Question

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3 on page 582. How are the results affected in each of the following cases?

a. The same constant is added to each sample value.

b. Each sample value is multiplied by the same nonzero constant.

c. The format of the table is transposed so that the row and column factors are interchanged.

d. The first sample value in the first cell is changed so that it becomes an outlier.

Short answer

a.The test statistics and the p-value do not change. Thus, the conclusion of the test also does not change.

b.The test statistics and the p-value do not change. Thus, the conclusion of the test also does not change.

c.The test statistics and the p-value do not change. Thus, the conclusion of the test also does not change.

d.The test statistics and the p-value change significantly. Thus, the conclusion of the test also changes.

Step-by-step solution

Given information

The data values are altered. The corresponding changes in the results of the two-way analysis of variance are studied.

Added by a constant

a.

If sample values are added by a constant, it results in a change of origin.

Under the two-way analysis of variance, the values of the test statistic and the p-values do not change if a constant value is added to the sample values. The test statistic and hence the p-value are independent of the change of scale.

Thus, the conclusion of the test does not change.

Multiplied by a constant

b.

If sample values are multiplied by a constant, it results in a change of scale.

Under the two-way analysis of variance, the values of the test statistic and the p-values do not change if a constant value is multiplied by the sample values. The test statistic and hence the p-value are independent of the change of scale.

Thus, the conclusion of the test does not change.

Interchange of factors

c.

Under the two-way analysis of variance, if the two factors are interchanged such that the row factor becomes the column factor, and the column factor becomes the row factor, the values of the test statistic and the p-values do not change.

Thus, the conclusion of the test does not change.

Presence of an outlier

d.

Under the two-way analysis of variance, if the first sample value is altered to become an outlier, the values of the test statistic and the p-values will change significantly as the presence of an outlier greatly affects the analysis of variance.

Thus, the conclusion of the test changes significantly.