12.3 Conducting a Two-Way Between-Groups ANOVA

image
Do We Remember the Medical Myth or the Fact? Skurnik and colleagues (2005) studied the factors that influence the misremembering of false medical claims as facts. They asked: When a physician tells a patient a false claim, then debunks it with the facts, does the patient remember the false claim or the facts? A source table examines each factor in the study and tells us how much of the variability in the dependent variable is explained by that factor.
Don Smetzer/Getty Images

Understanding interactions makes it easier to explain patterns in the data. For example, it is easy to understand why a ski resort would be tempted to exaggerate snowfall reports—especially right before a big, money-making weekend. Conducting a two-way ANOVA (with two independent variables) is the only way to test whether the effect of one independent variable depends on the levels of another independent variable.

Behavioral scientists explore interactions by using two-way ANOVAs. Fortunately, hypothesis testing for a two-way between-groups ANOVA uses the same logic as for a one-way between-groups ANOVA. For example, the null hypothesis is exactly the same: There are no mean differences between groups. Type I and Type II errors still pose the same threats to decision making. We compare an F statistic to a critical F value to make a decision. The main way that a two-way ANOVA differs from a one-way ANOVA is that three ideas are being tested and each idea is a separate source of variability.

The three ideas being tested in a two-way between-groups ANOVA are the main effect of the first independent variable, the main effect of the second independent variable, and the interaction effect of the two independent variables. A fourth source of variability in a two-way ANOVA is within-groups variance. Let’s learn how to separate and measure these four sources of variance by evaluating a commonly used educational method to improve public health: myth busting.

The Six Steps of Two-Way ANOVA

Two-way ANOVAs use the same six hypothesis-testing steps that you already know. The main difference is that you essentially perform most of the steps three times—once for each main effect and once for the interaction. Let’s look at an example.

341

EXAMPLE 12.4

Does myth busting really improve public health? Here are some myths and facts.

From the Web site of the Canadian Mental Health Association (2015):

Myth: “People who experience mental illnesses can’t work.”

Fact: “Whether you realize it or not, workplaces are filled with people who have experienced mental illnesses. Mental illnesses don’t mean that someone is no longer capable of working.”

From the Web site for the World Health Organization (2015):

Myth: “Disasters bring out the worst in human behavior.”

Fact: “Although isolated cases of antisocial behavior exist, the majority of people respond spontaneously and generously.”

A group of Canadian researchers examined the effectiveness of myth busting (Skurnik, Yoon, Park, & Schwarz, 2005). They wondered whether the effectiveness of debunking false medical claims depends on the age of the person targeted by the message. In one study, they compared two groups of adults: younger adults, ages 18–25, and older adults, ages 71–86. Participants were presented with a series of claims and were told that each claim was either true or false. (In reality, all claims were true, partly because researchers did not want to run the risk that participants would misremember false claims as being true.) In some cases, the claim was presented once, and in other cases, it was repeated three times. In either case, the accurate information was presented after each “false” statement. (Note that for our teaching purposes, we have slightly altered the study’s design.)

The two independent variables in this study were age, with two levels (younger, older), and number of repetitions, with two levels (once, three times). The dependent variable, proportion of responses that were wrong after a 3-day delay, was calculated for each participant. This was a two-way between-groups ANOVA—more specifically, a 2 × 2 between-groups ANOVA. From this name, we know that the table has four cells: (2 × 2) = 4. There were 64 participants—16 in each cell. But here we use an example with 12 participants—3 in each cell. Here are the data that we’ll use; they have similar means to those in the actual study, and the F statistics are similar as well.

Experimental Conditions Proportion of Responses That Were Wrong Mean
Younger, one repetition 0.25, 0.21, 0.14 0.20
Younger, three repetitions 0.07, 0.13, 0.16 0.12
Older, one repetition 0.27, 0.22, 0.17 0.22
Older, three repetitions 0.33, 0.31, 0.26 0.30

Let’s consider the steps of hypothesis testing for a two-way between-groups ANOVA in the context of this example.

