Analysis of variance

Analysis of variance is a method used extensively in evaluating the results of experiments. The general question involves determining the influence of a treatment on a dependent variable. The argument is that the total variance that exists among the data can be apportioned to specific factors by means of formal mathematical techniques. The amount of variance attributable to each factor is indicative of the factor's influence on the dependent variable.

EXAMPLE

As part of a new advertising campaign, a firm plans a full-page advertisement in either a news magazine or a sports magazine. The problem is to decide which magazine has the lowest ratio of full-page advertisements to the number of pages in the magazine (see Table 10.4).

Ratio of full-page advertisements to number of pages

Year

News magazine

Sports magazine

1989

0.79

0.60

0.80

0.46

0.54

0.38

0.72

0.42

1991

0.49

0.39

0.85

0.56

0.50

0.57

0.67

0.46

1993

0.36

0.37

0.65

0.64

0.47

0.52

0.68

0.41

Note: The advertisement ratio is calculated by dividing the number of full-page ads by the number of pages in the issue.

Note: The advertisement ratio is calculated by dividing the number of full-page ads by the number of pages in the issue.

It is also important to establish whether there has been a recent change in this ratio. (See Tables 10.5 and 10.6 for analysis of variance.) Three hypotheses are to be tested:

there is no interaction between year and magazine there is an interaction between year and magazine there is no difference in the average ad ratios for different years there is a difference in the average ad ratios for different years there is no difference in the average ad ratio when using different magazines there is a difference in the average ad ratio when using different magazines.

We reject the hypothesis when the probability is < = 0.05 (decision rule).

TABLE 10.5

Analysis of variance for advertisement ratio

TABLE 10.5

Analysis of variance for advertisement ratio

Source

DF SS MS

Magazine Year*Magazine Error Total

2 0.02386 0.01193 0.73 0.494

1 0.12615 0.12615 7.76 0.012

2 0.03753 0.01876 1.15 0.338 18 0.29265 0.01626

23 0.48018

Produced with the student edition of Minitab.

DF = degrees of freedom; SS = sum of squares; MS = mean square; F = statistic; P = probability.

Produced with the student edition of Minitab.

DF = degrees of freedom; SS = sum of squares; MS = mean square; F = statistic; P = probability.

Analysis of variance

Test

Effect

F ratio

P-value

Decision

1

Year*Magazine

1.15

0.338

Do not reject H0: do not conclude there is significant interaction between year and magazine

2

Year

0.73

0.494

Do not reject H0: the mean ad ratios are probably the same for the three years

3

Magazine

7.76

0.012

Reject H0: the mean ad ratios are not the same for the two magazines

For further details of analysis of variance, see McKenzie et al.2

For further details of analysis of variance, see McKenzie et al.2

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