Inference about conditional associations in 2 x 2 x k tables
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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. 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

Inference About Conditional Associations In 2 x 2 x K Tables

Demeke Kasaw

Gary Gongwer


An example from 2 3
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
Converting this to a 2 X 2 X 2 Table

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

Conditional Odds Ratios:



This can be generalized to k different levels
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)


Using logit models to test independence
Using logit Models to Test Independence

We wish to estimate the conditional probabilities

If Y depends on X, then

If Y and X are independent



Estimation of common odds ratio
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.


Testing homogeneity of odds ratios
Testing Homogeneity of Odds Ratios

Ha: At least one is different


Sas codes

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;


Partial outputs
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 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


    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


    • Thank you

    • Good luck with Prof. Trumbo’s Exam


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