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

The first step of hypothesis testing for a two-way between-groups ANOVA is very similar to that for a one-way between-groups ANOVA. First, we state the populations, but we specify that they are broken down into more than one category. In the current example, there are four populations, so there are four cells (as shown in Table 12-12). As we do the calculations, the first independent variable, age, appears in the rows of the table, and the second independent variable, number of repetitions, appears in the columns of the table.

342

image

There are four populations, each with labels representing the levels of the two independent variables to which they belong.

Population 1 (Y; 1): Younger adults who hear one repetition of a false claim

Population 2 (Y; 3): Younger adults who hear three repetitions of a false claim

Population 3 (O; 1): Older adults who hear one repetition of a false claim

Population 4 (O; 3): Older adults who hear three repetitions of a false claim

We next consider the characteristics of the data to determine the distributions to which we compare the sample. We have more than two groups, so we need to consider variances to analyze differences among means. Therefore, we use F distributions. Finally, we list the hypothesis test that we use for those distributions and check the assumptions for that test. For F distributions, we use ANOVA—in this case, two-way between-groups ANOVA.

The assumptions are the same for all types of ANOVA: The sample should be selected randomly; the populations should be distributed normally; and the population variances should be equal. Let’s explore that a bit further.

(1) These data were not randomly selected. Younger adults were recruited from a university, and older adults were recruited from the local community. Because random sampling was not used, we must be cautious when generalizing from these samples. (2) The researchers did not report whether they investigated the shapes of the distributions of their samples to assess the shapes of the underlying populations. (3) The researchers did not provide standard deviations of the samples as an indication of whether the population spreads might be approximately equal—the condition known as homoscedasticity, which we explored in Chapter 12. We typically explore these assumptions using the sample data.

Summary: Population 1 (Y; 1): Younger adults who hear one repetition of a false claim. Population 2 (Y; 3): Younger adults who hear three repetitions of a false claim. Population 3 (O; 1): Older adults who hear one repetition of a false claim. Population 4 (O; 3): Older adults who hear three repetitions of a false claim.

The comparison distributions will be F distributions. The hypothesis test will be a two-way between-groups ANOVA. Assumptions: (1) The data are not from random samples, so we must generalize only with caution. (2) From the published research report, we do not know if the underlying population distributions are normal. (3) We do not know if the population variances are approximately equal (homoscedasticity).

STEP 2: State the null and research hypotheses.

The second step, to state the null and research hypotheses, is similar to that for a one-way between-groups ANOVA, except that we now have three sets of hypotheses, one for each main effect and one for the interaction. Those for the two main effects are the same as those for the one effect of a one-way between-groups ANOVA (see the summary below). If there are only two levels, then we can simply say that the two levels are not equal; if there are only two levels and there is a statistically significant difference, the difference must be between those two levels. Note that because there are two independent variables, we clarify which variable we are referring to by using initial letters or abbreviations for the levels of each (e.g., Y for younger and O for older). If an independent variable has more than two levels, the research hypothesis would be that any two levels of the independent variable are not equal.

343

The hypotheses for the interaction are typically stated in words but not in symbols. The null hypothesis is that the effect of one independent variable is not dependent on the levels of the other independent variable. The research hypothesis is that the effect of one independent variable depends on the levels of the other independent variable. It does not matter which independent variable we list first (e.g., “The effect of age is not dependent. . .” or “The effect of number of repetitions is not dependent. . .”). Write the hypotheses in the way that makes the most sense to you.

Summary: The hypotheses for the main effect of the first independent variable, age, are as follows. Null hypothesis: On average, compared with older adults, younger adults have the same proportion of responses that are wrong when remembering which claims are myths—H0: µY = µO. Research hypothesis: On average, compared with older adults, younger adults have a different proportion of responses that are wrong when remembering which claims are myths—H1: µYµO.

