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Analysis of Variance |

The Linear Models task enables you to perform an analysis of variance when you have a continuous dependent variable with classification variables, quantitative variables, or both.

The data set Air, described in the section "The Air Quality Data Set", includes quantitative measures; for example, the variable wind represents wind speed, in knots. Suppose that you want to model ozone levels using the variables day (day of week), shift (factory workshift period), and wind (wind speed). Suppose that you also want your model to include the interaction between the variables day and shift. That is, you want to perform a simple two-way analysis of covariance with unequal slopes.

The following example fits this linear model and additionally requests a retrospective power analysis and a plot of the observed values versus the predicted values.

- Select
**Statistics****ANOVA****Linear Models**... - Select o3 as the dependent variable.
- Select shift and day as the class variables.
- Select wind as the quantitative variable.

Figure 10.19 displays the Linear Models dialog.

By default, the linear model analysis includes only the main effects specified in the main dialog: no interaction term is included.

- Click on the
**Model**button in the main dialog. - Highlight the variables shift and day.
- Click on the
**Cross**button. - Click
**OK**.

Note that you can build specific models with the **Add**, **
Cross**, and **Factorial** buttons, or you can select a model by
clicking on the **Standard Models** button and making a selection
from the pop-up list.

Figure 10.20 displays the Model dialog with the terms shift and day and the interaction term shift*day selected as effects in the model.

- Click on the
**Tests**button in the main dialog. - Click on the
**Power Analysis**tab. - Select
**Perform power analysis**.

To request power calculations for tests performed at
several
values, you can enter the
values, separated by a space, in the box labeled **Alphas**.
You can request power analysis for additional sample sizes in the
**Sample sizes** box. You can enter one or more specific values
for the sample sizes, or you can specify a series of sample sizes in
the boxes labeled **From:**, **To:**, and **By:**.

- Click
**OK**.

Figure 10.21 displays the **Power Analysis** tab, which requests
a retrospective power analysis with an alpha, or significance level,
of 0.05.

- Click on the
**Plots**button in the main dialog. - Click on the
**Predicted**tab. - Select
**Plot observed vs predicted**. - Click
**OK**.

Figure 10.22 displays the **Predicted** tab in the Plots
dialog.

Click **OK** in the Linear Models dialog to perform
the analysis.

Figure 10.23 displays the analysis of variance table, with an *
F* statistic of 19.44 and an associated *p*-value less than
0.0001. A *p*-value this small indicates that the model explains a
highly significant proportion of the variation in the dependent
variable.

The R-square value represents the proportion of variability accounted for by the independent variables. In this analysis, about 74% of the variation of the ozone level can be accounted for by the model (that is, by mean differences in day and shift, in conjunction with a linear dependence on wind speed).

The last table displayed in Figure 10.23 partitions the model sum
of squares into the separate contribution for each model
effect and tests for the significance of each effect. The main
effects and the interaction term are significant at the level
(that is, each *p*-value is less than 0.05).

Figure 10.24 displays the retrospective power analysis. The observed power is given for each effect in the linear model.

The column labeled Least Significant Number in Figure 10.24 displays the smallest number of observations required to determine that the effect is significant at the given value.

Figure 10.25 displays the plot of the observed values versus the predicted values from the model. If the model predicts the observed values perfectly, the points on the plot fall on a straight line with a slope of 1. This plot indicates reasonable prediction.

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