1. March 28 Analyses of binary outcomes 2 x 2 tables
Relative Risks and Relative Odds (Odds Ratio)
Lee 6.1 through 6.5
C & S Chapter 3 (G, I, K, L, M, O)

2. Estimating a Single Proportion
p = proportion in population with characteristic
Take random sample of size n
x = number in sample with characteristic
p = x/n estimate of p
SE(p) =
Assumptions:
n is large enough for CLT
Then p is normally distributed
95% CI for p: p ± 1.96 SE(p)
p – p0
Z =
SQRT(p(1-p) (1/n))

3. Example p = proportion favoring a certain candidate n = 625
x = 300 favor the candidate
p = 300/625 = 0.48 is estimate of p
SE(p) = = 0.020
95% CI for p: ± 1.96 (0.02)
0.48 ± 0.04
Note: Samplers use n=625 frequently

4. 1-Sample Z-Test: Matched Pair Data
Control Positive
Control Negative a b c d
Pos
Case
Neg
Analyses is done on discordant pairs
b and c
Called McNemar’s chi-square
Ho: p = 0.5 where n=b+c and x=b
Z = (b/(b+c) – 0.5)/sqrt(.5*.5/(b+c))
Z = (b-c)/sqrt(b+c)
c2 = (b-c)2/(b+c)

5. Example – Vitamin Use/Disease (440 Pairs)
Control Vitamin +
Control Vitamin -
100 50 90
200
Vit +
Case
Vit -
Ho: p = 0.5 where n=140 and b = 50
c2 = (50-90)2/(50+90) = (p=.0007)

6. Comparing Two Proportions
Ho: p1 = p2
Ha: p1 ≠ p2
p1 = x1/ n1
p2 = x2 / n2
p = (x1+x2)/(n1+n2) This is the pooled proportion
p2 – p1
Z =
p(1-p) (1/n1 + 1/n2)
Compare to standard normal distribution
Assume n1 and n2 are large enough to use normal approximation

7. CI: Difference in Proportions
95% CI for difference in proportions:

8. Example – Asthma and SES
Ho: p1 = p2
Ha: p1 ≠ p2
p1 = 30/ 160 = 0.188
p2 = 40 / 140 = 0.286
p = 70/300 = This is the pooled proportion
0.286 – 0.188
Z =
0.233(.767) (1/ /140)
c2 = Z2 = 4.03
X1 is number with asthma in high SES group
X2 is number with asthma in low SES group
= 0.098/0.049 = 2.01

9. 2 by 2 Table a b c d c2 = ( a + b + c + d ) (ad – bc)2
Factor Present
Factor Absent a b c d
Sample 1
n1 = a + b
Sample 2
n2 = c + d
a + c
b+ d
c2 = ( a + b + c + d ) (ad – bc)2
( a + c) (b + d) (a + b) (c + d)

10. 2 by 2 Table
Have Asthma
No Asthma 30
130 40
100
High SES
n1 =160
Low SES
n2 =140 70
230
c2 = ( ) (3000 – 5200)2
( 70) (230) (160) (140)
= 4.02

11. Relative Risks and Relative Odds
Factor Present
Factor Absent a b c d
Sample 1
n1 = a + b
Sample 2
n2 = c + d
a + c
b + d
RR = a/(a+b)
c/(c+d)
RO = a/b
c/d
= ad/bc
If a+b is approximately equal to b and
If c+d is approximately equal to d then RR will be close to RO

12. Calculation RR and RO 30 130 40 100 160 140 70 140 RR = 30/160 40/140
Have Asthma
No Asthma 30
130 40
100
High SES
160
Low SES
140 70
140
RR Asthma (High v Low SES)
RO Asthma (High v Low SES)
RR = 30/160
40/140
= 0.66
RO = 30/130
40/100
= 0.58
34% Lower risk of asthma in high SES compared to low SES

13. Confidence Interval for Relative Risk
This CI looks a little different from usual form
It is calculated on log scale
Distribution of RR possible values is skewed
Can’t be less than zero
Can be extremely large positive values
Usually transformed back when presented
Calculated automatically by SAS

14. Confidence Interval for Odds Ratio
Similar to CI for relative risk
Can calculate by hand easily;
SAS calculates automatically

15. Notes About CI for RR and RO
Confidence intervals are not symmetric around the point estimate
Cannot use RR ± SE notation
Point estimate: 0.66
95% CI (0.43 – .99)
0.23 below above

16. Odds Ratio Property 30 130 40 100 160 140 70 140 RO = 30/130 40/100
Have Asthma
No Asthma 30
130 40
100
High SES
160
Low SES
140 70
140
RO Asthma (High v Low SES)
RO High SES (Asthma v No Asthma)
RO = 30/130
40/100
= 0.58
RO = 30/40
130/100
= 0.58
Same Answer – Not true for RR

17. Cohort Versus Case Control Study
Cohort (Prospective)
Find a population of low SES persons and a population of high SES persons
For each person determine if he/she has asthma
Case-Control (Retrospective)
Find a group of persons with asthma and a group of persons without asthma.
Determine if person is of low or high SES

