16: Odds Ratios [from 16: Odds Ratios [from case-control studies]case-control studies]

Case-control studies get Case-control studies get around several limitations of around several limitations of

cohort studiescohort studies

Cohort Studies (Prior Cohort Studies (Prior Chapter)Chapter)

• Use incidences to assess riskUse incidences to assess risk• Exposed cohort Exposed cohort incidence incidence11

• Non-exposed cohort Non-exposed cohort incidence incidence00

• Compare incidences via risk ratio Compare incidences via risk ratio (())

0

1

incidence

incidenceˆ

Hindrances in Cohort Hindrances in Cohort StudiesStudies

• Long induction between exposure & Long induction between exposure & disease may cause delaysdisease may cause delays

• Study of rare diseases require large Study of rare diseases require large sample sizes to sample sizes to accrue sufficient numbersaccrue sufficient numbers

• When studying many people When studying many people information information by necessity can be limited in scope & by necessity can be limited in scope & accuracy accuracy

• Case-control studies were developed to Case-control studies were developed to help overcome some of these limitations help overcome some of these limitations

Levin et al. (1950) Levin et al. (1950) Historically important study (not in Reader)Historically important study (not in Reader)

• Selection criteriaSelection criteria• 236 lung cancer cases -- 156 (66%) 236 lung cancer cases -- 156 (66%)

smokedsmoked• 481 non-cancerous conditions 481 non-cancerous conditions

(“controls”) -- 212 (44%) smoked(“controls”) -- 212 (44%) smoked

• Although incidences of lung cancer Although incidences of lung cancer cannot be determined from data, we cannot be determined from data, we see an association between smoking see an association between smoking and lung cancerand lung cancer

How do we quantify risk How do we quantify risk from case-control data?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, 1269-1275.

• Miettinen, 1976Miettinen, 1976Miettinen, O. (1976). Estimability and estimation in case-referent studies. American Journal of Epidemiology, 103, 226-235.

Cornfield, 1951Cornfield, 1951

• Justified use of Justified use of odds ratioodds ratio as as estimate of relative risk estimate of relative risk

• Recognized potential bias in Recognized potential bias in selection of cases and controlsselection of cases and controls

Miettinen, 1976Miettinen, 1976

• Conceptualized case-control study Conceptualized case-control study as as nested in a populationnested in a population• all population cases studiedall population cases studied • sample of population non-cases sample of population non-cases

studiedstudied

Miettinen (1976) Density Miettinen (1976) Density SamplingSampling

• Imagine 5Imagine 5 people followed over timepeople followed over time• At time At time tt1 1 (shaded)(shaded), D occurs in person 1, D occurs in person 1• You select at random a non-cases at this timeYou select at random a non-cases at this time

Time

5

4

3

2

1

t1 t2

DD

Note: person #2 becomes a case later on but can still serve as a control at t1

How incidence density sampling How incidence density sampling worksworks

The ratio of exposed to non-exposed The ratio of exposed to non-exposed time in the controls estimates the ratio time in the controls estimates the ratio of exposed to non-exposed controls in of exposed to non-exposed controls in

the populationthe population(see EKS for details)(see EKS for details)

Data AnalysisData 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 ++

Disease -Disease - TotalTotal

Exposed +Exposed + AA11 BB11 NN11

Exposed -Exposed - AA00 BB00 NN00

TotalTotal MM11 MM00 NN

Calculate Odds Ratio (Calculate Odds Ratio (^))

Disease Disease ++

Disease -Disease - TotalTotal

Exposed +Exposed + AA11 BB11 NN11

Exposed -Exposed - AA00 BB00 NN00

TotalTotal MM11 MM00 NN

01

01

AB

BA

0

11 cases odds, exposureA

Ao

0

10 controls odds, exposureB

Bo

01

01

01

01

0

1

/

/ratio odds

AB

BA

BB

AA

o

o

Cross-product Cross-product ratioratio

Illustrative Example Illustrative Example (Breslow & Day, 1980)(Breslow & Day, 1980)

