Adjusting for extraneous factors Topics for today Stratified analysis of 2x2 tables Regression Readings Jewell Chapter 9

Berkeley Admissions Data 1973 study showed that 45% of 2691 male applicants were admitted, compared with only 30% of 1835 female applicants. The odds ratio is 1.84 with 95% confidence interval (1.62, 2.08). Is this evidence of sex bias? AdmitReject Male Female Log odds ratio = 95% conf interval:

Berkeley Admissions Data The picture changes completely once we look at admissions by department! Bickel, P.J., J.W. Hammel and J.W. O'Connell (1975) "Sex bias in graduate admissions: Data from Berkeley" in Science, 187: ) # applicants (% admit) DeptMaleFemale %10882% %2568% %59334% %37535% %39324% 63736%3417%

Stratified analysis Consider relationship between a disease outcome (D in Jewell, often Y in practice) and an exposure (E in Jewell, often X in practice), but we also want to adjust for an additional factor such as age or sex that can be divided up into I distinct strata. Suppose that the data from the ith stratum can be represented as follows: Jewell Tables 9.2 & 9.3 give two examples DiseasedNot Diseased Exposedaiai bibi Unexposedcici didi

What do we want to do? 1.Ask whether there is a significant association between disease (D) and exposure (E), after adjusting for the additional stratification factor 2.Estimate an adjusted odds ratio, that appropriately takes into account the stratification factor. Lets start with 1. but first, we need to quickly go over another way to assess whether there is a significant association for a 2x2 table

Assessing association - Berkeley Admissions again We already determined that there is a significant association in this 2x2 table, based on the 95% confidence interval for the odds ratio. An alternative approach is a chi-squared test There are several variations. But basic idea is to compare observed data to what would be expected if there were no association (see J p 69) Observed data AdmitReject Male Female Expected data AdmitReject Male Female

Chi-Squared test for a 2x2 table The test statistic is And its “significance” can be determined by looking up the chi-squared tables with 1 degree of freedom. For the Berkeley data, we get:

Back to the stratified analysis Cochran-Mantel-Haenszel test combines the differences between observed and expected values over all the strata. It focuses only on the “a” element of each 2x2 table Stratum iDNot D Eaiai bibi Not Ecici didi

Berkeley Admissions MaleFemale stratum a b c d

Estimating a common effect Wolf method (averages the log odds ratios) Mantel-Haenszel (averages the odds ratios) Regression-based

Wolf’s average log-odds ratio Can add.5 to cell entries if sample sizes are small

Applying Wolf method to Berkeley data stratumabcdlorvw=1/vw*lor Wolf estimate of LOR is.03, with variance What is 95% CI? Corresponding OR estimate is

Wolf’s average log-odds ratio Can add.5 to cell entries if sample sizes are small

Applying Wolf method to Berkeley data stratumabcdlorvw=1/vw*lor Wolf estimate of LOR is.03, with variance What is 95% CI? Corresponding OR estimate is

Mantel-Haenszel average odds ratio

Applying Wolf method to Berkeley data stratumabcdlorvw=1/vw*lor Wolf estimate of LOR is.03, with variance What is 95% CI? Corresponding OR estimate is

Regression-based analysis for Berkeley data data berkeley; input stratum male a b ; cards; run; data berkeley; set berkeley; n=a+b; Unstratified analysis; proc genmod; model a/n=male/dist=binomial; run; Code continued

Results of unstratified analysis Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square P Intercept <.0001 male <.0001 Scale Compare with our initial analysis

Stratified analysis proc genmod; class stratum; model a/n=male stratum/dist=binomial; run; Standard 95% Conf Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 male stratum <.0001 stratum <.0001 stratum <.0001 stratum <.0001 stratum <.0001 stratum Scale

More general modeling We can add additional factors into the logistic regression model so as to obtain an estimate of the log-odds ratio, adjusting for all these additional factors. Example, smoking in the Epilepsy study. Lets look in SAS: proc freq ; table one3*cig2 /chisq; run;

Epilepsy data in SAS

Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 DRUG DRUG DRUG Scale Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 DRUG DRUG DRUG CIG Scale

Why don’t drug estimates change much?? Hint – look at association between drug and smoking

proc freq ; table one3*cig2 /chisq; run;