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We begin with a discussion of the advantages of the two-way ANOVA design and illustrate these with some examples. Then we discuss the model and the conditions that justify two-way ANOVA.

Advantages of two-way ANOVA

In one-way ANOVA, we classify populations according to one categorical variable, or factor. In the two-way ANOVA model, there are two factors, each with several levels. When we are interested in the effects of two factors, a two-way design offers great advantages over two single-factor studies. We use several examples to illustrate these advantages.

Here is the design of the first experiment with the sample sizes:

And here is the second experiment:

In the first experiment, 20 stores were assigned to each level of the factor for a total of 60 stores. In the second experiment 15 stores were assigned to each level of the factor for a total of 60 stores. Each experiment lasts one week, so the total amount of time for the two experiments is two weeks.

Each experiment will be analyzed using one-way ANOVA. The factor in the first experiment is magazine layout with three levels, and the factor in the second experiment is magazine cover with four levels. Let’s now consider combining the two experiments into one.

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Here is a picture of the two-way design with the sample sizes:

Each combination of the factors in a two-way design corresponds to a cell. The ANOVA for the magazine experiment has 12 cells, each corresponding to a particular combination of layout and cover.

With the two-way design for layout and cover, notice that we have 20 stores assigned to each level, the same as we had for the one-way experiment for layout alone. Similarly, there are 15 stores assigned to each level of cover. Thus, the two-way design gives us the same amount of information for estimating the sales for each level of each factor as we had with the two one-way designs. The difference is that we can collect all the information in only one experiment. This experiment lasts one week (instead of two weeks) and involves a single observation from each of the 60 stores. By combining the two factors into one experiment, we have increased our efficiency by reducing the amount of data to be collected by half.

To design a study to answer this question, we first need to determine an appropriate target group. This will depend on the new product under consideration. Suppose we’re interested in introducing a new surround sound receiver and, therefore, decide to focus on college-age consumers of electronics. For our partnered products, we decide on a six-piece speaker system as the complementary product and a digital camera as the noncomplementary product. Because consumers have been shown to respond differently to products based on the product’s brand image, our design should also take that into account. Let’s consider a two-way ANOVA for this study.

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Here is a table that summarizes the design:

This example illustrates a second reason for using two-way designs. Although we’re primarily interested in the effect of the partnered product’s complementarity, we included the partnered product’s brand image because we thought there might be an effect. Consider an alternative, one-way design where we assign 200 consumers to the two levels of complementarity and ignore brand image. With this design, we have the same number of consumers at each of the complementarity levels; thus, in this way, it is similar to our two-way design. However, suppose that there is, in fact, an effect of brand image. In this case, the one-way ANOVA would assign this variation to the RESIDUAL (within groups) part of the model. In the two-way ANOVA, brand image is included as a factor, and therefore this variation is included in the FIT part of the model. Whenever we can move variation from RESIDUAL to FIT, we reduce the of our model and increase the power of our tests.

Suppose we wanted to compare an organized and disorganized shelf display across three different product quantity levels. Instead of three separate one-way ANOVAs (or two-sample tests), one for each quantity level, we make product quantity a factor in our design.

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Here is a table that summarizes the design with the sample sizes:

This example illustrates a third reason for using two-way designs. The difference between the two shelf displays may depend on the product quantity. We call this an interaction when it occurs. In contrast, the average values for the shelf display effect and the product quantity effect are represented as main effects. The two-way model represents FIT as the sum of a main effect for each of the two factors and an interaction. One-way designs that vary a single factor and hold other factors fixed cannot discover interactions. We discuss interactions more fully later in this section.

These examples illustrate several reasons why two-way designs are preferable to one-way designs.

These considerations also apply to study designs with more than two factors. We are content to explore only the two-way case. The choice of the design for data production (sample or experiment) is fundamental to any statistical study. Factors and levels must be carefully selected by an individual or team who understands both the statistical models and the issues that the study will address.

The two-way ANOVA model

When discussing two-way models in general, we use the labels A and B for the two factors. For particular examples and when using statistical software, it is better to use names for these categorical variables that suggest their meaning. Thus, in Example 15.2, we would say that the factors are layout and cover. The numbers of levels of the factors are often used to describe the model. Again referring to Example 15.2, we would call this a ANOVA. Similarly, Example 15.4 illustrates a ANOVA. In general, Factor A will have levels and Factor B will have levels. Therefore, we call the general two-way problem an ANOVA.

In a two-way design, every level of A appears in combination with every level of B, so that groups are compared. The sample size for level of Factor A and level of Factor B is . The total number of observations is⁵

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The FIT part of the model is the means , and the RESIDUAL part is the deviations of the individual observations from their group means. To estimate a population mean , we use the sample mean of the observations from this group:

The below the means that we sum the observations that belong to the th group.

