Odds Ratio

Lesson 14Lesson 14Lesson 14Lesson 14

Confidence Intervals of Confidence Intervals of Odds Ratio andOdds Ratio and

Relative RiskRelative RiskRelative RiskRelative Risk

Lesson 14 OutlineLesson 14 OutlineLesson 14 OutlineLesson 14 OutlineLesson 14 coversLesson 14 coversLesson 14 coversLesson 14 covers Confidence Interval of an Odds RatioConfidence Interval of an Odds Ratio

Review of Odds RatioReview of Odds Ratio Review of Odds RatioReview of Odds Ratio Sampling distribution of OR on natural log scaleSampling distribution of OR on natural log scale 95% Confidence Interval of OR example95% Confidence Interval of OR example 95% Confidence Interval of OR example95% Confidence Interval of OR example

Confidence Interval of Relative RiskConfidence Interval of Relative Risk Review of Relative RiskReview of Relative Risk Review of Relative RiskReview of Relative Risk Sampling distribution of RR on natural log scaleSampling distribution of RR on natural log scale 95% Confidence Interval of RR example95% Confidence Interval of RR example

2PubH 6414 Lesson 13

95% Confidence Interval of RR example95% Confidence Interval of RR example

Review: Odds Ratio and Relative Review: Odds Ratio and Relative Ri kRi kRiskRisk

Both Odds Ratio and Relative Risk areBoth Odds Ratio and Relative Risk are Both Odds Ratio and Relative Risk are Both Odds Ratio and Relative Risk are measures of association (or relationship) measures of association (or relationship) between two nominal variablesbetween two nominal variablesbetween two nominal variables. between two nominal variables.

The Odds Ratio is typically estimated from The Odds Ratio is typically estimated from data collected in a Casedata collected in a Case Control studyControl studydata collected in a Casedata collected in a Case--Control study.Control study.

The Relative Risk can be estimated from The Relative Risk can be estimated from d t ll t d i ti t dd t ll t d i ti t ddata collected in a prospective study. data collected in a prospective study.


Associations between 2 nominal Associations between 2 nominal i bli blvariablevariable

If there is no association between the two If there is no association between the two i bl th OR RR 1i bl th OR RR 1variables, the OR or RR = 1 variables, the OR or RR = 1

An OR or RR > 1.0 or < 1.0 indicates a An OR or RR > 1.0 or < 1.0 indicates a possiblepossiblet ti ti l l ti hi ( i ti ) b tt ti ti l l ti hi ( i ti ) b tstatistical relationship (or association) between statistical relationship (or association) between

the two variables.the two variables. Hypothesis tests for the OR and RR are not usedHypothesis tests for the OR and RR are not used Hypothesis tests for the OR and RR are not used Hypothesis tests for the OR and RR are not used

to determine statistical significance of the to determine statistical significance of the associationassociationassociationassociation

Instead, Confidence intervals of OR or RR are Instead, Confidence intervals of OR or RR are constructed and used to determine whether or not constructed and used to determine whether or not


the association is statistically significant.the association is statistically significant.

Confidence Interval Confidence Interval f Odd R tif Odd R tiof Odds Ratioof Odds Ratio

Review: Odds RatioReview: Odds RatioReview: Odds RatioReview: Odds Ratio

Odds of an event =Odds of an event = Odds of an event = Odds of an event = probability that the event occursprobability that the event occursprobability that the event does not occurprobability that the event does not occurprobability that the event does not occurprobability that the event does not occur

Odds RatioOdds Ratioodds of event for group 1odds of event for group 1g pg podds of event for group 2odds of event for group 2


Review: Disease OR for ‘Exposed’ Review: Disease OR for ‘Exposed’ d t ‘N t E d’d t ‘N t E d’compared to ‘Not Exposed’compared to ‘Not Exposed’

DiseaseDisease No DiseaseNo Disease TotalTotalDiseaseDisease No DiseaseNo Disease TotalTotal

Exposed GroupExposed Group aa bb a+ba+bNot Exposed Not Exposed GroupGroup

cc dd c+dc+d

Odds of Disease for Exposed = a/(a+b) b/(a+b) = a/b

Odds of Disease for Not Exposed = c/(c+d) d/(c+d) = c/dp ( ) ( )

