# The ORTHOREG Procedure

### Longley Data

The labor statistics data set of Longley (1967) is noted for being ill-conditioned. Both the ORTHOREG and GLM procedures are applied for comparison (only portions of the PROC GLM results are shown).

Note: The results from this example vary from machine to machine, depending on floating-point configuration.

The following statements read the data into the SAS data set `Longley`:

```title 'PROC ORTHOREG used with Longley data';
data Longley;
input Employment Prices GNP Jobless Military PopSize Year;
datalines;
60323  83.0 234289 2356 1590 107608 1947
61122  88.5 259426 2325 1456 108632 1948
60171  88.2 258054 3682 1616 109773 1949
61187  89.5 284599 3351 1650 110929 1950
63221  96.2 328975 2099 3099 112075 1951
63639  98.1 346999 1932 3594 113270 1952
64989  99.0 365385 1870 3547 115094 1953
63761 100.0 363112 3578 3350 116219 1954
66019 101.2 397469 2904 3048 117388 1955
67857 104.6 419180 2822 2857 118734 1956
68169 108.4 442769 2936 2798 120445 1957
66513 110.8 444546 4681 2637 121950 1958
68655 112.6 482704 3813 2552 123366 1959
69564 114.2 502601 3931 2514 125368 1960
69331 115.7 518173 4806 2572 127852 1961
70551 116.9 554894 4007 2827 130081 1962
;
```

The data set contains one dependent variable, `Employment` (total derived employment), and six independent variables: `Prices` (GNP implicit price deflator normalized to the value 100 in 1954), `GNP` (gross national product), `Jobless` (unemployment), `Military` (size of armed forces), `PopSize` (noninstitutional population aged 14 and over), and `Year` (year).

The following statements use the ORTHOREG procedure to model the Longley data by using a quadratic model in each independent variable, without interaction:

```proc orthoreg data=Longley;
model Employment = Prices   Prices*Prices
GNP      GNP*GNP
Jobless  Jobless*Jobless
Military Military*Military
PopSize  PopSize*PopSize
Year     Year*Year;
run;
```

Figure 72.1 shows the resulting analysis.

Figure 72.1: PROC ORTHOREG Results

 PROC ORTHOREG used with Longley data

The ORTHOREG Procedure

Dependent Variable: Employment

Source DF Sum of Squares Mean Square F Value Pr > F
Model 12 184864508.5 15405375.709 320.24 0.0003
Error 3 144317.49568 48105.831895
Corrected Total 15 185008826

 Root MSE 219.33 0.99922

Parameter DF Parameter Estimate Standard
Error
t Value Pr > |t|
Intercept 1 186931078.640216 154201839.66 1.21 0.3122
Prices 1 1324.50679362506 916.17455832 1.45 0.2440
Prices**2 1 -6.61923922845539 4.7891445654 -1.38 0.2609
GNP 1 -0.12768642156232 0.0738897784 -1.73 0.1824
GNP**2 1 3.1369569286212E-8 8.7167753E-8 0.36 0.7428
Jobless 1 -4.35507653558708 1.3851792402 -3.14 0.0515
Jobless**2 1 0.00022132944101 0.0001763541 1.26 0.2983
Military 1 4.91162014560828 1.826715856 2.69 0.0745
Military**2 1 -0.00113707146734 0.0003539971 -3.21 0.0489
PopSize 1 -0.0303997234299 5.9272538242 -0.01 0.9962
PopSize**2 1 -1.212511414607E-6 0.0000237262 -0.05 0.9625
Year 1 -194907.139041839 157739.28757 -1.24 0.3045
Year**2 1 50.8067603538501 40.279878943 1.26 0.2963

The estimates in Figure 72.1 compare very well with the best estimates available; for additional information, see Longley (1967) and Beaton, Rubin, and Barone (1976).

The following statements request the same analysis from the GLM procedure:

```proc glm data=Longley;
model Employment = Prices   Prices*Prices
GNP      GNP*GNP
Jobless  Jobless*Jobless
Military Military*Military
PopSize  PopSize*PopSize
Year     Year*Year;
ods select OverallANOVA
FitStatistics
ParameterEstimates
Notes;
run;
```

Figure 72.2 contains the overall ANOVA table and the parameter estimates produced by PROC GLM. Notice that the PROC ORTHOREG fit achieves a somewhat smaller root mean square error (RMSE) and also that the GLM procedure detects spurious singularities.

Figure 72.2: Partial PROC GLM Results

 PROC ORTHOREG used with Longley data

The GLM Procedure

Dependent Variable: Employment

Source DF Sum of Squares Mean Square F Value Pr > F
Model 11 184791061.6 16799187.4 308.58 <.0001
Error 4 217764.4 54441.1
Corrected Total 15 185008826.0

R-Square Coeff Var Root MSE Employment Mean
0.998823 0.357221 233.3262 65317.00

Parameter Estimate   Standard
Error
t Value Pr > |t|
Intercept -3598851.899 B 1327335.652 -2.71 0.0535
Prices 523.802   688.979 0.76 0.4894
Prices*Prices -2.326   3.507 -0.66 0.5434
GNP -0.138   0.078 -1.76 0.1526
GNP*GNP 0.000   0.000 0.24 0.8218
Jobless -4.599   1.459 -3.15 0.0344
Jobless*Jobless 0.000   0.000 1.14 0.3183
Military 4.994   1.942 2.57 0.0619
Military*Military -0.001   0.000 -3.15 0.0346
PopSize -4.246   5.156 -0.82 0.4565
PopSize*PopSize 0.000 B 0.000 0.81 0.4655
Year 0.000 B . . .
Year*Year 1.038   0.419 2.48 0.0683

 Note: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable.