### Hypothesis Tests with Monotonicity Constraints

Now consider a model with monotonicity constraints. This model has no counterpart in PROC REG. The following statements fit a monotone-spline model and produce Figure 97.72:

```title 'Monotone Splines';

proc transreg data=htex ss2 short;
model identity(y) = mspline(x1-x3 / nknots=3);
run;
```

The SHORT a-option is specified to suppress the iteration histories. Two ANOVA tables are displayed—one by using liberal degrees of freedom and one by using conservative degrees of freedom. All sums of squares and the R squares are the same for both tables. What differs are the degrees of freedom and statistics that use degrees of freedom. The liberal test has 8 model degrees of freedom and 11 error degrees of freedom, whereas the conservative test has 15 model degrees of freedom and only 4 error degrees of freedom. The true p-value is between 0.8462 and 0.9997, so clearly you would fail to reject the null hypothesis. Unfortunately, results are not always this clear. (See Figure 97.72.)

Figure 97.72: Monotone Spline Transformations

 Monotone Splines

The TRANSREG Procedure

Dependent Variable Identity(y)

 Number of Observations Read 20 20

Identity(y)
Algorithm converged.

The TRANSREG Procedure Hypothesis Tests for Identity(y)

Univariate ANOVA Table Based on Liberal Degrees of Freedom
Source DF Sum of Squares Mean Square F Value Liberal p
Model 8 58.0534 7.25667 0.48 >= 0.8462
Error 11 166.0904 15.09913
Corrected Total 19 224.1438

 Root MSE R-Square 3.88576 0.2590 0.8549 -0.2799 454.526

Univariate ANOVA Table Based on Conservative Degrees of Freedom
Source DF Sum of Squares Mean Square F Value Conservative p
Model 15 58.0534 3.87022 0.09 <= 0.9997
Error 4 166.0904 41.52261
Corrected Total 19 224.1438

 Root MSE R-Square 6.4438 0.2590 0.8549 -2.5197 753.746

Univariate Regression Table Based on Liberal Degrees of Freedom
Variable DF Coefficient Type II
Sum of
Squares
Mean Square F Value Liberal p
Intercept 1 4.8687676 54.7372 54.7372 3.63 >= 0.0834
Mspline(x1) 2 -0.6886834 12.1943 6.0972 0.40 >= 0.6773
Mspline(x2) 3 -1.8237319 46.3155 15.4385 1.02 >= 0.4199
Mspline(x3) 3 0.8646155 24.6840 8.2280 0.54 >= 0.6616

Univariate Regression Table Based on Conservative Degrees of Freedom
Variable DF Coefficient Type II
Sum of
Squares
Mean Square F Value Conservative p
Intercept 1 4.8687676 54.7372 54.7372 1.32 <= 0.3149
Mspline(x1) 5 -0.6886834 12.1943 2.4389 0.06 <= 0.9959
Mspline(x2) 5 -1.8237319 46.3155 9.2631 0.22 <= 0.9344
Mspline(x3) 5 0.8646155 24.6840 4.9368 0.12 <= 0.9809