OR = a/b c/d = ad / bc


Confidence Interval for an Confidence Interval for an Odds RatioOdds Ratio

The confidence interval for an Odds RatioThe confidence interval for an Odds Ratio The confidence interval for an Odds Ratio The confidence interval for an Odds Ratio has the same general formula as the has the same general formula as the Confidence Interval for a population mean Confidence Interval for a population mean p pp por population proportionor population proportion

ErrorStandard*t CoefficienConfidence EstimatePoint

The difference is that the confidence The difference is that the confidence interval for the Odds Ratio is calculated on interval for the Odds Ratio is calculated on the natural log (LN) scale and then the natural log (LN) scale and then

t d b k t th i i l lt d b k t th i i l l8PubH 6414 Lesson 13

converted back to the original scale. converted back to the original scale.

Sampling Distribution of the Sampling Distribution of the Odds RatioOdds Ratio

The sampling distribution of the Odds Ratio isThe sampling distribution of the Odds Ratio is The sampling distribution of the Odds Ratio is The sampling distribution of the Odds Ratio is positively skewed.positively skewed.

However It is approximately normallyHowever It is approximately normally However, It is approximately normally However, It is approximately normally distributed on the natural log scale. distributed on the natural log scale. Th fid i t l i l l t d thTh fid i t l i l l t d th The confidence interval is calculated on the The confidence interval is calculated on the natural log scale (LN)natural log scale (LN)

After finding the limits on the LN scale, use the After finding the limits on the LN scale, use the EXP function to find the limits on the original EXP function to find the limits on the original


scale. scale.

95% Confidence Interval of OR: 95% Confidence Interval of OR: hhthe stepsthe steps

1.1. Calculate the Odds Ratio from the dataCalculate the Odds Ratio from the data2.2. Find the Natural log of the OR using the LN Find the Natural log of the OR using the LN

function in Excelfunction in Excel3.3. The confidence coefficient is from the standard The confidence coefficient is from the standard

normal distribution: 1.96 for a 95% confidence normal distribution: 1.96 for a 95% confidence intervalintervalintervalinterval

4.4. Calculate the SE of LN(OR) Calculate the SE of LN(OR) –– see next slidesee next slide55 The lower and upper limits on the log scale =The lower and upper limits on the log scale =5.5. The lower and upper limits on the log scale = The lower and upper limits on the log scale =

LN(OR) LN(OR) ±± 1.96* SE LN(OR) = (LL, UL)1.96* SE LN(OR) = (LL, UL)6.6. Use the EXP function to find the CI limits on Use the EXP function to find the CI limits on


the original scale: EXP(LL), EXP(UL)the original scale: EXP(LL), EXP(UL)

SE of LN(OR)SE of LN(OR)SE of LN(OR)SE of LN(OR) The SE of LN(OR) is calculated from the The SE of LN(OR) is calculated from the

cell counts in the 2 X 2 table:cell counts in the 2 X 2 table:

CasesCases ControlsControlsCasesCases ControlsControls

ExposedExposed aa bbNot exposedNot exposed cc dd

SE of LN(OR) =SE of LN(OR) =d1

c1

b1

a1


dcba

95% Confidence Interval of Odds 95% Confidence Interval of Odds R iR iRatioRatio

First calculate the upper and lower limits on First calculate the upper and lower limits on ppppthe LN scalethe LN scale

dcbaOR 1111*96.1)ln(

After calculating the upper and lower limits After calculating the upper and lower limits

dcba

on the LN scale, use the exponential on the LN scale, use the exponential function (EXP) to find the lower and upper function (EXP) to find the lower and upper limits on the original scalelimits on the original scalegg

If the 95% Confidence Interval does not If the 95% Confidence Interval does not contain the value 1.0, the association is contain the value 1.0, the association is statistically significant at alpha = 0 05statistically significant at alpha = 0 05


statistically significant at alpha = 0.05statistically significant at alpha = 0.05

Example: Preterm deliveryExample: Preterm deliveryExample: Preterm delivery Example: Preterm delivery and Mother’s SES leveland Mother’s SES level

