Measuring Associations Between Exposure and Outcomes

Methods of analysis Crude Crude Adjusted Adjusted Stratification Stratification Standardization Standardization Stratification (Mantel Haenszel & Wolf) Stratification (Mantel Haenszel & Wolf) Modeling (multiple regression) Modeling (multiple regression) Linear regression Linear regression Logistic regression Logistic regression Cox regression Cox regression Poisson regression Poisson regression

Measures of Association can be based on: Absolute differences Between Groups (e.g., disease risk among exposed – disease risk among unexposed) Absolute differences Between Groups (e.g., disease risk among exposed – disease risk among unexposed) Relative differences or ratios Between Groups (e.g., disease risk ratio or relative risk: disease risk in exposed/disease risk in unexposed) Relative differences or ratios Between Groups (e.g., disease risk ratio or relative risk: disease risk in exposed/disease risk in unexposed)

Measure of Public Health Impact

Four closely related measure are used: Attributable Risk Attributable Risk Attributable( Risk) fraction Attributable( Risk) fraction Population Attributable Risk Population Attributable Risk Population Attributable (Risk) fraction Population Attributable (Risk) fraction

Attributable Risk (AR) The Incidence of disease in the Exposed population whose disease can be attributed to the exposure. The Incidence of disease in the Exposed population whose disease can be attributed to the exposure. AR=I e –I u AR=I e –I u

MI Free of MI Totals:Exposure High Blood Pressure NormalPressure AR= – 0.003= 0.015= 1.5% The cessation of the exposure would lower the risk in the exposed group from to

Vaccine Efficacy VE= I e /I u - I e /I u VE= I e /I u - I e /I u VE= RR-1

Attributable (Risk) Fraction (ARF) The proportion of disease in the exposed population whose disease can be attributed to the exposure. The proportion of disease in the exposed population whose disease can be attributed to the exposure. AR= (I e –I u )/I e AR= (I e –I u )/I e ARF=( RR-1)/RR ARF=( RR-1)/RR

ARF = – 0.003/ * 100 = 83.3% ARF = – 0.003/ * 100 = 83.3% RR=0.018/0.003 = 6 RR=0.018/0.003 = 6 ARF=( RR-1)/RR * 100=(6 – 1)/6 *100= 83.3% ARF=( RR-1)/RR * 100=(6 – 1)/6 *100= 83.3%

ARF= percent efficacy Risk of dis. In vaccinated group= 5% Risk of dis. In the placebo group= 15% ARF=Efficacy=((15 – 5) / 15) * 100 = 66.7% = (3-1)/3 * 100 = 66.7 %

Population Attributable Risk (PAR) The Incidence of disease in the total population whose disease can be attributed to the exposure. The Incidence of disease in the total population whose disease can be attributed to the exposure. PAR=I p –I u PAR=I p –I u

Population Attributable (Risk) Fraction (PARF) The proportion of disease in the total population whose disease can be attributed to the exposure. The proportion of disease in the total population whose disease can be attributed to the exposure. The PARF is defined as the fraction of all cases (exposed and unexposed) that would not have occurred if exposure had not occurred. The PARF is defined as the fraction of all cases (exposed and unexposed) that would not have occurred if exposure had not occurred. PARF= (I p –I u )/I p PARF= (I p –I u )/I p

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

2-Miettinen or case-based formula: 2-Miettinen or case-based formula: PARF=[(RR-1)/RR ]* CF PARF=[(RR-1)/RR ]* CF CF=number of exposed cases/overall number of cases CF=number of exposed cases/overall number of cases PAF has two Formula:

Relative differences or ratios For discrete variable For discrete variable To assess causal associations To assess causal associations Examples: Relative Risk/Rate, Relative odds Examples: Relative Risk/Rate, Relative odds

Cohort Study Cohort Study Disease d Non- disease d Totals: Risk odds Exposure Exposedaba+b a / a+b a / b Unexpose d cdc+d c /c+d c / d Totals: Disease a+cb+da+b+c+d

Odds in Exposed and Unexposed Odds in exposed=( a / a+b) / 1- (a / a+b ) Odds in exposed=( a / a+b) / 1- (a / a+b ) =(a / a+b) / (b / a+b) = a/b =(a / a+b) / (b / a+b) = a/b Odds in unexposed=( c / c+d) / 1- (c / c+d ) Odds in unexposed=( c / c+d) / 1- (c / c+d ) =(c / c+d) / (d / c+d) = c/d =(c / c+d) / (d / c+d) = c/d

