Value labels
1
Multiple Imputation for Missing Data
Any specific imputed values for missingness represent one set of an infinite number of plausible values that may have been imputed. 1. Determine prediction equation for imputation of missingness (e.g., Yˆ a b1 X1 b2 X 2 e ). 2. Consider sampling error of the bi in this equation and select another set of plausible bi (call this set of regression slopes bi* ). 3. Use the bi* to make a second prediction of each missing value. 4. Repeat steps 2 & 3 several more times, each imputation yielding a separate “complete” set of data. 5. Analyze each complete set of data and average the results (e.g., compute mean regression weight, mean structure coefficient, etc.). 6. Consider variability in these summary statistics across the multiple imputations (imputation error). Total error T2 = average estimation error E2 + imputation error I2
2
data one; input Y X1 X2; cards; 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 . . . 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 . 47.920 11.50 170 47.467 10.50 170 ; proc mi data=one out=imputed; proc print data=imputed; title 'Contents of Data = IMPUTED'; proc reg data=imputed outest=outreg covout noprint; model Y = X1 X2; by _Imputation_; proc print data=outreg; title ‘Output from PROC REG’; proc mianalyze data=outreg; modeleffects Intercept x1 x2; run;
3
Contents of Data = IMPUTED Obs 1 2 3 4
_Imputation_ 1 1 1 1
Y 44.6090 45.3130 54.2970 59.5710
X1
X2
11.3700 10.0700 8.6500 6.3158
178.000 185.000 156.000 179.597
32 33 34 35
2 2 2 2
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 8.0906
178.000 185.000 156.000 165.939
63 64 65 66
3 3 3 3
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 8.5059
178.000 185.000 156.000 169.985
94 95 96 97
4 4 4 4
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 6.6947
178.000 185.000 156.000 162.824
125 126 127 128
5 5 5 5
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 7.8401
178.000 185.000 156.000 170.157
4
Output from PROC REG Obs
_Imputation_
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
_MODEL_ MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1
_TYPE_ PARMS COV COV COV PARMS COV COV COV PARMS COV COV COV PARMS COV COV COV PARMS COV COV COV
_NAME_
Intercept X1 X2 Intercept X1 X2 Intercept X1 X2 Intercept X1 X2 Intercept X1 X2
_DEPVAR_ Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y
_RMSE_
Intercept
2.39810 2.39810 2.39810 2.39810 2.63360 2.63360 2.63360 2.63360 3.04577 3.04577 3.04577 3.04577 2.70800 2.70800 2.70800 2.70800 2.56819 2.56819 2.56819 2.56819
91.205 59.352 -0.882 -0.287 92.592 78.500 -0.715 -0.408 89.279 108.706 -0.844 -0.578 92.192 82.212 -0.214 -0.467 92.144 45.612 -0.997 -0.201
X1
X2
-3.04859 -0.88152 0.08155 0.00015 -3.21014 -0.71534 0.12431 -0.00347 -3.32479 -0.84410 0.17141 -0.00565 -2.81669 -0.21361 0.11445 -0.00578 -3.26296 -0.99681 0.11197 -0.00108
-0.06919 -0.28651 0.00015 0.00164 -0.06564 -0.40806 -0.00347 0.00257 -0.03803 -0.57770 -0.00565 0.00370 -0.09132 -0.46673 -0.00578 0.00309 -0.06185 -0.20100 -0.00108 0.00123
Y -1 . . . -1 . . . -1 . . . -1 . . . -1 . . .
