Epidemiology 9509 Wonders of Biostatistics Chapter 13 - Effect Measures John Koval Department of Epidemiology and Biostatistics University of Western Ontario
What is being covered 1. risk factors 2. risk differences 3. relative odds - odds ratio 4. relative risk - risk ratios
Risk factors factor which can lead to (bad) outcome Since risk and outcome are binary can think of risk as probability of presence of risk factor leading to bad outcome hence smoking is a risk factor for the outcome respiratory disease think of risk at two levels of smoking smokers, π 1 and non-smokers, π 2
Risk differences differences in risk for two levels of risk factor δ π = π 1 π 2 have already considered this 1. test of hypothesis (two-sided alternative) 1.1 Fisher exact test 1.2 test of association/independence with continuity correction S Y 2. test of hypothesis (one-sided alternative) 2.1 Fisher exact test 2.2 test of association/independence S Y 3. estimation 3.1 Wilson/Adjusted Wald estimators for π 1,π 2 3.2 then Newcombe combination of these two into estimator for π 1 π 2
Relative Odds odds ω = π 1 1 π 1 eg π 1 = 0.6, so that (1 π) = 0.4 odds ω = 1.5 often quoted as 3:2 relative odds φ = ω 1 ω 2 relative odds for group 1 compared to group 2 eg π 2 = 0.5, so (1 π 2 ) = 0.5 odds ω 2 = 1(1 : 1) relative odds φ = (1.5)/(1) = 1.5
Odds ratio - estimating the Relative Odds odds o i = p i 1 p i eg p 1 = 0.6, so that (1 p 1 ) = 0.4 odds o 1 = 1.5 often quoted as 3:2 odds ratio OR = o 1 o 2 odds ratio for group 1 compared to group 2 eg p 2 = 0.5, so (1 p 2 ) = 0.5 odds o 2 = 1(1 : 1) odds ratio OR = (1.5)/(1) = 1.5
shortcut computation of Odds Ratio if entries in 2x2 contingency table a, b, c,d p 1 = a/(a+b) p 2 = c/(c +d) so that o 1 = a a+b / b a+b = a b o 2 = c c+d / d c+d = c d then OR = o 1 o 2 = a b /c d = ad bc
Inference - test of hypothesis test of φ = 1 ie of ω 1 = ω 2 ie of π 1 = π 2 1. Fisher exact test 2. test of association S Y
inference - confidence interval can use odds ratio, OR, to estimate φ, the relative odds need standard error (1 se(or) = OR a + 1 b + 1 c + 1 ) d useful only for very large samples example, a=15, b=8, c=10,d=12 OR = ad bc = 15(12) 8(10 = 2.25 ( se(or) = OR 1 15 + 1 8 + 1 12 + 1 10 = 2.25 0.375 = 2.25(0.6124) = 1.3793 )
confidence interval (continued) 95% Confidence interval (2.25 ± 1.96(1.3793) = 2.25±2.70 = ( 0.45, 4.95) a very strange interval
confidence interval (better) use l = log(or) and its se l = log(or) = log(2.25) = 0.811 (1 se(l) = a + 1 b + 1 c + 1 ) d ( = 1 15 + 1 8 + 1 12 + 1 ) 10 = 0.6124 95% CI 0.811 ± 1.96(0.6124) = 0.811±1.200 = ( 0.389, 2.011) transform back (exponentiate) (0.68, 7.47)
Relative Risk if π 1 and π 2 are risks Relative Risk is π 1 π 2 if p 1 and p 2 are observed proportions Risk Ratio: RR = p 1 p 2 is point estimator of Relative Risk for example, for a,b,c,d p 1 = a a+b,p 2 = c c+d RR = a a+b / c c+d example RR = 15 = 1.4348 23 /12 22
Relative Risk: test of hypothesis test of H o : Relative Risk = 1 ie H o : π 1 π 2 = 1 can be rewritten as H o : π 1 = π 2 same hypothesis as for Risk Difference Hence use same tests: 1. Fisher s Exact Test 2. S Y, Yates continuity-corrected version of Pearson test
confidence interval for Relative Risk again using RR ± 1.96se(RR) produces strange interval for small samples use l RR = log(rr) and its standard error (1 p1 se(l RR ) = ) n 1 p 1 + 1 p 2 n 2 p 2 for contingency tables entries a,b,c,d ( ) se(l RR ) = b a(a+b) + d c(c+d)
example of RR estimation RR = 1.4388 l RR = log(1.4288) = 0.3610 ( ) se(l RR ) = 8 15(23) + 12 10(22) = 0.23188+0.54545 = 0.77733 95% confidence interval 0.3610±1.96 0.77733 = 0.3610±0.54646 = ( 0.18545, 0.90742) exponentiate to get 95% CI for relative risk (0.831,2.478)
summary of estimates Parameter point estimate interval estimate Risk difference 0.198 (-0.088,0.442) Relative odds 2.250 (0.68,7.47) Relative risk 1.435 (0.831,2.478)
SAS for effects title advanced contingency table ; DATA marj; INPUT r o freq; DATALINES; 0 0 15 0 1 8 1 0 10 1 1 12 ; PROC FREQ; WEIGHT freq; TABLES r*o/chisq RISKDIFF RELRISK NOROW NOCOL NOPERCENT; add RELRISK to get estimated of Relative odds AND Relative Risk
Output of SAS effects program The FREQ Procedure Table of r by o r o Frequency 0 1 Total 0 15 8 23 1 10 12 22 Total 25 20 45 Statistics for Table of r by o Statistic DF Value Prob ----------------------------------------------- Chi-Square 1 1.7787 0.1823 Likelihood Ratio Chi-Square 1 1.7900 0.1809 Continuity Adj. Chi-Square 1 1.0683 0.3013 Mantel-Haenszel Chi-Square 1 1.7391 0.1872 Phi Coefficient 0.1988 Contingency Coefficient 0.1950 Cramer s V 0.1988
Output of SAS effects program II Fisher s Exact Test ----------------------------------- Cell (1,1) Frequency (F) 15 Left-sided Pr <= F 0.9493 Right-sided Pr >= F 0.1507 Table Probability (P) 0.1000 Two-sided Pr <= P 0.2362
Output of SAS effects program III column 1 Risk Estimates (Asymptotic)95% Exact) 95% Risk ASE Confid Limits Confid Limits --------------------------------------------------- Row 1 0.6522 0.0993 0.4575 0.8468 0.4273 0.8362 Row 2 0.4545 0.1062 0.2465 0.6626 0.2439 0.6779 Total 0.5556 0.0741 0.4104 0.7007 0.4000 0.7036 Difference 0.1976 0.1454-0.0873 0.4825 Column 2 Risk Estimates (Asymptotic)95% Exact) 95% Risk ASE Confid Limits Confid Limits --------------------------------------------------- Row 1 0.3478 0.0993 0.1532 0.5425 0.1638 0.5727 Row 2 0.5455 0.1062 0.3374 0.7535 0.3221 0.7561 Total 0.4444 0.0741 0.2993 0.5896 0.2964 0.6000 Difference-0.1976 0.1454-0.4825 0.0873
Output of SAS effects program IV Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confid Limits ---------------------------------------------- Case-Control (Odds Ratio) 2.2500 0.6775 7.4720 Cohort (Col1 Risk) 1.4348 0.8307 2.4780 Cohort (Col2 Risk) 0.6377 0.3239 1.2553 Sample Size = 45