) is not significant (i.e., cannot rule out odds ratio parameter of 1 with 95% confidence

p value H 0 : ψ = 1 (“no association”) H 0 : ψ = 1 (“no association”) Use chi-square test (Pearson’s or Yates’) or Fisher’s test, as covered in prior chaptersUse chi-square test (Pearson’s or Yates’) or Fisher’s test, as covered in prior chapters Fisher’s exact test by computer

Chi-Square, Pearson OBSER VED D+D-Total E E Total EXPECT ED D+D-TotalE E Total  2 Pearson's = ( ) 2 / (109 – ) 2 / ( ) 2 / (666 – ) 2 / = =  = sqrt( ) =  off chart (way into tail)  p <.0001

Chi-Square, Yates OBSER VED D+D-Total E E Total EXPECT ED D+D-TotalE E Total  2 Pearson's = (| | - ½) 2 / (|109 – | - ½) 2 / (| | - ½) 2 / (|666 – | - ½) 2 / = =  = sqrt(108.22) =  p <.0001

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

Interpreting the p value "If the null hypothesis were correct, the probability of observing the data is p “"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 H 0 )e.g., p =.000 suggests association is unlikely due to chance (we can be confident in rejecting H 0 )

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

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

Tabulation & Notation ControlexposedControlnon-exposed Caseexposed tu Casenon-exposed vw Tabular display is optional Odds ratio for matched pair data:

Example (Matched Pairs) ControlexposedControlnon-exposed Caseexposed 530 Casenon-exposed 105

Confidence Interval for Matched Pairs

McNemar’s Test for Matched Pairs H 0 : ψ = 1 (“no association”)  chi-table  df = 1 for McNemar’s OK to convert to chi-statistic