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Have you ever participated in a taste test? If you have, then you were probably a participant in a within-groups experiment. About a decade ago, when pricier microbrew beers were becoming popular in North America, the journalist James Fallows, who loves beer, found himself spending increasingly more on a bottle of beer. He began to wonder whether he was getting his money’s worth for these more expensive beers. So, he recruited 12 colleagues, all self-professed beer snobs, to participate in a taste test to see whether they really could tell whether a beer was expensive or cheap (Fallows, 1999).

Fallows wanted to know whether his recruits could distinguish among widely available American beers that were categorized into three groups based on price—“high-end” beers like Sam Adams, “mid-range” beers like Budweiser, and “cheap” beers like Busch. All of these beers are lagers, a type of beer chosen because it can be found at every price point. Here are data—mean scores on a scale of 0–100 for each category of beer—for five of the participants. (Note: For teaching purposes, the means are slightly different and have been rounded to the nearest whole number; the take-home data story, however, remains the same.)

11-5: We use a one-way within-groups ANOVA when there is one independent variable with at least three levels, a scale dependent variable, and participants who are in every group.

11-6: The calculations for a one-way within-groups ANOVA are similar to those for a one-way between-groups ANOVA, but we now calculate a subjects sum of squares in addition to the between-groups, within-groups, and total sum of squares. The subjects sum of squares reduces the within-groups sum of squares by removing the variability associated with participants’ differences across groups.

STEP 1: Identify the populations, distribution, and assumptions.

The one-way within-groups ANOVA requires an additional assumption compared to the one-way between-groups ANOVA: We must be careful to avoid order effects. In the beer study, order may have influenced participants’ judgments because all participants tasted the beers in the same order: a mid-range beer, followed by a high-end beer, followed by a cheap beer, followed by another cheap beer, and so on. Perhaps the first sip of beer tastes the best, no matter what kind of beer is being tasted. Ideally, Fallows would have used counterbalancing, so that participants tasted the beers in different orders.

Summary: Population 1: People who drink cheap beer. Population 2: People who drink mid-range beer. Population 3: People who drink high-end beer.

The comparison distribution and hypothesis test: The comparison distribution is an F distribution. The hypothesis test is a one-way within-groups ANOVA.

Assumptions: (1) The participants were not selected randomly, so we must generalize with caution. (2) We do not know if the underlying population distributions are normal, but the sample data do not indicate severe skew. (3) After we calculate the test statistic, we will test the homoscedasticity assumption by checking to see whether the largest variance is more than twice the smallest. (4) The experimenter did not counterbalance, so there may be order effects.

STEP 2: State the null and research hypotheses.

This step is identical to that for a one-way between-groups ANOVA.

Summary: Null hypothesis: People who drink cheap, mid-range, and high-end beer rate their beverages the same, on average—H₀: μ₁ = μ₂ = μ₃. Research hypothesis: People who drink cheap, mid-range, and high-end beer do not rate their beverages the same, on average—H₁ is that at least one μ is different from another μ.

STEP 3: Determine the characteristics of the comparison distribution.

We state that the comparison distribution is an F distribution and determine the degrees of freedom. Instead of three, we now calculate four kinds of degrees of freedom—between-groups, subjects, within-groups, and total. The subjects degrees of freedom corresponds to a sum of squares for differences across participants: the subjects sum of squares, or SS_subjects. In a one-way within-groups ANOVA, we calculate between-groups degrees of freedom and subjects degrees of freedom first because we multiply these two together to calculate the within-groups degrees of freedom.

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We calculate the between-groups degrees of freedom exactly as before:

df_between = N_groups − 1 = 3 − 1 = 2

We next calculate the degrees of freedom that pairs with SS_subjects. Called df_subjects, it is calculated by subtracting 1 from the actual number of subjects, not from the number of data points. We use a lowercase n to indicate that this is the number of participants in a single sample (even though they’re all in every sample). The formula is:

df_subjects = n − 1 = 5 − 1 = 4

Once we know the between-groups degrees of freedom and the subjects degrees of freedom, we calculate the within-groups degrees of freedom by multiplying the first two:

df_within = (df_between)(df_subjects) = (2)(4) = 8

Note that the within-groups degrees of freedom is smaller than we would have calculated for a one-way between-groups ANOVA. For a one-way between-groups ANOVA, we would have subtracted 1 from each sample (5 − 1 = 4) and summed them to get 12. The within-groups degrees of freedom is smaller because we exclude variability related to differences among the participants from the within-groups sum of squares, and the degrees of freedom must reflect that.

