Measuring Associations Between Exposure and Outcomes

Measuring Measuring Associations Associations

Between Between Exposure and Exposure and

OutcomesOutcomes

Methods of analysisMethods of analysis

CrudeCrude AdjustedAdjusted

StratificationStratification StandardizationStandardization Stratification (Mantel Haenszel & Wolf)Stratification (Mantel Haenszel & Wolf)

Modeling (multiple regression)Modeling (multiple regression) Linear regressionLinear regression Logistic regressionLogistic regression Cox regressionCox regression Poisson regressionPoisson regression

Measures of Association Measures of Association can be based on:can be based on:

Absolute differences Between Absolute differences Between Groups (e.g., disease risk among Groups (e.g., disease risk among exposed – disease risk among exposed – disease risk among unexposed)unexposed)

Relative differences or ratios Relative differences or ratios Between Groups (e.g., disease risk Between Groups (e.g., disease risk ratio or relative risk: disease risk in ratio or relative risk: disease risk in exposed/disease risk in unexposed)exposed/disease risk in unexposed)

Measure of Measure of Public Health Public Health

ImpactImpact

Four closely related measure Four closely related measure are used:are used:

Attributable RiskAttributable Risk Attributable( Risk) fractionAttributable( Risk) fraction Population Attributable RiskPopulation Attributable Risk Population Attributable (Risk) Population Attributable (Risk)

fractionfraction

Attributable RiskAttributable Risk(AR)(AR)

The The IncidenceIncidence of disease in the of disease in the Exposed Exposed population whose disease population whose disease can be attributed to the exposure. can be attributed to the exposure.

AR=I AR=I ee –I –I uu

MIMIFree of MIFree of MITotals:Totals:

ExposureExposure

High BloodHigh Blood

PressurePressure 180180 982098201000010000

NormalNormal

PressurePressure 3030 997099701000010000

AR= 0.018 – 0.003= 0.015= 1.5% AR= 0.018 – 0.003= 0.015= 1.5% The cessation of the exposure would lower The cessation of the exposure would lower

the risk in the exposed group from 0.018 the risk in the exposed group from 0.018 to 0.0030to 0.0030

Vaccine EfficacyVaccine Efficacy

VE= I VE= I ee /I /I u u -- I I ee /I /I uu VE= RR-1

Attributable (Risk)Attributable (Risk)Fraction (ARF)Fraction (ARF)

TheThe proportion proportion of disease in the of disease in the exposed exposed population whose disease can be population whose disease can be attributed to the exposure.attributed to the exposure.

AR= (I AR= (I ee –I –I u u )/I )/I ee ARF=( RR-1)/RRARF=( RR-1)/RR

ARF = 0.018 – 0.003/ 0.018 * 100 = ARF = 0.018 – 0.003/ 0.018 * 100 = 83.3%83.3%

RR=0.018/0.003 = 6RR=0.018/0.003 = 6

ARF=( RR-1)/RR * 100=(6 – 1)/6 ARF=( RR-1)/RR * 100=(6 – 1)/6 *100= 83.3%*100= 83.3%

ARF= percent efficacyARF= percent efficacy Risk of dis. In vaccinated group= 5%Risk of dis. In vaccinated group= 5% Risk of dis. In the placebo group= 15%Risk of dis. In the placebo group= 15% ARF=Efficacy=((15 – 5) / 15) * 100 = ARF=Efficacy=((15 – 5) / 15) * 100 =

66.7% = (3-1)/3 * 100 = 66.7 % 66.7% = (3-1)/3 * 100 = 66.7 %

Population Attributable Population Attributable Risk (PAR)Risk (PAR)

The The IncidenceIncidence of disease in the of disease in the totaltotal population population whose disease can whose disease can be attributed to the exposure. be attributed to the exposure.

PAR=I PAR=I pp –I –I uu

Population Attributable Population Attributable (Risk) Fraction (PARF)(Risk) Fraction (PARF)

TheThe proportion proportion of disease in the of disease in the total total population whose disease can be population whose disease can be attributed to the exposure.attributed to the exposure.

