Original Program from program editor.
options ps=256 ls=99 nocenter nodate nonumber nolabel;
TITLE1 'Example of Multiple Regression (MLR)';
DATA SENIC; Infile cards missover;
TITLE2 'SENIC database from NKNW 1996 (Appendix C)';
INPUT IDNo LtofStay Age InfRisk CulRatio XRay NoBeds MedSch Region Census Nurses Services;
*** label IDNo = 'Identification number'
LtofStay = 'Length of stay (days)'
Age = 'Patient age (years)'
InfRisk = 'Average Infection risk (%)'
CulRatio = 'ratio cultures to patients w/o symptoms'
XRay = 'ratio xrays to patients w/o symptoms'
NoBeds = 'Average no. of beds in hosp.'
MedSch = 'Med School Affiliation'
Region = 'Region NE, NC, S, W'
Census = 'Average no. patients in hosp.'
Nurses = 'Av. no. nurses'
Services = '% of 35 potential service facilities';
CARDS; Run;
1 7.13 55.7 4.1 9.0 39.6 279 2 4 207 241 60.0
2 8.82 58.2 1.6 3.8 51.7 80 2 2 51 52 40.0
3 8.34 56.9 2.7 8.1 74.0 107 2 3 82 54 20.0
4 8.95 53.7 5.6 18.9 122.8 147 2 4 53 148 40.0
5 11.20 56.5 5.7 34.5 88.9 180 2 1 134 151 40.0
6 9.76 50.9 5.1 21.9 97.0 150 2 2 147 106 40.0
7 9.68 57.8 4.6 16.7 79.0 186 2 3 151 129 40.0
8 11.18 45.7 5.4 60.5 85.8 640 1 2 399 360 60.0
9 8.67 48.2 4.3 24.4 90.8 182 2 3 130 118 40.0
10 8.84 56.3 6.3 29.6 82.6 85 2 1 59 66 40.0
11 11.07 53.2 4.9 28.5 122.0 768 1 1 591 656 80.0
12 8.30 57.2 4.3 6.8 83.8 167 2 3 105 59 40.0
13 12.78 56.8 7.7 46.0 116.9 322 1 1 252 349 57.1
14 7.58 56.7 3.7 20.8 88.0 97 2 2 59 79 37.1
15 9.00 56.3 4.2 14.6 76.4 72 2 3 61 38 17.1
16 11.08 50.2 5.5 18.6 63.6 387 2 3 326 405 57.1
17 8.28 48.1 4.5 26.0 101.8 108 2 4 84 73 37.1
18 11.62 53.9 6.4 25.5 99.2 133 2 1 113 101 37.1
19 9.06 52.8 4.2 6.9 75.9 134 2 2 103 125 37.1
20 9.35 53.8 4.1 15.9 80.9 833 2 3 547 519 77.1
21 7.53 42.0 4.2 23.1 98.9 95 2 4 47 49 17.1
22 10.24 49.0 4.8 36.3 112.6 195 2 2 163 170 37.1
23 9.78 52.3 5.0 17.6 95.9 270 1 1 240 198 57.1
24 9.84 62.2 4.8 12.0 82.3 600 2 3 468 497 57.1
25 9.20 52.2 4.0 17.5 71.1 298 1 4 244 236 57.1
26 8.28 49.5 3.9 12.0 113.1 546 1 2 413 436 57.1
27 9.31 47.2 4.5 30.2 101.3 170 2 1 124 173 37.1
28 8.19 52.1 3.2 10.8 59.2 176 2 1 156 88 37.1
29 11.65 54.5 4.4 18.6 96.1 248 2 1 217 189 37.1
30 9.89 50.5 4.9 17.7 103.6 167 2 2 113 106 37.1
31 11.03 49.9 5.0 19.7 102.1 318 2 1 270 335 57.1
32 9.84 53.0 5.2 17.7 72.6 210 2 2 200 239 54.3
33 11.77 54.1 5.3 17.3 56.0 196 2 1 164 165 34.3
34 13.59 54.0 6.1 24.2 111.7 312 2 1 258 169 54.3
35 9.74 54.4 6.3 11.4 76.1 221 2 2 170 172 54.3
36 10.33 55.8 5.0 21.2 104.3 266 2 1 181 149 54.3
37 9.97 58.2 2.8 16.5 76.5 90 2 2 69 42 34.3
38 7.84 49.1 4.6 7.1 87.9 60 2 3 50 45 34.3
39 10.47 53.2 4.1 5.7 69.1 196 2 2 168 153 54.3
40 8.16 60.9 1.3 1.9 58.0 73 2 3 49 21 14.3
41 8.48 51.1 3.7 12.1 92.8 166 2 3 145 118 34.3
42 10.72 53.8 4.7 23.2 94.1 113 2 3 90 107 34.3
43 11.20 45.0 3.0 7.0 78.9 130 2 3 95 56 34.3
44 10.12 51.7 5.6 14.9 79.1 362 1 3 313 264 54.3
45 8.37 50.7 5.5 15.1 84.8 115 2 2 96 88 34.3
46 10.16 54.2 4.6 8.4 51.5 831 1 4 581 629 74.3
47 19.56 59.9 6.5 17.2 113.7 306 2 1 273 172 51.4
48 10.90 57.2 5.5 10.6 71.9 593 2 2 446 211 51.4
