Statistics In Evidence Based Medicine

www.epilepsy.va.gov/StatisticsStatistics in Evidence ksandLecture 2: Analysisof studyRelativedesigns forbinaryOdds Ratiosand Odds)outcomes (Proportions1

Course outlineUnderstanding logistic regression in fivelecturesDifference between relative risk and odds ratio,marginal and conditional odds ratios,terminology and interpretation of logisticregression, matched data analysisSuggested Book: Logistic Regression A SelfLearning Text by Kleinbaum & KleinThird Edition Springer2

Today’s Lecture Definitions Odds Ratios and Relative RiskUses in Study Designs Interpretation and appropriate useExamples3

Binary OutcomesA binary data takes only one of twovaluesExamples:Alive or dead, Sick or Well, Exposed orUnexposed etcWe can find proportions for binaryoutcomes4

Proportion (p)A proportion represents a situation where thenumerator and denominator both represent counts,and the numerator is a subset of the denominator.A proportion always lies between 0 and 15

An Example of ProportionsPhysicians’ Health Study 1989Aspirin for reduction of Myocardial InfarctionMI YesMI 0,79511,034Proportion of physicians who had an MI in aspirin group 139/11,037Proportion of physicians who had an MI in placebo group 239/11,034Proportions are risksHow to compare risks between groups?Basics and Clinical Biostatistics6

An Example of ProportionsPhysicians’ Health Study 1989Aspirin for reduction of Myocardial InfarctionMI YesMI 0,79511,034Proportion of physicians who had an MI in aspirin group 139/11,037Proportion of physicians who had an MI in placebo group 239/11,034Proportions are risks (probabilities)How to compare risks between groups?Basics and Clinical Biostatistics7

Absolute Risk Difference (ARD)Risk of an MI in aspirin group p1 139/11,037 0.0126Risk of an MI in placebo group p2 239/11,034 0.0217ARD p1-p2 0.0126 –0.0217 0.0091Interpretation per 10,000 persons:The risk of MI in aspirin group was 126 per10,000 people (0.0126 10,000) and risk ofMI in placebo group was 217 (0.0217 10,000).An extra 91(0.0091 10,000)people had an MIunder placebo.8

Determining Ratios of RisksRisk of an MI in aspirin group p1 139/11,037 0.0126Risk of an MI in placebo group p2 239/11,034 0.0217What if we divide the risks?Risk of an MI in placebo group/Risk of an MI in aspirin group p2 / p1 0.0217/0.0126 1.72How to interpret the number 1.72?Placebo group was 1.72 times more likely to have an MIthan the aspirin group. Placebo group was at a 72%increased risk of MI than aspirin group.9

Relative Risk (RR)Risk of an MI in aspirin group p1 139/11,037 0.0126Risk of an MI in placebo group p2 239/11,034 0.0217What if we divide risks the other way?Risk of an MI in aspirin group/Risk of an MI in placebo group 0.581Division of risks gives us Relative Risk (RR)How to interpret the number 0.581?The Relative Risk of MI in aspirin group compared withplacebo group is 0.581.10

Relative Risk ReductionWhen Relative Risk is less than 1RRR (p2 – p1) /p2 ARD/p2 1 – RRRelative Risk of MI with aspirin compared to placebo 0.581RRR 1 – 0.581 0.41942% reduced risk of MI in aspirin groupcompared with baseline group (placebo)11

Use of Relative Risk in Study Designs Summary statistics for binary data Clinical trials Cohort study design Case-control study design Cross-sectional study design 12

Clinical TrialIn a clinical trial an intervention such as newdrug (treatment group) is compared with aplacebo or standard therapy (control group)for an outcome (efficacy /safety) amonghumans.13

Clinical Trials14

Clinical Trial and Relative RiskOutcome YesOutcome NoTotalTreatment groupaba b N1Control groupcdc d N2Totala c M1b d M2T a b c dRisk in treatment group p1 a/N1Risk in control group p2 (baseline risk) c/N2p1 a/N1Relative Risk p 2 c/N215