Study background: A caseStudy background: A case--control study of control study of th id i l f t d lith id i l f t d lithe epidemiology of preterm delivery was the epidemiology of preterm delivery was undertaken at Yaleundertaken at Yale--New Haven Hospital in New Haven Hospital in Connecticut during 1977 The table on theConnecticut during 1977 The table on theConnecticut during 1977. The table on the Connecticut during 1977. The table on the following slide contains data for Mother’s following slide contains data for Mother’s socioeconomic status (SES) for those withsocioeconomic status (SES) for those withsocioeconomic status (SES) for those with socioeconomic status (SES) for those with (cases) and without (controls) preterm (cases) and without (controls) preterm delivery.delivery.


yy

Preterm DeliveryPreterm DeliveryyySocioSocio--EconomicEconomicStatusStatus CasesCases ControlsControlsUpperUpper 1111 4040Upper MiddleUpper MiddleMiddleMiddleL MiddlL Middl

141433335959

454564649191Lower MiddleLower Middle

LowerLowerUnknownUnknown

5959535355

9191585855UnknownUnknown 55 55

Construct a 95% Confidence Interval for the odds ratio


for Preterm delivery comparing Lower to Upper SES

Data table for Confidence Interval of Data table for Confidence Interval of Odds Ratio of preterm delivery Odds Ratio of preterm delivery comparing lower to upper SEScomparing lower to upper SES

Preterm DeliveryPreterm Delivery

SESSES Yes: CasesYes: Cases No: ControlsNo: Controls

LowerLower 5353 5858UpperUpper 1111 4040UpperUpper 1111 4040


1. Calculate the OR for Preterm Delivery: 1. Calculate the OR for Preterm Delivery: L SES d t U SESL SES d t U SESLower SES compared to Upper SESLower SES compared to Upper SES

Preterm DeliveryPreterm DeliveryPreterm DeliveryPreterm Delivery

SESSES Yes: CasesYes: Cases No: ControlsNo: Controls

LowerLower 5353 5858UpperUpper 1111 4040

Odds Ratio = (53/58) / (11/40) = 3.32


2. Calculate the LN (OR) and 2. Calculate the LN (OR) and 3 Fi d h fid ffi i3 Fi d h fid ffi i3. Find the confidence coefficient3. Find the confidence coefficient2 Calculate the LN(OR)2 Calculate the LN(OR)2. Calculate the LN(OR)2. Calculate the LN(OR)

LN(3.32) = 1.20LN(3.32) = 1.20Thi i th i t ti t f thThi i th i t ti t f thThis is the point estimate for the This is the point estimate for the confidence intervalconfidence interval

3. The confidence coefficient for the LN(OR) 3. The confidence coefficient for the LN(OR) is from the standard normal distribution. is from the standard normal distribution. For a 95% confidence interval, the For a 95% confidence interval, the coefficient = 1.96coefficient = 1.96


4. Calculate the SE of LN(OR)4. Calculate the SE of LN(OR)4. Calculate the SE of LN(OR)4. Calculate the SE of LN(OR)

Preterm DeliveryPreterm DeliveryPreterm DeliveryPreterm DeliverySESSES Yes: CasesYes: Cases No: ControlsNo: Controls

LL 5353 5858LowerLower 5353 5858UpperUpper 1111 4040pppp

1111 39.0401

111

581

531 ln(OR) SE


5. & 6. Find the Lower and Upper 5. & 6. Find the Lower and Upper limits on the LN scale and thelimits on the LN scale and thelimits on the LN scale and the limits on the LN scale and the

original scaleoriginal scale)97.1 ,44.0(39.0*96.120.1

5. The upper and lower limits of the 5. The upper and lower limits of the confidence interval on the LN scale areconfidence interval on the LN scale areconfidence interval on the LN scale areconfidence interval on the LN scale are(0.44, 1.97)(0.44, 1.97)

6. Use the EXP function in Excel to find 6. Use the EXP function in Excel to find the interval limits of the 95% CI of the the interval limits of the 95% CI of the


OR: OR: (1.55, 7.14)(1.55, 7.14)

Interpretation of the CI for Preterm Interpretation of the CI for Preterm Delivery and SESDelivery and SES

State the conclusion in the original scaleState the conclusion in the original scalegg The odds ratio for preThe odds ratio for pre--term delivery for Lower SES term delivery for Lower SES

compared to the Upper SES is 3.32 indicating compared to the Upper SES is 3.32 indicating increased odds of preincreased odds of pre--term delivery for mothers in term delivery for mothers in Lower SES Lower SES

The 95% Confidence Interval of the Odds Ratio The 95% Confidence Interval of the Odds Ratio (1.55 , 7.14) indicates that odds of pre(1.55 , 7.14) indicates that odds of pre--term delivery term delivery

i ifi tli ifi tl hi h f th L SEShi h f th L SESare are significantlysignificantly higher for the Lower SES group higher for the Lower SES group compared to the Upper SES group (at 0.05 compared to the Upper SES group (at 0.05 significance level) because the CI does not contain 1significance level) because the CI does not contain 1


significance level) because the CI does not contain 1.significance level) because the CI does not contain 1.