49 7.67 51.7 1.8 2.5 40.4 106 2 3 93 35 11.4
50 8.88 51.5 4.2 10.1 86.9 305 2 3 238 197 51.4
51 11.48 57.6 5.6 20.3 82.0 252 2 1 207 251 51.4
52 9.23 51.6 4.3 11.6 42.6 620 2 2 413 420 71.4
53 11.41 61.1 7.6 16.6 97.9 535 2 3 330 273 51.4
54 12.07 43.7 7.8 52.4 105.3 157 2 2 115 76 31.4
55 8.63 54.0 3.1 8.4 56.2 76 2 1 39 44 31.4
56 11.15 56.5 3.9 7.7 73.9 281 2 1 217 199 51.4
57 7.14 59.0 3.7 2.6 75.8 70 2 4 37 35 31.4
58 7.65 47.1 4.3 16.4 65.7 318 2 4 265 314 51.4
59 10.73 50.6 3.9 19.3 101.0 445 1 2 374 345 51.4
60 11.46 56.9 4.5 15.6 97.7 191 2 3 153 132 31.4
61 10.42 58.0 3.4 8.0 59.0 119 2 1 67 64 31.4
62 11.18 51.0 5.7 18.8 55.9 595 1 2 546 392 68.6
63 7.93 64.1 5.4 7.5 98.1 68 2 4 42 49 28.6
64 9.66 52.1 4.4 9.9 98.3 83 2 2 66 95 28.6
65 7.78 45.5 5.0 20.9 71.6 489 2 3 391 329 48.6
66 9.42 50.6 4.3 24.8 62.8 508 2 1 421 528 48.6
67 10.02 49.5 4.4 8.3 93.0 265 2 2 191 202 48.6
68 8.58 55.0 3.7 7.4 95.9 304 2 3 248 218 48.6
69 9.61 52.4 4.5 6.9 87.2 487 2 3 404 220 48.6
70 8.03 54.2 3.5 24.3 87.3 97 2 1 65 55 28.6
71 7.39 51.0 4.2 14.6 88.4 72 2 2 38 67 28.6
72 7.08 52.0 2.0 12.3 56.4 87 2 3 52 57 28.6
73 9.53 51.5 5.2 15.0 65.7 298 2 3 241 193 48.6
74 10.05 52.0 4.5 36.7 87.5 184 1 1 144 151 68.6
75 8.45 38.8 3.4 12.9 85.0 235 2 2 143 124 48.6
76 6.70 48.6 4.5 13.0 80.8 76 2 4 51 79 28.6
77 8.90 49.7 2.9 12.7 86.9 52 2 1 37 35 28.6
78 10.23 53.2 4.9 9.9 77.9 752 1 2 595 446 68.6
79 8.88 55.8 4.4 14.1 76.8 237 2 2 165 182 48.6
80 10.30 59.6 5.1 27.8 88.9 175 2 2 113 73 45.7
81 10.79 44.2 2.9 2.6 56.6 461 1 2 320 196 65.7
82 7.94 49.5 3.5 6.2 92.3 195 2 2 139 116 45.7
83 7.63 52.1 5.5 11.6 61.1 197 2 4 109 110 45.7
84 8.77 54.5 4.7 5.2 47.0 143 2 4 85 87 25.7
85 8.09 56.9 1.7 7.6 56.9 92 2 3 61 61 45.7
86 9.05 51.2 4.1 20.5 79.8 195 2 3 127 112 45.7
87 7.91 52.8 2.9 11.9 79.5 477 2 3 349 188 65.7
88 10.39 54.6 4.3 14.0 88.3 353 2 2 223 200 65.7
89 9.36 54.1 4.8 18.3 90.6 165 2 1 127 158 45.7
90 11.41 50.4 5.8 23.8 73.0 424 1 3 359 335 45.7
91 8.86 51.3 2.9 9.5 87.5 100 2 3 65 53 25.7
92 8.93 56.0 2.0 6.2 72.5 95 2 3 59 56 25.7
93 8.92 53.9 1.3 2.2 79.5 56 2 2 40 14 5.7
94 8.15 54.9 5.3 12.3 79.8 99 2 4 55 71 25.7
95 9.77 50.2 5.3 15.7 89.7 154 2 2 123 148 25.7
96 8.54 56.1 2.5 27.0 82.5 98 2 1 57 75 45.7
97 8.66 52.8 3.8 6.8 69.5 246 2 3 178 177 45.7
98 12.01 52.8 4.8 10.8 96.9 298 2 1 237 115 45.7
99 7.95 51.8 2.3 4.6 54.9 163 2 3 128 93 42.9
100 10.15 51.9 6.2 16.4 59.2 568 1 3 452 371 62.9
101 9.76 53.2 2.6 6.9 80.1 64 2 4 47 55 22.9
102 9.89 45.2 4.3 11.8 108.7 190 2 1 141 112 42.9
103 7.14 57.6 2.7 13.1 92.6 92 2 4 40 50 22.9
104 13.95 65.9 6.6 15.6 133.5 356 2 1 308 182 62.9
105 9.44 52.5 4.5 10.9 58.5 297 2 3 230 263 42.9
106 10.80 63.9 2.9 1.6 57.4 130 2 3 69 62 22.9
107 7.14 51.7 1.4 4.1 45.7 115 2 3 90 19 22.9
108 8.02 55.0 2.1 3.8 46.5 91 2 2 44 32 22.9
109 11.80 53.8 5.7 9.1 116.9 571 1 2 441 469 62.9
110 9.50 49.3 5.8 42.0 70.9 98 2 3 68 46 22.9
111 7.70 56.9 4.4 12.2 67.9 129 2 4 85 136 62.9
112 17.94 56.2 5.9 26.4 91.8 835 1 1 791 407 62.9
113 9.41 59.5 3.1 20.6 91.7 29 2 3 20 22 22.9
;
OPTIONS PS=50 LS=120;
