Transcript Inference About Conditional Associations In 2 x 2 x K Tables

Inference About Conditional
Associations In 2 x 2 x K Tables
Demeke Kasaw
Gary Gongwer
An Example from §2.3
Death Penalties in Florida for Multiple Murders,
1976-1987
Odds Ratio = 1.45
Defendant’s
Race
Death Penalty
Yes
No
Percent Yes
White
Black
53
15
11.0
7.9
430
176
Converting this to a 2 X 2 X 2 Table
We now have 2 Partial Tables, by race of the
victim
Conditional Odds Ratios:
ˆ( BV )  0
ˆ( WV)  0.43
Victim’s Race
Defendant’s
Race
White
White
Black
Death Penalty
Yes
No
53
414
11
37
Black
White
Black
0
4
16
139
Percent
Yes
11.3
22.9
0.0
2.8
Conditional and Marginal Odds
Ratios
 XY ( K )
11k  22k

12k  21k
 XY
11  22

12  21
This can be generalized to K
different levels
To study whether an association exists
between an explanatory and response
variable after controlling for a possibly
confounding variable
• Different medical centers
• Severity of Condition
• Age
• Different Studies of the same sort (Meta
Analysis)
Center Treatment
Response
Success
Failure
1
11
10
16
22
14
7
2
1
6
0
1
0
1
1
4
6
2
3
4
5
6
7
8
Drug
Control
Drug
Control
Drug
Control
Drug
Control
Drug
Control
Drug
Control
Drug
Control
Drug
Control
Odds Ratio
25
27
4
10
5
12
14
16
11
12
10
10
4
8
2
1
1.19
1.82
4.80
2.29
∞
∞
2.0
0.33
Using logit Models to Test
Independence
We wish to estimate the conditional probabilities
 ik  P(Y  1 / X  i, Z  k )
If Y depends on X, then
log it ( ik )    xi   k
If Y and X are independent
log it ( ik )     k
z
z
CMH Test for Conditional
Independence
11k  E (n11k )  n1k n1k / n  k
var( n11k )  n1 k n2 k n1k n 2 k / n
2
2
 k
(n  k  1)


 (n11k  11k )
k


2
CMH 
~  (1)
 var( n11k )
k
Estimation of Common Odds Ratio
When the association seems stable among the partial
tables, it is helpful to combine the K odds ratios into a
summary measure of conditional association.
ˆMH
n

n
11k
n22k / n  k
k
12 k
p

p
11|k
p 22|k n  k
12|k
p 21|k n  k
k
n21k / n  k
k
where pij|k  nijk / n  k
k
Testing Homogeneity of Odds
Ratios
H 0   XY (1)     XY ( k )
Ha: At least one is different
SAS CODES
data cmh;
input center $ treat response count ;
datalines;
a 1 1 11
a 1 2 25
a 2 1 10
h221
;
/*Consider 2x2xk*/
proc freq data = cmh;
weight count;
tables center*treat*response / cmh chisq All;
run;
/*Consider 2x2*/
proc freq data = cmh;
weight count;
tables treat*response / cmh chisq All;
run;
Partial outputs
Odds Ratio for calculated on each centers;
for center 1
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
95% Confidence Limits
Case-Control (Odds Ratio)
1.1880
Center 2
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
Case-Control (Odds Ratio)
Case-Control (Odds Ratio)
3.2766
95% Confidence Limits
1.8182
Center 3
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
0.4307
0.4826
6.8496
95% Confidence Limits
4.8000
1.2044
19.1292
Table 5 of treat by response
Controlling for center=e
treat
response
Frequency‚
Percent ‚
Row Pct ‚
Col Pct ‚
1‚
2‚ Total
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
1 ‚
6 ‚
11 ‚
17
‚ 20.69 ‚ 37.93 ‚ 58.62
‚ 35.29 ‚ 64.71 ‚
‚ 100.00 ‚ 47.83 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
2 ‚
0 ‚
12 ‚
12
‚
0.00 ‚ 41.38 ‚ 41.38
‚
0.00 ‚ 100.00 ‚
‚
0.00 ‚ 52.17 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Total
6
23
29
20.69
79.31
100.00
Table 6 of treat by response
Controlling for center=f
treat
response
Frequency‚
Percent ‚
Row Pct ‚
Col Pct ‚
1‚
2‚ Total
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
1 ‚
1 ‚
10 ‚
11
‚
4.76 ‚ 47.62 ‚ 52.38
‚
9.09 ‚ 90.91 ‚
‚ 100.00 ‚ 50.00 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
2 ‚
0 ‚
10 ‚
10
‚
0.00 ‚ 47.62 ‚ 47.62
‚
0.00 ‚ 100.00 ‚
‚
0.00 ‚ 50.00 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Total
1
20
21
4.76
95.24
100.00
Center 7
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
95% Confidence Limits
---------------------------------------------------------------------------------------------------------------------Case-Control (Odds Ratio)
2.0000
0.0976
41.0034
Center 8
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
95% Confidence Limits
---------------------------------------------------------------------------------------------------------------------Case-Control (Odds Ratio)
0.3333
0.0221
5.0271
Total
Type of Study Method
Value 95% Confidence Limits
--------------------------------------------------------------------------------------------------------------------------Case-Control
Mantel-Haenszel
2.1345
1.1776
3.8692
(Odds Ratio) Logit **
1.9497
1.0574
3.5949
Estimates of the Common Relative Risk (Row1/Row2)
Type of Study Method
Value 95% Confidence Limits
--------------------------------------------------------------------------------------------------------------------------Case-Control
Mantel-Haenszel
1.4979
0.9151
2.4518
(Odds Ratio) Logit
1.4979
0.9151
2.4518
Homogeneity test:
Breslow-Day Test for
Homogeneity of the Odds Ratios
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Chi-Square
7.9955
DF
7
Pr > ChiSq
0.3330
Total Sample Size = 273
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