Chapter 13 - Two-Factor Analysis of Variance

Chapter 13 - Two-Factor Analysis of Variance

Learning Outcomes

• Understand the logic of a two-factor study and a matrix of group means.
• Describe main effects and interactions from a pattern of group means in a two-factor ANOVA.
• Compute a two-factor ANOVA to evaluate means for a two-factor independent-measures study.
• Measure effect size, interpret results, and articulate assumptions for a two-factor ANOVA.

Tools You Will Need

• Independent-measures analysis of variance (discussed in Chapter 12).
• Understanding of individual differences (refer to page 347).

13-1 An Overview of the Two-Factor, Independent-Measures ANOVA

• Analysis of Variance (ANOVA):
  • Evaluates differences among two or more sample means.
  • In Chapter 12, ANOVA was a single factor limited to one independent variable (IV) or one quasi-independent variable.
• Independent Measures:
  • The study uses a separate sample for each of the different treatment conditions being compared.

Key Features of Two-Factor ANOVA

• Complex Analysis of Variance:

  • Two independent variables are manipulated; referred to as factorial ANOVA (only two-factor ANOVA will be covered in this text).
  • Both independent and quasi-independent variables may be used as factors in a two-factor ANOVA.
  • An independent variable (factor) is manipulated in an experiment.

• Quasi-Independent Variable:

  • Not manipulated but defines the groups of scores in a non-experimental study.
  • A two-factor ANOVA allows examination of three types of mean differences within one analysis.

• Factorial Designs:

  • Consider more than one factor.
  • This guide studies two-factor designs only.
  • Limited to situations with equal sample sizes (n's) in each group.
  • A joint impact of factors is considered, leading to:
  • Three hypotheses tested by three F-ratios.
  • Large F-ratio suggests greater treatment differences than expected with no treatment effects.

13-1.4 Structure of the F-Ratios

• Three F-ratios share a similar basic structure:
  • The numerator measures treatment mean differences.
  • The denominator measures treatment mean differences in the absence of a treatment effect.

Main Effects

• Mean differences among levels of one factor.
• Differences are tested for statistical significance.
• Each factor (Factor A and Factor B) is evaluated independently of the other factor(s) in the study.

Interactions

• The mean differences between individual treatment conditions, or cells, are different from what would be predicted based on the overall main effects of the factors.
• Hypotheses:
  • Null Hypothesis (H0): There is no interaction between factors A and B.
  • Alternative Hypothesis (H1): There is an interaction between factors A and B.

Understanding Interactions

• Interaction is defined as the unique effect of two factors working together.
• Alternative definitions include:
  • If the two factors are independent (one factor does not influence the effect of the other), then there is no interaction.
  • When one factor's effect depends on the other, interaction is present, observable in a graph.
• Dependence of Factors:
  • The effect of one factor depends on the level or value of the other factor.
  • Unique combinations of factors produce unique effects.
  • Interaction can be visualized in a graph with nonparallel lines (lines that cross, converge, or diverge) indicating interaction.
  • This is referred to as the A × B interaction or "the A by B" interaction.

13-1.5 Independence of Main Effects and Interactions

• Three distinct tests conducted:
  • Main effect of factor A.
  • Main effect of factor B.
  • Interaction of A and B.
• Each test is conducted separately, and the results of one are independent of the others.

Learning Check 1

• Question: Which of the following accurately describes an interaction between two variables?
  1. Both variables produce a change in the subjects’ scores.
  2. Both variables are equally influenced by a third variable.
  3. The two variables are differentially affected by a third variable.
  4. The effect of one variable depends on the levels of the second variable.

Learning Check 1 - Answer

• The effect of one variable depends on the levels of the second variable accurately describes an interaction between two variables.

13-2 Example of the Two-Factor ANOVA and Effect Size

• The A-effect: the main effect of Factor A typically evaluates the mean differences between rows.
• The B-effect: the main effect of Factor B typically evaluates the mean differences between columns.
• The A × B interaction: evaluates mean differences between treatment conditions not predicted from overall main effects.

Stages of Calculation

• First stage:

  • Identical to independent samples ANOVA:
  • Compute total sum of squares ($SS{total}$), between treatments ($SS{between}$), and within treatments ($SS_{within}$).

• Second stage:

  • Partition $SS_{between}$ into three components:
  • Attributed to Factor A.
  • Attributed to Factor B.
  • Remaining mean differences defining the interaction.

13-2.2 Summary Structure of the Two-Factor ANOVA

Stage 1 of Two-Factor Analysis

• Relationship of variance:
  • Total variance = Between-treatments variance + Within-treatments variance.
  • Denominator for all three F-ratios is the variance within treatments.

