One-factor ANOVA Practical

Module Overview

  • Welcome to Module three of the Analysis of Variance series.

  • Focus: One factor analysis of variance and conducting these analyses in SPSS.

Assumptions of Basic Analysis of Variance

  • Introduction to the four key assumptions:

    1. Dependent Variable Scale:

    • Must be of an interval or ratio scale.

    1. Normal Distribution:

    • Dependent variable scores should ideally be normally distributed.

    • Can be slightly violated if sample sizes exceed 30.

    1. Equivalent Group Sizes:

    • Group sizes should be roughly equivalent, with a tolerable ratio of about 1.5.

    1. Homogeneity of Variance:

    • Assumes the variance among populations is equal across group means, though sample variances may differ slightly.

Detailed Exploration of Homogeneity of Variance

  • Variance Definition:

    • Measures the spread of data relative to the mean.

  • Variance Example Calculation:

    • For sample variance:

    • Formula: ext{Variance (s^2)} = \frac{\sum (X - \text{mean})^2}{N - 1}

    • Example with data: 2, 3, 4

    • Mean = 3; Variance = 1.

    • Example with data: 0, 3, 6

    • Mean remains 3; Variance = 9.

  • Significance of Variance:

    • Variance calculations should be similar for all groups in ANOVA to meet the homogeneity assumption.

  • Implications of Violating Assumptions:

    • If homogeneity is violated, it necessitates conducting nonparametric tests as a last resort.

Analysis by Hand - One Way ANOVA

  • Review of calculating sum of squares by hand using:

    • Data from Module Two (as a reference).

  • Steps for calculating:

    1. Calculate totals for each group.

    2. Calculate sum of squares for each group and total sum of squares.

    3. Calculate the degrees of freedom.

  • Example calculations:

    • Total participants: 50

    • Total sum of squares result: 768.82.

    • Sum of squares between: 351.52 with 4 degrees of freedom.

    • Sum of squares within: Calculate as total - between.

Analysis of Variance Table

  • Structure includes:

    • Rows for between and within groups, total.

    • Calculation of mean square: \text{Mean Square} = \frac{\text{Sum of Squares}}{\text{Degrees of Freedom}}

    • F-value comparisons against critical values to test significance (with example values).

  • Result interpretation:

    • Example: F-value = 9.1 exceeds critical value = 2.61, thus null hypothesis rejected.

SPSS Analysis of Variance

  • Steps to initiate analysis in SPSS:

    • Input data under States of independent variable with levels (low, medium, high).

    • Steps to enter into SPSS for analysis of variance:

    1. Navigate to ANALYZE > General Linear Model > Univariate.

    2. Select factor and dependent variable.

    3. Enable options for descriptive statistics and homogeneity tests.

Output Interpretation from SPSS

  • Identification of descriptive statistics,

  • Levene's test results, indicating normality and significance.

  • ANOVA table including:

    • Significance of F = 112, p < 0.001.

    • Reporting format: F(2, 15) = 112.82, p < 0.001.

  • Reporting partial eta squared (93.8% variance explained).

  • Discussing observed power of the analysis.

Post Hoc Tests

  • Explanation of Bonferroni test to avoid family-wise error when performing multiple comparisons.

  • Results show significant differences across all tested conditions with relevant p-values.

  • Discuss the relevance of graphical representation for visualizing means among groups.

Concept of Family-Wise Error

  • Definition: Probability of making at least one type I error across multiple tests.

  • Explanation of how repeated t-tests without correction may lead to inflated error rates.

  • Summary of the Bonferroni correction process to maintain family-wise error rate.

Planned Comparisons and Contrast

  • Explanation of planned comparisons as specific comparisons made prior to analysis to control for family-wise error.

  • Types of contrasts differentiating orthogonal (independent) vs. non-orthogonal (dependent) comparisons.

Trend Analysis

  • Purpose: Exploring systematic changes in dependent variable across levels of independent variable.

  • Examples using temperature and anger levels to identify linear vs. quadratic trends.

Within Groups vs. Between Groups ANOVA

  • Review of within-subjects design focusing on participants experiencing all levels of IV.

  • Discuss covariates and sphericity adjustments in the composition of within ANOVA table outputs.

    • Details on Mauchly's test and corrections applicable if sphericity is violated.

Conclusion

  • Summary of key takeaways from one factor ANOVA, application in SPSS and introduction into future topics relevant to ANCOVA, analysis of covariance.