1. Lecture 10 Comparing 2xk Tables
Outline of Today
Comparing several proportions
Measuring Association in 2x2 Tables
12/9/2018
SA3202, Lecture 10

2. Comparing Several Proportions
Problem of Interest Suppose we are interested in comparing k groups (populations) with respect to a binary response variable R Again, the groups usually correspond to the k levels (categories) of an explanatory variable C, which is thought to affect R.
Sample Structure Sample j, drawn from Group j, size nj, positive response Xj, j=1,2,..k
2xk Contingency Table (Stratified sampling with groups as strata)
Response Group Group ………… Group k
Positive X X Xk
Negative n1-X n2-X nk-Xk
Total n n nk
12/9/2018
SA3202, Lecture 10

3. Probability
Pr( Positive | Group j)=pj, j=1,2, …,k
Distribution
Xj~ Binom(nj, pj), j=1,2,…k
Hypothesis of Independence H0: p1=p2=….=pk (C no effect on R)
Testing Procedure Applying a Chi-squared Test (see more details in Lecture 8).
12/9/2018
SA3202, Lecture 10

4. Measuring Association in 2x2 Tables
Problem of Interest Consider a 2x2 table, with observed cell frequencies Xij and cell probabilities pij. Let R and C denote the row and column variables. How do we measure the degree of association between R and C?
Using the Odds Ratio Regarding R as the response variable, with R=1 as “Positive response”, and C=j as Group j, the probabilities of Positive response in Group j (j=1,2) are
p1=Pr(R=1|C=1)= p2=Pr(R=1|C=2)=
1-p1=Pr(R=0|C=j)= p2=Pr(R=0|C=2)=
Thus the odds ratio is
=(p11p22)/(p12p21)
Which is known as the Cross Product Ratio, and is a measure of the degree of association between R and C. The range of the odds ratio is from 0 to infinity, with 1 representing independence between R and C.
12/9/2018
SA3202, Lecture 10

5. The Odds Ratio Estimation
The natural estimator of the odds ratio is the sample cross product ratio
which is approximately normal, with variance
This may be used to construct CI or conduct hypothesis testing about the odds ratio. In particular, the independence test is equivalent to testing if the odds ratio is 1.
12/9/2018
SA3202, Lecture 10

6. Using Log Odds Ratio To measure the association between R and C, it is more convenient to use the log odds ratio:
which has a range of the whole real line with the origin 0 representing independence. Its estimator is the sample log odds ratio:
which is approximately normal with variance
and the estimated standard error
12/9/2018
SA3202, Lecture 10