Interpretation of 95% Interpretation of 95% te p etat o o 95%te p etat o o 95%Confidence Interval of ORConfidence Interval of OR

An Odds Ratio = 1 indicates ‘no association’ An Odds Ratio = 1 indicates ‘no association’ between the exposure and the disease.between the exposure and the disease.

If the 95% confidence interval for the OR does If the 95% confidence interval for the OR does not contain 1.0 we can conclude that there is a not contain 1.0 we can conclude that there is a statisticall significant* association bet een thestatisticall significant* association bet een thestatistically significant* association between the statistically significant* association between the exposure and the disease.exposure and the disease.* at the 0 05 significance level* at the 0 05 significance level at the 0.05 significance level at the 0.05 significance level

If the 95% confidence interval for the OR If the 95% confidence interval for the OR contains 1.0, the association is contains 1.0, the association is not not significant at significant at


ggthe 0.05 level. the 0.05 level.

Confidence Interval Confidence Interval f R l ti Ri kf R l ti Ri kof Relative Riskof Relative Risk

Review: Relative RiskReview: Relative RiskReview: Relative RiskReview: Relative Risk

Risk of an event =Risk of an event = Risk of an event = Risk of an event = probability that the event occursprobability that the event occurs

Relative RiskRelative Riskprobability that event occurs for group 1probability that event occurs for group 1probability that event occurs for group 2probability that event occurs for group 2probability that event occurs for group 2probability that event occurs for group 2


Review: RR Calculated from a Review: RR Calculated from a tabletableDiseaseDisease No DiseaseNo Disease TotalTotalDiseaseDisease No DiseaseNo Disease TotalTotal

Exposed GroupExposed Group aa bb a+ba+bNot Exposed Not Exposed GroupGroup

cc dd c+dc+d

Risk of Disease for Exposed = a/(a+b)

Risk of Disease for Not exposed = c/(c+d)Risk of Disease for Not exposed = c/(c+d)

RR = a/(a+b) c/(c+d)


RR a/(a+b) c/(c+d)

Confidence Interval for Confidence Interval for Relative RiskRelative Risk

Like the Odds Ratio, the sampling Like the Odds Ratio, the sampling distribution of the Relative Risk is distribution of the Relative Risk is positively skewed but is approximately positively skewed but is approximately normally distributed on the natural log normally distributed on the natural log scalescalescale.scale.

Constructing a Confidence Interval for the Constructing a Confidence Interval for the Relative Risk is similar to constructing a CIRelative Risk is similar to constructing a CIRelative Risk is similar to constructing a CI Relative Risk is similar to constructing a CI for the Odds Ratio except that there is a for the Odds Ratio except that there is a different formula for the SEdifferent formula for the SE


different formula for the SE.different formula for the SE.

95% Confidence Interval of RR95% Confidence Interval of RR95% Confidence Interval of RR95% Confidence Interval of RR

1. Estimate the RR from the data1. Estimate the RR from the data2. Find the natural log of RR: LN(RR)2. Find the natural log of RR: LN(RR)3. The confidence coefficient is from the standard 3. The confidence coefficient is from the standard

normal distribution: 1.96 for a 95% confidence normal distribution: 1.96 for a 95% confidence intervalinterval

4 C l l t th SE f LN(RR)4 C l l t th SE f LN(RR) t lidt lid4. Calculate the SE of LN(RR) 4. Calculate the SE of LN(RR) –– see next slidesee next slide5. Calculate the lower and upper limits on the LN 5. Calculate the lower and upper limits on the LN

scale: LN(RR)scale: LN(RR) ±± 1 96* SE LN(RR)1 96* SE LN(RR)scale: LN(RR) scale: LN(RR) ±± 1.96 SE LN(RR)1.96 SE LN(RR)6. Use the EXP function to find the limits on the 6. Use the EXP function to find the limits on the

original scale: EXP(LL), EXP (UL)original scale: EXP(LL), EXP (UL)


g ( ), ( )g ( ), ( )

