Factorial ANOVA

FACTORIAL ANOVA NOTES

WHAT IS A FACTORIAL ANOVA?

  • Definition: Factorial ANOVA is employed when there are two or more factors (independent variables).
  • Analysis Types:
    • Univariate Analysis of Variance: When one dependent variable is analyzed.
    • Multivariate Analysis of Variance: When more than one dependent variable is analyzed.

PATHWAY TO KNOWLEDGE

  • Key Considerations in Research Design:

    • Are you examining relationships between variables or differences between groups of one or more variables?

    • If focusing on relationships:

      • How many variables?
      • If yes:
        • How many groups?
        • If two groups: Use t-test for the significance of the correlation coefficient.
        • If more than two: Utilize regression, factor analysis, or canonical analysis.

    • If examining differences:

      • Are the same participants tested more than once?
      • If no: How many groups?
        • If two groups: Use t-test for independent samples.
        • If more than two groups: Utilize factorial analysis.
      • If yes: How many groups?
        • If two groups: Use t-test for dependent samples.
        • If more than two groups: Implement repeated measures analysis of variance.

FACTORS AND LEVELS

  • Factors: Independent variables in a study.
  • Levels of Factors: Individual treatment conditions that make up a factor.
  • Notation:
    • Let k = number of levels of a factor (treatment conditions).
    • Let n = number of scores in each treatment level.
    • Let N = total number of scores in the entire study.

WHAT DOES A FACTORIAL ANOVA EXAMINE?

  • Involving two or more factors, factorial ANOVA allows researchers to assess:
    • Main Effects: The individual effects of each factor.
    • Interaction Effect: The simultaneous effects of both factors.
  • Examination include:
    • A test of the main effect for the first factor (IV).
    • A test of the main effect for the second factor (IV).
    • A test for interaction between the two factors (IV x IV).

MOST BASIC FACTORIAL ANOVA

  • Example:
    • Investigating sex (male or female) and treatment (high-impact or low-impact exercise program) on an outcome (weight loss).
    • This can be described as a 2 x 2 Factorial ANOVA or Two-way ANOVA.

MAIN EFFECT OF SEX ON WEIGHT LOSS

  • Examines how sex affects body mass loss, focusing on:
    • Sex (Male vs. Female) impacts weight loss.

MAIN EFFECT OF EXERCISE ON WEIGHT LOSS

  • Examines how different exercise types affect body mass, focusing on:
    • Exercise intensity (High vs. Low) affecting weight loss.

MAIN AND INTERACTION EFFECTS

  • Examines the combined effects of sex and exercise on body mass.
  • Factors include:
    • Sex (Male vs. Female) and Exercise (Low vs. High) impacting weight loss.

FACTORIAL DESIGN TABLE – 2 X 2

  • Factors:
    • Factor 1 = Exercise (High, Low)
    • Factor 2 = Sex (Males, Females)
  • Sample Sizes:
    • Male/High (n = 10)
    • Male/Low (n = 10)
    • Female/High (n = 10)
    • Female/Low (n = 10)
  • Dependent Variable (DV): Weight Loss

RESEARCH QUESTIONS FOR FACTORIAL ANOVA

  • Questions include:
    • Is there a difference in weight loss between high-impact and low-impact exercise?
    • Is there a difference in weight loss between males and females?
    • Does the impact of exercise on weight loss depend on sex?

NULL HYPOTHESES FOR FACTORIAL ANOVA

  • Null Hypotheses:
    • For the first factor main effect:
    • $H0$: m{High} = m_{Low}
    • For the second factor main effect:
    • $H0$: m{Male} = m_{Female}
    • For the interaction:
    • $H0$: m{High * Male} = m{High * Female} = m{Low * Male} = m_{Low * Female}

RESEARCH HYPOTHESES FOR FACTORIAL ANOVA

  • Research Hypotheses:
    • For the first factor main effect:
    • $H1$: M{High}
      eq M_{Low}
    • For the second factor main effect:
    • $H2$: M{Male}
      eq M_{Female}
    • For the interaction:
    • $H3$: M{High * Male}
      eq M{High * Female} eq M{Low * Male}
      eq M_{Low * Female}

SOURCE TABLES

  • Example Statistical Analysis:
    • Descriptive statistics and tests of between-subjects effects focused on the dependent variable: weight loss.
  • Presenting data includes:
    • Mean, Standard Deviation (SD), and number (N) for each condition:
    • High Impact Males: Mean = 73.70, SD = 6.667, N = 10.
    • High Impact Females: Mean = 79.40, SD = 11.890, N = 10.
    • Low Impact Males: Mean = 78.80, SD = 11.043, N = 10.
    • Low Impact Females: Mean = 64.00, SD = 11.235, N = 10.

