Logistic Regression A solution for imperfect binary data

Xue Yao, Lisa Lix

Department of Community Health Sciences

Winnipeg SAS Users Group

May 17, 2013

Outline

• Brief Review of Logistic Regression

• Logistic Regression in SAS

• Case Study: Utilizing logistic regression to deal with

imperfect binary data (i.e. missing and misclassification)

– SAS Global Forum Paper 283-2013: A Flexible Method to Apply

Multiple Imputation Using SAS/IML® Studio

• To model the PROBABILITY of the event of interest based on

the values of independent variables

• 𝑙𝑜𝑔𝑖𝑡 𝑌 = 1 𝑿 = log𝑃

1−𝑃= 𝜷𝑿

where 𝑃 = 𝑃𝑟𝑜𝑏 𝑌 = 1 𝑿 = 𝐸(𝑌 = 1|𝑿)

• 𝑃 =exp(𝜷𝑿)

1+exp(𝜷𝑿)

• o𝑑𝑑𝑠 =𝑃

1−𝑃= exp 𝜷𝑿

Logistic Regression

Logistic Regression

M1 is a continuous

independent variable

M2 is a binary

independent variable

Logistic Regression in SAS

• CATMOD, GENMOD, PROBIT and LOGISTIC procedures perform

logistic regression in SAS

• LOGISTIC Procedure Syntax:

PROC LOGISTIC ;

CLASS discrete variable ;

MODEL variable= ;

OUTPUT ;

Case Study: Utilizing logistic regression to

deal with imperfect binary data

• Problem: Misclassification of disease status (0 or 1) results in

bias of either descriptive or inferential analysis based on

disease status using administrative health data

• Solution: Using logistic regression as predictive model for

multiple imputation method

• Data Sources: Validation dataset (e.g. medical chart) includes

accurate measures which can be linked to the administrative

health data

Illustration of Data

Using Logistic Predictive Model For Multiple

Imputation

Figure 1. A Schematic Diagram of the Multiple Imputation Method

Logistic predictive model

Step 1: Prepare the Data

Step 2: Build the Logistic Predictive Model

Step 3: Generate Multiple Parameters

Step 4: Create Multiple Datasets

Step 5: Analyze the Multiple Complete Datasets

Using Logistic Predictive Model For Multiple

Imputation

Building Logistic Predictive Model

𝑙𝑜𝑔𝑖𝑡 𝑌 = 1 𝑴 = 𝛽0 + 𝛽1𝑀1 + 𝛽2𝑀2

• To estimate the parameters of the model using LOGISTIC

procedure in SAS IML studio

Using Logistic Predictive Model For Multiple

Imputation

• To generate the multiple coefficients of the logistic predictive

model for multiple imputation based on the estimated

coefficients and covariance matrix from LOGISTIC procedure

Using Logistic Predictive Model For Multiple

Imputation

• To predict/impute the disease status (1 or 0) multiple times

using the generated coefficients and logistic predictive model

• To save the dataset that contains the variable of imputed

disease status and the number of imputations for further

analysis

Using Logistic Predictive Model For Multiple

Imputation

• To estimate the disease prevalence

– PROC UNIVARIATE to estimate prevalence of each dataset

– PROC MIANALYZE to combine the estimates from each dataset

Using Logistic Predictive Model For Multiple

Imputation

• Outputs of PROC UNIVARIATE and MIANALYZE

Results

• Multiple imputation based on logistic model improves the

accuracy of disease prevalence estimate

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

True Prev Obs Prev Imp Prev

Conclusions

• Imputation of missing data is better than discarding incomplete

observations. – Frank E. Harrell

• Misclassification of binary data can be treated as a missing data

problem

• For monotone missing data pattern, the binary data can be

imputed using PROC MI with logistic model

• For arbitrary missing data pattern, the proposed approach can

be used to impute more than one binary variables

simultaneously

Thank you!

Your comments and questions are valued and encouraged, please

contact

Xue.Yao@med.umanitoba.ca

mailto:Xue.Yao@med.umanitoba.ca