Analysis of Variance

One-Way Analysis of Variance

The One-Way ANOVA task enables you to perform an analysis of variance when you have a continuous dependent variable and a single classification variable.

For example, consider the data set on air quality (Air), described in the preceding section. Suppose you want to compare the ozone level corresponding to each of the three factory workshift periods.

Request the One-Way ANOVA Task

To request the one-way ANOVA task, follow these steps:

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

Figure 10.4 defines the one-way ANOVA model.

Figure 10.4: One-Way ANOVA Dialog

Request a Means Comparison Test

The analysis of variance performed in the One-Way ANOVA task indicates whether the means of the groups are different; it does not indicate which particular means are different. To generate more detailed information about the differences between the means, follow these steps:

Click on the Means button in the main dialog. The resulting window displays the Comparisons tab.
Click on the arrow adjacent to the Comparison method list.
Select Tukey's HSD.
Highlight the variable shift in the Main Effects: box.
Click on the Add button.

You can click on the arrow next to Significance level: to select a significance level, or you can type in the desired value.

Click OK.

Figure 10.5 specifies Tukey's studentized range (HSD) means comparison test at the 0.05 significance level.

Figure 10.5: One-Way ANOVA: Means Dialog

Request a Box-and-Whisker Plot

To request a box-and-whisker plot in addition to the analysis, follow these steps:

Click on the Plots button in the main dialog.
Select Box-&-whisker plot.
Click OK.

Figure 10.6 displays the Plots dialog with the Box-&-whisker plot selected.

Figure 10.6: One-Way ANOVA: Plots Dialog

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

Review the Results

This analysis tests whether the independent variable (shift) is a significant factor in accounting for the variation in ozone levels. Figure 10.7 displays the analysis of variance table, with an F statistic of 31.93 and an associated p-value that is less than 0.0001. The small p-value indicates that the model explains a highly significant proportion of the variation present in the dependent variable.

Figure 10.7: One-Way ANOVA: Analysis Results

The R-square value, which follows the ANOVA table in Figure 10.7, represents the proportion of variability accounted for by the independent variable. Approximately 28% of the variability in the ozone level can be accounted for by differences between shifts.

Information detailing which particular means are different is available in the multiple comparison test, as displayed in Figure 10.8. The means comparison output provides the alpha value, error degrees of freedom, and error mean square.

Figure 10.8: One-Way ANOVA: Multiple Comparisons Results

In the "Tukey Grouping" table, means with the same letter are not significantly different. The analysis shows that the daytime shift is associated with ozone levels that are significantly different from the other two shifts. The early and late shifts cannot be statistically distinguished on the basis of mean ozone level.

Figure 10.9: One-Way ANOVA: Box-and-Whisker Plot

The box-and-whisker plot displayed in Figure 10.9 provides a graphical view of the multiple comparison results. The variance among the ozone levels may be unequal: subsequent analyses may include a test for homogeneity of variance or a transformation of the response variable, o3.

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