BIOSTATISTICS 590
Multiple Imputation (MI) Technique
Using a Sequence of Regression
Models
OJOC Cohort 15
Veronika N. Stiles, BSDH
University of Michigan
September’2012
Basis for Presentation
• This presentation is based on an article by:
• T.E. Raghunathan
• J.M. Lepkowski
• J.V. Hoewyk
• P. Solenberger
“A multivariate Technique for Multiply Imputing
Missing Values Using a Sequence of Regression
Models”
Survey Methodology, June 2001
Vol. 27, No. 1, pp. 85-95
Rationale for Multiple Imputation
• Incomplete data is a common problem
• Allows to use an existing complete-data software, once
the missing values have been imputed
Basic Definitions
• “Imputation” is the placement of one or more estimated answers into a field
of a data record that previously had NO data
• Draws from a predictive distribution
Basic Strategy
• To create imputations through fitting a sequence of multiple regressions
• Regressions use the variable with missing data as the outcome (Y) variable
• Regression models based on complete data are used to make predictions of
Y when Y is missing
• To draw values from the predictive distributions
• Cyclical manner
• The type of regression model varies by imputed variable
(Example is coming up in future slides)
Types of Regression Models Used
1.
2.
3.
4.
5.
Linear
Logistic
Poisson
Generalized logit
Mixture of the above
Remember! The type of regression model depends on the
type of imputed variable!
Assumptions in MI
Technique
• Population is infinite
• Sample is SRS
• Variables are one of the following:
• Continuous
• Binary
• Categorical
• Counts
• Mixed
Advantages of Multiple
Imputation
+
+
+
+
+
Method for imputation is known;
Analyses are based on the same # of cases;
All data provided is used in each analysis;
Allows for multiple predictors;
Valid points and interval estimates under a
general set of conditions are obtained
 by repeatedly applying the complete
data software
Imputation Method
• Each imputation consists of “rounds”
• Start round 1 by regressing the variable with fewest # of
missing values
• Remember! Imputations for missing values in Y are draws
from the predictive distribution
(Use predicted mean Y + a random draw from the normal
error distribution)
• Then, update X by replacing missing Y with the imputed value
• X=full matrix with all variables (including Y)
Lesion Location
Temporal
Etiology
Lobectomy
Lesion Size
2.72
Chronicity
89.3
Occipital
Temporal
Stroke
Hemorrhage
.
.
36.3
55.3
Imputation Method
• Move on to the next Y with fewest missing
values
• Repeat MI using updated X as predictors
until all variables have been imputed
 Run the process M times;
 Yield M entire datasets;
 Each dataset has different set of imputed
values, but the same data for complete values
Example Time
Effect of Smoking on Primary Cardiac Arrest
(CA)
• Case-control study
• Examine relationship between smoking and
CA
Means and Proportions of Key
Variables and Percent Missing
Variable
Control (n = 551)
Cases (n = 347)
% Missing
Mean (SD)
% Missing
Mean (SD)
Age
0
58.4 (10.4)
0
59.4 (9.9)
BMI
8.2
25.8 (4.1)
2.6
26.4 (4.6)
Years Smoked
16.8
24.8 (14.7)
5.4
31.7 (13.8)
Proportion
Female
>= High School
Smoking Status
Proportion
0
23.2
0
19.9
0
76.8
0
61.9
0
Never Smoked
0
47.2
0
27.3
Former Smoker
0
42.1
0
38.2
Current Smoker
0
10.7
0
34.5
Intuitively…
• What variables might predict missing data?
• Could age, education, smoking status
predict BMI?
• Could age predict years smoked?
• However, years smoked can only be
imputed for current and former
smokers!
• Some values may need to be fixed post-MI
Multiple Imputation Process in
CA Study
• Log (BMI) has fewest missing values
• Regress Log (BMI) on age, female,
education, Years_Smoked, smoking status,
and cardiac arrest through normal linear
model
• Cardiac Arrest IS included in the imputation
model
• Predicted values of log (BMI) are saved to
the dataset, replacing the missing values
Multiple Imputation Process in
CA Study
• Next, Years Smoked was regressed on all of the
variables above+ log (BMI)
(Please note that the regression excludes ‘never-smokers’)
• Predicted values of Years Smoked are saved to the
dataset, replacing the missing values
• M=25 imputations
(Note: many researchers use M=5 or 5<M<10)
• Original logistic regression model was fit for each
MI data set
How were estimates of coefficients and
covariance matrices obtained?
• IVEware software performs calculations, using
estimates and covariance matrix
• Combines the results from 5-25 regressions
• Combines both within-regression and betweenregression error
• IVEware:
Imputation and Variance Estimation Software
http://www.isr.umich.edu/src/smp/ive/
• Developed by our own Dr. Raghunathan &
researchers at the Survey Methodology Program
Complete-Case Analysis vs MI
Predictor Variables
Complete Case
SRMI
(n = 795)
Method 1 (n = 898)
Estimate (SE)
Estimate (SE)
Intercept
-2.922
(0.791)
-2.61
(0.757)
Age
0.015
(0.009)
0.015
(0.009)
Female
-0.007
(0.203)
-0.115
(0.189)
Education
-0.448
(0.173)
-0.467
(0.166)
BMI
0.056
(0.018)
0.049
(0.013)
Current Smoker
1.693
(0.569)
2.001
(0.543)
Former Smoker
0.003
(0.284)
-0.029
(0.262)
Current Smoker x Yrs Smoked
-0.003
(0.015)
-0.008
(0.013)
Former Smoker x Yrs Smoked
0.019
(0.009)
0.014
(0.009)
Results of the Multiple
Imputations
• MI standard errors are smaller:
• due to additional subjects in imputed data
• Modest changes in relationship between
smoking and CA
• Years Smoked in Former Smokers is a
significant predictor of cardiac arrest in the
complete-case analysis, but NOT in the MI
analysis (!!!)
Additional Variables MI
Approach
• Additional variables NOT in the substantive analysis
can be used
• Prediction for missing values in each variable
borrows strength from all other variables
• In our cardiac arrest example, impute dataset
+50 additional variables  SE are smaller
• Improved efficiency vs. variables in model only
In Addition…
IVEware performs…
1. Single or multiple imputations
2. Analyses accounting for:
•
•
Clustering
Stratification
Weighting
3. Combines information from multiple sources
•
(+some other functions beyond the scope of this
presentation)
Critique
• This article might be too challenging and complicated as an
entry-level description of multiple imputation
• Some of the foundational concepts from this article have not
been covered thus far in OJOC program
• nonignorable missing mechanism
RECOMMENDATION
Start with “Survey Methodology” (2nd edition) by R.M. Groves,
F. J. Fowler, Jr., M.P. Couper, J.M. Lepkowski, E. Singer, R.
Tourangeau. Wiley Series in Survey Methodology, A John
Wiley & Sons, Inc., Publication, 2009, p. 356.
Thank You for Your Attention!