Lecture 5 Agenda Basic Contingency Table Analysis - PowerPoint Presentation

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Lecture 5

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Agenda

Basic Contingency Table Analysis

RxC

Contingency Tables

Pearson Chi square test of association

Stratified 2x2 tables

Cochran Mantel

Haenszel

test of association

Breslow Day test of interaction

Simple Logistic Regression

Modeling dichotomous outcomes

Odds Ratios and Logistic Regression

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Western Collaborative Group Study (WCGS)

Large scale prospective cohort study designed to examine risk factors for cardiovascular disease.

Main outcome is coronary heart disease (chd69)

1 = yes, 0 = no

Primary risk factor is personality type (

dibpat

)

1 = type A, 0 = type B

Other risk factors collected as well:

Blood pressure, cholesterol, smoking, age,

arcus

senilis

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Contingency table analysis

Historical precursor to logistic regression models.

What variables are associated with coronary heart disease?

Examples

:

Arcus

senilis

(1 = present, 0 = not present)

Cholesterol (1 = < 200, 2 = 200 to 240, 3 = > 240)

Smoking (1 = 0 cigarettes/day, 2 = 1-10, 3= 11-20, 4= > 20)

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Chi-square Test of Independence

Test of association between categorical variables based on Pearson Chi-square statistic.

r = # of rows (levels of variable 1)

c = # of columns (levels of variable 2)

Compares observed cell count (

O

ij

) to cell count that would be observed if the variables were independent (

Eij). H0: variables are independent ↔ no associationH1: variables are not independent ↔ association

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Basic 2 x 2 Table

Is

arcus

senilis

associated with CHD?

H

0

: arcus senilis and CHD are independentH1: arcus senilis and CHD are dependent

Chi-square

test with 1

d.f.

Reject H0: conclude that CHD and arcus are not independent.

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Example: 2 x 2 Table

Odds ratio interpretation

Odds of CHD are 1.63 times higher for those with presence of arcus

senilis

compared to those without presence.

Relative risk interpretation

Risk (probability) of CHD is 1.56 times higher for those with presence of arcus

senilis

compared to those without presence.

Use

relrisk

option in

proc freq for this table.

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Stratified 2x2 Table

Arcus

senilis

is a condition associated with fatty deposits in the eye.

Also caused by widening of the eye vessels with age which makes it easier for fat to deposit.

May be of interest, therefore, to control for cholesterol and age.

Stratification allows us to divide a 2x2 table with respect to a third (and possibly 4

th

variable).

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Stratified 2x2 Table

Below we stratify by cholesterol group

Chol < 200, 200 <=

chol

< 240,

chol

> 240

The Cochran Mantel

Haenszel test allows us to compute the OR and RR adjusting for the stratifying variable (cholesterol group). Use proc freq with cmh option to calculate the adjusted analysis statistics.

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Stratified 2x2 Table

To stratify by cholesterol group, add it to table statement in the order below.

Adjusted test of association in Cochran-Mantel-

Hanszel

statistics table.

H

0

: zero association between arcus and CHD after adjusting for cholesterol group

H1: nonzero association between arcus and CHD after adjusting for cholesterol group

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Stratified 2x2 Table

Since p = .004 < .05, reject H0 and conclude there is enough evidence to suggest arcus and CHD are associated after adjusting for cholesterol.

All three statistics above are the same for stratified 2x2 tables.

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Stratified 2x2 Table

Adjusted odds ratio and relative risk suggest elevated risk of CHD with presence of arcus.

Effect is a little smaller than the unadjusted effect.

After adjusting for cholesterol group, odds of CHD are 1.47 (1.13, 1.93) times higher for arcus vs. no arcus

After adjusting for cholesterol group, risk of CHD is 1.42 (1.12, 1.80) times higher for arcus vs. no arcus

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Stratified 2x2 Table

In order to conduct the previous analysis, we must assume that the OR (or RR) is the same in all strata.

Breslow Day Test of interaction between stratum and group

H

0

: OR

1

= OR

2H1: OR1 ≠ OR2Breslow Day test p-value = .8288 > .05. Fail to reject H0 and conclude that we cannot reject the hypothesis of equal odds ratios across strata.

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Stratified 2x2 Table: Flowchart

Breslow Day test of Interaction

p < . 05

p ≥ . 05

Interpret the common OR

or

RR

Report the CMH test of association

Interpret the OR (or RR) separately within each stratum

Report the test of association for each stratum

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More General Contingency Tables

Cholgrp

= 1,2, or 3 based on categorization of a subject’s cholesterol level.

Is cholesterol group associated with CHD?

H

0

: cholesterol group and CHD are independent

H

1: cholesterol group and CHD are dependent

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Example: 3 x 2 table

Reject H

0

– cholesterol group is not independent of coronary heart disease.

Summarize this by calculating odds ratios relative to reference group.

OR 2 vs. 1 = (84/1121)/(31/800) = 1.93

Odds of CHD are twice as high for

cholgrp

= 2 vs. 1

OR 3 vs. 1 = (142/964)/(31/800) = 3.80

Odds of CHD are 3.8 times higher for

cholgrp

= 3 vs. 1

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Example: 3 x 4 table

Is cholesterol group associated with CHD type?

H

0

: cholesterol group and CHD type are independent

H

1

: cholesterol group and CHD type are dependent

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Contingency Table Analysis

Contingency table analysis is useful for descriptive purposes.