Relative Risk RR= a / a+b / c / c+d RR= a / a+b / c / c+d OR= a / b / c / d = a*d / b*c OR= a / b / c / d = a*d / b*c Odds ratio is a cross-product ratio Odds ratio is a cross-product ratio

Rare Disease - MI MIFree of MITotals: Exposure High Blood Pressure Normal Pressure

Probability + =q + = 180/10000 = Probability + =q + = 180/10000 = Probability - = q - = 30/10000 = Probability - = q - = 30/10000 = Odds dis + = 180/9820 = Odds dis + = 180/9820 = Odds dis - = 30/9970 = Odds dis - = 30/9970 = RR=6 RR=6 OR=6.09 OR=6.09

Common Disease – Vaccine Reactions Local Reactions Free of Reactions Totals: Exposure Vaccinated Placebo

RR = 650 / 2570 / 170 / 2410 = / = 3.59 RR = 650 / 2570 / 170 / 2410 = / = 3.59 OR = 650 / 1920 / 170 / 2240 = / = 4.46 OR = 650 / 1920 / 170 / 2240 = / = 4.46

Built – in bias OR = ( q + / 1 - q + ) / ( q - / 1 - q – ) OR = ( q + / 1 - q + ) / ( q - / 1 - q – ) = q + / q - * ( 1 - q - / 1- q + ) = q + / q - * ( 1 - q - / 1- q + ) = RR * ( 1 - q - / 1- q + ) = RR * ( 1 - q - / 1- q + )

Built – in bias Use of the odds ratio as an estimate of the relative risk biases it in a direction opposite to the null hypothesis. Use of the odds ratio as an estimate of the relative risk biases it in a direction opposite to the null hypothesis. (1 - q - / 1- q + ) defines the bias responsible for the discrepancy between the RR & OR. (1 - q - / 1- q + ) defines the bias responsible for the discrepancy between the RR & OR.

When the disease is relatively rare, this bias is negligible. When the disease is relatively rare, this bias is negligible. When the incidence is high, the bias can be substantial. When the incidence is high, the bias can be substantial.

OR is a valuable measure of association : 1. It can be measured in case – control studies. 1. It can be measured in case – control studies. 2. It is directly derived from logistic regression models 2. It is directly derived from logistic regression models 3. The OR of an event is the exact reciprocal of the OR of the nonevent. (survival or death OR both are informative) 3. The OR of an event is the exact reciprocal of the OR of the nonevent. (survival or death OR both are informative) 4. when the baseline risk is not very small, RR can be meaningless. 4. when the baseline risk is not very small, RR can be meaningless.

Case-Control Study The OR of disease and the OR of exposure are mathematically equivalent. The OR of disease and the OR of exposure are mathematically equivalent. In case control study we calculate the OR of exposure as it’s algebraically identical to the OR of disease. In case control study we calculate the OR of exposure as it’s algebraically identical to the OR of disease. OR exp = a /c / b/ d = a*d/ b*c = a / b / c / d = OR dis OR exp = a /c / b/ d = a*d/ b*c = a / b / c / d = OR dis

Case-Control Study The fact that the OR exp is identical to the OR dis explains why the interpretation of the odds ratio in case control studies is prospective. The fact that the OR exp is identical to the OR dis explains why the interpretation of the odds ratio in case control studies is prospective.

Odds Ratio as an Estimate of the Relative Risk: The disease under study has low Incidence thus resulting in a small built-in bias : OR is an estimate of RR The disease under study has low Incidence thus resulting in a small built-in bias : OR is an estimate of RR The case – cohort approach allows direct estimation of RR by OR and does not have to rely on rarity assumption. The case – cohort approach allows direct estimation of RR by OR and does not have to rely on rarity assumption. When the OR is used as a measure of association in itself, this assumption is obviously is not needed When the OR is used as a measure of association in itself, this assumption is obviously is not needed

Calculation of the OR when there are more then two exposure categories To calculate the OR for different exposure categories, one is chosen as the reference category (biologically or largest sample size) To calculate the OR for different exposure categories, one is chosen as the reference category (biologically or largest sample size)