The hypotheses for the main effect of the second independent variable, number of repetitions, are as follows. Null hypothesis: On average, those who hear one repetition have the same proportion of responses that are wrong when remembering which claims are myths compared with those who hear three repetitions—H0: µ1 = µ3. Research hypothesis: On average, those who hear one repetition have a different proportion of responses that are wrong when remembering which claims are myths compared with those who hear three repetitions—H1: µ1µ3.

The hypotheses for the interaction of age and number of repetitions are as follows. Null hypothesis: The effect of number of repetitions is not dependent on the levels of age. Research hypothesis: The effect of number of repetitions depends on the levels of age.

STEP 3: Determine the characteristics of the comparison distribution.

The third step is similar to that of a one-way between-groups ANOVA, except that there are three comparison distributions, all of them F distributions. We need to provide the appropriate degrees of freedom for each of these: two main effects and one interaction. As before, each F statistic is a ratio of between-groups variance and within-groups variance. Because there are three effects, there are three between-groups variance estimates, each with its own degrees of freedom. There is only one within-groups variance estimate, with its degrees of freedom for all three.

MASTERING THE FORMULA

12-1: The formula for the between-groups degrees of freedom for the independent variable in the rows of the table of cells is:

dfrows = Nrows – 1.

We subtract 1 from the number of rows, representing levels, for that variable.

For each main effect, the between-groups degrees of freedom is calculated as for a one-way ANOVA: the number of groups minus 1. The first independent variable, age, is in the rows of the table of cells, so the between-groups degrees of freedom is:

dfrows(age) = Nrows − 1 = 2 − 1 = 1

MASTERING THE FORMULA

12-2: The formula for the between-groups degrees of freedom for the independent variable in the columns of the table of cells is:

dfcolumns = Ncolumns – 1.

We subtract 1 from the number of columns, representing levels, for that variable.

The second independent variable, number of repetitions, is in the columns of the table of cells, so the between-groups degrees of freedom is:

dfcolumns(reps) = Ncolumns − 1 = 2 − 1 = 1

MASTERING THE FORMULA

12-3: The formula for the between groups degrees of freedom for the interaction is:

dfinteraction = (dfrows)(dfcolumns).

We multiply the degrees of freedom for each of the independent variables.

344

We now need a between-groups degrees of freedom for the interaction, which we calculate by multiplying the degrees of freedom for the two main effects:

dfinteraction = (dfrows(age))(dfcolumns(reps)) = (1)(1) = 1

The within-groups degrees of freedom is calculated like that for a one-way between-groups ANOVA, the sum of the degrees of freedom in each of the cells. In the current example, there are three participants in each cell, so the within-groups degrees of freedom is calculated as follows, with N representing the number in each cell:

dfY,1 = N − 1 = 3 − 1 = 2

dfY,3 = N − 1 = 3 − 1 = 2

dfO,1 = N − 1 = 3 − 1 = 2

dfO,3 = N − 1 = 3 − 1 = 2

dfwithin = dfY,1 + dfY,3 + dfO,1 + dfO,3 = 2 + 2 + 2 + 2 = 8

MASTERING THE FORMULA

12-4: To calculate the within-groups degrees of freedom, we first calculate degrees of freedom for each cell. That is, we subtract one from the number of participants in each cell. We then sum the degrees of freedom for each cell. For a study in which there are four cells, we’d use this formula:

dfwithin = dfcell 1 + dfcell 2 + dfcell 3 + dfcell 4.

For a check on our work, we calculate the total degrees of freedom just as we did for the one-way between-groups ANOVA. We subtract 1 from the total number of participants:

dftotal = Ntotal − 1 = 12 − 1 = 11

We now add up the three between-groups degrees of freedom and the within-groups degrees of freedom to see if they equal 11. In this case, they match:

11 = 1 + 1 + 1 +8

MASTERING THE FORMULA

12-5: There are two ways to calculate the total degrees of freedom. We can subtract 1 from the total number of participants in the entire study:

dftotal = Ntotal − 1.

