Chapter 14

• 14.1 A factorial ANOVA is a statistical analysis used with one scale dependent variable and at least two nominal (or sometimes ordinal) independent variables (also called factors).
• 14.2 A statistical interaction occurs in a factorial design when the two independent variables have an effect in combination that we do not see when we examine each independent variable on its own.
• 14.3
  • a. There are two factors: diet programs and exercise programs.
  • b. There are three factors: diet programs, exercise programs, and metabolism type.
  • c. There is one factor: gift certificate value.
  • d. There are two factors: gift certificate value and store quality.
• 14.4
  • a. The participants are the stocks themselves.
  • b. One independent variable is the type of ticker-code name, with two levels: pronounceable and unpronounceable. The second independent variable is time lapsed since the stock was initially offered, with four levels: 1 day, 1 week, 6 months, and 1 year.
  • c. The dependent variable is the stocks’ selling price.
  • d. This would be a two-way mixed-design ANOVA.
  • e. This would be a 2 × 4 mixed-design ANOVA.
  • f. This study would have eight cells: 2 × 4 = 8. We multiplied the numbers of levels of each of the two independent variables.
• 14.5 A quantitative interaction is an interaction in which one independent variable exhibits a strengthening or weakening of its effect at one or more levels of the other independent variable, but the direction of the initial effect does not change. More specifically, the effect of one independent variable is modified in the presence of another independent variable. A qualitative interaction is a particular type of quantitative interaction of two (or more) independent variables in which one independent variable reverses its effect depending on the level of the other independent variable. In a qualitative interaction, the effect of one variable doesn’t just become stronger or weaker; it actually reverses direction in the presence of another variable.
• 14.6 An interaction indicates that the effect of one independent variable depends on the level of the other independent variable(s). The main effect alone cannot be interpreted because the effect of that one variable depends on another.
• 14.7
  • a. There are four cells.
    

    		IV 2
    		LEVEL A	LEVEL B
    IV 1	LEVEL A
    	LEVEL B

  • b.
    
  • c. Because the sample size is the same for each cell, we can compute marginal means as simply the average between cell means.
    

    		IV 2
    		LEVEL A	LEVEL B	MARGINAL MEANS
    IV 1	LEVEL A	1.75	2.75	2.25
    	LEVEL B	4	2.5	3.25
    	Marginal means	2.875	2.625

  • d.
    
• 14.8
  • a.
    • i. Independent variables: student (Caroline, Mira); class (philosophy, psychology)
    • ii. Dependent variable: performance in class
    • iii.

      	CAROLINE	MIRA
      Philosophy class
      Psychology class

    • iv. This describes a qualitative interaction because the direction of the effect reverses. Caroline does worse in philosophy class than in psychology class, whereas Mira does better.
  • b.
    • i. Independent variables: game location (home, away); team (own conference, other conference)
    • ii. Dependent variable: number of wins
    • iii.

      	HOME	AWAY
      Own conference
      Other conference

    • iv. This describes a qualitative interaction because the direction of the effect reverses. The team does worse at home against teams in the other conference but does well against those teams while away; the team does better at home against teams in its own conference, but performs poorly against teams in its own conference when away.
  • c.
    • i. Independent variables: amount of caffeine (caffeine, none); exercise (worked out, did not work out)
    • ii. Dependent variable: amount of sleep
    • iii.

      	CAFFEINE	NO CAFFEINE
      Working out
      Not working out

    • iv. This describes a quantitative interaction because the effect of working out is particularly strong in the presence of caffeine versus no caffeine (and the presence of caffeine is particularly strong in the presence of working out versus not). The direction of the effect of either independent variable, however, does not change depending on the level of the other independent variable.
• 14.9 Hypothesis testing for a one-way between-groups ANOVA evaluates only one idea. However, hypothesis testing for a two-way between-groups ANOVA evaluates three ideas: two main effects and an interaction. So, the two-way ANOVA requires three hypotheses, three comparison distributions, three critical F values, three F statistics, and three conclusions.
• 14.10 Variability is associated with the two main effects, the interaction, and the within-groups component.
• 14.11 df_IV1 = df_rows = N_rows − 1 = 2 − 1 = 1
  df_IV2 = df_columns = N_columns − 1 = 2 − 1 = 1
  df_interaction = (df_rows)(df_columns) = (1)(1) = 1
  
