epidemiology 9509 - wonders of biostatistics chapter 13...

Epidemiology 9509 effect measures

Epidemiology 9509Wonders of Biostatistics

Chapter 13 - Effect Measures

John Koval

Department of Epidemiology and BiostatisticsUniversity of Western Ontario

1


What is being covered

1. risk factors

2. risk differences

3. relative odds - odds ratio

4. relative risk - risk ratios

2


Risk factors

factor which can lead to (bad) outcomeSince risk and outcome are binarycan think of risk asprobability of presence of risk factorleading to bad outcome

hence smoking is a risk factorfor the outcome respiratory disease

think of risk at two levels of smokingsmokers, π1and non-smokers, π2

3


Risk differences

differences in risk for two levels of risk factorδπ = π1 − π2have already considered this

1. test of hypothesis (two-sided alternative)

1.1 Fisher exact test1.2 test of association/independence

with continuity correction SY

2. test of hypothesis (one-sided alternative)

2.1 Fisher exact test2.2 test of association/independence SY

3. estimation

3.1 Wilson/Adjusted Wald estimators for π1, π2

3.2 then Newcombe combination of these twointo estimator for π1 − π2

4


Relative Odds

◮ odds

ω = π11−π1

eg π1 = 0.6, so that (1− π) = 0.4odds ω = 1.5often quoted as 3:2

◮ relative odds

φ = ω1ω2

relative odds for group 1 compared to group 2eg π2 = 0.5, so (1− π2) = 0.5odds ω2 = 1(1 : 1)relative odds φ = (1.5)/(1) = 1.5

5


Odds ratio - estimating the Relative Odds

◮ odds

oi =pi

1−pieg p1 = 0.6, so that (1− p1) = 0.4odds o1 = 1.5often quoted as 3:2

◮ odds ratio

OR = o1o2

odds ratio for group 1 compared to group 2eg p2 = 0.5, so (1− p2) = 0.5odds o2 = 1(1 : 1)odds ratio OR = (1.5)/(1) = 1.5

6


shortcut computation of Odds Ratio

if entries in 2x2 contingency tablea, b, c ,dp1 = a/(a + b)p2 = c/(c + d)

so thato1 =

aa+b

/ ba+b

= ab

o2 =c

c+d/ dc+d

= cd

thenOR = o1

o2= a

b/ cd= ad

bc

7


Inference - test of hypothesis

test of φ = 1ie of ω1 = ω2

ie of π1 = π2

1. Fisher exact test

2. test of association SY

8


inference - confidence interval

can use odds ratio, OR,to estimate φ, the relative odds

need standard error

se(OR) = OR

√

(

1a+ 1

b+ 1

c+ 1

d

)

useful only for very large samples

example, a=15, b=8, c=10,d=12OR = ad

bc= 15(12)

8(10 = 2.25

se(OR) = OR

√

(

115 + 1

8 +112 + 1

10

)

= 2.25√0.375 = 2.25(0.6124) = 1.3793

9


confidence interval (continued)

95% Confidence interval

(2.25 ± 1.96(1.3793)= 2.25 ± 2.70

= (−0.45, 4.95)a very strange interval

10


confidence interval (better)

use l = log(OR) and its se

l = log(OR) = log(2.25) = 0.811

se(l) =√

(

1a+ 1

b+ 1

c+ 1

d

)

=√

(

115 +

18 + 1

12 +110

)

= 0.6124

95% CI0.811 ± 1.96(0.6124)= 0.811 ± 1.200= (−0.389, 2.011)

transform back (exponentiate)(0.68, 7.47)

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Relative Risk

if π1 and π2 are risksRelative Risk is π1

π2

if p1 and p2 are observed proportionsRisk Ratio: RR = p1

p2is point estimator of Relative Risk

for example, for a,b,c,dp1 =

aa+b

,p2 =c

c+d

RR = aa+b

/ cc+d

exampleRR = 15

23/1222

= 1.4348

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Relative Risk: test of hypothesis

test ofHo : Relative Risk = 1ieHo : π1

π2= 1

can be rewritten asHo : π1 = π2

same hypothesis as for Risk DifferenceHence use same tests:

1. Fisher’s Exact Test

2. SY , Yates continuity-corrected version of Pearson test

13


confidence interval for Relative Risk

again using RR ± 1.96se(RR)produces strange interval for small samples

use lRR = log(RR) and its standard error

se(lRR) =

√

(

1−p1n1p1

+ 1−p2n2p2

)

for contingency tables entries a,b,c,d

se(lRR) =

√

(

ba(a+b) +

dc(c+d)

)

14


example of RR estimation

RR = 1.4388lRR = log(1.4288) = 0.3610

se(lRR) =

√

(

815(23) +

1210(22)

)

=√0.23188 + 0.54545 =

√0.77733

95% confidence interval0.3610 ± 1.96

√0.77733

= 0.3610 ± 0.54646= (−0.18545, 0.90742)

exponentiate to get 95% CI for relative risk(0.831,2.478)

15


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)

16


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 oddsAND Relative Risk

17


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 0.198818


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

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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

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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

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epidemiology 9509 - wonders of biostatistics chapter 13...

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