18. What Can Be Estimated Cohort (Prospective)
Can estimate probability of asthma for both low and high SES groups
Can compute relative risk of asthma (high versus low SES)
Case-Control (Retrospective)
Can not estimate probability of asthma
Can not estimate relative risk of asthma (H versus L SES)
Can estimate relative odds (H versus L SES)
If disease is fairly rare the RO can estimate RR

19. Cohort Versus Case-Control
Cohort (Prospective)
May not be possible to do
Case-Control (Retrospective)
May be only way to assess risk factors for disease

20. INPUT ses asthma count; DATALINES; 1 1 30 1 2 130 2 1 40 2 2 100 ;
USING SAS
DATA asthma;
INFILE DATALINES;
INPUT ses asthma count;
DATALINES;
1 1 30
2 1 40 ;
PROC FREQ DATA=asthma;
TABLES ses*asthma/CHISQ RELRISK;
WEIGHT count;
RUN;
Insert 2 x 2 table. The variable count contains the number in each cell of the table
Get c2 value
Get RR and RO
Very important statement !

21. Table of ses by asthma
ses asthma
Frequency|
Percent |
Row Pct |
Col Pct | | | Total
1 | | |
| | |
| | |
| | |
2 | | |
| | |
| | |
| | |
Total
Ho: p1 = p2
p1 = 30/160 =
p2 = 40/140 =
Statistic DF Value Prob
Chi-Square
Likelihood Ratio Chi-Square
Continuity Adj. Chi-Square
Mantel-Haenszel Chi-Square

22. Table of ses by asthma
ses asthma
Frequency|
Percent |
Row Pct |
Col Pct | | | Total
1 | | |
| | |
| | |
| | |
2 | | |
| | |
| | |
| | |
Total
Ho: RR =1 or RO = 1
RR = / =
RO = (30/130) / (40/100) =
Row1 = High SES
Row2 = Low SES
Col1 = Have asthma
Estimates of the Relative Risk (Row1/Row2)
Type of Study Value % Confidence Limits
Case-Control (Odds Ratio)
Cohort (Col1 Risk)
Cohort (Col2 Risk)

23. Adjusting for Other Factors
Other factors must be categorical
Estimated RR and RO are a pooled estimate across all combinations of adjustment variables
Analyses called Mentel-Haenszel c2
PROC FREQ DATA=asthma;
WEIGHT count;
TABLES gender*ses*asthma/CHISQ CMH;
RUN;
Dependent variable
Risk factor of interest
Adjustment Variable (s)

24. Adjusting for Other Factors
Perhaps some or all of the SES/ASTHMA relationship is due to sex/gender
PROC FREQ DATA=asthma;
WEIGHT count;
TABLES gender*ses*asthma/CHISQ CMH;
RUN;
Dependent variable
Risk factor of interest
Adjustment Variable (s)

25. INPUT gender ses asthma count; DATALINES; 1 1 1 10 1 1 2 70 1 2 1 10
USING SAS
DATA asthma;
INFILE DATALINES;
INPUT gender ses asthma count;
DATALINES; ;
PROC FREQ DATA=asthma;
TABLES gender*ses*asthma/CHISQ CMH;
WEIGHT count;
2 x 2 table for men
2 x 2 table for women

26. Analysis for men ses asthma Frequency| Percent | Row Pct |
Table of ses by asthma
ses asthma
Frequency|
Percent |
Row Pct |
Col Pct | | | Total
1 | | |
| | |
| | |
| | |
2 | | |
| | |
| | |
| | |
Total
Estimates of the Relative Risk (Row1/Row2)
Type of Study Value % Confidence Limits
Case-Control (Odds Ratio)
Cohort (Col1 Risk)
Analysis for men

27. Analysis for women Table of ses by asthma ses asthma Frequency|
Percent |
Row Pct |
Col Pct | | | Total
1 | | |
| | |
| | |
| | |
2 | | |
| | |
| | |
| | |
Total
Estimates of the Relative Risk (Row1/Row2)
Type of Study Value % Confidence Limits
Case-Control (Odds Ratio)
Cohort (Col1 Risk)
Analysis for women

28. Tests if Odds Ratio is same for men and women
POOLED ANALYSES
Estimates of the Common Relative Risk (Row1/Row2)
Type of Study Method Value % Confidence Limits
Case-Control Mantel-Haenszel
(Odds Ratio) Logit
Cohort Mantel-Haenszel
(Col1 Risk) Logit
Breslow-Day Test for
Homogeneity of the Odds Ratios
Chi-Square DF
Pr > ChiSq
Pooled Analyses
Tests if Odds Ratio is same for men and women

29. Class Exercise
Among the 668 patients in TOMHS randomized to active treatment 74 experienced a CVD event during the study. Among the 234 patients randomized to placebo 38 had a CVD event.
Compute the RR of CVD, for active versus placebo
Compute the RO of CVD, for active versus placebo
Use SAS to create the 2 x 2 table
Using SAS compute the c2 statistic
Using SAS compute the 95% for the RR and RO above