Some limitations

As dimensions increase it becomes harder to summarize the

direction

of association

Cannot estimate association of categorical and continuous variables (must categorize)

Multivariable modeling beyond one or two stratifying variables is cumbersome

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Logistic Regression

Linear Regression describes how the mean of a continuous outcome is affected by independent predictors.

Mean is directly related to covariates.

A dichotomous outcome (Y) takes on values of either 0 or 1 with probabilities 1-p and p respectively.

 

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Logistic Regression

Mean of a dichotomous outcome is equal to the probability of a “positive” outcome: p = P(Y=1).

μ

= 1*p + 0*(1-p) = p

Cannot use linear regression.

0 ≤ p ≤ 1

Linear regression may estimate p > 1 or p < 0 since there is no constraint

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Logistic Regression

Instead of directly modeling p, we model the log-odds of positive outcome.

Parameters can be interpreted in terms of odds ratios.

 

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Logistic Regression

Re-expressing previous equation in terms of p, we can see that 0 < p < 1 is guaranteed.

)

 

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Logistic Regression

β

j

is the adjusted log-odds ratio comparing unit differences in x

j

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Example:

Arcus

Senilis

vs. CHD

Using logistic regression define:

1 if subject

i has arcus senilis presentX = 0 if subject i does not have arcus senilis present

The model then defines log-odds of CHD as:

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Logistic Regression

OR = e

.4918

= 1.635

Interpretation:

Odds of CHD 1.63 times higher among subjects with

arcus

compared to those without

arcus

.

63% increased odds of CHD among subjects with

arcus.Statistically significant (p=.0002)

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Logistic Regression

Important considerations in

proc

logistic

Descending:

defines numerator of odds to be P(Y=1)

Descending

 Odds = p/(1-p) Param=ref  use indicator variables for categorical independent variables. Ref= last sets last alphanumeric category as reference groupRef = first sets first alphanumeric category as reference group

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Logistic Regression

Polychotomous

Predictors

Pick a baseline (reference) group

Set up a series of indicators for all other groups

Should have k-1 Odds ratios comparing each group to baseline group.

Continuous Predictors

Compute Odds for two levels that differ by 1-unit, odds ratio is then the

exponentiated coefficient for the predictor. Odds Ratio comparing c-unit differences in the predictor is ORc where OR is the 1-unit odds ratio. Confidence interval for c-unit OR comparisons are 1-unit endpoints raised to the cth

power.

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Example

: Cholesterol Group vs. CHD

Cholgrp

2 vs. 1



1.933 = exp(.6592)

Odds of CHD for cholesterol between 200 and 240 is approximately twice as high as odds of CHD for cholesterol < 200 (statistically significant p = .0022).

Cholgrp

3 vs. 1  3.80 = exp(1.3351) Odds of CHD for cholesterol > 240 is 3.8 times higher than odds of CHD for cholesterol < 200 (statistically significant p < .0001).

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Example

: Cholesterol vs. CHD

1-unit OR = exp(.0124) = 1.0125

Odds of CHD are 1.2% higher per unit increase in total cholesterol (statistically significant p < .0001)

30-unit OR = exp(30*.0124) = 1.0125^30 = 1.45

Odds of CHD are 45% higher per 30 unit increase in total cholesterol (statistically significant p < .0001)

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Multivariable Logistic Regression

Basic principles of covariate adjustment and effect modification from linear regression carry over to logistic regression.

Use additional covariates to:

Control for potential confounders, other covariates

Build a stronger predictive model for the outcome

Describe effect modification (interaction)

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Example

: Adjust for Categorical Confounder

Suppose we now want to adjust for cholesterol group.

Arcus

OR = 1.475 = exp(.3888)

Odds of CHD among subjects with presence of

arcus

senilis is approximately 50% higher than among subjects without presence adjusting for cholesterol group statistically significant (p=.0042).

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Example

: Adjust for Categorical Confounder

For cholesterol group, we must first look at the type III p-value to determine overall significance.

Overall joint effect of cholesterol group is significant (p < .0001).

OR

cholgrp

2 vs. 1 = 1.884 = exp(.6332)

 significant (p=.0033)

OR cholgrp 3 vs. 1 = 3.563 = exp(1.2706)  significant (p < .0001

)

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Example

: Adjust for Categorical Confounder

Is cholesterol group a potential confounder?

Adjusted estimate = .3888

Unadjusted estimate = .4918

% change = (.3888-.4918)/.3888 = 26.5%

Suggests it is important to adjust for cholesterol group.

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Example

: Adjust for Continuous Confounder

Adjusting for cholesterol as a continuous covariate is an alternative option.

Requires assumption that log-odds is linearly associated with cholesterol.

Odds of CHD among subjects with presence of

arcus

senilis

is approximately 44% higher than among subjects without presence adjusting for cholesterol level. Effect is significant (p=.0076)

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Example

: Adjust for Continuous Confounder

Is cholesterol group a potential confounder?

Adjusted estimate = .3658

Unadjusted estimate = .4918

% change = (.3658-.4918)/.3658 = 34.4%

Suggests it is important to adjust for cholesterol.

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Example

: Adjust for Multiple Confounders

Now suppose we wish to adjust for cholesterol, age, and smoking status (1=

ncigs

> 0, 0 =

ncigs

=0).

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Example

: Adjust for Multiple Confounders

Arcus

% change = (.1699-.3888)/.1699 = 128.8%

Important to adjust for these covariates.