This example illustrates the pattern-mixture model approach to multiple imputation under the MNAR assumption by adjusting imputed classification levels.
Carpenter and Kenward (2013, pp. 240–241) describe an implementation of sensitivity analysis that adjusts an imputed missing covariate, where the covariate is a nominal classification variable.
Suppose a high school class is conducting a study to analyze the effects of an extra web-based study class and grade level on the improvement of test scores. The regression model that is used for the study is
where Grade
is the grade level (with the values 6 to 8), Study
is an indicator variable (with the values 1 for “completes the study class” and 0 for “does not complete the study class”), Score0
is the current test score, and Score
is the test score for the subsequent test.
Also suppose that Study
, Score0
, and Score
are fully observed and the classification variable Grade
contains missing grade levels. Output 61.17.1 lists the first 10 observations in the data set Mono2
.
Output 61.17.1: Student Test Data
First 10 Obs in the Student Test Data |
Obs | Grade | Score0 | Score | Study |
---|---|---|---|---|
1 | 6 | 64.4898 | 68.8210 | 1 |
2 | 6 | 72.0700 | 76.5328 | 1 |
3 | 6 | 65.7766 | 75.5567 | 1 |
4 | . | 70.2853 | 76.0180 | 1 |
5 | 6 | 74.3388 | 80.0617 | 1 |
6 | 6 | 70.2207 | 76.1606 | 1 |
7 | 6 | 68.6904 | 77.9770 | 1 |
8 | . | 72.6758 | 79.6895 | 1 |
9 | 6 | 64.8939 | 69.3889 | 1 |
10 | 6 | 66.6038 | 72.7793 | 1 |
The following statements use the MONOTONE and MNAR statements to impute missing values for Grade
under the MNAR assumption:
proc mi data=Mono2 seed=34857 nimpute=10 out=outex17; class Study Grade; monotone logistic (Grade / link=glogit); mnar adjust( Grade (event='6') /shift=2); var Study Score0 Score Grade; run;
The LINK=GLOGIT suboption specifies that the generalized logit function be used in fitting the logistic model for Grade
. The ADJUST option specifies a shift parameter that is applied to the generalized logit model function values for the response level GRADE=6. This assumes that students
who have a missing grade level are more likely to be students in grade 6.
The “Model Information” table in Output 61.17.2 describes the method that is used in the multiple imputation process.
Output 61.17.2: Model Information
Model Information | |
---|---|
Data Set | WORK.MONO2 |
Method | Monotone |
Number of Imputations | 10 |
Seed for random number generator | 34857 |
The “Monotone Model Specification” table in Output 61.17.3 describes methods and imputed variables in the imputation model. The MI procedure uses the logistic regression method (generalized
logit model) to impute the variable Grade
.
Output 61.17.3: Monotone Model Specification
Monotone Model Specification | |
---|---|
Method | Imputed Variables |
Regression | Score0 Score |
Logistic Regression | Grade |
The “Missing Data Patterns” table in Output 61.17.4 lists distinct missing data patterns and their corresponding frequencies and percentages.
Output 61.17.4: Missing Data Patterns
Missing Data Patterns | ||||||||
---|---|---|---|---|---|---|---|---|
Group | Study | Score0 | Score | Grade | Freq | Percent | Group Means | |
Score0 | Score | |||||||
1 | X | X | X | X | 128 | 85.33 | 70.418230 | 74.469573 |
2 | X | X | X | . | 22 | 14.67 | 69.338503 | 73.666293 |
The “MNAR Adjustments to Imputed Values” table in Output 61.17.5 lists the adjustment parameter for the 10 imputations.
Output 61.17.5: MNAR Adjustments to Imputed Values
MNAR Adjustments to Imputed Values |
||
---|---|---|
Imputed Variable |
Event | Shift |
Grade | 6 | 2.0000 |
The following statements list the first 10 observations of the data set Outex17
in Output 61.17.6:
proc print data=outex17(obs=10); var _Imputation_ Grade Study Score0 Score; title 'First 10 Observations of the Imputed Student Test Data Set'; run;
Output 61.17.6: Imputed Data Set
First 10 Observations of the Imputed Student Test Data Set |
Obs | _Imputation_ | Grade | Study | Score0 | Score |
---|---|---|---|---|---|
1 | 1 | 6 | 1 | 64.4898 | 68.8210 |
2 | 1 | 6 | 1 | 72.0700 | 76.5328 |
3 | 1 | 6 | 1 | 65.7766 | 75.5567 |
4 | 1 | 6 | 1 | 70.2853 | 76.0180 |
5 | 1 | 6 | 1 | 74.3388 | 80.0617 |
6 | 1 | 6 | 1 | 70.2207 | 76.1606 |
7 | 1 | 6 | 1 | 68.6904 | 77.9770 |
8 | 1 | 6 | 1 | 72.6758 | 79.6895 |
9 | 1 | 6 | 1 | 64.8939 | 69.3889 |
10 | 1 | 6 | 1 | 66.6038 | 72.7793 |