Absolute Risk Difference vs. Relative Risk Reduction Absolute Risk Difference provides number needed totreat (NNT), so may be more helpful.NNT 1/ARD If RRR is reported, multiply this with risk in control group(p2) to get ARD.RRR ARD/p2ARD RRR p216

Prospective Cohort study and Relative RiskIn a cohort study, a group without thedisease is followed up to see who develops it,and disease incidence in persons with acharacteristic (risk factor) is compared withincidence in persons without thecharacteristic (risk factor).Statistical Analysis: Risk Ratio or Relative Risk17

Cohort Study18

Cohort Study and Relative RiskOutcome (Disease)YesExposure(Risk Factor)YesNoNoTotalaba b N1cdc d N2b d M2T a b c dTotal a c M1Risk of disease among exposed pE a/N1Risk of disease among unexposed pNE (baseline risk) c/N2a/N 1Relative Risk(RR) c/N 219

A Cohort StudyEating Broiled Fish Reduces the Risk of StrokeIntake of animal products and stroke mortality inthe Hiroshima/Nagasaki Life Span StudySauvaget C, Nagano J, Allen N, et al. (2003)International Journal of EpidemiologyRisk factor present Not eating broiled fish at allRisk factor absent Eating broiled fish20

Eating Fish Reduces the Risk of StrokeStroke YesStroke NoTotalRisk factor Yes(Don’t eat fish)8215491631Risk factor No(Eat fish)23779802Total1052328243382 0.0503163123Risk of stroke amongunexposed pNE 0.02878020.0503RR 1.750.0287Risk of stroke among exposed pE 21

Interpretation of Relative RiskRR 1.75Those who never eat fish are 1.75 timesmore likely to have stroke than thosewho eat fish almost daily.22

Case-Control StudyA case-control study is one in which “cases”(persons with disease) are identified,“controls” (similar to cases but disease free)are identified, and the two groups arecompared with respect to prior exposure torisk factor.23

Case-Control StudyWe can’t compute Relative Risk directly in acase-control study. Instead we calculateOdds Ratio.24

Validity of Relative Risk in a Case-Control StudyCaseControlTotalExposedaba bNot exposedcdc dTotala cb da b c daRisk of diseaseamong exposedRelative Risk a bRisk of diseaseamongunexposed cc dWhat if we double controls?CaseControlTotalExposeda2ba 2bNot exposedc2dc 2dTotala c2b 2da 2b c 2daRelative Risk a 2bcc 2d25

Odds (o) The odds (O) of an event are the likelihood of anevent occurring divided by the likelihood of eventnot occurring For a 2 2 table divide the counts of occurrence of anevent by counts of non occurrence of an eventOdds can lie between zero and infinityOdds are ratios of proportions26

Relationship between odds and probability To calculate the odds (o) from Probability(p)𝑝Odds 1 𝑝 To calculate the probability from Odds𝑜Probability 1 𝑜27

Example of OddsMale Lung Cancer & Smoking (Doll and Hill 1950)Lung 622Odds of smokingamong Controls 27Odds of smokingamong Cases Odds of lung cancer among Smokers 6476222Odds of lung cancer amongNon - smokers 27an introduction to medical statistics: martin bland28

Comparing Groups With Respect to SmokingOdds of smoking in cases 647:2 647/2Odds of smoking in controls 622:27 622/27We obtain Odds Ratio by division of oddsOdds of smoking in cases/ Odds of smoking in controls 647 / 2 647 27Odds Ratio 14.04622 / 272 622The odds of smoking in lung cancer patients were 14 times theodds of smoking in controls29

Comparing Groups With Respect to Lung CancerOdds of lung cancer in smokers 647:622 647/622Odds of lung cancer in controls 2:27 2/27What if we divide again? We obtain odds RatioOdds of lung cancer in smokers/ Odds of lung cancer in non-smokers 647 / 622 647 27 14.042 / 272 622The odds of lung cancer in smokers were 14 times the odds of lungcancer in non-smokers30