• Dataset = bd1.sav Dataset = bd1.sav • Exposure variable (Exposure variable (alc2alc2) = Alcohol use dichotomized) = Alcohol use dichotomized• Disease variable (Disease variable (casecase) = Esophageal cancer) = Esophageal cancer

AlcoholAlcohol CaseCase ControlControl TotalTotal

80 g/day80 g/day 96 109 205

< 80 g/day< 80 g/day 104 666 770

TotalTotal 200 775 975

64.5)104)(109(

)666)(96(ˆ

Interpretation of Odds Interpretation of Odds RatioRatio

• Odds ratios are Odds ratios are relative risk relative risk estimatesestimates• Risk multiplierRisk multiplier

• e.g., odds ratio of 5.64 suggests 5.64× risk e.g., odds ratio of 5.64 suggests 5.64× risk with exposure with exposure

• Percent relative risk difference Percent relative risk difference = = (odds ratio – 1) × 100%(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) Percent relative risk difference = (5.64 – 1)

× 100% = 464%× 100% = 464%

95% Confidence 95% Confidence IntervalInterval

• CalculationsCalculations• Convert Convert ψψ^ to ln scale^ to ln scale• seselnlnψψ^ = sqrt( = sqrt(AA11

-1-1 + + AA00-1-1 + + BB11

-1-1 + + BB00-1-1))

• 95% CI for ln95% CI for lnψ ψ = (ln = (lnψψ^) ± (1.96)(se)^) ± (1.96)(se)• Exponentiate limitsExponentiate limits

• Illustrative exampleIllustrative example• ln(ln(ψψ^) = ln(5.640) = 1.730^) = ln(5.640) = 1.730• seselnlnψψ^ = sqrt(96 = sqrt(96-1-1 + 104 + 104-1-1 + 109 + 109-1-1 + 666 + 666-1-1) = 0.1752) = 0.1752• 95% CI for ln95% CI for lnψ = ψ = 1.730 ± (1.96)(0.1752) = (1.387, 1.730 ± (1.96)(0.1752) = (1.387,

2.073)2.073)• 95% CI for 95% CI for ψ = eψ = e(1.387, 2.073)(1.387, 2.073) = (4.00, 7.95) = (4.00, 7.95)

SPSS OutputSPSS Output

Ignore “For cohort” lines when data are case-control

Odds ratio point estimate and confidence limits

Interpretation of the Interpretation of the 95% CI95% CI

• Locates odds ratio parameter (Locates odds ratio parameter (ψψ) ) with 95% confidence with 95% confidence

• Illustrative example: 95% Illustrative example: 95% confident odds ratio confident odds ratio parameter parameter is is no less than 4.00 and no more no less than 4.00 and no more than 7.95than 7.95

• Confidence interval width provides Confidence interval width provides information about precisioninformation about precision

Testing Testing HH00: : ψψ = 1 with = 1 with the Confidence Intervalthe Confidence Interval

• 95% CI corresponds to 95% CI corresponds to = .05 = .05• If 95% CI for odds ratio excludes 1 If 95% CI for odds ratio excludes 1 odds odds

ratio is significant ratio is significant • e.g., (95% CI: 4.00, 7.95) is a significant positive e.g., (95% CI: 4.00, 7.95) is a significant positive

associationassociation• e.g., (95% CI: 0.25, 0.65) is a significant negative e.g., (95% CI: 0.25, 0.65) is a significant negative

associationassociation• If 95% CI includes 1 If 95% CI includes 1 odds ratio NOT odds ratio NOT

significantsignificant• e.g., (95% CI: 0.80, 1.15) is not significant (i.e., e.g., (95% CI: 0.80, 1.15) is not significant (i.e.,

cannot rule out odds ratio parameter of 1 with cannot rule out odds ratio parameter of 1 with 95% confidence95% confidence

p p valuevalue

• HH00: : ψψ = 1 (“no association”) = 1 (“no association”)

