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Analysis of Variance |

In statistical inference, or hypothesis testing, the traditional tests are called parametric tests because they depend on the specification of a probability distribution (such as the normal) except for a set of free parameters. Parametric tests are said to depend on distributional assumptions. Nonparametric tests, on the other hand, do not require distributional assumptions. Even if the data are distributed normally, nonparametric methods are often almost as powerful as parametric methods.

The Nonparametric One-Way ANOVA task enables you to perform nonparametric
tests for location and scale when you have a continuous dependent variable
and a single independent classification variable. You can perform a
nonparametric one-way ANOVA using Wilcoxon (Kruskal-Wallis), median, Van der
Waerden, and Savage scores. In addition, you can test for scale differences
across levels of the independent variable using Ansari-Bradley,
Siegal-Tukey, Klotz, and Mood scores.
The Nonparametric One-Way ANOVA task provides asymptotic and exact
*p*-values
for all tests for location and scale.

For example, consider the air quality data set (Air), described in the section "The Air Quality Data Set". Suppose that you want to perform a nonparametric one-way ANOVA and also test for scale differences for ozone levels across shift periods.

- Select
**Statistics****ANOVA****Nonparametric One-Way ANOVA**... - Select o3 as the dependent variable.
- Select shift as the independent variable.

Figure 10.10 defines the nonparametric one-way ANOVA model.

The box-and-whisker plot in Figure 10.9 indicates that ozone levels may be more variable during the daytime shift than during the early shift or at night. You can use the Ansari-Bradley test to test for scale differences across shifts.

To request the Kruskal-Wallis and Ansari-Bradley tests, follow these steps:

- Click on the
**Tests**button in the main dialog. - Select
**Wilcoxon (Kruskal-Wallis test)**in the**Location test scores**. - Select
**Ansari-Bradley**in the**Dispersion test scores**box.

Figure 10.11 displays the Tests dialog with the **Wilcoxon
(Kruskal-Wallis)** and **Ansari-Bradley** tests selected.

Click **OK** in the Nonparametric One-Way ANOVA dialog to perform
the analysis.

Figure 10.12 displays the Wilcoxon scores and Kruskal-Wallis test results.
The table labeled
"Wilcoxon Scores (Rank Sums) for Variable o3" contains the sum
of the rank scores, expected sum, and mean score for each shift. The daytime
shift has a mean score of 117.77, which is higher than the mean scores of
both the early and late shift. The "Kruskal-Wallis Test" table
displays the results of the Kruskal-Wallis test. The test statistic of 40.75
indicates that there is a significant difference in ozone levels across
shift times (the *p*-value is less than 0.0001).

Figure 10.13 displays the results of the Ansari-Bradley test. The
Ansari-Bradley test chi-square has the value of 5.80 with 2 degrees of
freedom, which is not significant at the level. Since
the *p*-value is just slightly higher than 0.05, there is moderate
evidence of scale differences across shift times.

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