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Education Expenditure data, table 7.3, p. 189.
data p189;
length state $ 2 ;
input State Y X1 X2 X3 Region ;
label x1 = 'Income'
x2 = 'Residents under 18'
x3 = 'Residents in Urban Areas'
y = 'Expenditure';
cards;
ME 235 3944 325 508 1
NH 231 4578 323 564 1
VT 270 4011 328 322 1
MA 261 5233 305 846 1
RI 300 4780 303 871 1
CT 317 5889 307 774 1
NY 387 5663 301 856 1
NJ 285 5759 310 889 1
PA 300 4894 300 715 1
OH 221 5012 324 753 2
IN 264 4908 329 649 2
IL 308 5753 320 830 2
MI 379 5439 337 738 2
WI 342 4634 328 659 2
MN 378 4921 330 664 2
IA 232 4869 318 572 2
MO 231 4672 309 701 2
ND 246 4782 333 443 2
SD 230 4296 330 446 2
NB 268 4827 318 615 2
KS 337 5057 304 661 2
DE 344 5540 328 722 3
MD 330 5331 323 766 3
VA 261 4715 317 631 3
WV 214 3828 310 390 3
NC 245 4120 321 450 3
SC 233 3817 342 476 3
GA 250 4243 339 603 3
FL 243 4647 287 805 3
KY 216 3967 325 523 3
TN 212 3946 315 588 3
AL 208 3724 332 584 3
MS 215 3448 358 445 3
AR 221 3680 320 500 3
LA 244 3825 355 661 3
OK 234 4189 306 680 3
TX 269 4336 335 797 3
MT 302 4418 335 534 4
ID 268 4323 344 541 4
WY 323 4813 331 605 4
CO 304 5046 324 785 4
NM 317 3764 366 698 4
AZ 332 4504 340 796 4
UT 315 4005 378 804 4
NV 291 5560 330 809 4
WA 312 4989 313 726 4
OR 316 4697 305 671 4
CA 332 5438 307 909 4
AK 546 5613 386 484 4
HI 311 5309 333 831 4
;
run;
Table 7.4, fig. 7.3-7.7, p. 191-192.
goptions reset = all; symbol v=dot h=.8 c=blue; proc reg data = p189; var region; model y = x1 x2 x3; plot student.*p. student.*region student.*x1 student.*x2 student.*x3; run; quit;
The REG Procedure
Model: MODEL1
Dependent Variable: Y Expenditure
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 109020 36340 22.19 <.0001
Error 46 75348 1637.99091
Corrected Total 49 184368
Root MSE 40.47210 R-Square 0.5913
Dependent Mean 284.60000 Adj R-Sq 0.5647
Coeff Var 14.22070
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 -556.56804 123.19525 -4.52 <.0001
X1 Income 1 0.07239 0.01160 6.24 <.0001
X2 Residents under 18 1 1.55205 0.31467 4.93 <.0001
X3 Residents in Urban Areas 1 -0.00427 0.05139 -0.08 0.9342
Table 7.5 and fig. 7.8-7.9, p. 193-194. The where statement omits Alaska from the analysis.
symbol v=dot h=.8 c=blue; proc reg data = p189; where State ~= 'AK'; var region; model y = x1 x2 x3; output out=outr r=resid ; plot student.*p. student.*region; run; quit;
The REG Procedure
Model: MODEL1
Dependent Variable: Y Expenditure
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 56944 18981 14.80 <.0001
Error 45 57700 1282.21687
Corrected Total 48 114644
Root MSE 35.80806 R-Square 0.4967
Dependent Mean 279.26531 Adj R-Sq 0.4631
Coeff Var 12.82224
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 -277.57731 132.42286 -2.10 0.0417
X1 Income 1 0.04829 0.01215 3.98 0.0003
X2 Residents under 18 1 0.88693 0.33114 2.68 0.0103
X3 Residents in Urban Areas 1 0.06679 0.04934 1.35 0.1826
By looking at the plot of standardized residuals against region we can observe that the variance for each region is different (the points for each category of region have a different range). In order to fix this problem we create by the appropriate weights and run the regression model again the resulting residuals should have the same variance.Generating the weights to be used in the WLS, p. 190.
proc sql;
create table outresid as
select *, mean(resid*resid) as ratio
from outr
group by region;
quit;
proc sql;
create table outresid1 as
select *, 1/sqrt(ratio/mean(ratio)) as c
from outresid;
quit;
proc sql;
create table weighted as
select *, y*c as ty,
x1*c as tx1, c*x2 as tx2, c*x3 as tx3
from outresid1;
quit;
Table 7.6, p. 195.
proc sql; select distinct region, count(region) as n, ratio as sigma2, 1/c as cj from weighted group by region; quit;
Region n sigma2 cj
----------------------------------------
1 9 1451.113 1.110099
2 12 2436.978 1.438589
3 16 249.4333 0.460244
4 12 950.4233 0.8984
The WLS part of table 7.7, p. 195 and fig. 7.8-7.9, p. 196.
symbol v=dot h=.8 c=blue; proc reg data = weighted; var region; model ty = c tx1 tx2 tx3/ noint; plot student.*p. student.*region; run; quit;
The REG Procedure
Model: MODEL1
Dependent Variable: ty
NOTE: No intercept in model. R-Square is redefined.
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 4 7186453 1796613 1570.69 <.0001
Error 45 51473 1143.83478
Uncorrected Total 49 7237926
Root MSE 33.82063 R-Square 0.9929
Dependent Mean 355.52063 Adj R-Sq 0.9923
Coeff Var 9.51299
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
c 1 -315.51704 78.17764 -4.04 0.0002
tx1 1 0.06227 0.00787 7.92 <.0001
tx2 1 0.87426 0.20026 4.37 <.0001
tx3 1 0.02936 0.03425 0.86 0.3958
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