PLOTTING AND IDENTIFYING MAIN EFFECTS

  • Estimated Marginal Means of Loss:
    • Analysis of main effects includes the estimated marginal means for weight loss across exercise treatments and gender.

MAIN EFFECT OF EXERCISE ON LOSS

  • Results indicate High and Low exercise impact different genders with means provided:
    • Males: Male/High Mean = 76.55, Male/Low Mean = 78.80.
    • Females: Female/High Mean = 79.40, Female/Low Mean = 64.

IS THERE A DIFFERENCE BETWEEN THE TWO LEVELS OF EXERCISE ON WEIGHT LOSS?

  • Result:
    • Non-significant difference found:
    • F(1, 36) = 2.44, p = .127
    • Conclusion: Mean loss scores between high-impact and low-impact exercises are not statistically different; the null hypothesis failed to be rejected.

MAIN EFFECT OF SEX ON LOSS

  • Males vs. Females Mean Weight Loss:
    • Males: Mean = 76.25
    • Females: Mean = 71.70

IS THERE A DIFFERENCE BETWEEN THE TWO LEVELS OF SEX ON WEIGHT LOSS?

  • Result:
    • Non-significant difference found:
    • F(1, 36) = 1.91, p = .176
    • Conclusion: Mean loss scores between genders are not statistically different; the null hypothesis failed to be rejected.

PLOTTING INTERACTION EFFECT

  • Examination of interaction effects through graphical representations of results supports understanding of significant differences in group impacts.

FACTORIAL DESIGN TABLE – 2 X 2

  • Presenting detailed conditions and means for weight loss across various groups:
    • Males: Male/High (M = 73.70, SD = 6.67), Male/Low (M = 78.80, SD = 11.04)
    • Females: Female/High (M = 79.40, SD = 11.89), Female/Low (M = 64, SD = 11.24)
    • Dependent Variable: Weight Loss

DOES THE IMPACT OF EXERCISE ON WEIGHT LOSS DEPEND ON SOMEONE’S SEX?

  • Significant interaction found:
    • F(1, 36) = 9.68, p = .004
    • Interpretation: The impact of exercise on weight loss depends on sex:
    • Women lose less weight with low-impact exercise, while men lose more weight with low-impact exercise.
    • The null hypothesis is rejected.

ANOTHER FACTORIAL ANOVA EXAMPLE

  • Research Example:
    • Investigating the impact of posing with cats on female perceptions of male dateability.
    • Factors:
    • Factor 1: Cat Presence (Male alone vs. Male with cat)
    • Factor 2: Pet Preference (Cat Person, Dog Person, Both, Neither)
    • Dependent Variable: Dateability Rating
    • Design: 2 x 4

NUTS AND NURTURE: INVESTIGATING INFLUENCE OF NUT TYPE AND SQUIRREL SEX ON FORAGING PREFERENCE

  • Research Example:
    • Inquiry on how nut type and sex of the squirrel affect foraging preference.
    • Factors:
    • Factor 1: Nut Type (5 types of nuts e.g., acorns, hazelnuts)
    • Factor 2: Sex of Squirrel (Female vs. Male)
    • Dependent Variable: Foraging Preference
    • Design: 5 x 2

INFLUENCE OF CLOTHING AND POSTURE ON PERCEPTIONS OF MASCULINITY

  • Research Example:
    • Investigating how clothing style and posture affect masculinity ratings.
    • Factors:
    • Factor 1: Posture (Legs Crossed, Legs Spread, Legs in front)
    • Factor 2: Clothing (Dress, T-shirt/Jean, Suit)
    • Dependent Variable: Masculine Rating
    • Design: 3 x 3

DESIGN A FACTORIAL ANOVA STUDY

  • Class discussion on how to conceptualize and implement a factorial ANOVA study.