# Conditional Logistic Regression for Matched Data HRP 261 02/25/04 reading: Agresti chapter 9.2 - PowerPoint PPT Presentation

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Conditional Logistic Regression for Matched Data HRP 261 02/25/04 reading: Agresti chapter 9.2. Recall: Matching. Matching can control for extraneous sources of variability and increase the power of a statistical test.

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Conditional Logistic Regression for Matched Data HRP 261 02/25/04 reading: Agresti chapter 9.2

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## Conditional Logistic Regression for Matched DataHRP 261 02/25/04reading: Agresti chapter 9.2

### Recall: Matching

• Matching can control for extraneous sources of variability and increase the power of a statistical test.

• Match M controls to each case based on potential confounders, such as age and gender.

### Recall: Agresti example, diabetes and MI

Match each MI case to an MI control based on age and gender.

Diabetes

No Diabetes

9

37

16

82

MI controls

MI cases

46

Diabetes

No diabetes

98

25

119

144

=the probability of observing a case-control pair with only the case exposed

=the probability of observing a case-control pair with only the control exposed

P(“favors” case/discordant pair) =

Diabetes

No Diabetes

9

37

16

82

MI controls

MI cases

46

Diabetes

No diabetes

98

25

119

144

odds(“favors” case/discordant pair) =

### Logistic Regression for Matched Pairs option 1:the logistic-normal model

• Mixed model; logit=i+x

• Where irepresents the “stratum effect”

• (e.g. different odds of disease for different ages and genders)

• Example of a “random effect”

• Allow i’s to follow a normal distribution with unknown mean and standard deviation

• Gives “marginal ML estimate of ”

or, prospectively:

### option 2: Conditional Logistic Regression

The conditional likelihood is based on….

The conditional probability (for pair-matched data):

P(“favors” case/discordant pair) =

### The Conditional Likelihood: each discordant stratum (rather than individual) gets 1 term in the likelihood

Note: the marginal probability of disease may differ in each age-gender stratum, but we assume that the (multiplicative) increase in disease risk due to exposure is constant across strata.

### Recall probability terms:

Each age-gender stratum has the same baseline odds of disease; but these baseline odds may differ across strata

### Conditional Logistic Regression

Example:Prenatal ultrasound examinations and risk of childhood leukemia: case-control study BMJ 2000;320:282-283

• Could there be an association between exposure to ultrasound in utero and an increased risk of childhood malignancies?

• Previous studies have found no association, but they have had poor statistical power to detect an association.

• Swedish researchers performed a nationwide populationbased case-control study using prospectively assembled data onprenatal exposure toultrasound.

### Example:Prenatal ultrasound examinations and risk of childhood leukemia: case-control study BMJ 2000;320:282-283

• 535 cases: all children born and diagnosed as having myeloid leukemia between 1973 and 1989 in Swedish registers of birth, cancer, and causesof death.

• 535 matched controls: 1 control was randomly selectedfor each case from the Swedish Birth Registry, matched by sex and year and month of birth.

Ultrasound

No Ultrasound

Myeloid leukemia controls

Leukemia cases

200

Ultrasound

No ultrasound

335

215

320

535

115

85

100

235

But this type of analysis is limited to single dichotomous exposure…

• Used conditional logistic regression to look at dose-response with number of ultrasounds:

• Results:

• Reference OR = 1.0; no ultrasounds

• OR =.91 for 1-2 ultrasounds

• OR=.64 for >=3 ultrasounds

• Conclusion: no evidence of a positive association between prenatal ultrasound and childhood leukemia; even evidence of inverse association (which could be explained by reasons for frequent ultrasound)

Extension: 1:M matching

• Each term in the likelihood represents a stratum of 1+M individuals

• More complicated likelihood expression! See: 02/02/04 lecture

• http://www.stanford.edu/class/hrp223/2003/Lecture15/Lecture15_223_2003.ppt

Available here:

-SAS tips, explanations and code

-SAS macro that generates automatic logit plots (under “Lecture 15” at: http://www.stanford.edu/class/hrp223/) to check if predictor is linear in the logit.

Put the values in the IsOUTCOME variable here that are the controls. Typically this is just the value 0.

This is the switch requesting a m:n CLR.

This is the m:n matching variable.

### M:N Matching Syntax

• The basic syntax is shown here.

procphregdata=BLAH;

model WEIRD*IsOUTCOME(Censor_v)= PREDICTORS /ties=discrete;

strata STRATA_VARS;

run;

Courtesy: Ray Balise

## Part II: Rater agreement: Cohen’s KappaAgresti, Chapter 9.5

### Cohen’s Kappa

Actual agreement = sum of the proportions found on the diagonals.

Cohen: Compare the actual agreement with the “chance agreement” (which depends on the marginals).

Normalize by its maximum possible value.

Rating by supervisor 2

Rating by supervisor 1

Authoritarian

Democratic

Permissive

Totals

Authoritarian

17

4

8

29

Democratic

5

12

0

17

Permissive

10

3

13

26

Totals

32

19

21

72

### Example: student teacher ratings

Null hypothesis: Kappa=0 (no agreement beyond chance)

Interpretation: achieved 36.2% of maximum possible improvement over that expected by chance alone

### Example: student teacher ratings

Null hypothesis: Kappa=0 (no agreement beyond chance)