Imagine that Dr. Chung, a psychologist who studies learning and motivation, designed a study to learn whether rats could discriminate among different types of food and if this discrimination influenced their behavior. He built a large and complex maze and trained 10 rats to run it. A rat would be placed in the start box and had to find its way to the goal box. To motivate the rats, food was placed in the goal box. Dr. Chung trained the rats with three different types of food: a low-calorie food, a normal-calorie food, and a high-calorie food. Each rat received an equal number of trials with each food and the rats were trained until they all ran the maze equally quickly.

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Dr. Chung then put all the rats on a diet until they lost 10% of their normal body weight. The purpose of this was to increase their motivation to find food. Up until this point, all the rats had received the same treatment. Now, Dr. Chung implemented his experimental manipulation.

He randomly assigned the rats to three groups based on the type of food they would find in the goal box the next time they ran the maze: low-calorie, medium-calorie, or high-calorie. Each rat, individually, was placed in the maze and allowed to approach the goal box. When the rat got there, however, it found a screen that prevented it from entering. The rat could see and smell the food through the screen, but it couldn’t get to the food. The purpose of this was to inform the rat of the type of food that awaited it.

Dr. Chung picked up the rat, removed the screen, and placed the rat in the start box. He then timed, in seconds, how long it took the rat to get to the goal box. This is the dependent variable in his study and the fewer seconds it took the rat to get to the goal box, the faster the rat traveled the maze. Dr. Chung figured that if the different caloric contents of the foods had been recognized by the rats and if hungry rats were more motivated by higher-calorie foods, then the time taken to run the maze should differ among the three groups. The data and the means for the three groups are shown in Table 10.3.

The average was 31.00 seconds to get to the low-calorie food, 28.00 seconds for the medium-calorie food, and 25.00 seconds for the high-calorie food (Figure 10.6). It appears as if mean speed increases as calorie content goes up, but the effect is not dramatic. It seems possible that the differences among the three groups may be explained by sampling error. It is also possible that the calorie content of the food in the goal box has had an effect. A statistical test is needed to decide between these two options.

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Step 1 Pick a Test

The first step of hypothesis testing is picking the right test. This study compares the means of three independent samples. There is one independent variable (calorie content), which has three levels (low, medium, and high). The dependent variable, seconds, is measured at the ratio level, so means can be calculated. This study calls for a between-subjects, one-way ANOVA.

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Step 2 Check the Assumptions

The assumptions for the between-subjects, one-way ANOVA, listed in Table 10.4, are the same as they are for an independent-samples t test: (1) each sample should be a random sample from its population, (2) the cases should be independent of each other, (3) the dependent variable should be normally distributed in each population, and (4) each population should have the same degree of variability. The robustness of the assumptions is the same as for t. The only nonrobust assumption is the second one, the assumption of independence of observations within each group.

Here is an evaluation of the assumptions for the maze data:

With no nonrobust assumptions violated, Dr. Chung can proceed with the planned between-subjects, one-way ANOVA.

Step 3 List the Hypotheses

The hypotheses are statements about the populations. In Dr. Chung’s study, there are three samples, with each one thought of as having come from a separate population. His hypotheses will be about the population of rats that finds low-calorie food in the goal box, the population of rats that finds medium-calorie food, and the population of rats that finds high-calorie food.

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Look at Figure 10.5, the example of a sampling distribution for an F ratio, and notice two things—there are no negative values on the x-axis and the sampling distribution is positively skewed. It looks like there is only one “tail” to the sampling distribution. This doesn’t mean that all ANOVAs are one-tailed; it means just the opposite. Both “tails” are wrapped into this one side, so ANOVA always has nondirectional hypotheses. One doesn’t have to decide between a one-tailed or two-tailed test. ANOVA is always two-tailed.

The nondirectional null hypothesis will state that no difference exists in the means of the populations. The generic form of this, where there are k samples, is

H₀: μ₁ = μ₂ = . . . = μ_k

For the maze study, with three populations, Dr. Chung’s null hypothesis is

H₀: μ₁ = μ₂ = μ₃

The alternative hypothesis, also nondirectional, says not all population means are equal. It is not written as μ₁ ≠ μ₂ ≠ μ₃ because that doesn’t cover options like Populations 1 and 2 having different means but Populations 2 and 3 don’t. The alternative hypothesis states that at least two of the population means are different, and maybe all three are. There is no easy way to write this mathematically, so Dr. Chung will write it as

H₁: At least one population mean is different from the others.

Step 4 Set the Decision Rule

This step finds the critical value of F, abbreviated F_cv, the value that separates the rare zone from the common zone of the sampling distribution of the F ratio. Figure 10.7 is an example of a sampling distribution of F with the rare and common zones marked.

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The decision rule will sound familiar as it is similar to what we used for the z test and t tests: if the calculated value of the test statistic, F, falls on the line or in the rare zone, the null hypothesis is rejected.

To obtain the value of F_cv, use Appendix Table 4. It provides two F_cv tables: one for critical values of F with a 5% chance of making a Type I error (i.e., α = .05) and one for a 1% chance of making a Type I error (α = .01). A portion of the table with the most commonly used alpha level, .05, is shown in Table 10.5.