We can also add the three between-groups degrees of freedom and the within-groups degrees of freedom. It’s a good idea to calculate it both ways as a check on our work.

Finally, for this step, we list the distributions with their degrees of freedom for the three effects. Note that, although the between-groups degrees of freedom for the three effects are the same in this case, they are often different. For example, if one independent variable had three levels and the other had four, the between-groups degrees of freedom for the main effects would be 2 and 3, respectively, and the between-groups degrees of freedom for the interaction would be 6.

Summary: Main effect of age: F distribution with 1 and 8 degrees of freedom. Main effect of number of repetitions: F distribution with 1 and 8 degrees of freedom. Interaction of age and number of repetitions: F distribution with 1 and 8 degrees of freedom. (Note: It is helpful to include all degrees of freedom calculations in this step.)

STEP 4: Determine the critical values, or cutoffs.

Again, this step for the two-way between-groups ANOVA is just an expansion of that for the one-way version. We now need three critical values but we determine them just as we determined them before. We use the F table in Appendix B.

For each main effect and for the interaction, we look up the within-groups degrees of freedom, which is always the same for each effect, along the left-hand side and the appropriate between-groups degrees of freedom across the top of the table. The place on the grid where this row and this column intersect contains three numbers. From top to bottom, the table provides cutoffs for p levels of 0.01, 0.05, and 0.10. As usual, we typically use 0.05. In this instance, it happens that the critical value is the same for all three effects because the between-groups degrees of freedom is the same for all three. But when the between-groups degrees of freedom are different, there are different critical values. Here, we look up the between-groups degrees of freedom of 1, within-groups degrees of freedom of 8, and p level of 0.05. The cutoff for all three is 5.32, as seen in Figure 12-8.

image
Figure 12.20: FIGURE 12-8
Determining Cutoffs for an F Distribution
We determine the critical values for an F distribution for a two-way between-groups ANOVA just as we did for a one-way between-groups ANOVA, except that we calculate three cutoffs, one for each main effect and one for the interaction. In this case, the between-groups degrees of freedom are the same for all three, so the cutoffs are the same.

345

Summary: There are three critical values (which in this case are all the same), as seen in the curve in Figure 12-8. The critical F value for the main effect of age is 5.32. The critical F value for the main effect of number of repetitions is 5.32. The critical F value for the interaction of age and number of repetitions is 5.32.

STEP 5: Calculate the test statistic.

As with the one-way between-groups ANOVA, the fifth step for the two-way between-groups ANOVA is the most time consuming. As you might guess, it’s similar to what we already learned, but we have to calculate three F statistics instead of one. We learn the logic and the specific calculations for this step in the next section.

STEP 6: Make a decision.

This step is the same as for a one-way between-groups ANOVA, except that we compare each of the three F statistics to its appropriate cutoff F statistic. If the F statistic is beyond the critical value, then we know that it is in the most extreme 5% of possible test statistics if the null hypothesis is true. After making a decision for each F statistic, we present the results in one of three ways.

First, if we are able to reject the null hypothesis for the interaction, then we draw a specific conclusion with the help of a table and graph. Because we have more than two groups, we use a post hoc test, such as the one that we learned in Chapter 11. When there are three effects, post hoc tests are typically implemented separately for each main effect and for the interaction (Hays, 1994). If the interaction is statistically significant, then it might not matter whether the main effects are also significant; if they are also significant, then those findings are usually qualified by the interaction, and are not described separately. The overall pattern of cell means tells the whole story.

Second, if we are not able to reject the null hypothesis for the interaction, then we focus on any significant main effects, drawing a specific directional conclusion for each. In this study, each independent variable has only two levels, so there is no need for a post hoc test. If there were three or more levels, however, then each significant main effect would require a post hoc test to determine exactly where the differences lie.

