The Fitness
data described in the REG procedure are measurements of 31 individuals in a physical fitness course. See Chapter 85: The REG Procedure, for more information. The Fitness1
data set is constructed from the Fitness
data set and contains three variables: Oxygen
, RunTime
, and RunPulse
. Some values have been set to missing, and the resulting data set has an arbitrary pattern of missingness in these three
variables.
*----------------- Data on Physical Fitness -----------------* | These measurements were made on men involved in a physical | | fitness course at N.C. State University. | | Only selected variables of | | Oxygen (oxygen intake, ml per kg body weight per minute), | | Runtime (time to run 1.5 miles in minutes), and | | RunPulse (heart rate while running) are used. | | Certain values were changed to missing for the analysis. | *------------------------------------------------------------*; data Fitness1; input Oxygen RunTime RunPulse @@; datalines; 44.609 11.37 178 45.313 10.07 185 54.297 8.65 156 59.571 . . 49.874 9.22 . 44.811 11.63 176 . 11.95 176 . 10.85 . 39.442 13.08 174 60.055 8.63 170 50.541 . . 37.388 14.03 186 44.754 11.12 176 47.273 . . 51.855 10.33 166 49.156 8.95 180 40.836 10.95 168 46.672 10.00 . 46.774 10.25 . 50.388 10.08 168 39.407 12.63 174 46.080 11.17 156 45.441 9.63 164 . 8.92 . 45.118 11.08 . 39.203 12.88 168 45.790 10.47 186 50.545 9.93 148 48.673 9.40 186 47.920 11.50 170 47.467 10.50 170 ;
Suppose that the data are multivariate normally distributed and that the missing data are missing at random (see the "Statistical
Assumptions for Multiple Imputation" section in the chapter "The MI Procedure" for a description of these assumptions). The
following statements use the MI procedure to impute missing values for the Fitness1
data set:
proc mi data=Fitness1 seed=3237851 noprint out=outmi; var Oxygen RunTime RunPulse; run;
The MI procedure creates imputed data sets, which are stored in the Outmi
data set. A variable named _Imputation_
indicates the imputation numbers. Based on m imputations, m different sets of the point and variance estimates for a parameter can be computed. In this example, m = 5 is the default.
The following statements generate regression coefficients for each of the five imputed data sets:
proc reg data=outmi outest=outreg covout noprint; model Oxygen= RunTime RunPulse; by _Imputation_; run;
The following statements display (in Figure 64.1) output parameter estimates and covariance matrices from PROC REG for the first two imputed data sets:
proc print data=outreg(obs=8); var _Imputation_ _Type_ _Name_ Intercept RunTime RunPulse; title 'Parameter Estimates from Imputed Data Sets'; run;
Figure 64.1: Parameter Estimates
Parameter Estimates from Imputed Data Sets |
Obs | _Imputation_ | _TYPE_ | _NAME_ | Intercept | RunTime | RunPulse |
---|---|---|---|---|---|---|
1 | 1 | PARMS | 86.544 | -2.82231 | -0.05873 | |
2 | 1 | COV | Intercept | 100.145 | -0.53519 | -0.55077 |
3 | 1 | COV | RunTime | -0.535 | 0.10774 | -0.00345 |
4 | 1 | COV | RunPulse | -0.551 | -0.00345 | 0.00343 |
5 | 2 | PARMS | 83.021 | -3.00023 | -0.02491 | |
6 | 2 | COV | Intercept | 79.032 | -0.66765 | -0.41918 |
7 | 2 | COV | RunTime | -0.668 | 0.11456 | -0.00313 |
8 | 2 | COV | RunPulse | -0.419 | -0.00313 | 0.00264 |
The following statements combine the five sets of regression coefficients:
proc mianalyze data=outreg; modeleffects Intercept RunTime RunPulse; run;
The "Model Information" table in Figure 64.2 lists the input data set(s) and the number of imputations.
The "Variance Information" table in Figure 64.3 displays the between-imputation, within-imputation, and total variances for combining complete-data inferences. It also displays the degrees of freedom for the total variance, the relative increase in variance due to missing values, the fraction of missing information, and the relative efficiency for each parameter estimate.
Figure 64.3: Variance Information Table
Variance Information | |||||||
---|---|---|---|---|---|---|---|
Parameter | Variance | DF | Relative Increase in Variance |
Fraction Missing Information |
Relative Efficiency |
||
Between | Within | Total | |||||
Intercept | 45.529229 | 76.543614 | 131.178689 | 23.059 | 0.713777 | 0.461277 | 0.915537 |
RunTime | 0.019390 | 0.106220 | 0.129487 | 123.88 | 0.219051 | 0.192620 | 0.962905 |
RunPulse | 0.001007 | 0.002537 | 0.003746 | 38.419 | 0.476384 | 0.355376 | 0.933641 |
The "Parameter Estimates" table in Figure 64.4 displays a combined estimate and standard error for each regression coefficient (parameter). Inferences are based on t distributions. The table displays a 95% confidence interval and a t test with the associated p-value for the hypothesis that the parameter is equal to the value specified with the THETA0= option (in this case, zero by default). The minimum and maximum parameter estimates from the imputed data sets are also displayed.
Figure 64.4: Parameter Estimates
Parameter Estimates | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Parameter | Estimate | Std Error | 95% Confidence Limits | DF | Minimum | Maximum | Theta0 | t for H0: Parameter=Theta0 |
Pr > |t| | |
Intercept | 90.837440 | 11.453327 | 67.14779 | 114.5271 | 23.059 | 83.020730 | 100.839807 | 0 | 7.93 | <.0001 |
RunTime | -3.032870 | 0.359844 | -3.74511 | -2.3206 | 123.88 | -3.204426 | -2.822311 | 0 | -8.43 | <.0001 |
RunPulse | -0.068578 | 0.061204 | -0.19243 | 0.0553 | 38.419 | -0.112840 | -0.024910 | 0 | -1.12 | 0.2695 |