proc plot data=senic; PLOT InfRisk*ltofstay;
TITLE3 'Scatter plot on one of the independent variables';
run;
OPTIONS PS=256 LS=99;
proc reg data=SENIC LINEPRINTER;
TITLE3'PROC REG analysis with selected variables';
model InfRisk = LtofStay Age Nurses;
output out=next1 p=YHat r=e;
run;
OPTIONS PS=50 LS=120; PLOT RESIDUAL.*PREDICTED. / VREF=0;
Run; OPTIONS PS=256 LS=99;
proc univariate data=next1 normal plot; var e;
TITLE4 'Univariate analysis of residuals';
run;
OPTIONS PS=50 LS=120;
proc plot data=next1; PLOT e*YHat / VREF=0;
TITLE4 'Residual plot';
run; OPTIONS PS=256 LS=99;
proc reg data=SENIC;
TITLE3 'Stepwise regression - backward selection';
model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services
/ selection=backward;
Run;
proc reg data=SENIC;
TITLE3 'Stepwise regression - stepwise selection';
model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services
/ selection=stepwise;
Run;
proc reg data=SENIC;
TITLE3 'Multiple regression - RSquare option';
model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services
/ selection=rsquare start=3 stop=6 best=8;
Run;
proc glm data=SENIC;
TITLE3 'Full Model done in GLM';
model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services;
RUN;
QUIT;
Proc reg data=senic;
TITLE3 'Extra sum of squares';
model InfRisk = LtofStay / ss1 ss2;
model InfRisk = Age / ss1 ss2;
model InfRisk = Nurses / ss1 ss2;
model InfRisk = LtofStay Age / ss1 ss2;
model InfRisk = LtofStay Nurses / ss1 ss2;
model InfRisk = Age Nurses / ss1 ss2;
model InfRisk = LtofStay Age Nurses / ss1 ss2;
run;
Below is output from the SAS log (bold) and output from the SAS Output window.
1 options ps=256 ls=99 nocenter nodate nonumber nolabel;
2 TITLE1 'Example of Multiple Regression (MLR)';
3
4 DATA SENIC; Infile cards missover;
5 TITLE2 'SENIC database from NKNW 1996 (Appendix C)';
6 INPUT IDNo LtofStay Age InfRisk CulRatio XRay NoBeds MedSch Region Census Nurses
6 ! Services;
7 *** label IDNo = 'Identification number'
8 LtofStay = 'Length of stay (days)'
9 Age = 'Patient age (years)'
10 InfRisk = 'Average Infection risk (%)'
11 CulRatio = 'ratio cultures to patients w/o symptoms'
12 XRay = 'ratio xrays to patients w/o symptoms'
13 NoBeds = 'Average no. of beds in hosp.'
14 MedSch = 'Med School Affiliation'
15 Region = 'Region NE, NC, S, W'
16 Census = 'Average no. patients in hosp.'
17 Nurses = 'Av. no. nurses'
18 Services = '% of 35 potential service facilities';
19 CARDS;
NOTE: The data set WORK.SENIC has 113 observations and 12 variables.
NOTE: DATA statement used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
19 ! Run;
133 ;
134 OPTIONS PS=50 LS=120;
135 proc plot data=senic; PLOT InfRisk*ltofstay;
136 TITLE3 'Scatter plot on one of the independent variables';
137 run;
138 OPTIONS PS=256 LS=99;
139
NOTE: There were 113 observations read from the data set WORK.SENIC.
NOTE: The PROCEDURE PLOT printed page 1.
NOTE: PROCEDURE PLOT used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Scatter plot on one of the independent variables
Plot of InfRisk*LtofStay. Legend: A = 1 obs, B = 2 obs, etc.