The PARF is defined as the fraction The PARF is defined as the fraction of all cases (exposed and unexposed) of all cases (exposed and unexposed) that would not have occurred if that would not have occurred if exposure had not occurred.exposure had not occurred.

PARF= (I PARF= (I pp –I –I u u )/I )/I pp

PARF= (I PARF= (I pp –I –I u u )/I )/I pp

P=exposure prevalence=0.4P=exposure prevalence=0.4 Ie = 0.2Ie = 0.2 Iu = 0.15Iu = 0.15 I I pp = (Ie *0.4)+(Iu *0.6) =0.17 = (Ie *0.4)+(Iu *0.6) =0.17 PAF = (0.17 – 0.15) / 0.17 = 0.12PAF = (0.17 – 0.15) / 0.17 = 0.12

2-Miettinen or case-based 2-Miettinen or case-based formula:formula:

PARF=[(RR-1)/RR ]* CFPARF=[(RR-1)/RR ]* CF CF=number of exposed CF=number of exposed

cases/overall number of casescases/overall number of cases

PAF has two Formula:PAF has two Formula:

Relative differences or Relative differences or ratiosratios

For discrete variableFor discrete variable To assess causal associationsTo assess causal associations Examples: Relative Examples: Relative

Risk/Rate, Relative oddsRisk/Rate, Relative odds

Cohort StudyCohort Study

Diseased

Non-diseased

Totals: Risk odds

Exposure

Exposedaba+b a / a+b a / b

Unexposed

cdc+d c /c+d c / d

Totals:

Disease

a+cb+da+b+c+d

Odds in Exposed and Odds in Exposed and UnexposedUnexposed

Odds in exposed=( a / a+b) / 1- (a / Odds in exposed=( a / a+b) / 1- (a / a+b )a+b )

=(a / a+b) / (b / =(a / a+b) / (b / a+b) = a+b) = a/ba/b

Odds in unexposed=( c / c+d) / 1- Odds in unexposed=( c / c+d) / 1- (c / c+d )(c / c+d )

=(c / c+d) / (d / =(c / c+d) / (d / c+d) = c+d) = c/dc/d

Relative RiskRelative Risk

RR= a / a+bRR= a / a+b / / c / c+d c / c+d

OR= a / bOR= a / b / / c / d = a*d c / d = a*d / / b*cb*c Odds ratio is a cross-product Odds ratio is a cross-product

ratioratio

Rare Disease - MIRare Disease - MI

MIFree of MITotals:

Exposure

High Blood

Pressure

180 982010000

Normal

Pressure

30 997010000

ProbabilityProbability ++ =q =q ++ = 180/10000 = = 180/10000 = 0.01800.0180

ProbabilityProbability -- = q = q -- = 30/10000 = 0.0030 = 30/10000 = 0.0030

Odds Odds dis dis ++

= 180/9820 = 0.01833= 180/9820 = 0.01833

Odds Odds dis dis -- = 30/9970 = 0.00301= 30/9970 = 0.00301

RR=6RR=6 OR=6.09OR=6.09

Common Disease – Vaccine Common Disease – Vaccine ReactionsReactions

Local

Reactions

Free of

Reactions

Totals:

Exposure

Vaccinated 65019202570

Placebo17022402240

RR = 650 / 2570 / 170 / 2410 = RR = 650 / 2570 / 170 / 2410 = 0.2529 / 0.0705 = 0.2529 / 0.0705 = 3.593.59

OR = 650 / 1920 / 170 / 2240 = OR = 650 / 1920 / 170 / 2240 = 0.3385 / 0.0759 = 0.3385 / 0.0759 = 4.464.46

Built – in biasBuilt – in bias

OR =OR =(( q q ++ / 1 - q / 1 - q ++)) / (/ (q q -- / 1 - q / 1 - q ––))

= q = q ++ / q / q -- * (* (1 - q 1 - q -- / 1- q / 1- q ++ ) ) = RR = RR * (* (1 - q 1 - q -- / 1- q / 1- q ++ ) )

Built – in biasBuilt – in bias

Use of the odds ratio as an Use of the odds ratio as an estimate of the relative risk estimate of the relative risk biases it in a direction opposite biases it in a direction opposite to the null hypothesis.to the null hypothesis.

