ANOVA Flashcards
ANOVA (Analysis of Variance)
ANOVA is a statistical method used to compare means between three or more groups to see if they’re significantly different.
It helps determine if there are real differences in the means of the groups or if those differences could have occurred by chance.
between three or more groups.. e.g. treatments or conditions
Key idea of ANOVA
ANOVA tells us if there’s something going on between the groups, but it doesn’t pinpoint exactly which groups are different. That’s where post-hoc tests come in.
One-Way ANOVA
Compares means of three or more independent groups to determine if they are significantly different.
Formula: F = Between-group variability / Within-group variability.
A significant F-value indicates at least one group mean is significantly different from the others.
Two way ANOVA
Looks at effects of two factors on a variable.
Checks if one factor’s effect changes with another factor.
Main Effect
The overall effect of one independent variable on the dependent variable, averaging across the levels of other independent variables.
Indicates whether there is a significant difference in the dependent variable across the levels of one factor.
Interaction effect
The effect of one independent variable depends on the level of another independent variable.
Indicates whether the effect of one factor varies depending on the levels of another factor.
Assumptions of ANOVA
Independence**: Observations within each group are independent.
- Homogeneity of Variances: Variances of the dependent variable are equal across all groups.
- Normality: Residuals are normally distributed within each group.
Post-Hoc Tests
Used after ANOVA to determine which specific groups differ from each other.
Tukey’s HSD, Bonferroni, Scheffé, and Dunnett’s tests are common post-hoc tests.
to find which specific groups are different
Repeated Measures ANOVA
Analyzes data where the same subjects are measured at multiple time points or under different conditions.
Determines whether there are significant differences between the means of repeated measures.
Orthogonal Contrasts in ANOVA
Comparisons of group means in a way that each comparison is independent of the others.
If the sum of products of corresponding coefficients equals zero, it shows orthogonality.
Why are Orthogonal Contrasts important?
They help compare specific combinations of group means without influencing each other, making interpretation clearer and more reliable.
Comparing Group A to the average of Groups B and C.
c1 = 1, c2 = -0.5, c3 = -0.5
c1 x mean(Group A) + c2 x Mean(Group B) + c3 x Mean(Group C)
Why do we use Levene’s test
Before we run the ANOVA, we first perform Levene’s test. If the p-value from Levene’s test is greater than 0.05 (our chosen significance level), it suggests that the variability in test scores across the three teaching methods is roughly the same.
Levene’s Test in ANOVA
A test that checks if the variances of the groups are equal.
Why use it?: Helps make sure that the groups are comparable.
Example: If we’re comparing test scores between classes, Levene’s test helps us see if the variability in scores is about the same for each class.
Two-Way Design
Imagine a study comparing the effects of different teaching methods (Factor A) and student engagement levels (Factor B) on exam scores.
Design: This is a two-way design because it involves two independent variables (teaching methods and engagement levels) and their interactions.
Mixed Design
A mixed design combines elements of both between-subjects and within-subjects designs. It involves at least one between-subjects factor and one within-subjects factor.
Example of Mixed Design ANOVA
In the context of our previous example, a mixed design could involve studying the effects of different teaching methods (between-subjects factor) on students’ performance, measured by pre- and post-tests (within-subjects factor).
Analysis: Researchers conduct an ANOVA that accounts for both between-subjects and within-subjects effects. They analyze the main effects of each factor and their interaction effect, considering both the differences between groups and the changes within each individual over time.
Factor A and B
Factor A: This is one thing we’re looking at, like different types of teaching methods or levels of a drug.
Factor B: This is another thing we’re looking at, maybe different times of day or different age groups, it could represent different environmental conditions, participant characteristics, or experimental conditions.
Advantages of Within-Subjects Designs
Higher Statistical Power: Within-subjects designs can detect smaller effects because each participant serves as their own control, reducing variability.
Smaller Sample Sizes: Fewer participants are needed compared to between-subjects designs, saving time and resources.
Disadvantages of Within-Subjects Designs
Potential for Sequencing Effects: Repeated exposure to conditions may influence participants’ responses, affecting the validity of the results.
Carryover Effects: Effects from one condition may carry over and affect responses in subsequent conditions, confounding the results.
Visual Representation of 2x2 Factorial ANOVA:
Interaction plot showing means of the dependent variable for each combination of levels of the independent variables.
Non-parallel lines indicate a significant interaction.
Importance of Follow-Up Tests in Factorial ANOVA
After finding significant interactions, follow-up tests help understand why. They show which groups differ from each other and why.
If two treatments affect people differently based on age, follow-up tests reveal which ages benefit most from each treatment.
follow-up tests: pairwise comparisons and simple effects tests
Pairwise Comparisons
Pairwise Comparisons: Compare specific groups or levels of one factor against each other to identify where differences lie. For example, comparing the mean scores of two treatment groups to see if one is significantly higher than the other.
specific groups
Simple Effects Tests
Investigate how one factor affects the dependent variable at specific levels of another factor. They examine the effect of one factor within each level of another factor. For example, testing if the effect of a teaching method differs between boys and girls by looking at their test scores separately.
specific conditions
Cohen’s d Effect Sizes
Small: 0.2
Medium: 0.5
Large: 0.8
Post Hoc Power Analysis
An analysis conducted after a study to assess the statistical power of the observed effect.
To determine if the study had sufficient power to detect the observed effect, retrospectively.
Caution about Post Hoc Power
Post hoc power analysis can be misleading and should not be used to justify nonsignificant results or make retrospective interpretations.
Post hoc power analysis relies on observed data, which may be influenced by sample size, effect size, and variability. It cannot account for factors that were not measured or controlled during the study.
A Priori Power Analysis
Conducted before the study to determine the required sample size for achieving adequate statistical power to detect hypothesized effects.
Advantages of A Priori Analysis
Guides study design by ensuring sufficient sample size, minimizing the risk of Type II errors (false negatives), and maximizing the likelihood of detecting true effects.
Provides a proactive approach to study planning, helping researchers make informed decisions before data collection begins.
Allocation Ratio
he ratio of participants assigned to different experimental conditions or groups in a study.
Allocation ratios can influence statistical power, the precision of effect estimates, and the generalizability of study findings.
Imbalanced allocation rations
may affect the study’s ability to detect treatment effects and lead to unequal group sizes, potentially affecting the reliability and validity of the results.
The allocation ratio should be carefully chosen based on study objectives, ethical considerations, and practical constraints to ensure a balanced and efficient distribution of participants across conditions.
How One - Way and Two - Way ANOVA differ
Number of Factors: One-Way ANOVA involves analyzing the effect of a single factor on the dependent variable, while Two-Way ANOVA involves analyzing the effects of two factors simultaneously.
Main Effects and Interaction Effect: One-Way ANOVA examines only the main effect of the single factor on the dependent variable. In contrast, Two-Way ANOVA examines both the main effects of each factor and the interaction effect between the factors.
Interpretation: In One-Way ANOVA, significant differences indicate that at least one group mean is significantly different from the others. In Two-Way ANOVA, significant main effects indicate that there are differences in the dependent variable based on the levels of each factor, while a significant interaction effect indicates that the effect of one factor depends on the levels of the other factor.