InfRisk |
8 +
| A A
| A
|
|
7 +
|
| A A
| A A A
| A A
6 + A
| A BA A
| A A A A AA A
| AA A A A
| AAA A
5 + A B AA B
| A A A A A A
| A A A BAA AA A
| B A A A AA AAA A A
| A AA AC A A
4 + A A A A
| A A AB
| B
| A A
| A A A
3 + A B B A
| A A A
| A A
| A
| A
2 + A A
| A A
| A
| A A A
|
1 +
|
---+------------+------------+------------+------------+------------+------------+------------+--
6 8 10 12 14 16 18 20
LtofStay
140 proc reg data=SENIC LINEPRINTER;
141 TITLE3'PROC REG analysis with selected variables';
142 model InfRisk = LtofStay Age Nurses;
143 output out=next1 p=YHat r=e;
144 run;
NOTE: 113 observations read.
NOTE: 113 observations used in computations.
145 OPTIONS PS=50 LS=120; PLOT RESIDUAL.*PREDICTED. / VREF=0;
146 Run;
146 ! OPTIONS PS=256 LS=99;
147
NOTE: The data set WORK.NEXT1 has 113 observations and 14 variables.
NOTE: The PROCEDURE REG printed pages 2-3.
NOTE: PROCEDURE REG used (Total process time):
real time 0.04 seconds
cpu time 0.04 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
PROC REG analysis with selected variables
The REG Procedure
Model: MODEL1
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 68.46806 22.82269 18.72 <.0001
Error 109 132.91176 1.21937
Corrected Total 112 201.37982
Root MSE 1.10425 R-Square 0.3400
Dependent Mean 4.35487 Adj R-Sq 0.3218
Coeff Var 25.35675
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 1.89700 1.29112 1.47 0.1446
LtofStay 1 0.32864 0.05968 5.51 <.0001
Age 1 -0.02057 0.02412 -0.85 0.3958
Nurses 1 0.00220 0.00080714 2.73 0.0074
------+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+-------
RESIDUAL | |
| |
3 + +
| |
| 1 1 1 |
| |
| 1 |
2 + 1 1 1 +
| 1 |
| 1 11 1 |
| |
| 1 1 |
1 + 1 1 1 +
R | 1 11 11 111 2 2 |
e | 1 1 1 1 2 1 1 |
s | 1 1 1 1 1 11 1 |
i | 11 11 1 1 1 1 1 |
d 0 + 1 1 1 21 1 1 +
u | 1 1 1 1 1 1 11 1 1 |
a | 1 1 1 11 1 |
l | 2 1 1 |
| 1 1 1 1 1 |
-1 + 111 1 1 1 1 1 +
| 1 1 1 1 |
| 1 1 1 |
| 1 1 1 |
| 1 1 1 1 |
-2 + 1 1 +
| 1 |
| 1 |
| |
| |
-3 + +
| |
| |
------+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+-------
3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75
Predicted Value of InfRisk PRED
148 proc univariate data=next1 normal plot; var e;
149 TITLE4 'Univariate analysis of residuals';
150 run;
NOTE: The PROCEDURE UNIVARIATE printed page 4.
NOTE: PROCEDURE UNIVARIATE used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
PROC REG analysis with selected variables
Univariate analysis of residuals
The UNIVARIATE Procedure
Variable: e
Moments
N 113 Sum Weights 113
Mean 0 Sum Observations 0
Std Deviation 1.0893632 Variance 1.18671217
Skewness 0.12446139 Kurtosis -0.1681454
Uncorrected SS 132.911763 Corrected SS 132.911763
Coeff Variation . Std Error Mean 0.10247867
Basic Statistical Measures
Location Variability
Mean 0.000000 Std Deviation 1.08936
Median 0.034938 Variance 1.18671
Mode . Range 5.11840
Interquartile Range 1.52582
Tests for Location: Mu0=0n
Test -Statistic- -----p Value------
Student's t t 0 Pr > |t| 1.0000
Sign M 2.5 Pr >= |M| 0.7069
Signed Rank S -34.5 Pr >= |S| 0.9218
Tests for Normality
Test --Statistic--- -----p Value------
Shapiro-Wilk W 0.989495 Pr < W 0.5355
Kolmogorov-Smirnov D 0.054321 Pr > D >0.1500
Cramer-von Mises W-Sq 0.036762 Pr > W-Sq >0.2500
Anderson-Darling A-Sq 0.282125 Pr > A-Sq >0.2500
Quantiles (Definition 5)
Quantile Estimate
100% Max 2.6675258
99% 2.6081781
95% 1.9417749
90% 1.5398900
75% Q3 0.7101761
50% Median 0.0349382
25% Q1 -0.8156473
10% -1.3721342
5% -1.8199663
1% -2.1133088
0% Min -2.4508710
Extreme Observations
------Lowest----- -----Highest-----
Value Obs Value Obs
-2.45087 93 2.00204 13
-2.11331 2 2.10713 63
-2.07257 40 2.51020 10
-2.06595 81 2.60818 53
-1.82215 107 2.66753 54
Stem Leaf Boxplot
26 17 2 |
24 1 1 |
22 |
20 01 2 |
18 24 2 |
16 900 3 |
14 47 2 |
12 23 2 |
10 47 2 |
8 01456805 8 |
6 312368 6 +-----+
4 235773366899 12 | |
2 88013 5 | |
0 013440345779 12 *--+--*
-0 965420 6 | |
-2 7738554321 10 | |
-4 1987 4 | |
-6 51622 5 | |
-8 98774211972 11 +-----+
-10 |
-12 7758742 7 |
-14 3 1 |
-16 833 3 |
-18 220 3 |
-20 177 3 |
-22 |
-24 5 1 |
----+----+----+----+
Multiply Stem.Leaf by 10**-1
Normal Probability Plot
2.7+ * +*
| * +
| ++
| * *+
| *++
| **+
| *+
| *
| +*
| ****
| ***
| ****
| **
0.1+ ***
| **
| ***
| **
| +**
| ****
| ++
| ****
| +*
| +**
| +**
| * **
| ++
-2.5+* +
+----+----+----+----+----+----+----+----+----+----+
-2 -1 0 +1 +2
151 OPTIONS PS=50 LS=120;
152 proc plot data=next1; PLOT e*YHat / VREF=0;
153 TITLE4 'Residual plot';
154 run;
154 ! OPTIONS PS=256 LS=99;
155
NOTE: There were 113 observations read from the data set WORK.NEXT1.