SE of LN(RR)SE of LN(RR)SE of LN(RR)SE of LN(RR) The SE of LN(RR) is calculated from the cell The SE of LN(RR) is calculated from the cell

t i th 2 X 2 t blt i th 2 X 2 t blcounts in the 2 X 2 table:counts in the 2 X 2 table:DiseaseDisease No DiseaseNo Disease TotalTotal

Exposed GroupExposed Group aa bb a+ba+bNot Exposed Not Exposed cc dd c+dc+dGroupGroup

cc dd c dc d

dbd)c(c

db)a(a

bln(RR) SE


This formula is algebraically equivalent to the formula in text on pg. 201

95% Confidence Interval of 95% Confidence Interval of R l i Ri kR l i Ri kRelative RiskRelative Risk

First calculate the upper and lower limits on the First calculate the upper and lower limits on the ppppLN scaleLN scale

*961)l ( dbRR )()(

*96.1)ln(dccbaa

RR

After calculating the upper and lower limits on After calculating the upper and lower limits on the LN scale, use the exponential function to find the LN scale, use the exponential function to find the limits on the original scalethe limits on the original scalethe limits on the original scalethe limits on the original scale

If the 95% Confidence Interval does not contain If the 95% Confidence Interval does not contain the value 1.0, the association is statistically the value 1.0, the association is statistically significant at alpha = 0 05significant at alpha = 0 05


significant at alpha = 0.05significant at alpha = 0.05

Example: 95% CI for RR of Example: 95% CI for RR of Myocardial Infarction (MI) Myocardial Infarction (MI)

Background: Physicians enrolled in the Physician’s Health Background: Physicians enrolled in the Physician’s Health g y yg y yStudy were randomly assigned to take a daily aspirin or Study were randomly assigned to take a daily aspirin or placebo. The table provides the number with MI in each placebo. The table provides the number with MI in each group.group.g pg p

Group Yes No

Myocardial Infarction (MI)

Group Yes NoAspirin 139 10898 11037

Placebo 239 10795 11034378 21693 22071

Calculate the RR and 95% CI for the RR for MI (aspirin Calculate the RR and 95% CI for the RR for MI (aspirin

378 21693 22071


( p( pgroup compared to placebo group)group compared to placebo group)

1.1. Calculate the RR for MI:Calculate the RR for MI:aspirin group compared to aspirin group compared to

placeboplaceboplaceboplaceboMyocardial

Infarction (MI)Group Yes No

Aspirin 139 10898 11037Placebo 239 10795 11034

Infarction (MI)

Placebo 239 10795 11034378 21693 22071

RR = (139/11037) / (239/11034) = 0.581


2. Calculate the LN (RR) and 2. Calculate the LN (RR) and 3 Fi d h fid ffi i3 Fi d h fid ffi i3. Find the confidence coefficient3. Find the confidence coefficient2 Calculate the LN(RR)2 Calculate the LN(RR)2. Calculate the LN(RR)2. Calculate the LN(RR)

LN(0.581) = LN(0.581) = --0.5430.543Thi i th i t ti t f thThi i th i t ti t f thThis is the point estimate for the This is the point estimate for the confidence intervalconfidence interval

3. The confidence coefficient for the LN(RR) 3. The confidence coefficient for the LN(RR) is from the standard normal distribution. is from the standard normal distribution. For a 95% confidence interval, the For a 95% confidence interval, the coefficient = 1.96coefficient = 1.96


4. Calculate the SE LN(RR)4. Calculate the SE LN(RR)4. Calculate the SE LN(RR) 4. Calculate the SE LN(RR)

Group Yes No

Myocardial Infarction (MI)

Group Yes NoAspirin 139 10898 11037

Placebo 239 10795 11034378 21693 22071378 21693 22071

1079510898 106.011034*239

1079511037*139

10898 ln(RR) SE


5. & 6. Find the Lower and 5. & 6. Find the Lower and Upper limits on the LN scale and Upper limits on the LN scale and

the original scalethe original scalethe original scalethe original scale)336.0,75.0(106.0*96.1543.0

