factorial anova

Definitions
- Factorial analysis of variance
- Factor
- Main effect
- Interactions
- Advantages of Factorial ANOVA
- Research Design & Hypothesis Tests for Factorial ANOVA
- Assumptions
- Calculations:
- Sums of squares
- Degrees of freedom
- Mean squares
- F-ratios
- Interpretation of overall factorial ANOVA
- Factorial ANOVA in SPSS

Up to this point, discussions have revolved around one-way ANOVA with a single independent variable (IV).
Factorial analysis of variance (Factorial ANOVA) is intended for scenarios involving more than one IV.
The IVs in this study design are referred to as factors.
Each factor possesses multiple levels, and all combinations of the levels from each factor must be present in the study—this configuration is known as factorial design.
- Example: A design that includes every combination of geographical location and gender for psychological testing.

Goal: The primary aim of a factorial ANOVA is to:
- Assess the isolated effects of each factor independently.
- Investigate the potential interactions between the factors.

Generalizability: Facilitates more flexible generalization to larger populations.
Interactions: Enables the examination of how one IV influences another throughout the study.
Economy: Utilizes fewer participants compared to multiple successive one-way ANOVAs to address multiple IVs.

Factorial designs are classified based on the count of factors involved in the study:
- Two IVs = two-way factorial design
- Three IVs = three-way factorial design
- Notation can also express the structure of the design, e.g., a design with 2 levels of one IV and 4 levels of another is labeled as 2 x 4.

Main Effect: Refers to the impact of one IV on the dependent variable, disregarding the influence of other IVs.
Interaction: The degree to which the effect of one IV depends on the level of another IV.

Simplest design involves two IVs with equal sample sizes in each experimental group - this investigates:
- Effect of Factor A on outcomes.
- Effect of Factor B on outcomes.
- Interaction between Factors A and B.

All standard assumptions for factorial ANOVA apply:
- Null Hypothesis ($H0$): $ ext{μ}{11} = ext{μ}{12} = … = ext{μ}{1i}$ AND $ ext{μ}{21} = ext{μ}{22} = … = ext{μ}_{2i}$\
- This implies that the IVs exert no influence and samples are drawn from the identical population.
- Alternative Hypothesis ($H_A$): At least one mean differs, which leads to several possible alternatives:
- $ ext{μ}{11} eq ext{μ}{12}
  eq …$ OR
- $ ext{μ}{21} eq ext{μ}{22}
  eq …$ OR BOTH.
- There is also a hypothesis regarding the interaction: systematic differences between one factor cause systematic differences in the other.
Assumptions include:
- Normal distribution of data.
- Data being interval or ratio level.
- Independence of observations.

Study Context: Craik and Lockhart (1972) developed a memory processing model indicating that memory retention is linked to the information processing depth.
Study Design from Eysenck (1974): Eysenck focused on evaluating how memorization techniques affect recall while also considering age differences.
- Independent Variables:
- Age with two levels: Young (18-30 years) and Old (55-65 years)
- Learning Condition with five levels:
  1. Counting
  2. Rhyming
  3. Adjective Usage
  4. Imagery
  5. Intentional Learning (i.e., rote memorization)

Determine if memorization technique influences word recall across five conditions.
Assess whether older participants recall fewer words than younger ones, particularly in complex processing conditions.
Examine potential interactions between age and memorization techniques.

Sample Size: N = 100
Recalls and Table Structure: A detailed table recording the number of words recalled based on age and memorization conditions is provided, showcasing distinct performance metrics under different conditions.

Initial calculations require the determination of Sums of Squares (SS) across six sources of variability, notably:
- Main effects, cell effects, interaction effects, error effects, and total SS need to be calculated systematically.
Degrees of Freedom (df) calculation includes:
- Main effects
- Interaction
- Error
Example of computation for SS using the formula:
$SS = ext{Sum of squares of observations}$

Proceed to calculate F-ratios by dividing the mean squares by error mean squares, ensuring that observer effects are significant among factors:
- F-ratios for each main effect and interaction need to be tabulated and interpreted against critical values obtained from relevant statistical tables.
- An interpretation process includes asserting significant differences identified and implications drawn, thereby validating or rejecting initial hypotheses outlined.

Significant distinctions were detected for both age effects and conditions.
- Younger participants generally recalled more than older participants, particularly on complex and deep processing conditions.
- Insights derived from interaction results elucidate further dependencies between the factors tested.

Factorial ANOVA can be executed efficiently using SPSS:
- Necessary steps include structuring the data per specified variables and selecting appropriate tests (i.e., Tukey's HSD for post hoc tests).
- Analysis should prioritize both fixed factors, and interpretation of graphical outputs should also reflect relationships and marginal means across levels.

Craik, F. I. M., & Lockhart, R. S. (1972). Levels of processing: A framework for memory research. Journal of Verbal Learning and Verbal Behavior, 11, 671–684.
Eysenck, M. W. (1974). Age differences in incidental learning. Developmental Psychology, 10, 936–941.
Levene, H. (1960). Robust tests for the equality of variance. In I. Olkin (Ed.), Contributions to probability and statistics. Palo Alto, CA: Stanford University Press.