NOTE: The PROCEDURE PLOT printed page 5.
NOTE: PROCEDURE PLOT used (Total process time):
real time 0.00 seconds
cpu time 0.00 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
PROC REG analysis with selected variables
Residual plot
Plot of e*YHat. Legend: A = 1 obs, B = 2 obs, etc.
3 +
|
| A A
| A
|
| A
2 + A A A
|
| A A A
| A A
|
| A A
1 + A A A
| A A A BA B A
| A AA A A A A
e | A A A A B A A A
| A A A A A
| A A A A A A A
0 +--------------A----------A--A-----BA-------------------A-----A-----------------------------------------------------
| A A A A A AA A A
| A A A A
| A A A A
| A A A A A
| A A A A A
-1 + AB A A A
| A A
| A A A A A
| A
| A A
| A A A A
-2 + A A
| A
|
| A
|
|
-3 +
---+----------+----------+----------+----------+----------+----------+----------+----------+----------+----------+--
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
Yhat
156 proc reg data=SENIC;
157 TITLE3 'Stepwise regression - backward selection';
158 model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services
159 / selection=backward;
160 Run;
NOTE: 113 observations read.
NOTE: 113 observations used in computations.
161
NOTE: The PROCEDURE REG printed page 6.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Stepwise regression - backward selection
The REG Procedure
Model: MODEL1
Dependent Variable: InfRisk
Backward Elimination: Step 0
All Variables Entered: R-Square = 0.5251 and C(p) = 9.0000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 8 105.74000 13.21750 14.37 <.0001
Error 104 95.63982 0.91961
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept -0.74726 1.20763 0.35211 0.38 0.5374
LtofStay 0.17693 0.06906 6.03648 6.56 0.0118
Age 0.01621 0.02226 0.48809 0.53 0.4679
CulRatio 0.04699 0.01075 17.57678 19.11 <.0001
XRay 0.01204 0.00549 4.42782 4.81 0.0304
NoBeds -0.00145 0.00271 0.26220 0.29 0.5945
Census 0.00072796 0.00347 0.04044 0.04 0.8343
Nurses 0.00191 0.00175 1.08719 1.18 0.2794
Services 0.01628 0.01019 2.34765 2.55 0.1131
Bounds on condition number: 34.702, 674.59
---------------------------------------------------------------------------------------------
Backward Elimination: Step 1
Variable Census Removed: R-Square = 0.5249 and C(p) = 7.0440
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 7 105.69957 15.09994 16.57 <.0001
Error 105 95.68026 0.91124
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept -0.76082 1.20039 0.36606 0.40 0.5276
LtofStay 0.18363 0.06095 8.27183 9.08 0.0032
Age 0.01557 0.02195 0.45899 0.50 0.4795
CulRatio 0.04665 0.01057 17.73508 19.46 <.0001
XRay 0.01196 0.00545 4.39134 4.82 0.0303
NoBeds -0.00094901 0.00130 0.48680 0.53 0.4665
Nurses 0.00199 0.00170 1.24827 1.37 0.2445
Services 0.01614 0.01012 2.31695 2.54 0.1138
Bounds on condition number: 7.7057, 162.09
----------------------------------------------------------------------------------------------
Backward Elimination: Step 2
Variable Age Removed: R-Square = 0.5226 and C(p) = 5.5431
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 6 105.24057 17.54010 19.34 <.0001
Error 106 96.13925 0.90697
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept -0.01195 0.57099 0.00039725 0.00 0.9833
LtofStay 0.19699 0.05783 10.52453 11.60 0.0009
CulRatio 0.04448 0.01010 17.59563 19.40 <.0001
XRay 0.01189 0.00543 4.33823 4.78 0.0309
NoBeds -0.00103 0.00129 0.57917 0.64 0.4260
Nurses 0.00200 0.00170 1.26422 1.39 0.2404
Services 0.01637 0.01009 2.38761 2.63 0.1077
Bounds on condition number: 7.6446, 129.79
----------------------------------------------------------------------------------------------
Backward Elimination: Step 3
Variable NoBeds Removed: R-Square = 0.5197 and C(p) = 4.1729
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 5 104.66141 20.93228 23.16 <.0001
Error 107 96.71842 0.90391
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept 0.08173 0.55788 0.01940 0.02 0.8838
LtofStay 0.18315 0.05508 9.99423 11.06 0.0012
CulRatio 0.04567 0.00997 18.96154 20.98 <.0001
XRay 0.01247 0.00538 4.86433 5.38 0.0223
Nurses 0.00093910 0.00105 0.72307 0.80 0.3731
Services 0.01400 0.00963 1.91090 2.11 0.1489
Bounds on condition number: 2.6532, 46.546
