Case Control Study : Analysis

Odds and Probability

what is odds? Let’s say that probability of success is 0.8, thus P =.8 Then the probability of failure is = 1-p = q =.2 The odds of success are The odds of failure would be Odds of <1 mean the outcome occurs less than half the time Odds of 1 mean the outcome occurs half the time Odds of >1 mean the outcome occurs more than half the time

odds ratio= The odds of success are 16 times greater for failure.

Odds of disease among exposed = a/ b Odds of disease among unexposed = c/d Exposure Case Control +-+- a+c b+d dc ba ODDS RATIO: a+b c+d

Example : The following data refer to a study that investigated the relationship between MI and smoking.The first column refers to 262 young and middle - aged women admitted to 30 coronary care units with acute MI. Cross classification of smoking status and MI MI Controls Ever smokers yes no

The odds of having MI is 3.82 times more for smokers as compared nonsmokers.

MI Yes No Group placebo aspirin Rare disease example:

Example of OR with common outcome Stroke unit Yes No Control The frequency of poor outcome (i.e., mortality in the control group) was very high (55%) The OR underestimates the RR by 19% Mortality

TESTING HYPOTHESIS USING CI 01 

ISSUES IN APPROXIMATING ‘OR’ INTO ‘RR’ When the risks (or odds) in the two groups being compared are both small (say less than 20%) then the OR will approximate to the RR The discrepancy between odds ratio and relative risk depends on the risk (odds) in both groups (i.e., the discrepancy depends on the initial risk and the OR itself ) OR may be non-intuitive in interpretation, but in almost all realistic cases interpreting them as though they were RRs is unlikely to change any qualitative assessment of the study findings

Since the odds ratio is difficult to interpret, why is it so widely used? Odds ratio is valuable in case control studies where events are usually rare The odds ratio remains especially useful when researchers need to adjust for other variables, for which LR is usual approach Odds ratios are a common way of presenting the results of a Meta - Analysis - a statistical analysis for combining the results of several studies, used within systematic reviews

Examples of Confounding and Effect Modification ____________________________________________________ RR (exposed vs non exposed)control var is str1str2str3Total(crude)ConfEM Example A YN Example B NN Example C NY Example D YY ___________________________________________________ Level of Control Variable:

Twenty-Four-Hour Mortality After Coronary Artery Bypass Surgery, by Sex At 24 hours Sex Total Dead Dead % OR RR (95% CI) Female Male ( ) ( ) Total

Twenty-Four-Hour Mortality After Coronary Artery Bypass Surgery, by Body Surface Area (BSA) At 24 hours Total Dead Dead % OR RR (95% CI) (95% CI) Low BSA High BSA ( )( ) Total

Twenty-Four-Hour Mortality After Coronary Artery Bypass Surgery, by Sex within Each BSA Category At 24 hours Total Dead Dead % OR RR (95% CI) (95% CI) Low BSA Female Male ( ) ( High BSA Female Male ( ) (

Calculation of Mantel-Haenzel Chi-Square Dead Sex Yes No Total a i E(a i )V(a i ) Low BSA Female Male Total High BSA Female Male Total

Calculation of Mantel-Haenzel Adjusted Measures Dead ExposureYes No Total OR RR Low BSA Female /7.88= /8.11=1.06 Male Total High BSA Female /0.61= /0.62=1.58 Male Total Adjusted Measure 9.32/4.89= /8.73=1.1

Study design procedure: select referent group comparable to index group on one or more matching factors Basics for matching

Matching factor = age referent group constraint to have same age structure Case-control study: referent = controls index = cases Follow-up study: referent = unexposed index = exposed

Category Matching Factor A: A1, A2, A3 Factor B: B1, B2, B3 Factor C: C1, C2, C3 Example: AGE: 20-39, 30-39, 40-49, 50-59, GENDER: Male, Female STAGE: I, II, III

Category Matching Factor A: A1, A2, A3 Factor B: B1, B2, B3 Factor C: C1, C2, C3 Example: AGE: 20-39, 30-39, 40-49, 50-59, GENDER: Male, Female STAGE: I, II, III Control has same age-gender-stage combination as Case

Category Matching Factor A: A1, A2, A3 Factor B: B1, B2, B3 Factor C: C1, C2, C3 Example: AGE: 20-39, 30-39, 40-49, 50-59, GENDER: Male, Female STAGE: I, II, III Control has same age-gender-stage combination as Case

Category Matching Factor A: A1, A2, A3 Factor B: B1, B2, B3 Factor C: C1, C2, C3 Example: AGE: 20-39, 30-39, 40-49, 50-59, GENDER: Male, Female STAGE: I, II, III

CaseNumber ofType Controls RR- to or pair matching R may vary from case to case e.gR=3 for some cases R=2 for other cases R=1 for other cases Not always possible to find exactly R controls for each case.

Usual Display of Matched Case-Control Data with Dichotomous Exposure Controls Exposure Cases ExposurePresentAbsentTotal Presentfgf+g Absenthjh+j Totalf+hg+jn

Analysis of matched data Control +-Total Case+aba+b -cdc+d Totala+cb+dn Comparison: proportion of exposed cases vs proportion of exposed controls Information regarding differential exposure is given by the discordant pairs

Odds ratio “Inference regarding the difference in proportions in matched pairs is made solely on the basis of discordant pairs” McNemar (1947) Estimate of OR conditional on the number of discordant pairs is given as Kraus (1960), Cox (1958)

To match or not to match Advantages: Matching can be statistically efficient i,.e may gain precision using confidence interval Disadvantages: Matching is costly To find matches Information loss due to discarding controls

Match on strong risk factors expected to be confounders. MatchingNo matching Correct estimate?YES Appropriate analysis?YES Matched Standard stratified (Stratified)OR1, OR2, OR3 Combine Validity is not an important reason for matching Match to gain efficiency or precession

Matched Analysis using Stratification: Strata = matched sets Special case: case-control study 100 matched pairs, n= strata = 100 matched pairs 2 observations per stratum 1 st pair 2 nd pair 100th pair E E E E E E DDDD DDDD DDDD

E D W + X + Y + Z = total number of pairs DDDD DDDD DDDD W pairs Z pairs Y pairs X pairs

E D 2. McNemar chi-square test Ch-square = (X-Y)2/(X-Y) = (30-10)2/(30+10) = 10.0 (df=1, p<0.01) McNemar’s test = MH test for pair matching MOR = X/Y = DDDD DDDD DDDD W pairs (30) Z pairs (30) Y pairs (10) X pairs (30) W Y X Z E E D E D

Example: W = 30, X=30, Y =10, Z =30 W + X + Y + Z = 100 Analysis : Two equivalent ways 1.Mantel -Haenzel Chi-square test Stratum 1 Stratum 2 Stratum 100 Compute Mantel-Haenzel chi-square and MOR

Analysis for R-to-1 and mixed matching : Use stratified analysis Example: If, R = 4 E E DDDD Each stratum contains five subjects Compute MH chi-square and MOR based on stratified data