Analyses of Covariance Comparing k means adjusting for 1 or more other variables (covariates) Ho: u 1 = u 2 = u 3 (Adjusting for X) Combines ANOVA and regression –u y = u k +  X Assumptions are same as ANOVA + regression –Y normally distributed, constant variance across groups –Slope  the same for each group

SAS Code PROC GLM; CLASS group; MODEL chol12 = group cholbl/SS3 SOLUTION; MEANS group; LSMEANS group; ESTIMATE ‘Adjusted Mean Dif' group 1 -1; RUN;

Adjusted Means Computation Observations YBAR(A) i = YBAR i –  XBAR i – XBAR) 1)If  then adjusted mean equals unadjusted mean 2)If mean of X is same for all group then adjusted mean equals unadjusted mean

Computing the Adjusted Means 12-mo Avg.Baseline Avg. Diuretic Placebo Total227.0  =0.894Regression slope of 12-month cholesterol on baseline cholesterol YBAR(A) (Diur)= – (230.7 – 227.0) = – (3.7) = YBAR(A) (Plac)= – (224.9 – 227.0) = – (-3.7) =

Summary of Analyses of Continuous Variables Y is continuous variable Estimate a single mean  Compare 2 means     Compare k means  1,  2,  3, …  k Model means as function of 1 or more variables (LR) –Y =     X  +   X 

Summary of Analyses of Continuous Variables Hypothesis testing Ho:  0 Ho:     Ho:        k Ho:  j = 0

Summary of Analyses of Continuous Variables Confidence intervals

Analyses of Binary Outcomes Much of bio-medical data relates to analyses of binary outcomes: –Cancer (yes/no) –Survival (yes/no) –Had side-effect (yes/no) –Currently smoke cigarettes Social Sciences: –Divorced (yes/no) –Return to prison (yes/no) Political: –Favor a candidate (yes/no) –State has capital punishment (yes/no)

Analyses of Binary Variables Y has two outcomes (yes/no or 1/0) Estimate a single proportion  Compare 2 proportions     Compare k proportions  1,  2,  3, …  k Model probability as function of 1 or more variables –Y =     X  +   X 

Binary Outcomes Binary outcomes (Y=0 or 1) can be thought of in terms of probabilities: P (Y=1) =  P (Y=0) = (1 –  The ratio of the P(Y=1) to P(Y=0) is the odds  Odds (Y=1 versus Y = 0) = P(Y=1)/P(Y=0) =  (1 – 

Example Y = 1 indicates your horse winning the race P (Y=1) = 0.20 P (Y=0) = (1 – 0.20) = 0.80 What is the Odds of winning versus losing. Odds = P(Winning)/P(Losing) = 0.20/0.80 = 0.25 or ¼ In gambling terms the odds are 4 to 1.

Relationship Between Probability and Odds  Odds (o=  For small values the probability and the odds are close in value

Comparing Two Groups With Binary Outcomes  1 = probability of Y=1 for group 1  2 = probability of Y=1 for group 2 Ways to summarize the probability differences: 1)  1 -  2 difference in probabilities 2)  1 /  2 ratio of probabilities (Relative Risk) 3) (  1 /(1-  1 )/ ratio of odds (Relative Odds) (  2 /(1-  2 )

Example Group 1: Smokers Group 2: Non-smokers Y = 1 indicates cough upon awakening    0.30    0.20     = 0.10     = 0.30/0.20 = 1.50     = (.30/.70)/(.20/.80) = 0.429/0.250 = 1.71

Interpretation of Relative Risks Group 1: Smokers Group 2: Non-smokers RR = 1.50 There is a 50% increased risk of cough for smokers compared to non-smokers. Smokers are at a 50% increased risk of cough compared to non-smokers

Interpretation of Relative Risks Changing the Reference Group Group 1: Non-Smokers Group 2: Smokers RR = 0.67 (1/1.50 or.20/.30) There is a 33% decreased risk of cough for non- smokers compared to smokers. Non-smokers are at a 33% lower risk of cough compared to smokers.

Results: During follow-up, 477 major cardiovascular events were confirmed in the aspirin group, as compared with 522 in the placebo group, for a nonsignificant reduction in risk with aspirin of 9 percent (relative risk, 0.91; 95 percent confidence interval, 0.80 to 1.03; P=0.13). With regard to individual end points, there was a 17 percent reduction in the risk of stroke in the aspirin group, as compared with the placebo group (relative risk, 0.83; 95 percent confidence interval, 0.69 to 0.99; P=0.04), owing to a 24 percent reduction in the risk of ischemic stroke (relative risk, 0.76; 95 percent confidence interval, 0.63 to 0.93; P=0.009) and a nonsignificant increase in the risk of hemorrhagic stroke (relative risk, 1.24; 95 percent confidence interval, 0.82 to 1.87; P=0.31). As compared with placebo, aspirin had no significant effect on the risk of fatal or nonfatal myocardial infarction (relative risk, 1.02; 95 percent confidence interval, 0.84 to 1.25; P=0.83) or death from cardiovascular causes (relative risk, 0.95; 95 percent confidence interval, 0.74 to 1.22; P=0.68). Gastrointestinal bleeding requiring transfusion was more frequent in the aspirin group than in the placebo group (relative risk, 1.40; 95 percent confidence interval, 1.07 to 1.83; P=0.02). NEJM March 2005: A Randomized Trial of Low-Dose Aspirin in the Primary Prevention of Cardiovascular Disease in Women

Relationship Between Relative Risk and Relative Odds RO = RR x (1-  2 ) / (1-  1 ) If  1 and  2 are small (<0.10) then –RO ~ RR –Because of this relative risk and relative odds are sometimes interpreted in the same way

Example RR = 2.0 and RR=0.5 RR = 2.0  1  2 Odds Ratio RR = 0.5  1  2 Odds Ratio

Why Use Ratios In most cases the probability of an event is dependent on length of time  time) Using ratios removes time as a factor    t  prob. of developing lung cancer for smokers    t  prob. of developing lung cancer for non-smokers –RR =    t    t  Using differences does not remove time as a factor DIF =    t    t 

Comparing Studies With Different Follow-up Time Study 1 follows patients for 5 years:    5  prob. of developing lung cancer for smokers    prob. of developing lung cancer for non-smokers –RR =  Study 2 follow patients for 30 years:    30  prob. of developing lung cancer for smokers    30  prob. of developing lung cancer for non- smokers –RR = 

Hypothesis Testing Confidence Intervals Ho:  1 =  2 Ha:  1 ≠  2 Estimate  1 with p 1 = number with condition/total in group 1 Estimate  2 with p 2 = number with condition/total in group 2 p 1 -p 2 is point estimate of  1 -  2

Proportions for two groups 95% CI for difference in proportions:

Proportions for two groups Example 50 men with 13 smokers 50 women with 10 smokers p 1 = 13/50 = 0.26, p 2 = 10/50 = 0.20 SE = sqrt( ) = % CI for difference = 0.06 ± 1.96* ± (-0.105, 0.225) We do not have evidence that the proportion of smokers is different for men and women

Estimation a Single Proportion Example N=625 sampled; X=# favor X = 300 p = 300/625 = 0.48 SE = SQRT( (0.48)(0.52)/625) = % CI: =0.48 ± 1.96* ± 0.04 (0.44, 0.52)