----------------------------------------------------------------------------------------------
Backward Elimination: Step 4
Variable Nurses Removed: R-Square = 0.5161 and C(p) = 2.9592
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 4 103.93833 25.98458 28.80 <.0001
Error 108 97.44149 0.90224
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept -0.06358 0.53321 0.01283 0.01 0.9053
LtofStay 0.18841 0.05471 10.69867 11.86 0.0008
CulRatio 0.04645 0.00992 19.76510 21.91 <.0001
XRay 0.01205 0.00535 4.57751 5.07 0.0263
Services 0.02047 0.00635 9.37912 10.40 0.0017
Bounds on condition number: 1.3578, 20.506
----------------------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination
Variable Number Partial Model
Step Removed Vars In R-Square R-Square C(p) F Value Pr > F
1 Census 7 0.0002 0.5249 7.0440 0.04 0.8343
2 Age 6 0.0023 0.5226 5.5431 0.50 0.4795
3 NoBeds 5 0.0029 0.5197 4.1729 0.64 0.4260
4 Nurses 4 0.0036 0.5161 2.9592 0.80 0.3731
162 proc reg data=SENIC;
163 TITLE3 'Stepwise regression - stepwise selection';
164 model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services
165 / selection=stepwise;
166 Run;
NOTE: 113 observations read.
NOTE: 113 observations used in computations.
167
NOTE: The PROCEDURE REG printed page 7.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Stepwise regression - stepwise selection
The REG Procedure
Model: MODEL1
Dependent Variable: InfRisk
Stepwise Selection: Step 1
Variable CulRatio Entered: R-Square = 0.3127 and C(p) = 41.5161
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 62.96314 62.96314 50.49 <.0001
Error 111 138.41668 1.24700
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept 3.19790 0.19377 339.64906 272.37 <.0001
CulRatio 0.07326 0.01031 62.96314 50.49 <.0001
Bounds on condition number: 1, 1
----------------------------------------------------------------------------------------------
Stepwise Selection: Step 2
Variable LtofStay Entered: R-Square = 0.4504 and C(p) = 13.3525
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 90.70199 45.35099 45.07 <.0001
Error 110 110.67784 1.00616
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept 0.80549 0.48776 2.74400 2.73 0.1015
LtofStay 0.27547 0.05246 27.73885 27.57 <.0001
CulRatio 0.05645 0.00980 33.39688 33.19 <.0001
Bounds on condition number: 1.1195, 4.4779
----------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Services Entered: R-Square = 0.4934 and C(p) = 5.9368
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 99.36082 33.12027 35.39 <.0001
Error 109 102.01900 0.93595
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept 0.49133 0.48164 0.97402 1.04 0.3099
LtofStay 0.22391 0.05337 16.47665 17.60 <.0001
CulRatio 0.05420 0.00948 30.59828 32.69 <.0001
Services 0.01963 0.00645 8.65884 9.25 0.0029
Bounds on condition number: 1.2451, 10.57
----------------------------------------------------------------------------------------------
Stepwise Selection: Step 4
Variable XRay Entered: R-Square = 0.5161 and C(p) = 2.9592
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 4 103.93833 25.98458 28.80 <.0001
Error 108 97.44149 0.90224
Corrected Total 112 201.37982
Parameter Standard
Variable Estimate Error Type II SS F Value Pr > F
Intercept -0.06358 0.53321 0.01283 0.01 0.9053
LtofStay 0.18841 0.05471 10.69867 11.86 0.0008
CulRatio 0.04645 0.00992 19.76510 21.91 <.0001
XRay 0.01205 0.00535 4.57751 5.07 0.0263
Services 0.02047 0.00635 9.37912 10.40 0.0017
Bounds on condition number: 1.3578, 20.506
----------------------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.1500 significance level for entry into the model.
Summary of Stepwise Selection
Variable Variable Number Partial Model
Step Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F
1 CulRatio 1 0.3127 0.3127 41.5161 50.49 <.0001
2 LtofStay 2 0.1377 0.4504 13.3525 27.57 <.0001
3 Services 3 0.0430 0.4934 5.9368 9.25 0.0029
4 XRay 4 0.0227 0.5161 2.9592 5.07 0.0263
168 proc reg data=SENIC;
169 TITLE3 'Multiple regression - RSquare option';
170 model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services
171 / selection=rsquare start=3 stop=6 best=8;
172 Run;
NOTE: 113 observations read.