Third, if we do not reject the null hypothesis for either main effect or the interaction, then we can only conclude that there is insufficient evidence from this study to support the research hypotheses. We will complete step 6 of hypothesis testing for this study in the next section, after we consider the calculations of the source table for a two-way between-groups ANOVA.

Identifying Four Sources of Variability in a Two-Way ANOVA

In this section, we complete step 5 for a two-way between-groups ANOVA. The calculations are similar to those for a one-way between-groups ANOVA, except that we calculate three F statistics. We use a source table with elements like those shown in Table 12-18 on page 349.

MASTERING THE FORMULA

12-6: We calculate the total sum of squares using the following formula:

SStotal =∑(XGM )2.

We subtract the grand mean from every score to create deviations, then square the deviations, and finally sum the squared deviations.

First, we calculate the total sum of squares (Table 12-13). We calculate this number in exactly the same way as we do for a one-way ANOVA. We subtract the grand mean, in this case 0.21, from every score to create deviations, then square the deviations, and finally sum the squared deviations:

SStotal = ∑(XGM )2 = 0.0672

346

image

We now calculate the between-groups sums of squares for the two main effects. Both are calculated similarly to the between-groups sum of squares for a one-way between-groups ANOVA. Table 12-14 shows the cell means, marginal means, and grand mean. The between-groups sum of squares for the main effect of the independent variable age would be the sum, for every score, of the marginal mean minus the grand mean, squared. We list all 12 scores in Table 12-15, marking the divisions among the cells. For each of the six younger participants, those in the top six rows of Table 12-15, we subtract the grand mean, 0.21, from the marginal mean, 0.16. For the six older participants, those in the bottom six rows, we subtract 0.21 from the marginal mean, 0.26. We square all of these deviations and then add them to calculate the sum of squares for the rows, the independent variable of age:

image
image

MASTERING THE FORMULA

12-7: We calculate the between-groups sum of squares for the first independent variable, that in the rows of the table of cells, using the following formula:

SSbetween(rows) =∑(MrowGM )2.

For every participant, we subtract the grand mean from the marginal mean for the appropriate row for that participant. We square these deviations, and sum the squared deviations.

SSbetween(rows) =∑(Mrow(age)GM )2 = 0.03

We repeat this process for the second possible main effect, that of the independent variable in the columns (Table 12-16). The between-groups sum of squares for number of repetitions, then, would be the sum, for every score, of the marginal mean minus the grand mean, squared. We again list all 12 scores, marking the divisions among the cells. For each of the six participants who had one repetition, those in the left-hand column of Table 12-14 and in rows 1–3 and 7–9 of Table 12-16, we subtract the grand mean, 0.21, from the marginal mean, 0.21. For each of the six participants who had three repetitions, those in the right-hand column of Table 12-14 and in rows 4–6 and 10–12 of Table 12-16, we subtract 0.21 from the marginal mean, 0.21. (Note: It is a coincidence that in this case the marginal means are exactly the same.) We square all of these deviations and add them to calculate the between-groups sum of squares for the columns, the independent variable of number of repetitions. Again, the calculations for the between-groups sum of squares for each main effect are just like the calculations for a one-way between-groups ANOVA:

MASTERING THE FORMULA

12-8: We calculate the between-groups sum of squares for the second independent variable, that in the columns of the table of cells, using the following formula:

SSbetween(columns) =∑(McolumnGM )2.

For every participant, we subtract the grand mean from the marginal mean for the appropriate column for that participant. We square these deviations, and then sum the squared deviations.

SSbetween(columns) =∑(Mcolumns(reps)GM )2 = 0

347

image

348

The within-groups sum of squares is calculated in exactly the same way as for the one-way between-groups ANOVA (Table 12-17). The cell mean is subtracted from each of the 12 scores. The deviations are squared and summed:

image

SSwithin = ∑(XMcell)2 = 0.018

MASTERING THE FORMULA

12-9: We calculate the within-groups sum of squares using the following formula:

SSwithin =?∑(XMcell)2.

