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# Inference About Conditional Associations In 2 x 2 x K Tables - PowerPoint PPT Presentation

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. Converting this to a 2 X 2 X 2 Table. We now have 2 Partial Tables, by race of the victim

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### 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

We now have 2 Partial Tables, by race of the victim

Conditional Odds Ratios:

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)

We wish to estimate the conditional probabilities

If Y depends on X, then

If Y and X are independent

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.

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

h 2 2 1

;

/*Consider 2x2xk*/

procfreq data = cmh;

weight count;

tables center*treat*response / cmh chisq All;

run;

/*Consider 2x2*/

procfreq data = cmh;

weight count;

tables treat*response / cmh chisq All;

run;

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 0.4307 3.2766

Center 2

Estimates of the Relative Risk (Row1/Row2)

Type of Study Value 95% Confidence Limits

Case-Control (Odds Ratio) 1.8182 0.4826 6.8496

Center 3

Estimates of the Relative Risk (Row1/Row2)

Type of Study Value 95% Confidence Limits

Case-Control (Odds Ratio) 4.8000 1.2044 19.1292

• 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

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

• Thank you

• Good luck with Prof. Trumbo’s Exam