ANOVA

Slide 1

Analysis of variance frequently is used to analyze experimental data. It’s reasonable to assume that you’ve seen analysis of variance in your statistics courses. Therefore, my goal is merely to remind you about the topic, to allow you to review the basic equations for analysis of variance, and to provide you some examples that reinforce how analysis of variance works.

Slide 2

In this schematic here, we see four different groups with four different means on a single variable. Analysis of variance can evaluate whether or not the mean scores for these four groups differ statistically from one another; in particular, if the difference between group 1 and group 3 is sufficient to justify a differential marketing strategy for each group.

Slide 3

The next dozen or so slides present the basic equations and statistical tests associated withanalysis of variance. Analysis of variance is analogous to market segmentation. What are marketers doing when they segmentmarkets? Ideally, they’re trying to identify homogenous groups of people who on average differ as much between groups as possible. Marketers want people within a market segment to be as similar as possible with one another and the averages between segments to be as different as possible. If the difference between segments is trivial, then a mass marketing rather than target marketing approach is most cost effective. Analogously, for analysis of variance researchers who want to reject the null hypothesis would want the differences in within-group scores to berelatively smaller than differences inbetween-group scores. In other words, within-group variability should be meaningfully less than between group variability. If true, thengroups differ meaningfully from one another because they vary from each other far more than the natural variability within a group.

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Slide 12

The F test truly drives home this analogy between analysis of variance and market segmentation. The F test, which controls for degrees of freedom, takes the ratio of variance-between-groups to variance-within-groups. If the between-group variability is meaningfully greater than the within group-variability, then the F ratio is statistically significant.

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Slide 18

Assume this example depicts three different ad copy tests. To make it simple, also assume five study participants saw each ad. In situation A, the five people who saw ad #1eachrated it a 20, the five different people who saw ad #2 rated it a 30, and the five still different people who saw ad #3 and rated it 40. Mean scores for the 20 ratings (group 1) is 20, for the 30 ratings (group 2) is 30, and for the 40 ratings (group 3) is 40. The overall mean is 30. A sum-of-squaresanalysis would determine if the three means—of 20, 30, and 40—differ meaningfully from one another. The within-group sum-of-squares is zero (0) because everyone in each group answered identically. The between-group sum-of-squares, which relates to the group means, is 1000. The total sum-of-squares also is 1000. In situation B, the group that saw ad #1 rated it between 10 and 30, the group that saw ad #2 rated it between a 10 and 50, and the group that saw ad #3 rated it between a 10 and a 70. Notice that the means of those groups are identical to the corresponding means in for situation A. In situations A and B, the mean for group 1 is 20, for group 2 is 30, and for group 3 is 40. The group means are equally different in both cases and the overall mean is the same in both cases, but the sum of squares are vastly different. The total sum of squares in situation B is 4850 as opposed to 1000. The sum of squares between groups is identical in both situations. In situation A, the sum of squares is 0 because everyone within each group responded identically. In situation B, everyone within the group responded differently. What does this mean for whether or not these groups differ meaningfully from one another?

Slide 19

The F ratio tells us is that in situation A the group means differ meaningfully from one another. In fact, the F ratio is infinite because the denominator iszero (0). However, there’s so much within-group variability in situation B that the group means don’t differ meaningfully from one another. Although the within-group means and overall mean are the same for situations A and B, in situation A the means differ meaningfully and in situation Bthey don’t differ meaningfully.

Slide 20

The next two slides show two pricing experiments. In both cases, the data analysis lends itself to an analysis of variance approach. The first slide shows three different price points:$0.39, $0.44, and $0.49. The researcher wants to relate these price points to different quantities sold at different stores. The slide shows that 50 units were sold at the $0.39 price point, 40 units were sold at the $0.44 price point, and 35 units were sold at the$0.49 price point. Do the units sold at each price point differ meaningfully from one another? I’ll leave you to perform the calculations and make the assessment.

Slide 21

Here’s another pricing experiment that would lend itself to an analysis of variance. In this case, the regular price, a reduced price, and a cents-off coupon are compared relative to their influence on sales. Again, I’ll leave you to conduct the analysis and decide if the mean difference in sales, across these three different conditions, is meaningful.

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