In this example we reconsider the `ZelazoKolb1972`

dataset discussed
in the example about ordered means (Ex.1). As a reminder, the purpose of the
study was to test the claim that the walking exercises are associated with a
reduction in the mean age at which children start to walk. The results confirmed
strong evidence in favor of the order-constrained hypothesis $$ H_1: \mu_1 \leq \mu_2 \leq \mu_3 \leq \mu_4 $$.
The second question we like to answer is whether the differences between the means
are relevant. To answer this question we use the popular effect-size measure
Cohen's $d$ and is given by:

$$ d = \frac{\mu_{max} - \mu_{min}}{\sigma}, $$

where $\mu_{max}$ is the largest of the $k$ means and $\mu_{min}$ is the smallest of $k$ means and $\sigma$ is the pooled standard deviation within the populations.

Hypothesis $H_1$ can be reformulated such that the effect-sizes are included. According to Cohen, values of 0.2, 0.5 and 0.8 indicate a small, medium and large effect, respectively. The within group standard deviation ($\sigma$) equals 1.516:

\begin{equation*} \begin{array}{l} H_1 =\ \end{array} \begin{array}{l} (\mu_2 - \mu_1) / 1.516 \geq 0.2 \\ (\mu_3 - \mu_2) / 1.516 \geq 0.2 \\ (\mu_4 - \mu_3) / 1.516 \geq 0.2. \end{array} \end{equation*}

This hypothesis states that $\mu_2$ is at least 0.2 standard deviations larger than $\mu_1$. The other two constraints have analogue interpretations.

In what follows, we describe all steps to test this order constrained hypothesis.

```
library(restriktor)
```

The `ZelazoKolb1972`

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

An ANOVA model is just a special case of the linear model. Therefore, we can make
use of the built-in linear model `lm()`

function in R. The intercept (-1)
is removed, such that the regression coefficients reflect the group means. To fit
the unrestricted linear model type

```
fit.ANOVAd <- lm(Age ~ -1 + Group, data = ZelazoKolb1972)
```

The constraint syntax can be constructed by using the factor level names preceded by the factor name. To get the correct names, type

```
names(coef(fit.ANOVAd))
```

```
[1] "GroupActive" "GroupControl" "GroupNo" "GroupPassive"
```

Based on these factor level names, we can construct the constraint syntax:

```
myConstraints <- ' (GroupPassive - GroupActive) / 1.516 > 0.2
(GroupControl - GroupPassive) / 1.516 > 0.2
(GroupNo - GroupControl) / 1.516 > 0.2 '
# note that the constraint syntax is enclosed within single quotes.
```

The restricted ANOVA model is estimated using the restriktor() function. The first argument to restriktor() is the fitted linear model and the second argument is the constraint syntax.

```
restr.ANOVAd <- restriktor(fit.ANOVAd, constraints = myConstraints)
```

```
summary(restr.ANOVAd)
```

```
Call:
conLM.lm(object = fit.ANOVAd, constraints = myConstraints)
Restriktor: restricted linear model:
Residuals:
Min 1Q Median 3Q Max
-2.7083 -0.8500 -0.3500 0.6375 3.6250
Coefficients:
Estimate Std. Error t value Pr(>|t|)
GroupActive 10.12500 0.61907 16.355 1.191e-12 ***
GroupControl 11.70833 0.61907 18.913 8.774e-14 ***
GroupNo 12.35000 0.67815 18.211 1.736e-13 ***
GroupPassive 11.37500 0.61907 18.375 1.478e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.5164 on 19 degrees of freedom
Standard errors: standard
Multiple R-squared remains 0.986
Generalized Order-Restricted Information Criterion:
Loglik Penalty Goric
-40.0142 3.0976 86.2236
```

```
## by default the iht() function prints a summary of all available
## hypothesis tests. A more detailed overview of the separate hypothesis
## tests can be requested by specifying the test = "" argument, e.g.,
## iht(restr.ANOVAd, test = "A").
iht(fit.ANOVAd, constraints = myConstraints)
```

```
Restriktor: restricted hypothesis tests ( 19 residual degrees of freedom ):
Multiple R-squared remains 0.986
Constraint matrix:
GroupActive GroupControl GroupNo GroupPassive op rhs active
1: -0.6596 0 0 0.6596 >= 0.2 no
2: 0 0.6596 0 -0.6596 >= 0.2 no
3: 0 -0.6596 0.6596 0 >= 0.2 no
Overview of all available hypothesis tests:
Global test: H0: all parameters are restricted to be equal (==)
vs. HA: at least one inequality restriction is strictly true (>)
Test statistic: 2.3840, p-value: 0.1689
Type A test: H0: all restrictions are equalities (==)
vs. HA: at least one inequality restriction is strictly true (>)
Test statistic: 2.3840, p-value: 0.1689
Type B test: H0: all restrictions hold in the population
vs. HA: at least one restriction is violated
Test statistic: 0.0000, p-value: 1
Type C test: H0: at least one restriction is false or active (==)
vs. HA: all restrictions are strictly true (>)
Test statistic: 0.0344, p-value: 0.4865
Note: Type C test is based on a t-distribution (one-sided),
all other tests are based on a mixture of F-distributions.
```

In Ex.1 we found a significant result for the informative hypothesis. However, if we include a small effect-size in the hypothesis test, the mean differences become irrelevant. Although, the results from hypothesis test Type B show that all order-constraints are in line with the data ($\bar{\text{F}}^{\text{B}}_{(0,1,2,3; 19)}$ = 0, p = 1), the results from hypothesis test Type A show that the null-hypothesis is not rejected in favor of the order-constrained hypothesis ($\bar{\text{F}}^{\text{A}}_{(0,1,2,3; 19)}$ = 2.384, p = .1689).