For every participant, we subtract the appropriate cell mean from that participant’s score. We square these deviations, and sum the squared deviations.

All we need now is the between-groups sum of squares for the interaction. We calculate this by subtracting the other between-groups sums of squares (those for the two main effects) and the within-groups sum of squares from the total sum of squares. The between-groups sum of squares for the interaction is essentially what is left over when the main effects are accounted for. Mathematically, any variability that is predicted by these variables, but is not directly predicted by either independent variable on its own, is attributed to the interaction. The formula is:

SSbetween(interaction) = SStotal − (SSbetween(rows) + SSbetween(columns) + SSwithin)

The calculations are:

SSbetween(interaction) = 0.0672 − (0.03 + 0 + 0.018) = 0.0192

MASTERING THE FORMULA

12-10: To calculate the between-groups sum of squares for the interaction, we subtract the two between-groups sums of squares for the independent variables and the within-groups sum of squares from the total sum of squares. The formula is: SSbetween(interaction) = SStotal − (SSbetween(rows) + SSbetween(columns) + SSwithin).

Now we complete step 6 of hypothesis testing by calculating the F statistics using the formulas in Table 12-18. The results are in the source table (Table 12-19). The main effect of age is statistically significant because the F statistic, 13.04, is larger than the critical value of 5.32. The means tell us that older participants tend to make more mistakes, remembering more medical myths as true, than do younger participants. The main effect of number of repetitions is not statistically significant, however, because the F statistic of 0.00 is not larger than the cutoff of 5.32. It is unusual to have an F statistic of 0.00. Even when there is no statistically significant effect, there is usually some difference among means due to random sampling. The interaction is also statistically significant because the F statistic of 8.35 is larger than the cutoff of 5.32. Therefore, we construct a bar graph of the cell means, as seen in Figure 12-9, to interpret the interaction.

349

image
image

MASTERING THE FORMULA

12-11: The formulas to calculate the four mean squares are in Table 12-18. There are three between-groups mean squares—one for each main effect and one for the interaction—and one within-groups mean square. For each mean square, we divide the appropriate sum of squares by its related degrees of freedom. The formulas for the three F statistics, one for each main effect and one for the interaction, are also in Table 12-18. For each of the three effects, we divide the appropriate between-groups mean square by the within-group mean square. The denominator is the same in all three cases.

In Figure 12-9, the lines are not parallel; in fact, they intersect without even having to extend them beyond the graph. We see that among younger participants, the proportion of responses that were incorrect was lower, on average, with three repetitions than with one repetition. Among older participants, the proportion of responses that were incorrect was higher, on average, with three repetitions than with one repetition. Does repetition help? It depends. It helps for younger people but is detrimental for older people. Specifically, repetition tends to help younger people distinguish between myth and fact. But the mere repetition of a medical myth tends to lead older people to be more likely to view it as fact. The researchers speculate that older people remember that they are familiar with a statement but forget the context in which they heard it. This is a qualitative interaction because the direction of the effect of repetition reverses from one age group to another.

image
Figure 12.28: FIGURE 12-9
Interpreting the Interaction
The nonparallel lines demonstrate the interaction. The bars tell us that, on average, repetition decreases errors for younger people but increases them for older people. Because the direction reverses, this is a qualitative interaction.

350

Effect Size for Two-Way ANOVA

With a two-way ANOVA, as with a one-way ANOVA, we calculate R2 as the measure of effect size. As before, we use sums of squares as indicators of variability. For each of the three effects—the two main effects and the interaction—we divide the appropriate between-groups sum of squares by the total sum of squares minus the sums of squares for both of the other effects. We subtract the sums of squares for the other two effects from the total so that we isolate the effect size for a single effect at a time. For example, if we want to determine effect size for the main effect in the rows, we divide the sum of squares for the rows by the total sum of squares minus the sum of squares for the column and the sum of squares for the interaction.