  df_total = N_total − 1 = 16 − 1 = 15
  We can also verify that this calculation is correct by adding all of the other degrees of freedom together: 1 + 1 + 1 + 12 = 15.
• 14.12 The critical value for the main effect of the first independent variable, based on a between-groups degrees of freedom of 1 and a within-groups degrees of freedom of 12, is 4.75. The critical value for the main effect of the second independent variable, based on 1 and 12 degrees of freedom, is 4.75. The critical value for the interaction, based on 1 and 12 degrees of freedom, is 4.75.
• 14.13
  • a. Population 1 is students who received an initial grade of C who received e-mail messages aimed at bolstering self-esteem. Population 2 is students who received an initial grade of C who received e-mail messages aimed at bolstering their sense of control over their grades. Population 3 is students who received an initial grade of C who received e-mails with just review questions. Population 4 is students who received an initial grade of D or F who received e-mail messages aimed at bolstering self-esteem. Population 5 is students who received an initial grade of D or F who received e-mail messages aimed at bolstering their sense of control over their grades. Population 6 is students who received an initial grade of D or F who received emails with just review questions.
  • b. Step 2: Main effect of first independent variable—initial grade:
    Null hypothesis: The mean final exam grade of students with an initial grade of C is the same as that of students with an initial grade of D or F. H₁: μ_C = μ_D/F. Research hypothesis: The mean final exam grade of students with an initial grade of C is not the same as that of students with an initial grade of D or F. H₀: μ_C ≠ μ_D/F.
    Main effect of second independent variable—type of email:
    Null hypothesis: On average, the mean exam grades among those receiving different types of e-mails are the same—H₀: μ_SE = μ_CG = μ_TR. Research hypothesis: On average, the mean exam grades among those receiving different types of e-mails are not the same.
    Interaction: Initial grade × type of e-mail:
    Null hypothesis: The effect of type of e-mail is not dependent on the levels of initial grade. Research hypothesis: The effect of type of e-mail depends on the levels of initial grade.
  • c. Step 3: df_{between/grade} + N_groups − 1 = 2 − 1 = 1
    df_{between/e-mail} = N_groups − 1 = 3 − 1 = 2
    df_interaction = (df_{between/grade})(df_{between/e-mail}) = (1)(2) = 2
    df_C,SE = N − 1 = 14 − 1 = 13
    df_C,C = N − 1 = 14 − 1 = 13
    df_C,TR = N − 1 = 14 − 1 = 13
    df_D/F,SE = N − 1 = 14 − 1 = 13
    df_D/F,C = N − 1 = 14 − 1 = 13
    df_D/F,TR = N − 1 = 14 − 1 = 13
    df_within = df_C,SE + df_C,C + df_C,TR + df_D/F,SE + df_D/F,C + df_D/F,TR = 13 + 13 + 13 + 13 + 13 + 13 = 78
    Main effect of initial grade: F distribution with 1 and 78 degrees of freedom
    Main effect of type of e-mail: F distribution with 2 and 78 degrees of freedom
    Main effect of interaction of initial grade and type of e-mail: F distribution with 2 and 78 degrees of freedom
  • d. Step 4: Note that when the specific degrees of freedom is not in the F table, you should choose the more conservative—that is, larger—cutoff. In this case, go with the cutoffs for a within-groups degrees of freedom of 75 rather than 80. The three cutoffs are:
    Main effect of initial grade: 3.97
    Main effect of type of e-mail: 3.12
    Interaction of initial grade and type of e-mail: 3.12
  • e. Step 6: There is a significant main effect of initial grade because the F statistic, 20.84, is larger than the critical value of 3.97. The marginal means, seen in the accompanying table, tell us that students who earned a C on the initial exam have higher scores on the final exam, on average, than do students who earned a D or an F on the initial exam. There is no statistically significant main effect of type of e-mail, however. The F statistic of 1.69 is not larger than the critical value of 3.12. Had this main effect been significant, we would have conducted a post hoc test to determine where the differences were. There also is not a significant interaction. The F statistic of 3.02 is not larger than the critical value of 3.12. (Had we used a cutoff based on a p level of 0.10, we would have rejected the null hypothesis for the interaction. The cutoff for a p level of 0.10 is 2.38.) If we had rejected the null hypothesis for the interaction, we would have examined the cell means in tabular and graph form.
    

    	SELF-ESTEEM	TAKE RESPONSIBILITY	CONTROL GROUP	MARGINAL MEANS
    C	67.31	69.83	71.12	69.42
    D/F	47.83	60.98	62.13	56.98
    Marginal means	57.57	65.41	66.63