Cases of Craniosynostosis and normal Control according to maternal age Matern al age CasesControl s Odds exp in case Odds exp in control OR < /1289/ /12242/ /12255/ > /12173/892.49

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

Types of Variables Discrete/categorical Dichotomous, binary Absolute Difference? Relative Difference Continuous Difference between means

Methods of analysis Crude Crude Stratification Stratification Standardization Standardization Stratification (Mantel Haenszel & Wolf) Stratification (Mantel Haenszel & Wolf) Modeling (multiple regression) Modeling (multiple regression) Linear regression Linear regression Logistic regression Logistic regression Cox regression Cox regression Poisson regression Poisson regression

Confounding 8262female 6888male controlcase Crude 310female 1553male controlcase Outdoor occupation 7952female 5335male controlcase Indoor occupation OR = 1.71 OR = 1.06 OR = 1.00

Standardization Direct standardization Direct standardization Using standard population Indirect standardization Indirect standardization Using standard rates

AgeCasesPopulationRateCasesPopulationRate ,5233,145, ,904741, ,9283,075, ,421275, ,1041,294, ,456 59, Total

AgeCasesPopulationRateCasesPopulationRate ,5233,145, ,904741, ,9283,075, ,421275, ,1041,294, ,456 59, Total73,5557,514, ,7811,075,

Direct Adjustment Rate PanamaSweden 1,075,0007,7817,514,00073,555Total 59,0002,4561,294,00059, ,0001,4213,075,00010, ,0003,9043,145,000 3, PopulationCasesPopulationCasesAge Crude mortality rate in Sweden = 97.9 / 10,000 Crude mortality rate in Panama = 72.4 / 10,000 Crude Rate ratio = 97.9 / 72.4 = 1.35

Direct Adjustment 161,404153, ,881137,025 18,08512,436 18,4383,920 Expected 10,000,000 3,000,000 3,500,000 Population PanamaSweden Total Rate Age Age-adjusted mortality rate in Sweden = Age-adjusted mortality rate in Panama = Age-adjusted rate ratio = 153.4/10, /10,

Standardization Direct standardization Direct standardization Using standard population Indirect standardization Indirect standardization Using standard rates

Stratification When we have : Few confounders - Direct adjustment when : Study populations are large Study populations are large Comparing two group ( absolute or relative differences ) Comparing two group ( absolute or relative differences ) Indirect adjustment when : Indirect adjustment when : Populations are small Populations are small Strata with cells with zero contents Strata with cells with zero contents Rates of standard population exists Rates of standard population exists

Confounding 8262female 6888male controlcase Crude 310female 1553male controlcase Outdoor occupation 7952female 5335male controlcase Indoor occupation OR = 1.71 OR = 1.06 OR = 1.00

Mantel-Haenszel summary measure CaseControl Exposure +ab Exposure -cd b c a d Crude OR =

Mantel-Haenszel summary measure CaseControl Exposure +a1b1 Exposure -c1d1 N1 CaseControl Exposure +aibi Exposure -cidi Nk Stratum 1 Stratum K ∑ 1 ∑ 1 Ni bi ci k Ni ai di k OR MH =

Mantel-Haenszel summary measure OR MH = ∑ bi ci * ai di = ∑ wi * ORi Nibi ci ∑ ∑ wi Ni

Mantel Haenszel summary measure for cohort CasePerson time Exposure +a1ia1iy1iy1i Exposure -a0ia0iy0iy0i Ti Stratum i ∑ 1 ∑ 1 Ti a0i y1i k Ti a1i y0i k ORMH =

Woolf summary measure Variance LnORi : (1/ai + 1/bi + 1/ci + 1/di) Wi = 1 / variance LnORi ∑ ∑ wi LnORi * wi LnOR woolf =

Confidence interval of Woolf summary measure Var LnOR = 1 ∑ wi Confidence Interval 95% : LnOR +/ √( 1/ ∑ wi )

Test for interaction 1 ∑ = Var LnORik-1 (LnORi – LnOR)^2 k 22 OR MH OR4 OR3 ORi OR1 OR2

Methods of analysis Crude Crude Adjusted Adjusted Stratification Stratification Standardization Standardization Stratification (Mantel Haenszel & Wolf) Stratification (Mantel Haenszel & Wolf) Modeling (multiple regression) Modeling (multiple regression) Linear regression Linear regression Logistic regression Logistic regression Cox regression Cox regression Poisson regression Poisson regression