measuring associations between exposure and outcomes
TRANSCRIPT
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