628 CHAPTER 12 Analysis of Variance Two-Way Analysis of Variance Objective With sample data categorized with two factors (a row variable and a column variable), use two-way analysis of variance to conduct the following three tests: 1. Test for an effect from an interaction between the row factor and the column factor. 2. Test for an effect from the row factor. 3. Test for an effect from the column factor. Requirements 1. Normality For each cell, the sample values come from a population with a distribution that is approximately normal. (This procedure is robust against reasonable departures from normal distributions.) 2. Variation The populations have the same variance s 2 (or standard deviation s). (This procedure is robust against reasonable departures from the requirement of equal variances.) 3. Sampling The samples are simple random samples of quantitative data. 4. Independence The samples are independent of each other. (This procedure does not apply to samples lacking independence.) 5. Two-Way The sample values are categorized two ways. (This is the basis for the name of the method: two-way analysis of variance.) 6. Balanced Design All of the cells have the same number of sample values. (This is called a balanced design. This section does not include methods for a design that is not balanced.) Procedure for Two-Way ANOVA (See Figure 12-4) Step1: Interaction Effect: In two-way analysis of variance, begin by testing the null hypothesis that there is no interaction between the two factors. Use technology to find the P-value corresponding to the following test statistic: F = MS1interaction2 MS1error2 Conclusion: • Reject: If the P-value corresponding to the above test statistic is small (such as less than or equal to 0.05), reject the null hypothesis of no interaction. Conclude that there is an interaction effect. • Fail to Reject: If the P-value is large (such as greater than 0.05), fail to reject the null hypothesis of no interaction between the two factors. Conclude that there is no interaction effect. Step 2: Row, Column Effects: If we conclude that there is an interaction effect, then we should stop now; we should not proceed with the two additional tests. (If there is an interaction between factors, we shouldn’t consider the effects of either factor without considering those of the other.) If we conclude that there is no interaction effect, then we should proceed with the following two hypothesis tests. Row Factor For the row factor, test the null hypothesis H0: There are no effects from the row factor (that is, the row values are from populations with the same mean). Find the P-value corresponding to the test statistic F = MS1row2>MS1error2. KEY ELEMENTS Instead of relying only on subjective judgments made by examining the means in Table 12-4 and the interaction graph in Figure 12-3, we will proceed with the more objective procedure of two-way analysis of variance. Here are the requirements and basic procedure for two-way analysis of variance (ANOVA). The procedure is also summarized in Figure 12-4, which follows the Key Elements box.
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