NOTE: 113 observations used in computations.
173
NOTE: The PROCEDURE REG printed page 8.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Multiple regression - RSquare option
The REG Procedure
Model: MODEL1
Dependent Variable: InfRisk
R-Square Selection Method
Number in
Model R-Square Variables in Model
3 0.4934 LtofStay CulRatio Services
3 0.4852 LtofStay CulRatio Nurses
3 0.4736 LtofStay CulRatio NoBeds
3 0.4735 LtofStay CulRatio Census
3 0.4696 LtofStay CulRatio XRay
3 0.4630 CulRatio XRay Services
3 0.4619 CulRatio XRay Census
3 0.4538 CulRatio XRay Nurses
----------------------------------------------------------------------------------
4 0.5161 LtofStay CulRatio XRay Services
4 0.5102 LtofStay CulRatio XRay Nurses
4 0.5000 LtofStay CulRatio XRay Census
4 0.4997 LtofStay CulRatio XRay NoBeds
4 0.4956 LtofStay CulRatio Nurses Services
4 0.4956 LtofStay Age CulRatio Services
4 0.4935 LtofStay CulRatio NoBeds Services
4 0.4934 LtofStay CulRatio Census Services
----------------------------------------------------------------------------------
5 0.5197 LtofStay CulRatio XRay Nurses Services
5 0.5183 LtofStay Age CulRatio XRay Services
5 0.5166 LtofStay CulRatio XRay Census Services
5 0.5163 LtofStay CulRatio XRay NoBeds Services
5 0.5130 LtofStay Age CulRatio XRay Nurses
5 0.5107 LtofStay CulRatio XRay NoBeds Nurses
5 0.5106 LtofStay CulRatio XRay Census Nurses
5 0.5033 LtofStay Age CulRatio XRay Census
----------------------------------------------------------------------------------
6 0.5226 LtofStay CulRatio XRay NoBeds Nurses Services
6 0.5225 LtofStay Age CulRatio XRay Nurses Services
6 0.5216 LtofStay CulRatio XRay Census Nurses Services
6 0.5191 LtofStay Age CulRatio XRay Census Services
6 0.5187 LtofStay Age CulRatio XRay NoBeds Services
6 0.5169 LtofStay CulRatio XRay NoBeds Census Services
6 0.5134 LtofStay Age CulRatio XRay NoBeds Nurses
6 0.5132 LtofStay Age CulRatio XRay Census Nurses
174 proc glm data=SENIC;
175 TITLE3 'Full Model done in GLM';
176 model InfRisk = LtofStay Age CulRatio XRay NoBeds Census Nurses Services;
177 RUN;
178 QUIT;
NOTE: The PROCEDURE GLM printed pages 9-10.
NOTE: PROCEDURE GLM used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Full Model done in GLM
The GLM Procedure
Number of observations 113
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Full Model done in GLM
The GLM Procedure
Dependent Variable: InfRisk
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 8 105.7400031 13.2175004 14.37 <.0001
Error 104 95.6398199 0.9196137
Corrected Total 112 201.3798230
R-Square Coeff Var Root MSE InfRisk Mean
0.525077 22.02053 0.958965 4.354867
Source DF Type I SS Mean Square F Value Pr > F
LtofStay 1 57.30510979 57.30510979 62.31 <.0001
Age 1 2.07505963 2.07505963 2.26 0.1361
CulRatio 1 31.45734721 31.45734721 34.21 <.0001
XRay 1 3.84764876 3.84764876 4.18 0.0433
NoBeds 1 6.51641887 6.51641887 7.09 0.0090
Census 1 0.17435655 0.17435655 0.19 0.6642
Nurses 1 2.01641616 2.01641616 2.19 0.1417
Services 1 2.34764618 2.34764618 2.55 0.1131
Source DF Type III SS Mean Square F Value Pr > F
LtofStay 1 6.03647547 6.03647547 6.56 0.0118
Age 1 0.48809372 0.48809372 0.53 0.4679
CulRatio 1 17.57677710 17.57677710 19.11 <.0001
XRay 1 4.42781800 4.42781800 4.81 0.0304
NoBeds 1 0.26219566 0.26219566 0.29 0.5945
Census 1 0.04043524 0.04043524 0.04 0.8343
Nurses 1 1.08719258 1.08719258 1.18 0.2794
Services 1 2.34764618 2.34764618 2.55 0.1131
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept -.7472552477 1.20762993 -0.62 0.5374
LtofStay 0.1769313961 0.06905830 2.56 0.0118
Age 0.0162135960 0.02225515 0.73 0.4679
CulRatio 0.0469933765 0.01074904 4.37 <.0001
XRay 0.0120368799 0.00548557 2.19 0.0304
NoBeds -.0014471797 0.00271027 -0.53 0.5945
Census 0.0007279559 0.00347158 0.21 0.8343
Nurses 0.0019062110 0.00175316 1.09 0.2794
Services 0.0162795745 0.01018895 1.60 0.1131
180 Proc reg data=senic;
181 TITLE3 'Extra sum of squares';
182 model InfRisk = LtofStay / ss1 ss2;
183 model InfRisk = Age / ss1 ss2;
184 model InfRisk = Nurses / ss1 ss2;
185 model InfRisk = LtofStay Age / ss1 ss2;
186 model InfRisk = LtofStay Nurses / ss1 ss2;
187 model InfRisk = Age Nurses / ss1 ss2;
188 model InfRisk = LtofStay Age Nurses / ss1 ss2;
189 run;
NOTE: 113 observations read.
