We illustrate one-way ANOVA first with an example showing calculations done by hand and then using computer analysis. Suppose that only one factor, namely in-store promotion, was manipulated, that is, let us ignore couponing for the purpose of this illustration. The department store is attempting to determine the effect of in-store promotion (X) on sales (Y). For the purpose of illustrating hand calculations, the data of Table 16.2 are transformed in Table 16.3 to show the store (Y v> for each level of promotion.

The null hypothesis is that the category means are equal:

Hο:µ¹ = µ² = µ³

To test the null hypothesis, the various sums of squares are computed as follows:

= (3.933)2 + (2.933)2 + (3.933)2 + (1.933)2 + (2.933)2

+ (1.933)2 + (2.933)2 + (0.933)2 + (0.933)2 + (–0.067)2

+ (1.933)2 + (1.933)2 + (0.933)2 + (2.933)2 + (–0.067)2

+ (-2.067)2 + (-1.067)2 + (-1.067)2 + (–0.067)2 + (-2.067)2

+ (-1.067)2 + (0.933)2 + (–0.067)2 + (-2.067)2 + (-1.067)2

+ (-4.067)2 + (-3.067)2 + (-4.067)2 + (-5.067)2 + (-4.067)2

=185.867

SSerror = 10(8.3 – 6.067)2 + \0(6.2 – 6.067)2 + 10(3.7 – 6.067)2

=10(2.233)2 + \0(0.\33)2 + 10(-2.367)2

=106.067

SSerror =(10 – 8.3)2 + (9 – 8.3)2 + (10 – 8.3)2 + (8 – 8.3)2 + (9- 8.3)2

+ (8 – 8.3)2 + (9 – 8.3)2 + (7 – 8.3)2 + (7 – 8.3)2 + (6 – 8.3)2

+ (8 – 6.2)2 + (8 – 6.2)2 + (7 – 6.2)2 + (9 – 6.2)2 + (6 – 6.2)2

+ (4 – 6.2)2 + (5 – 62)2 + (5 – 6.2)2 + (6 – 6.2)2 + (4 – 6.2)2

+ (5 – 3.7)2 + (7 – 3.7)2 + (6 – 3.7)2 + (4 – 3.7)2 + (5 – 3.7)2

+ (2 – 3.7)2 + (3 – 3.7)2 + (2 – 3.7)2 + (1 – 3.7)2 + (2 – 3.7)2

= (1.7)2 + (0.7)2 + (1.7)2 + (–0.3)2 + (0.7)2

+ (–0.3)2 + (0.7)2 + (-1.3)2 + (-1.3)2 + (-2.3)2

+ (1.8)2 + (1.8)2 + (0.8)1 + (2.8)2 + (–0.2)2

+ (-2.2)2 + (-1.2)2 + (-1.2)2 + (–0.2)2 + (-22)2

+ (1.3)2 + (3.3)2 + (2.3)2 + (0.3)2 + (1.3)2

+ (-1.7)2 + (–0.7)2 + (-1.7)2 + (-2.7)2 + (-1.7)2

= 79.80

It can be verified that

SSγ = SSx + SSerror

as follows: 185.867 = 106.067 + 79.80

The strength of the effects of X on Y are measured as follows:

In other words, 57.1 percent of the variation in sales (Y) is accounted for by in-store promotion (X), indicating a modest effect. The null hypothesis may now be tested.

From Table 5 in the Statistical Appendix, we see that for 2 and 27 degrees of freedom, the critical value of F is 3.35 for a = 0.05, Because the calculated value of F is greater than the critical value, we reject the null hypothesis. We conclude that the population means for the three levels of in-store promotion are indeed different. The relative magnitudes of the means for the three categories indicate that a high level of in-store promotion leads to significantly higher sales.

We now illustrate the analysis-or-variance procedure using a computer program, The results of conducting the same analysis by computer are presented in Table 16.4. The value of SSx denoted by between groups is 106.067 with 2 df; that of SSerror denoted by within groups is 79.80 with 27 df. Therefore, MSx = 106.067 n = 53.033, and MSerror = 79.8027 = 2.956. The value of F = 53.033 n.956 = 17.944 with 2 and 27 degrees of freedom, resulting in a probability of 0.000. Because the associated probability is less than the significance level of 0.05, the null hypothesis of equal population means is rejected. Alternatively. it can be seen from

Table 5 in the Statistical Appendix that the critical value of F for 2 and 27 degrees of freedom is 3.35. Because the calculated value of F (17.944) is larger than the critical value, the null hypothesis is rejected. As can be seen from Table 16.4, the sample means, with values of 8.3, 6.2, and 3.7, are quite different. Stores with a high level of in-store promotion have the highest average sales (8.3) and stores with a low level of in-store promotion have the lowest average sales (3.7). Stores with a medium level of in-store promotion have an intermediate level of average sales (6.2). These findings seem plausible. Instead of 30 stores, if this were a large and representative sample, the implications would be that management seeking to increase sales should emphasize in-store promotion.