• Use chi-square test (Pearson’s or Use chi-square test (Pearson’s or Yates’) or Fisher’s test, as covered Yates’) or Fisher’s test, as covered in prior chaptersin prior chapters

i

iiPearson E

EO 22 )(

i

iiYates E

EO 22 )5.0|(|

Fisher’s exact test by computerFisher’s exact test by computer

Chi-Square, PearsonChi-Square, PearsonOBSEOBSERVEDRVED

D+D+ D-D- TotalTotal

E+E+ 96 109 205

E-E- 104 666 770

TotalTotal 200 775 975

EXPECEXPECTEDTED D+D+ D-D- TotalTotal

E+E+42.051

162.949

205

E-E-157.949

612.051

770

TotalTotal 200 775 975

2Pearson'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

= sqrt(110.256) = 10.50 off chart (way into tail) p < .0001

Chi-Square, YatesChi-Square, YatesOBSEOBSERVEDRVED

D+D+ D-D- TotalTotal

E+E+ 96 109 205

E-E- 104 666 770

TotalTotal 200 775 975

EXPECEXPECTEDTED D+D+ D-D- TotalTotal

E+E+42.051

162.949

205

E-E-157.949

612.051

770

TotalTotal 200 775 975

2Pearson'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

= sqrt(108.22) = 10.40 p < .0001

SPSS OutputSPSS Output

Pearson = uncorrected Yates = continuity correctedFisher’s unnecessary here

Linear-by-linear not covered

Interpreting the Interpreting the p p valuevalue

• "If the null hypothesis were "If the null hypothesis were correct, the probability of correct, the probability of observing the data is observing the data is pp““

• e.g., e.g., p p = .000 suggests association = .000 suggests association is unlikely due to chance (we can is unlikely due to chance (we can be confident in rejecting be confident in rejecting HH00))

Validity! Validity!

• Before you get too carried away with Before you get too carried away with the odds ratio (or any other statistic), the odds ratio (or any other statistic), remember they assume validityremember they assume validity• No info bias (exposure and disease No info bias (exposure and disease

accurately classified)accurately classified)• No selection bias (cases and controls are No selection bias (cases and controls are

fair reflection of population analogues)fair reflection of population analogues)• No confoundingNo confounding

Matched-PairsMatched-Pairs• Matching can be employed to help Matching can be employed to help

control for confoundingcontrol for confounding• e.g., matching on age and sex e.g., matching on age and sex

• Each Each pairpair represents an observation represents an observation• Classify each Classify each pairpair

• Concordant pairsConcordant 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 pairsDiscordant 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 Tabulation & Notation

Control Control

exposedexposedControl Control

non-exposednon-exposedCase Case

exposedexposedt u

Case Case

non-non-exposedexposed

v w

v

u

Tabular display is optionalTabular display is optional

Odds ratio for matched Odds ratio for matched pair data:pair data:

Example (Matched Example (Matched Pairs)Pairs)

Control Control

exposedexposedControl Control

non-exposednon-exposedCase Case

exposedexposed5 30

Case Case

non-non-exposedexposed

10 5

00.310

30ˆ

Confidence Interval for Confidence Interval for Matched PairsMatched Pairs

3651.010

1

30

1ˆ se

)14.6 ,47.1(for CI%95 )8142.1 ,3838.0( e

00.310

30ˆ

)8142.1 ,3838.0(

7156.01.0986

651)(1.96)(0.3)00.3ln(

)(1.96)(ˆlnlnfor CI%95 ˆln

se

McNemar’s Test for Matched McNemar’s Test for Matched PairsPairs

HH00: : ψψ = 1 (“no association”) = 1 (“no association”)

025.91030

)1|1030(|)1|(| 222

vu

vuMcN

00.3025.9 0027.p chi-table chi-table

df = 1 for McNemar’sdf = 1 for McNemar’sOK to convert to chi-statisticOK to convert to chi-statistic