5. The upper and lower limits of the 5. The upper and lower limits of the confidence interval on the LN scale areconfidence interval on the LN scale are

)(

confidence interval on the LN scale areconfidence interval on the LN scale are((--0.750, 0.750, --0.336)0.336)

6. Use the EXP function in Excel to find 6. Use the EXP function in Excel to find the interval limits of the 95% CI of the the interval limits of the 95% CI of the


RR: RR: (0.472, 0.715)(0.472, 0.715)

Interpretation of 95% CI of RR Interpretation of 95% CI of RR for MIfor MI

The Relative Risk estimate = 0.58 which The Relative Risk estimate = 0.58 which indicates that physicians in the aspirin indicates that physicians in the aspirin

h d l i k f MI thh d l i k f MI thgroup had a lower risk of MI than group had a lower risk of MI than physicians in the placebo group.physicians in the placebo group.Th 95% fid i t l i di t th tTh 95% fid i t l i di t th t The 95% confidence interval indicates that The 95% confidence interval indicates that the decreased risk related to daily aspirin the decreased risk related to daily aspirin use isuse is significantsignificant (at alpha = 0 05) since(at alpha = 0 05) sinceuse is use is significant significant (at alpha = 0.05) since (at alpha = 0.05) since the interval (0.472, 0.715) does not the interval (0.472, 0.715) does not contain 1.contain 1.


contain 1.contain 1.

General Interpretation of 95% General Interpretation of 95% Ge e a te p etat o o 95%Ge e a te p etat o o 95%Confidence Interval of RRConfidence Interval of RR

A Relative Risk = 1.0 indicates ‘no association’ A Relative Risk = 1.0 indicates ‘no association’ between the exposure and the disease.between the exposure and the disease.

If the 95% confidence interval for the RR does If the 95% confidence interval for the RR does not contain 1.0 we can conclude that there is a not contain 1.0 we can conclude that there is a statisticall significant* association bet een thestatisticall significant* association bet een thestatistically significant* association between the statistically significant* association between the exposure and the disease.exposure and the disease.* at the 0 05 significance level* at the 0 05 significance level at the 0.05 significance level at the 0.05 significance level

If the 95% confidence interval for the RR If the 95% confidence interval for the RR contains 1.0, the association is contains 1.0, the association is notnot significant at significant at


ggthe 0.05 level. the 0.05 level.

Notes on the OR and RR CI:Notes on the OR and RR CI:Notes on the OR and RR CI:Notes on the OR and RR CI: The RR estimate will not be exactly in the middle The RR estimate will not be exactly in the middle yy

of the confidence interval on the original scaleof the confidence interval on the original scale The LN(RR) is exactly in the middle of the interval on The LN(RR) is exactly in the middle of the interval on

the LN scalethe LN scalethe LN scalethe LN scale The OR estimate will not be exactly in the middle The OR estimate will not be exactly in the middle

of the confidence interval on the original scaleof the confidence interval on the original scaleof the confidence interval on the original scaleof the confidence interval on the original scale The LN(OR) is exactly in the middle of the interval on The LN(OR) is exactly in the middle of the interval on

the LN scalethe LN scaleCh k th t th RR OR ti t i t i dCh k th t th RR OR ti t i t i d Check that the RR or OR estimate is contained Check that the RR or OR estimate is contained in the final interval in the final interval –– it’s easy to forget the EXP it’s easy to forget the EXP stepstep


stepstep

Readings and AssignmentsReadings and AssignmentsReadings and AssignmentsReadings and Assignments Reading: Chapter 8 pgs 200Reading: Chapter 8 pgs 200--201201 Reading: Chapter 8 pgs 200Reading: Chapter 8 pgs 200 201 201 Work through the Lesson 14 practice exercisesWork through the Lesson 14 practice exercises Excel Module 14 works through the examples inExcel Module 14 works through the examples in Excel Module 14 works through the examples in Excel Module 14 works through the examples in

this lesson for the confidence intervals of OR and this lesson for the confidence intervals of OR and RRRR Excel functions: LN and EXPExcel functions: LN and EXP There is no CI function for the RR and OR There is no CI function for the RR and OR –– you just you just

work through the stepswork through the steps Complete Homework 10Complete Homework 10


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Odds Ratio