### Example 61.2 Monotone Propensity Score Method

This example uses the propensity score method to impute missing values for variables in a data set with a monotone missing pattern. The following statements invoke the MI procedure and request the propensity score method. The resulting data set is named `Outex2`.

```proc mi data=Fish1 seed=899603 out=outex2;
monotone propensity;
var Length1 Length2 Length3;
run;
```

Note that the VAR statement is required and the data set must have a monotone missing pattern with variables as ordered in the VAR statement.

The Model Information  table in Output 61.2.1 describes the method and options used in the multiple imputation process. By default, five imputations are created for the missing data.

Output 61.2.1: Model Information

The MI Procedure

Model Information
Data Set WORK.FISH1
Method Monotone
Number of Imputations 5
Seed for random number generator 899603

When monotone methods are used in the imputation, MONOTONE is displayed as the method. The Monotone Model Specification table in Output 61.2.2 displays the detailed model specification. By default, the observations are sorted into five groups based on their propensity scores.

Output 61.2.2: Monotone Model Specification

Monotone Model Specification
Method Imputed Variables
Propensity( Groups= 5) Length2 Length3

Without covariates specified for imputed variables `Length2` and `Length3`, the variable `Length1` is used as the covariate for `Length2`, and the variables `Length1` and `Length2` are used as covariates for `Length3`.

The Missing Data Patterns table in Output 61.2.3 lists distinct missing data patterns with corresponding frequencies and percentages. Here, values of X and '.' indicate that the variable is observed or missing, respectively, in the corresponding group. The table confirms a monotone missing pattern for these three variables.

Output 61.2.3: Missing Data Patterns

Missing Data Patterns
Group Length1 Length2 Length3 Freq Percent Group Means
Length1 Length2 Length3
1 X X X 30 85.71 30.603333 33.436667 38.720000
2 X X . 3 8.57 29.033333 31.666667 .
3 X . . 2 5.71 27.750000 . .

For the imputation process, first, missing values of `Length2` in group 3 are imputed using observed values of `Length1`. Then the missing values of `Length3` in group 2 are imputed using observed values of `Length1` and `Length2`. And finally, the missing values of `Length3` in group 3 are imputed using observed values of `Length1` and imputed values of `Length2`.

After the completion of m imputations, the Variance Information table in Output 61.2.4 displays the between-imputation variance, within-imputation variance, and total variance for combining complete-data inferences. It also displays the degrees of freedom for the total variance. The relative increase in variance due to missingness, the fraction of missing information, and the relative efficiency for each variable are also displayed. A detailed description of these statistics is provided in the section Combining Inferences from Multiply Imputed Data Sets.

Output 61.2.4: Variance Information

Variance Information
Variable Variance DF Relative
Increase
in Variance
Fraction
Missing
Information
Relative
Efficiency
Between Within Total
Length2 0.001500 0.465422 0.467223 32.034 0.003869 0.003861 0.999228
Length3 0.049725 0.547434 0.607104 27.103 0.108999 0.102610 0.979891

The Parameter Estimates table in Output 61.2.5 displays the estimated mean and standard error of the mean for each variable. The inferences are based on the t distributions. For each variable, the table also displays a 95% mean confidence interval and a t statistic with the associated p-value for the hypothesis that the population mean is equal to the value specified in the MU0= option, which is 0 by default.

Output 61.2.5: Parameter Estimates

Parameter Estimates
Variable Mean Std Error 95% Confidence Limits DF Minimum Maximum Mu0 t for H0:
Mean=Mu0
Pr > |t|
Length2 33.006857 0.683537 31.61460 34.39912 32.034 32.957143 33.060000 0 48.29 <.0001
Length3 38.361714 0.779169 36.76328 39.96015 27.103 38.080000 38.545714 0 49.23 <.0001

The following statements list the first 10 observations of the data set `Outex2`, as shown in Output 61.2.6. The missing values are imputed from observed values with similar propensity scores.

```proc print data=outex2(obs=10);
title 'First 10 Observations of the Imputed Data Set';
run;
```

Output 61.2.6: Imputed Data Set

 First 10 Observations of the Imputed Data Set

Obs _Imputation_ Length1 Length2 Length3
1 1 23.2 25.4 30.0
2 1 24.0 26.3 31.2
3 1 23.9 26.5 31.1
4 1 26.3 29.0 33.5
5 1 26.5 29.0 38.6
6 1 26.8 29.7 34.7
7 1 26.8 29.0 35.0
8 1 27.6 30.0 35.0
9 1 27.6 30.0 35.1
10 1 28.5 30.7 36.2