Stage 2 of Two-Factor Analysis

• Degrees of Freedom:
  • $df_{total} = N - 1$.
  • $df{within ext{ treatments}} = ext{Sum of } df{inside ext{ each treatment}}$.
  • $df_{between ext{ treatments}} = k - 1$.
  • $df_{A} = ext{number of rows} - 1$.
  • $df_{B} = ext{number of columns} - 1$.
  • $df{error} = df{within ext{ treatments}} - df_{between ext{ subjects}}$.

Mean Squares and F-Ratios for the Two-Factor ANOVA

• Formulas for mean squares (MS) and F-ratios:
  • $MS<em>A = \frac{SS</em>A}{df_A}$
  • $MS<em>B = \frac{SS</em>B}{df_B}$
  • $MS<em>{A imes B} = \frac{SS</em>{A imes B}}{df_{A imes B}}$
  • $F<em>A = \frac{MS</em>A}{MS_{within}}$
  • $F<em>B = \frac{MS</em>B}{MS_{within}}$
  • $F<em>{A imes B} = \frac{MS</em>{A imes B}}{MS_{within}}$

Summary Table for the Two-Factor ANOVA

• Example data:
  

• Follows the same four-step procedure:
  1. State the hypotheses and select an alpha level.
  2. Locate the critical region.
  3. Compute the F-ratios.
  4. Make a decision based on the comparison of computed F to the critical value.

Measuring Effect Size for the Two-Factor ANOVA

• To compute effect size, calculate B \eta^2 (eta squared), indicating the percentage of variance explained by treatment effects.

Reporting the Results of a Two-Factor ANOVA

• Report means and standard deviations (often presented in tables or graphs due to complexity).
• Include results from all three hypothesis tests (main effects for A & B; interaction of A × B).
• For each test, report:
  • F-value
  • df
  • p-value
  • B \eta^2
  • Example: F(1, 16) = 9.00, p < .05, B \eta^2 = 0.36.

Interpreting Results from a Two-Factor ANOVA

• Focus on the overall result patterns.
• Significant interactions necessitate careful attention as they indicate more than might be understood from main effects alone.
• Requires practice to clearly articulate results, especially with significant interactions.

13-3 More about the Two-Factor ANOVA

Testing Simple Main Effects

• Conduct separate analyses for each of the individual columns or rows to test the significance of mean differences.
• This separates the two-factor experiment into a series of single-factor experiments, especially when a significant interaction is found.

Reducing Variance Caused by Individual Differences

• Individual characteristics (e.g. age, gender) can vary significantly.
• High variance reduces the likelihood of significant mean differences.
• Solution: Utilize a second factor associated with consistent individual differences to mitigate variance.

Example Sliding Groups

• Instead of one group in each treatment:
  • Divide participants into separate groups within each treatment based on a second factor (e.g., gender or age).

Assumptions for the Two-Factor ANOVA

• Three key assumptions:
  1. Observations within each sample must be independent.
  2. Populations must be normally distributed from which samples are drawn.
  3. Populations must have equal variances (homogeneity of variance).

Learning Check 3 - Understanding the Interaction

• Question: If a two-factor ANOVA produces a statistically significant interaction, then you can conclude that:
  • a. Either the main effect for Factor A or the main effect for Factor B is also significant.
  • b. Neither the main effect for Factor A nor the main effect for Factor B is significant.
  • c. Both the main effect for Factor A and the main effect for Factor B are significant.
  • d. The significance of the main effects is not related to the significance of the interaction.

Learning Check 3 - Answer

• The significance of the main effects is not related to the significance of the interaction indicates that interaction can exist independently of main effects.

Additional True or False Statements

• True/False: Two separate single-factor ANOVAs provide exactly the same information as a two-factor ANOVA.
• True/False: A disadvantage of combining two factors is that you cannot determine each factor’s effect independently.

Answers to Statements

• False: Although main effects in a two-factor ANOVA are similar to two one-way ANOVAs, a two-factor ANOVA reveals interaction results.
• False: A two-factor ANOVA allows determination of one variable's effect while controlling for the other.

Final Structure of an Independent-Measures Two-Factor Design ANOVA

Summary of Formulae

• Total sum of squares:

  • $SS_{total} = ext{Σ}X^2 - \frac{G^2}{N}$

• Between treatments:

  • $SS_{between} = ext{Σ}\frac{T^2}{n} - \frac{G^2}{N}$

• Within treatments:

  • Total within treatments = Σ (sum of squares for each cell).

• Degrees of Freedom:

  • $df_{total} = N - 1$.
  • $df_{between} = (k - 1)$ where k = number of treatment groups.

Closing Questions

• Review concepts and equations to ensure a comprehensive understanding of Two-Factor ANOVA and its applications and implications.