NOTE: 113 observations used in computations.
NOTE: The PROCEDURE REG printed pages 11-17.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
NOTE: SAS Institute Inc., SAS Campus Drive, Cary, NC USA 27513-2414
NOTE: The SAS System used:
real time 1.23 seconds
cpu time 0.39 seconds
Example of Multiple Regression (MLR)
SENIC database from NKNW 1996 (Appendix C)
Extra sum of squares
The REG Procedure
Model: MODEL1
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 57.30511 57.30511 44.15 <.0001
Error 111 144.07471 1.29797
Corrected Total 112 201.37982
Root MSE 1.13929 R-Square 0.2846
Dependent Mean 4.35487 Adj R-Sq 0.2781
Coeff Var 26.16119
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 0.74430 0.55386 1.34 0.1817 2143.03018 2.34406
LtofStay 1 0.37422 0.05632 6.64 <.0001 57.30511 57.30511
The REG Procedure
Model: MODEL2
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 0.00024065 0.00024065 0.00 0.9908
Error 111 201.37958 1.81423
Corrected Total 112 201.37982
Root MSE 1.34693 R-Square 0.0000
Dependent Mean 4.35487 Adj R-Sq -0.0090
Coeff Var 30.92939
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 4.33738 1.52379 2.85 0.0053 2143.03018 14.69937
Age 1 0.00032854 0.02853 0.01 0.9908 0.00024065 0.00024065
The REG Procedure
Model: MODEL3
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 31.25844 31.25844 20.40 <.0001
Error 111 170.12139 1.53263
Corrected Total 112 201.37982
Root MSE 1.23799 R-Square 0.1552
Dependent Mean 4.35487 Adj R-Sq 0.1476
Coeff Var 28.42779
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 3.69766 0.18639 19.84 <.0001 2143.03018 603.19681
Nurses 1 0.00379 0.00083997 4.52 <.0001 31.25844 31.25844
The REG Procedure
Model: MODEL4
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 59.38017 29.69008 23.00 <.0001
Error 110 141.99965 1.29091
Corrected Total 112 201.37982
Root MSE 1.13618 R-Square 0.2949
Dependent Mean 4.35487 Adj R-Sq 0.2820
Coeff Var 26.08990
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 2.26591 1.32115 1.72 0.0891 2143.03018 3.79730
LtofStay 1 0.38792 0.05720 6.78 <.0001 57.30511 59.37993
Age 1 -0.03107 0.02450 -1.27 0.2075 2.07506 2.07506
The REG Procedure
Model: MODEL5
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 67.58193 33.79096 27.78 <.0001
Error 110 133.79789 1.21634
Corrected Total 112 201.37982
Root MSE 1.10288 R-Square 0.3356
Dependent Mean 4.35487 Adj R-Sq 0.3235
Coeff Var 25.32523
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 0.89701 0.53873 1.67 0.0987 2143.03018 3.37223
LtofStay 1 0.31685 0.05798 5.46 <.0001 57.30511 36.32349
Nurses 1 0.00231 0.00079582 2.91 0.0044 10.27682 10.27682
The REG Procedure
Model: MODEL6
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 31.48971 15.74485 10.19 <.0001
Error 110 169.89011 1.54446
Corrected Total 112 201.37982
Root MSE 1.24276 R-Square 0.1564
Dependent Mean 4.35487 Adj R-Sq 0.1410
Coeff Var 28.53729
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 3.14892 1.43036 2.20 0.0298 2143.03018 7.48526
Age 1 0.01022 0.02641 0.39 0.6995 0.00024065 0.23127
Nurses 1 0.00382 0.00084613 4.52 <.0001 31.48947 31.48947
The REG Procedure
Model: MODEL7
Dependent Variable: InfRisk
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 68.46806 22.82269 18.72 <.0001
Error 109 132.91176 1.21937
Corrected Total 112 201.37982
Root MSE 1.10425 R-Square 0.3400
Dependent Mean 4.35487 Adj R-Sq 0.3218
Coeff Var 25.35675
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 1.89700 1.29112 1.47 0.1446 2143.03018 2.63234
LtofStay 1 0.32864 0.05968 5.51 <.0001 57.30511 36.97835
Age 1 -0.02057 0.02412 -0.85 0.3958 2.07506 0.88613
Nurses 1 0.00220 0.00080714 2.73 0.0074 9.08789 9.08789