The RESIDUAL part of the model contains the unknown . We calculate the sample variances for each SRS and, provided the rule of thumb for equal standard deviations is met, pool these to estimate :

Just as in one-way ANOVA, the numerator in this fraction is SSE and the denominator is DFE. Also as in the one-way analysis, DFE is the total number of observations minus the number of groups. That is, . The estimator of is , the pooled standard error.

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Main effects and interactions

Because we have independent samples from each of groups, we can first think of the two-way ANOVA as a one-way ANOVA with groups. Each population mean is estimated by the corresponding sample mean , and we can calculate sums of squares and degrees of freedom as in one-way ANOVA. Thus, SSG is the group sum of squares constructed from deviations of the form , where is the average of all the observations and is the mean of the th group. Similarly, DFG is simply .

The first important distinction between one-way and two-way ANOVA is that in two-way ANOVA we break down the FIT part of the model (the population means ) in a way that reflects the presence of two factors. This means the terms SSG and DFG are broken down into terms corresponding to a main effect for A, a main effect for B, and an AB interaction. Each of SSG and DFG is then a sum of terms:

and

The term SSA represents variation among the means for the different levels of Factor A. Because there are such means, degrees of freedom. Similarly, SSB represents variation among the means for the different levels of Factor B, with .

Interactions are a bit more involved. We can see that SSAB, which is , represents the variation in the group means that is not accounted for by the main effects. By subtraction we see that its degrees of freedom are

There are many kinds of interactions. The easiest way to study them is through some examples.

The table also includes averages of the means in the rows and columns (rounded to the nearest dollar). For example, the second entry in the far-right margin is the average of the per capita income for individuals who claim to be of white or Asian descent in the Northeast:

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Similarly, the average per capita income of a white individual in the two regions is

These averages are called marginal means because of their location at the margins of the table. The grand mean (31,295 in this case) can be obtained by averaging either set of marginal means.

It is clear from the marginal means that white individuals have a higher per capita income than Asian individuals and that individuals in the Northeast have a higher per capita income than those in the Midwest. These are main effects for the two factors. We can describe the main effects by the differences between the marginal means. On average, a person of white descent has a per capita income $1129 higher than someone of Asian descent and a person in the Northeast has a per capita income that is $4897 more than an individual in the Midwest.

What about the interaction between region and ethnicity? An interaction is present if the main effects provide an incomplete description of the data. That is, if the ethnicity earnings gap is different in the two regions, then ethnicity and region interact. In this survey, the earnings gap is much larger in the Northeast:

Figure 15.1(a) is a plot of the four group means. Because the difference between the per capita incomes is different in the two regions, the gap between the lines increases from left to right. That is, the white and Asian lines are not parallel.

How would the plot look if there were no interaction? No interaction says that the regional earnings gap is the same in both ethnicities. That is, the effect of ethnicity does not depend on region. Suppose that the gap were $2897 in both regions. Figure 15.1(b) plots the means. The white and Asian lines are now parallel. Interaction between the factors is visible as lack of parallelism in a plot of the group means.

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To examine two-way ANOVA data for a possible interaction, always construct a plot similar to Figure 15.1. Profiles that are roughly parallel imply that there is no clear interaction between the two factors. When no interaction is present, the marginal means provide a reasonable description of the two-way table of means.

In this case, it is clear that the two profiles (the collections of marginal means for a given region) are not parallel. When there is an interaction, the marginal means do not tell the whole story. For example, with these data, the marginal mean difference between regions is $4897. This is larger than the difference for Asian individuals ($3129) and smaller than the difference for white individuals ($6664).

Interactions come in many forms. When we find them, a careful examination of the means is needed to properly interpret the data. Simply stating that interactions are significant tells us little. Plots of the group means are very helpful.

When the partnered product’s brand image is low, perceived quality of the surround sound receiver is about the same. When the brand image is high, the perceived quality increases regardless of complementarity. However, the perceived quality increases more when the partnered product’s complementarity is high (speaker system) than when it is low (camera).

In a statistical analysis, the pattern of means shown in Figure 15.3 produced significant main effects for brand image and complementarity in addition to a brand image–by–complementarity interaction. The main effects record that perceived quality is higher when the partnered product has high brand image and when the partnered product complements the new product. This clearly does not tell the whole story. We need to discuss the complementarity effect in each of the brand image levels to fully understand how perceived quality is affected.

A different kind of interaction is present in the next example. Here, we must be very cautious in our interpretation of the main effects as one of them can lead to a distorted conclusion.

What catches our eye in the plot is that the lines cross. The purchase intention is much higher for the self-benefit advertisement in the private accountability condition, but the purchase intention was much higher for the other-benefit advertisement in the public accountability condition. This interaction between accountability and benefit is the important result. The main effects for accountability and appeal are not practically meaningful because of this interaction. Both factors have effects, but we must know which type of accountability a person is under in order to say which type of appeal results in higher purchase intent.