The procedure for conducting one-way analysis of variance and the illustrative application help us understand the assumptions involved.

**ACTIVE RESEARCH**

**Assumptions in Analysis of Variance**

The salient assumptions in analysis of variance can be summarized as follows.

1. Ordinarily, the categories of the independent variable are assumed to be fixed. Inferences are made only to the specific categories considered. This is referred to as the fixed-effects model, Other models are also available. In the random-effects model, the categories or treatments are considered to be random samples from a universe of treatments. Inferences are made to other categories not examined in the analysis. A mixed-effects model results if some treatments are considered fixed and others random.

2. The error term is normally distributed with a zero mean and a constant variance. The error is not related to any of the categories of X. Modest departures from these assumptions do not seriously affect the validity of the analysis. Furthermore, the data can be transformed to satisfy the assumption of normality or equal variances.

3. The error terms are uncorrected. If the error terms are correlated (i.e., the observations are not independent), the F ratio can be seriously distorted. In many data analysis situations, these assumptions are reasonably met. Analysis of variance is therefore a common procedure, as illustrated by the following example.

**Real Research**

Viewing Ethical Perceptions from Different Lenses

A survey was conducted to examine differences in perceptions of ethical issues.The data were obtained from 31 managers, 21 faculty, 97 undergraduate business students. and 48 MBA students. As part of the survey, respondents were required to rate five ethical items on a scale of 1 = strongly agree and 5 = strongly disagree with 3 representing a neutral response. The means for each group are shown. One-way analysis of variance was conducted to examine the significance of differences between groups for each survey item, and the F and P values obtained are also shown.

The findings indicating significant differences on three of the five ethics items point to the need for more communication among the four groups so as to better align perceptions of ethical issues-in management education.

**N-Way Analysis of Variance**

In marketing research, one is often concerned with the effect of more than one factor simultaneously For example:

• How do the consumers’ intentions to buy a brand vary with different levels of price and different levels of distribution?

• How do advertising levels (high, medium, and low) interact with price levels (high, medium, and low) to influence a brand’s sales?

• Do educational levels (less than high school, high school graduate, some college, and college graduate) and age (less than 35, 35-55, more than 55) affect consumption of a brand?

• What is the effect of consumers’ familiarity with a department store (high, medium,and low) and store image (positive, neutral, and negative) on preference for the store?

In determining such effects, n-way analysis of variance can be used. A major advantage of this technique is that it enables the researcher to examine interactions between the factors, Interactions occur when the effects of one factor on the dependent variable depend on the level (category) of the other factors. The procedure for conducting n-way analysis of variance is similar to that one-way analysis of variance. The statistics associated with n-way analysis of variance are also defined similarly. Consider the simple case of two factors, Xl and X2, having categories eland c2. The total variation in this case is partitioned as follows:

SStotal = SS due to X1 + SS due to X2 + SS due to interaction of X1 and X2 + SS within

SSγ = SSx1+SSx2+SSx1x2+SSerror

A larger effect of X1 will be reflected in a greater mean difference in the levels of XI and a larger SSx,. The same is true for the effect of X2. The larger the interaction between X1 andX2, the larger SSx, X2 will be. On the other hand, if X1 and X2 are independent, the value of SSx, X1 will be close to zero.?

The strength of the joint effect of two factors, called the overall effect, or multiple Π², is measured as follows

Multiple Π² = (SSx¹ + SSx² + SSx¹x²)

SSγ

The significance of the overall effect may be tested by an F test, as follows:

F=(SSx¹ + SSx² + SSx¹x²)/dfn

SSerror/dfd

If the.overall effect is significant, the next step is to examine the significance of the interaction effect. Under the null hypothesis of no interaction, the appropriate F test is:

The foregoing analysis assumes that the design was orthogonal, or balanced (the number of cases in each cell was the same). If the cell size varies, the analysis becomes more complex.