To determine the critical value of F, find the value at the intersection of the column for the degrees of freedom in the numerator and the row for the degrees of freedom in the denominator. Equation 10.1 shows how to calculate the different degrees of freedom needed for a between-subjects, one-way ANOVA. Recall the formula for F ratio from the previous section:

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For the maze data, there are three groups (low-, medium-, and high-calorie), so k = 3. There are a total of 10 participants in the study, so N = 10. Given these values, degrees of freedom are calculated as

df_Between = k – 1

= 3 – 1

= 2

df_Within = N – k

= 10 – 3

= 7

df_Total = N – 1

= 10 – 1

= 9

To find the critical value of F, Dr. Chung needs numerator degrees of freedom (df_Between = 2) and denominator degrees of freedom (df_Within = 7). The intersection of the column for 2 degrees of freedom and the row for 7 degrees of freedom leads to a critical value of F of 4.737. That means the decision rule for the maze data can now be written as:

The sampling distribution of F with degrees of freedom of 2 (numerator) and 7 (denominator) is shown in Figure 10.8. The critical value of F, F_cv = 4.737, is used to separate the rare zone from the common zone. If the observed value of F, the value of the test statistic calculated in Step 5, falls on the line or in the rare zone, the null hypothesis is rejected. If it falls in the common zone, Dr. Chung will fail to reject the null hypothesis.

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Step 5 Calculate the Test Statistic

To complete the ANOVA, Dr. Chung’s next step is to analyze the variability in the maze data. This means taking account of variability both within and between groups:

To understand why analysis of variance is called analysis of variance, it is useful to review the variance formula from Chapter 3. Equation 3.6 says:

The numerator in the variance formula is calculated by following three steps: (1) calculating deviation scores by subtracting the mean from each score, (2) squaring all the deviation scores, and (3) adding up all the squared deviation scores. This numerator, a sum of squared deviation scores, is called a sum of squares, abbreviated as SS. Calculating sums of squares for the two sources of variability, between-groups and within-groups, is necessary to find the between- and within-group variances that are analyzed in an analysis of variance.

Actually, there are three sums of squares that are calculated. In addition to sum of squares between-groups (SS_Between) and sum of squares within groups (SS_Within), sum of squares total (SS_Total) is also calculated. Sum of squares total represents all the variability in the data. Between-subjects, one-way ANOVA divides SS_Total into between-groups sum of squares (SS_Between), which measures variability due to treatment, and within-groups sum of squares (SS_Within), which measures variability due to individual differences. In other words, SS_Total = SS_Between + SS_Within.

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Sum of squares total is calculated by treating all the cases as if they belong in one group: the grand mean, the mean of all the scores, is subtracted from each score, these deviation scores are squared, and the squared deviation scores are added up. Voilà, a sum of squares, the numerator for a variance.

Sum of squares between represents the variability between groups. It is calculated by subtracting the grand mean from each group mean, squaring these deviation scores, multiplying each squared deviation score by the number of cases in the group, and then adding them up. The final variance numerator, the one that represents variability within groups, sum of squares within, is calculated by taking each score, subtracting from it its group mean, squaring each deviation score, and adding them all up.

The calculations just described are what are called definitional formulas because the calculations explain what the value being calculated is. Unfortunately, definitional formulas often are not straightforward mathematically. In contrast, computational formulas are designed to be easier to use. Equation 10.2 is a computational formula for sum of squares total.

This formula says:

The easiest way to do this is to take Table 10.3 and add another column to it, one for squared scores. This can be seen in Table 10.6. From the table, we can see that the sum of squared scores for Step 1 is 7,900.00.

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Step 2 is next.

Finally,

Next, let’s use Equation 10.3 to calculate sum of squares between groups.

Here’s how the formula works:

Again, most of the work has already been done in Table 10.6. Here is Step 1:

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And, Step 2:

Finally,

The final sum of squares to calculate is the sum of squares within. Equation 10.4 covers this.

As with the other sums of squares, Table 10.6 has the components already prepared:

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Earlier in the chapter, it was stated that SS_Total = SS_Between + SS_Within. Let’s see if that is true. SS_Between is 54.00 and SS_Within is 6.00. They sum to 60.00, which is what we calculated SS_Total to be. This is shown visually in Figure 10.9.

Now that the degrees of freedom and sums of squares have been calculated, Dr. Chung can go on to calculate the F ratio. By organizing the sums of squares and the degrees of freedom into an ANOVA summary table, the F ratio will almost calculate itself.

Table 10.7 is a template of a summary table for a between-subjects, one-way ANOVA. It is important to note the order in which it is laid out—three rows and five columns. Always arrange a summary table for a between-subjects, one-way ANOVA in exactly this way:

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Dr. Chung already has enough information to fill in the second and third columns. Mean square, the fourth column, sounds like something new, but it is a variance by another name. To calculate the mean square, one divides the sum of squares in a row by the degrees of freedom for that row.

To see how a mean square is a variance, let’s revisit the variance formula from Chapter 3 one more time:

The numerator in this formula is the same as the sum of squares total. And, df_Total is N – 1. Thus, if mean square total were calculated, it would be exactly the same as the variance for all the cases. Analysis of variance really does analyze variances.

The only mean squares needed to complete a between-subjects, one-way ANOVA are between-groups mean square (MS_Between) and within-groups mean square (MS_Within). As the mean square total is not needed, the convention is not to calculate it and to leave the space blank where mean square total would go.

The formula for calculating the between-groups mean square is given in Equation 10.5.

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For the maze data, where SS_Between = 54.00 and df_Between = 2, the between-groups mean square would be calculated:

Equation 10.6 offers the formula for the within-groups mean square.

For the maze data, SS_Within = 6.00 and df_Within = 7, so MS_Within would be calculated:

Once the two mean squares are calculated, it is time to calculate the F ratio. F is the ratio of variability due to between-group factors and individual differences divided by the variability due to individual differences. F is calculated as shown in Equation 10.7.

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For the maze data, where MS_Between = 27.00 and MS_Within = 0.86, F is calculated as

The complete ANOVA summary table for the maze data is shown in Table 10.8. Table 10.9 is a summary table that provides the organization and formulas for completing a between-subjects, one-way ANOVA.

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