Finally, we calculate total degrees of freedom using either method we learned earlier. We can sum the other degrees of freedom:

df_total = df_between + df_subjects + df_within = 2 + 4 + 8 = 14

Alternatively, we can use the second formula we learned before, treating the total number of participants as every data point, rather than every person. We know, of course, that there are just five participants and that they participate in all three levels of the independent variable, but for this step, we count the 15 total data points:

df_total = N_total − 1 = 15 − 1 = 14

We have calculated the 4 degrees of freedom that we will include in the source table. However, we only report the between-groups and within-groups degrees of freedom at this step.

Summary: We use the F distribution with 2 and 8 degrees of freedom.

STEP 4: Determine the critical values, or cutoffs.

The fourth step is identical to that for a one-way between-groups ANOVA. We use the between-groups degrees of freedom and within-groups degrees of freedom to look up a critical value on the F table in Appendix B.

Summary: The critical value for the F statistic for a p level of 0.05 and 2 and 8 degrees of freedom is 4.46.

STEP 5: Calculate the test statistic.

As before, we calculate the test statistic in the fifth step. To start, we calculate four sums of squares—one each for between-groups, subjects, within-groups, and total. For each sum of squares, we calculate deviations between two different types of means or scores, square the deviations, and then sum the squared differences. We calculate a squared deviation for every score; so for each sum of squares in this example, we sum 15 squared deviations.

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As we did with the one-way between-groups ANOVA, let’s start with the total sum of squares, SS_total. We calculate this exactly as we calculated it previously:

SS_total = Σ(X − GM)² = 2117.732

Next, we calculate the between-groups sum of squares. It, too, is the same as for a one-way between-groups ANOVA:

SS_between = Σ(M − GM)² = 1092.130

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So far, the calculations of the sums of squares for a one-way within-groups ANOVA have been the same as they were for a one-way between-groups ANOVA. We left the subjects sum of squares and within-groups sum of squares for last. Here is where we see some changes. We want to remove the variability caused by participant differences from the estimate of variability across conditions. So we calculate the subjects sum of squares separately from the within-groups sum of squares. To do that, we subtract the grand mean from each participant’s mean for all of his or her scores. We first have to calculate a mean for each participant across the three conditions. For example, the first participant had ratings of 40 for cheap beers, 30 for mid-range beers, and 53 for high-end beers. This participant’s mean is 41.

So the formula for the subjects sum of squares is:

SS_subjects = Σ(M_participant − GM)² = 729.738

We only have one sum of squares left to go. To calculate the within-groups sum of squares from which we’ve removed the subjects sum of squares, we take the total sum of squares and subtract the two others that we’ve calculated so far—the between-groups sum of squares and the subjects sum of squares. The formula is:

SS_within = SS_total − SS_between − SS_subjects

= 2117.732 − 1092.130 − 729.738 = 295.864

We now have enough information to fill in the first three columns of the source table—the source, SS, and df columns. We calculate the rest of the source table as we did for a one-way between-groups ANOVA. For each of the three sources—between-groups, subjects, and within-groups—we divide the sum of squares by the degrees of freedom to get its variance, MS.

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We then calculate two F statistics—one for between-groups and one for subjects. For the between-groups F statistic, we divide its MS by the within-groups MS. For the subjects F statistic, we divide its MS by the within-groups MS.

The completed source table is shown here:

Here is a recap of the formulas used to calculate a one-way within-groups ANOVA:

We calculated two F statistics, but we’re really only interested in the between-groups F statistic, 14.766, that tells us whether there is a statistically significant difference between groups.

Summary: The F statistic associated with the between-groups difference is 14.77.

STEP 6: Make a decision.

This step is identical to that for the one-way between-groups ANOVA.

Summary: The F statistic, 14.77, is beyond the critical value, 4.46. We reject the null hypothesis. It appears that mean ratings of beers differ based on the type of beer in terms of price category, although we cannot yet know exactly which means differ. We report the statistics in a journal article as F (2,8) = 14.77, p < 0.05. (Note: If we used software, we would report the exact p value.)

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Solutions to these Check Your Learning questions can be found in Appendix D.