MASTERING THE FORMULA

12-12: To calculate effect size for two-way ANOVA, we make three R2 calculations, one for each main effect and one for the interaction. In each case, we divide the appropriate between-groups sum of squares by the total sum of squares minus the sums of squares for the other two effects. For example, the effect size for the interaction is calculated using this formula: Rinteraction = image

For the first main effect, the one in the rows of the table of cells, the formula is:

image

For the second main effect, the one in the columns of the table of cells, the formula is:

image

For the interaction, the formula is:

image

EXAMPLE 12.5

Let’s apply this to the ANOVA we just conducted. We use the statistics in the source table shown in Table 12-19 to calculate R2 for each main effect and the interaction. Here are the calculations for the main effect for age:

image

Here are the calculations for the main effect for repetitions:

image

351

Here are the calculations for the interaction:

image

The conventions are the same as those presented in Chapter 11, shown again here in Table 12-20. From this table, we can see that the R2 of 0.63 for the main effect of age and 0.52 for the interaction are very large. The R2 of 0.00 for the main effect of repetitions indicates that there is no observable effect in this study.

image
Figure 12.29:

CHECK YOUR LEARNING

Reviewing the Concepts
  • The six steps of hypothesis testing for a two-way between-groups ANOVA are similar to those for a one-way between-groups ANOVA.

  • Because we have the possibility of two main effects and an interaction, each step is broken down into three parts, with three sets of hypotheses, comparison distributions, critical F values, F statistics, and conclusions.

  • An expanded source table helps us to keep track of the calculations.

  • Statistically significant F statistics require post hoc tests to determine where differences lie when there are more than two groups.

  • We calculate a measure of effect size, R2, for each main effect and for the interaction.

Clarifying the Concepts 12-9 What is the basic difference between the six steps of hypothesis testing for a two-way between-groups ANOVA and a one-way between-groups ANOVA?
12-10 What are the four sources of variability in a two-way ANOVA?
Calculating the Statistics 12-11 Compute the three between-groups degrees of freedom (both main effects and the interaction), the within-groups degrees of freedom, and the total degrees of freedom for the following data:

IV 1, level A; IV 2, level A: 2, 1, 1, 3

IV 1, level B; IV 2, level A: 5, 4, 3, 4

IV 1, level A; IV 2, level B: 2, 3, 3, 3

IV 1, level B; IV 2, level B: 3, 2, 2, 3

12-12 Using the degrees of freedom you calculated in Check Your Learning 12-11, determine critical values, or cutoffs, using a p level of 0.05, for the F statistics of the two main effects and the interaction.
Applying the Concepts 12-13 Researchers studied the effect of e-mail messages on students’ final exam grades (Forsyth & Kerr, 1999; Forsyth et al., 2007). To test for possible interactions, participants included students whose first exam grade was either (1) a C, or (2) a D or an F. Participants were randomly assigned to receive several e-mails in one of three conditions: e-mails intended to bolster their self-esteem, e-mails intended to enhance their sense of control over their grades, and e-mails that just included review questions (control group). The accompanying table shows the cell means for the final exam grades (note that some of these are approximate, but all represent actual findings). For simplicity, assume there were 84 participants in the study evenly divided among cells.
Self-esteem
(SE)
Take responsibility
(TR)
Control group
(CG)
C 67.31 69.83 71.12
D/F 47.83 60.98 62.13
  1. From step 1 of hypothesis testing, list the populations for this study.

  2. Conduct step 2 of hypothesis testing.

  3. Conduct step 3 of hypothesis testing.

  4. Conduct step 4 of hypothesis testing.

  5. The F statistics are 20.84 for the main effect of the independent variable of initial grade, 1.69 for the main effect of the independent variable of type of e-mail, and 3.02 for the interaction. Conduct step 6 of hypothesis testing.

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

352