5
The MIANALYZE Procedure Model Information Data Set Number of Imputations
WORK.OUTREG 5
Multiple Imputation Variance Information
Parameter Intercept x1 x2
-----------------Variance----------------Between Within Total 1.776494 0.041700 0.000362
74.876345 0.120737 0.002444
77.008138 0.170777 0.002878
DF
Relative Increase in Variance
Fraction Missing Information
Relative Efficiency
5219.7 46.588 175.59
0.028471 0.414460 0.177759
0.028055 0.321531 0.160438
0.994420 0.939579 0.968910
Multiple Imputation Parameter Estimates Parameter
Estimate
Std Error
Intercept x1 x2
91.482275 -3.132633 -0.065207
8.775428 0.413252 0.053648
95% Confidence Limits 74.27876 -3.96418 -0.17108
108.6858 -2.3011 0.0407
DF
Minimum
Maximum
Theta0
t for H0: Parameter=Theta0
Pr > |t|
5219.7 46.588 175.59
89.278516 -3.324786 -0.091324
92.591804 -2.816686 -0.038031
0 0 0
10.42 -7.58 -1.22
Multiple Imputation for Missing Data
Any specific imputed values for missingness represent one set of an infinite number of plausible values that may have been imputed. 1. Determine prediction equation for imputation of missingness (e.g., Yˆ a b1 X1 b2 X 2 e ). 2. Consider sampling error of the bi in this equation and select another set of plausible bi (call this set of regression slopes bi* ). 3. Use the bi* to make a second prediction of each missing value. 4. Repeat steps 2 & 3 several more times, each imputation yielding a separate “complete” set of data. 5. Analyze each complete set of data and average the results (e.g., compute mean regression weight, mean structure coefficient, etc.). 6. Consider variability in these summary statistics across the multiple imputations (imputation error). Total error T2 = average estimation error E2 + imputation error I2
2
data one; input Y X1 X2; cards; 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 . . . 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 . 47.920 11.50 170 47.467 10.50 170 ; proc mi data=one out=imputed; proc print data=imputed; title 'Contents of Data = IMPUTED'; proc reg data=imputed outest=outreg covout noprint; model Y = X1 X2; by _Imputation_; proc print data=outreg; title ‘Output from PROC REG’; proc mianalyze data=outreg; modeleffects Intercept x1 x2; run;
3
Contents of Data = IMPUTED Obs 1 2 3 4
_Imputation_ 1 1 1 1
Y 44.6090 45.3130 54.2970 59.5710
X1
X2
11.3700 10.0700 8.6500 6.3158
178.000 185.000 156.000 179.597
32 33 34 35
2 2 2 2
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 8.0906
178.000 185.000 156.000 165.939
63 64 65 66
3 3 3 3
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 8.5059
178.000 185.000 156.000 169.985
94 95 96 97
4 4 4 4
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 6.6947
178.000 185.000 156.000 162.824
125 126 127 128
5 5 5 5
44.6090 45.3130 54.2970 59.5710
11.3700 10.0700 8.6500 7.8401
178.000 185.000 156.000 170.157
4
Output from PROC REG Obs
_Imputation_
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
_MODEL_ MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1 MODEL1
_TYPE_ PARMS COV COV COV PARMS COV COV COV PARMS COV COV COV PARMS COV COV COV PARMS COV COV COV
_NAME_
Intercept X1 X2 Intercept X1 X2 Intercept X1 X2 Intercept X1 X2 Intercept X1 X2
_DEPVAR_ Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y
_RMSE_
Intercept
2.39810 2.39810 2.39810 2.39810 2.63360 2.63360 2.63360 2.63360 3.04577 3.04577 3.04577 3.04577 2.70800 2.70800 2.70800 2.70800 2.56819 2.56819 2.56819 2.56819
91.205 59.352 -0.882 -0.287 92.592 78.500 -0.715 -0.408 89.279 108.706 -0.844 -0.578 92.192 82.212 -0.214 -0.467 92.144 45.612 -0.997 -0.201
X1
X2
-3.04859 -0.88152 0.08155 0.00015 -3.21014 -0.71534 0.12431 -0.00347 -3.32479 -0.84410 0.17141 -0.00565 -2.81669 -0.21361 0.11445 -0.00578 -3.26296 -0.99681 0.11197 -0.00108
-0.06919 -0.28651 0.00015 0.00164 -0.06564 -0.40806 -0.00347 0.00257 -0.03803 -0.57770 -0.00565 0.00370 -0.09132 -0.46673 -0.00578 0.00309 -0.06185 -0.20100 -0.00108 0.00123
Y -1 . . . -1 . . . -1 . . . -1 . . . -1 . . .
5
The MIANALYZE Procedure Model Information Data Set Number of Imputations
WORK.OUTREG 5
Multiple Imputation Variance Information
Parameter Intercept x1 x2
-----------------Variance----------------Between Within Total 1.776494 0.041700 0.000362
74.876345 0.120737 0.002444
77.008138 0.170777 0.002878
DF
Relative Increase in Variance
Fraction Missing Information
Relative Efficiency
5219.7 46.588 175.59
0.028471 0.414460 0.177759
0.028055 0.321531 0.160438
0.994420 0.939579 0.968910
Multiple Imputation Parameter Estimates Parameter
Estimate
Std Error
Intercept x1 x2
91.482275 -3.132633 -0.065207
8.775428 0.413252 0.053648
95% Confidence Limits 74.27876 -3.96418 -0.17108
108.6858 -2.3011 0.0407
DF
Minimum
Maximum
Theta0
t for H0: Parameter=Theta0
Pr > |t|
5219.7 46.588 175.59
89.278516 -3.324786 -0.091324
92.591804 -2.816686 -0.038031
0 0 0
10.42 -7.58 -1.22