(1 - q - / 1- q + ) defines the bias (1 - q - / 1- q + ) defines the bias responsible for the discrepancy responsible for the discrepancy between the RR & OR.between the RR & OR.

When the disease is relatively When the disease is relatively rare , this bias is negligible.rare , this bias is negligible.

When the incidence is high, the When the incidence is high, the bias can be substantial.bias can be substantial.

OR is a valuable measure of OR is a valuable measure of association :association :

1. It can be measured in case – control 1. It can be measured in case – control studies.studies.

2. It is directly derived from logistic 2. It is directly derived from logistic regression modelsregression models

3. The OR of an event is the exact 3. The OR of an event is the exact reciprocal of the OR of the nonevent. reciprocal of the OR of the nonevent. (survival or death OR both are (survival or death OR both are informative)informative)

4. when the baseline risk is not very 4. when the baseline risk is not very small, RR can be meaningless.small, RR can be meaningless.

Case-Control StudyCase-Control Study

The OR of disease and the OR of The OR of disease and the OR of exposure are mathematically exposure are mathematically equivalent.equivalent.

In case control study we calculate the In case control study we calculate the OR of exposure as it’s algebraically OR of exposure as it’s algebraically identical to the OR of disease.identical to the OR of disease.

OR OR expexp = a /c / b/ d = a*d/ b*c = a / b / = a /c / b/ d = a*d/ b*c = a / b / c / d = OR c / d = OR disdis

Case-Control StudyCase-Control Study

The fact that the OR The fact that the OR expexp is identical to is identical to the OR the OR dis dis explains why the explains why the interpretation of the odds ratio in interpretation of the odds ratio in case control studies is prospective.case control studies is prospective.

Odds Ratio as an Estimate Odds Ratio as an Estimate of the Relative Risk:of the Relative Risk:

The disease under study has low The disease under study has low Incidence thus resulting in a small Incidence thus resulting in a small built-in bias : OR is an estimate of RRbuilt-in bias : OR is an estimate of RR

The case – cohort approach allows The case – cohort approach allows direct estimation of RR by OR and does direct estimation of RR by OR and does not have to rely on rarity assumption.not have to rely on rarity assumption.

When the OR is used as a measure of When the OR is used as a measure of association in itself, this assumption is association in itself, this assumption is obviously is not neededobviously is not needed

Calculation of the OR when Calculation of the OR when there are more then two there are more then two

exposure categoriesexposure categories To calculate the OR for different To calculate the OR for different

exposure categories , one is chosen exposure categories , one is chosen as the reference category as the reference category (biologically or largest sample size)(biologically or largest sample size)

Cases of Craniosynostosis and Cases of Craniosynostosis and normal Control according to normal Control according to

maternal agematernal age

Maternal age

CasesControls

Odds exp in case

Odds exp in control

OR

<20 128912/1289/891

20-244724247/12242/891.44

25-295625556/12255/891.63

>295817358/12173/892.49

When the multilevel exposure When the multilevel exposure variable is ordinal, it may be variable is ordinal, it may be of interest to perform a trend of interest to perform a trend testtest

Types of VariablesTypes of Variables

Discrete/categoricalDiscrete/categorical Dichotomous, Dichotomous,

binarybinary Absolute Absolute

Difference?Difference? Relative DifferenceRelative Difference

ContinuousContinuous Difference Difference

between meansbetween means

Methods of analysisMethods of analysis

CrudeCrude StratificationStratification

StandardizationStandardization Stratification (Mantel Haenszel & Wolf)Stratification (Mantel Haenszel & Wolf)

Modeling (multiple regression)Modeling (multiple regression) Linear regressionLinear regression Logistic regressionLogistic regression Cox regressionCox regression Poisson regressionPoisson regression

Confounding Confounding

8262female

6888male

controlcase

Crude

310female

1553male

controlcase

Outdoor occupation

7952female

5335male

controlcaseIndoor occupation

OR = 1.71

OR = 1.06

OR = 1.00

StandardizationStandardization

Direct standardizationDirect standardization

Using standard populationUsing standard population Indirect standardizationIndirect standardization

