### Hypothesis Tests with One-Way ANOVA

One-way ANOVA models are fit with either an explicit or implicit intercept. In implicit intercept models, the ANOVA table of PROC TRANSREG is the correct table for a model with an intercept, and the regression table is the correct table for a model that does not have a separate explicit intercept. The PROC TRANSREG implicit intercept ANOVA table matches the PROC REG table when the NOINT a-option is not specified, and the PROC TRANSREG implicit intercept regression table matches the PROC REG table when the NOINT a-option is specified. The following statements illustrate this relationship and produce Figure 97.74:

```data oneway;
input y x \$;
datalines;
0 a
1 a
2 a
7 b
8 b
9 b
3 c
4 c
5 c
;
```
```title 'Implicit Intercept Model';

proc transreg ss2 data=oneway short;
model identity(y) = class(x / zero=none);
output out=oneway2;
run;

proc reg data=oneway2;
model y = xa xb xc;         /* Implicit Intercept ANOVA      */
model y = xa xb xc / noint; /* Implicit Intercept Regression */
run; quit;
```

Figure 97.74: Implicit Intercept Model

 Implicit Intercept Model

The TRANSREG Procedure

Dependent Variable Identity(y)

Class Level Information
Class Levels Values
x 3 a b c

Number of Observations Read 9 9

The TRANSREG Procedure Hypothesis Tests for Identity(y)

Univariate ANOVA Table Based on the Usual Degrees of Freedom
Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 74.00000 37.00000 37.00 0.0004
Error 6 6.00000 1.00000
Corrected Total 8 80.00000

 Root MSE R-Square 1 0.9250 4.33333 0.9000 23.0769

Univariate Regression Table Based on the Usual Degrees of Freedom
Variable DF Coefficient Type II
Sum of
Squares
Mean Square F Value Pr > F Label
Class.xa 1 1.00000000 3.000 3.000 3.00 0.1340 x a
Class.xb 1 8.00000000 192.000 192.000 192.00 <.0001 x b
Class.xc 1 4.00000000 48.000 48.000 48.00 0.0004 x c

 Implicit Intercept Model

The REG Procedure
Model: MODEL1
Dependent Variable: y

 Number of Observations Read 9 9

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 2 74.00000 37.00000 37.00 0.0004
Error 6 6.00000 1.00000
Corrected Total 8 80.00000

 Root MSE R-Square 1 0.9250 4.33333 0.9000 23.0769

 Note: Model is not full rank. Least-squares solutions for the parameters are not unique. Some statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.

 Note: The following parameters have been set to 0, since the variables are a linear combination of other variables as shown.

xc = Intercept - xa - xb

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept Intercept B 4.00000 0.57735 6.93 0.0004
xa x a B -3.00000 0.81650 -3.67 0.0104
xb x b B 4.00000 0.81650 4.90 0.0027
xc x c 0 0 . . .

 Implicit Intercept Model

The REG Procedure
Model: MODEL2
Dependent Variable: y

 Number of Observations Read 9 9

 Note: No intercept in model. R-Square is redefined.

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 3 243.00000 81.00000 81.00 <.0001
Error 6 6.00000 1.00000
Uncorrected Total 9 249.00000

 Root MSE R-Square 1 0.9759 4.33333 0.9639 23.0769

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
xa x a 1 1.00000 0.57735 1.73 0.1340
xb x b 1 8.00000 0.57735 13.86 <.0001
xc x c 1 4.00000 0.57735 6.93 0.0004