In this example, we demonstrate the generalized order-restricted information criterion (GORIC), which is a modification of the Akaike information criterion (AIC). The GORIC can be used to evaluate competing hypotheses based on their fit (i.e., likelihood) and complexity (i.e., (in)equality constraints).

Consider the order-constrained hypothesis $H_1: \mu_{\text{No}} \leq {\mu_{\text{Physical}} = \mu_{\text{Behavioral}}} \leq \mu_{\text{Both}}$. To test this informative hypothesis, we evaluated it against the competing unconstrained hypothesis. However, instead of using the unconstrained-hypothesis as competing hypothesis, it is also possible to specify other order-constrained hypotheses. The GORIC can be used to evaluate a set of informative hypotheses. Suppose, we want to evaluate the following set of informative hypotheses:

\begin{align*} H_0&: \mu_{\text{No}} = \mu_{\text{Physical}} = \mu_{\text{Behavioral}} = \mu_{\text{Both}} \\ H_1&: \mu_{\text{No}} \leq {\mu_{\text{Physical}} = \mu_{\text{Behavioral}}} \leq \mu_{\text{Both}} \\ H_2&: \mu_{\text{No}} \leq \mu_{\text{Physical}} \leq \mu_{\text{Behavioral}} \leq \mu_{\text{Both}} \\ H_{\text{u}}&: \mu_{\text{No}} ~,~ \mu_{\text{Physical}} ~,~ \mu_{\text{Behavioral}} ~,~ \mu_{\text{Both}}. \end{align*}

Note that it is recommended to also include the unconstrained hypothesis $H_\text{u}$ in the set, to avoid drawing wrong conclusions. The model with the lowest GORIC value is the preferred one. To improve the interpretation, we will also compute the GORIC weights, which are comparable to the Akaike weights and reflect the support for each model in the set.

```
library(restriktor)
```

The `AngerManagement`

dataset is available in restriktor and can be
called directly, see next step.

First, we fit the unconstrained model using the \texttt{lm()} function. The model is identical to the one discussed in the introduction:

```
fit.ANOVA <- lm(Anger ~ -1 + Group, data = AngerManagement)
```

First, we construct the syntax for each hypothesis, except for the unconstrained hypothesis:

```
myConstraints1 <- ' GroupNo = GroupPhysical = GroupBehavioral = GroupBoth '
myConstraints2 <- ' GroupNo < GroupPhysical = GroupBehavioral < GroupBoth '
myConstraints3 <- ' GroupNo < GroupPhysical < GroupBehavioral < GroupBoth '
```

```
restr1.ANOVA <- restriktor(fit.ANOVA, constraints = myConstraints1)
restr2.ANOVA <- restriktor(fit.ANOVA, constraints = myConstraints2)
restr3.ANOVA <- restriktor(fit.ANOVA, constraints = myConstraints3)
restr4.ANOVA <- restriktor(fit.ANOVA)
```

The fourth model is the unrestricted model. It is recommended to include this model to avoid selecting the wrong model.

Finally, we use the `goric()`

function to compute the log-likelihood,
penalty (complexity), GORIC value, and the GORIC weight for each model. The input
are the four fitted restriktor objects:

```
goric(restr1.ANOVA, restr2.ANOVA, restr3.ANOVA, restr4.ANOVA)
```

```
model loglik penalty goric goric.weights
1 restr1.ANOVA -93.40117 2.00000 190.80234 0.00001
2 restr2.ANOVA -84.16211 2.89183 174.10788 0.02598
3 restr3.ANOVA -80.48390 3.08324 167.13427 0.84912
4 restr4.ANOVA -80.48390 5.00000 170.96780 0.12489
```

The model with the lowest GORIC value is the preferred one. Note that the GORIC value for the unconstrained model renders to the AIC. The last column, shows the GORIC weights and reflect the support of each model in the set. If we want to compare model `restr2_ANOVA`

with model `restr3_ANOVA`

we can examine the ratio of the two corresponding GORIC weights: 0.849/0.026 = 32.654. This means that model `restr3_ANOVA`

is about 32.654 times more likely than model `restr2_ANOVA`

. In addition, model `restr2_ANOVA`

is about 6.792 (0.849/0.125) more likely than the unconstrained model `restr4_ANOVA`

. Hence, we can concluded that model `restr3_ANOVA`

is the preferred one.