One-Way Analysis of Variance (ANOVA)
Purpose
One-Way ANOVA tests the null hypothesis that the means of three or more independent
groups are equal, focusing on one factor's impact on a continuous dependent variable.
Model Components
Factor (Independent Variable): A categorical variable with two or more levels
(e.g., types of airlines: Lufthansa, Malaysia Airlines, Cathay Pacific).
Dependent Variable: A continuous variable affected by the factor (e.g., Flight
Service Rating).
Assumptions
1. Normality: The data in each group should follow a normal distribution.
2. Equal Variances: Variances among groups should be approximately equal
(homogeneity of variances).
3. Independence: Observations must be independent of one another.
Hypotheses
Null Hypothesis (H₀): The group means are equal (μ₁ = μ₂ = μ₃).
Alternative Hypothesis (H₁): At least one group mean is different.
F Statistic Calculation
The F statistic is calculated as the ratio of between-group variance to within-group
variance:
F=Mean Square Between / Mean Square Within
Where:
Mean Square Between (MSB): Variance explained by the treatment (between the
groups).
Mean Square Within (MSW): Variance due to error (within the groups).
Example Summary Table
Source d.f. Sum of Squares Mean Square F Value p Value
Model (Airline) 2 11644.033 5822.017 28.304 0.0001
Residual (Error) 57 11724.550 205.694
Total 59 23368.583
Interpretation
If the calculated F value (e.g., 28.304) is greater than the critical value from the F-
distribution table, we reject H₀, concluding that there are significant differences
among the group means.
Post Hoc Analysis
After finding significant results, post hoc tests (e.g., Scheffé’s, Tukey's HSD) can identify
specific group differences while controlling for Type I error.
Two-Way Analysis of Variance (ANOVA)
Purpose
Two-Way ANOVA examines the effect of two independent categorical variables
(factors) on a continuous dependent variable and evaluates any interaction between them.
Model Components
Factors:
o Factor A: Airline (3 levels: Lufthansa, Malaysia Airlines, Cathay Pacific).
o Factor B: Seat Selection (2 levels: Economy, Business).
Dependent Variable: Flight Service Rating.
Assumptions
Similar to One-Way ANOVA, including normality, equal variances, and independence.
Hypotheses
1. Main Effect of Airline: H₀: μ_A1 = μ_A2 = μ_A3
2. Main Effect of Seat Selection: H₀: μ_B1 = μ_B2
3. Interaction Effect: H₀: There is no interaction between the two factors.
F Statistic Calculation
The F statistic for each main effect and the interaction is calculated similarly:
F=Mean Square Effect \ Mean Square Error
Example Summary Table
Source d.f. Sum of Squares Mean Square F Value p Value
Airline 2 11644.033 5822.017 39.178 0.0001
Seat Selection 1 3182.817 3182.817 21.418 0.0001
Airline × Seat Selection 2 517.033 258.517 1.740 0.1853
Residual 54 8024.700 148.606
Interpretation
Significant main effects indicate that both the airline and seat selection
independently influence service ratings.
If the interaction is not significant, we can interpret the main effects
independently; if it is significant, the interaction must be considered first.
Visualization
Interaction plots can help visualize how the levels of one factor affect the levels of
another factor, allowing for an intuitive understanding of the data. If the lines in an
interaction plot cross, this suggests an interaction effect.
Conclusion
ANOVA is a powerful statistical tool for comparing group means and understanding the
effects of one or more factors on a dependent variable. It helps researchers draw
conclusions about group differences and interactions, guiding decision-making in various
fields, including marketing, healthcare, and social sciences.