Odds Ratio for a Case-Control Studyodds of exposure in cases a/codds of exposure in controls b/daadcOdds Ratio bbcdOdds Ratio is symmetrical31

Odds Ratio Approximates Relative Risk for aRare DiseaseWhen a disease is rareaacc & a b bc d daaada bbRelative Risk Odds Ratioccbcc dd32

Cross-Sectional StudyIn a cross-sectional study people are observedat a single point in time. We inquire what ishappening right now? We can investigateprevalence of disease and exposure to riskfactors.Examples; Surveys, Registries reports etc33

Cross-Sectional Study34

Analysis of a Cross-Sectional StudyM1TNPrevalence of Exposure 1TadOdds Ratio bca(c d)Relative Risk c(a b)Prevalence of disease 35

What Summary Statistic to use? For prospective studies (clinical trials/cohort)quote Relative Risk. For case-control studies quote Odds Ratio. Odds Ratio approximates Relative Risk for a raredisease in case-control studies. For a cross-sectional study one has a choicebetween Odds Ratio and Relative Risk.36

Example: Lung cancer among womenin the USIn the US, the estimated annual probability that a womenover the age of 35 dies of lung cancer equals 0.001304 forcurrent smokers and 0.000121 for non smokers( M Paganoand K. Gauvreau, principles of Biostatistics, 1993, p.134).a. Find and interpret the difference of proportions and therelative risk. Which measure is more informative for thesedata and why?b. Find and interpret the odds ratio. Is odds ratio almostequal to relative risk? Why?Categorical Data Analysis (2002 ) Agresti37

Example: Lung cancer among womenin the USIn the US, the estimated annual probability that a women over the age of35 dies of lung cancer equals 0.001304 for current smokers and 0.000121for non smokers( M Pagano and K. Gauvreau, principles of Biostatistics,1993, p.134).ARD 0.001304-0.000121 0.001183RR 0.001304/0.000121 10.777RR is more informative than ARD38

Example: Lung cancer among womenin the USIn the US, the estimated annual probability that a women over the age of 35dies of lung cancer equals 0.001304 for current smokers and 0.000121 for nonsmokers( M Pagano and K. Gauvreau, principles of Biostatistics, 1993, p.134).0.001304/(1 0.001304)Odds Ratio 10.790.000121/(1 0.000121)Odds Ratio Relative Riskbecause both probabilities are very small39

Misinterpretation of Odds RatiosA research study estimated that under a certain condition,the probability that a subject would be referred for heartcatheterization was 0.906 for whites and 0.847 for blacks. Apress release about the study stated that the odds of referralfor cardiac catheterization for blacks are 60% of the odds forwhites. Explain how they obtained 60% (more accurately 57%)𝟎.𝟖𝟒𝟕/(𝟏 𝟎.𝟖𝟒𝟕) 𝟎.𝟖𝟒𝟕(𝟏 𝟎.𝟗𝟎𝟔) 0.574𝟎.𝟗𝟎𝟔(𝟏 𝟎.𝟗𝟎𝟔) 𝟎.𝟗𝟎𝟔(𝟏 𝟎.𝟖𝟒𝟕)40For details see New Engl. J. Med. 341: 279-283, 1999

MisinterpretationAn associated press story later described the study and said.“Doctors were only 60% as likely to order cardiaccatheterization for blacks as for whites”.What is wrong with above interpretation? What is thecorrect percentage for this interpretation?News story interpreted odds ratio as relative risk.For correct interpretation𝟎.𝟖𝟒𝟕RR 93.488%𝟎.𝟗𝟎𝟔41

Uses of Odds Ratios Odds Ratios are used in all kind of studies Odds Ratios have nice mathematical properties Odds Ratios are results of logistic regression. Logisticregression adjusts for confounding A common way to present results of a meta analysis42

zwana.Rehman@va.gov(919) 286-0411 ext: 5024Thank you for being patient !43

smokers( M Pagano and K. Gauvreau, principles of Biostatistics, 1993, p.134). Odds Ratio 0.001304/ :1