Using standard ratesUsing standard rates

AgeCasesPopulationRateCasesPopulationRate

0-29 3,5233,145,000.0011203,904741,000.005268

30-5910,9283,075,000.0035531,421275,000.005167

60+59,1041,294,000.0456752,456 59,000.041627

Total

AgeCasesPopulationRateCasesPopulationRate

0-29 3,5233,145,000.0011203,904741,000.005268

30-5910,9283,075,000.0035531,421275,000.005167

60+59,1041,294,000.0456752,456 59,000.041627

Total73,5557,514,000.0097897,7811,075,000.007238

Direct Adjustment Direct Adjustment

.007238.009789

.041627.045675

.005167.003553

.005268.001120

RateRate

PanamaSweden

1,075,0007,7817,514,00073,555Total

59,0002,4561,294,00059,10460+

275,0001,4213,075,00010,92830-59

741,0003,9043,145,000 3,5230-29

PopulationCasesPopulationCasesAge

Crude mortality rate in Sweden = 97.9 / 10,000Crude mortality rate in Panama = 72.4 / 10,000Crude Rate ratio = 97.9 / 72.4 = 1.35

Direct Adjustment Direct Adjustment

161,404153,381

124,881137,025

18,08512,436

18,4383,920

ExpectedExpected

10,000,00010,000,000

3,000,0003,000,000

3,500,0003,500,000

3,500,0003,500,000

PopulationPopulation

PanamaSweden

.007238.009789Total

.041627.04567560+

.005167.00355330-59

.005268.0011200-29

RateRateAge

Age-adjusted mortality rate in Sweden =

Age-adjusted mortality rate in Panama = Age-adjusted rate ratio =

153.4/10,000161.4/10,000

0.95

StandardizationStandardization

Direct standardizationDirect standardization

Using standard populationUsing standard population Indirect standardizationIndirect standardization

Using standard ratesUsing standard rates

StratificationStratification

When we have : Few confoundersWhen we have : Few confounders- Direct adjustment when :Direct adjustment when :

Study populations are largeStudy populations are large Comparing two group ( absolute or Comparing two group ( absolute or

relative differences )relative differences ) Indirect adjustment when :Indirect adjustment when :

Populations are smallPopulations are small Strata with cells with zero contentsStrata with cells with zero contents Rates of standard population existsRates of standard population exists

Confounding Confounding

8262female

6888male

controlcase

Crude

310female

1553male

controlcase

Outdoor occupation

7952female

5335male

controlcaseIndoor occupation

OR = 1.71

OR = 1.06

OR = 1.00

Mantel-Haenszel summary Mantel-Haenszel summary measuremeasure

Case Control

Exposure +ab

Exposure -cd

b c

a dCrude OR =

Mantel-Haenszel summary Mantel-Haenszel summary measuremeasure

Case Control

Exposure +a1b1

Exposure -c1d1

N1

Case Control

Exposure +aibi

Exposure -cidi

Nk

Stratum 1

Stratum K ∑

1

∑

1

Ni

bi ci

k

Ni

ai di

k

ORMH =

Mantel-Haenszel summary Mantel-Haenszel summary measuremeasure

ORMH =

∑bi ci*ai di

=

∑ wi * ORiNibi ci

∑bi ci

∑ wiNi

Woolf summary measureWoolf summary measure

Variance LnORi : (1/ai + 1/bi + 1/ci + 1/di)

Wi = 1 / variance LnORi

∑

∑ wi

LnORi * wiLnOR woolf =

Confidence interval of Woolf Confidence interval of Woolf summary measuresummary measure

Var LnOR =1

∑ wi

Confidence Interval 95% :

LnOR +/- 1.96 √( 1/ ∑ wi )

Test for interactionTest for interaction

1

∑=Var LnORik-1

(LnORi – LnOR)^2

k

2

ORMH OR4

OR3

ORi

OR1

OR2

Methods of analysisMethods of analysis

CrudeCrude AdjustedAdjusted

StratificationStratification StandardizationStandardization Stratification (Mantel Haenszel & Wolf)Stratification (Mantel Haenszel & Wolf)

Modeling (multiple regression)Modeling (multiple regression) Linear regressionLinear regression Logistic regressionLogistic regression Cox regressionCox regression Poisson regressionPoisson regression