Epidemiology Wonders of Biostatistics Chapter 13 - Effect Measures. John Koval

Transcription

1 Epidemiology 9509 Wonders of Biostatistics Chapter 13 - Effect Measures John Koval Department of Epidemiology and Biostatistics University of Western Ontario

2 What is being covered 1. risk factors 2. risk differences 3. relative odds - odds ratio 4. relative risk - risk ratios

3 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

4 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 then Newcombe combination of these two into estimator for π 1 π 2

5 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

6 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

7 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

8 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

9 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 = = 2.25(0.6124) = )

10 confidence interval (continued) 95% Confidence interval (2.25 ± 1.96(1.3793) = 2.25±2.70 = ( 0.45, 4.95) a very strange interval

11 confidence interval (better) use l = log(or) and its se l = log(or) = log(2.25) = (1 se(l) = a + 1 b + 1 c + 1 ) d ( = ) 10 = % CI ± 1.96(0.6124) = 0.811±1.200 = ( 0.389, 2.011) transform back (exponentiate) (0.68, 7.47)

12 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 = /12 22

13 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

14 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 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)

15 example of RR estimation RR = l RR = log(1.4288) = ( ) se(l RR ) = 8 15(23) (22) = = % confidence interval ± = ± = ( , ) exponentiate to get 95% CI for relative risk (0.831,2.478)

16 summary of estimates Parameter point estimate interval estimate Risk difference (-0.088,0.442) Relative odds (0.68,7.47) Relative risk (0.831,2.478)

17 SAS for effects title advanced contingency table ; DATA marj; INPUT r o freq; DATALINES; ; PROC FREQ; WEIGHT freq; TABLES r*o/chisq RISKDIFF RELRISK NOROW NOCOL NOPERCENT; add RELRISK to get estimated of Relative odds AND Relative Risk

18 Output of SAS effects program The FREQ Procedure Table of r by o r o Frequency 0 1 Total Total Statistics for Table of r by o Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Continuity Adj. Chi-Square Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer s V

19 Output of SAS effects program II Fisher s Exact Test Cell (1,1) Frequency (F) 15 Left-sided Pr <= F Right-sided Pr >= F Table Probability (P) Two-sided Pr <= P

20 Output of SAS effects program III column 1 Risk Estimates (Asymptotic)95% Exact) 95% Risk ASE Confid Limits Confid Limits Row Row Total Difference Column 2 Risk Estimates (Asymptotic)95% Exact) 95% Risk ASE Confid Limits Confid Limits Row Row Total Difference

21 Output of SAS effects program IV Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confid Limits Case-Control (Odds Ratio) Cohort (Col1 Risk) Cohort (Col2 Risk) Sample Size = 45