11.4 Post-hoc Analysis for Two-Way ANOVA
4 min read•july 30, 2024
Two-Way ANOVA helps us spot differences between groups, but it doesn't tell us which ones. That's where post-hoc analysis comes in. It lets us compare specific groups and figure out exactly where the differences lie.
Post-hoc tests are crucial because they control for errors when making multiple comparisons. They help us avoid false positives and draw more accurate conclusions about our data. Understanding post-hoc analysis is key to getting the most out of Two-Way ANOVA results.
Post-Hoc Analysis in ANOVA
Purpose and Importance
Top images from around the web for Purpose and Importance
r - How to obtain the results of a Tukey HSD post-hoc test in a table showing grouped pairs ... View original
Is this image relevant?
Question about Tukey post-hoc ANOVA test results - Cross Validated View original
Is this image relevant?
How to Perform ANOVA in Python View original
Is this image relevant?
r - How to obtain the results of a Tukey HSD post-hoc test in a table showing grouped pairs ... View original
Is this image relevant?
Question about Tukey post-hoc ANOVA test results - Cross Validated View original
Is this image relevant?
1 of 3
Top images from around the web for Purpose and Importance
r - How to obtain the results of a Tukey HSD post-hoc test in a table showing grouped pairs ... View original
Is this image relevant?
Question about Tukey post-hoc ANOVA test results - Cross Validated View original
Is this image relevant?
How to Perform ANOVA in Python View original
Is this image relevant?
r - How to obtain the results of a Tukey HSD post-hoc test in a table showing grouped pairs ... View original
Is this image relevant?
Question about Tukey post-hoc ANOVA test results - Cross Validated View original
Is this image relevant?
1 of 3
- Conducted after a significant main effect or interaction effect is found in a two-way ANOVA to determine which specific group means differ significantly from each other
- Control the familywise error rate, the probability of making at least one Type I error (false positive) when conducting multiple pairwise comparisons
- Essential for identifying the specific differences between group means that contribute to the overall significant effect found in the two-way ANOVA
- Without post-hoc analysis, researchers cannot determine which specific group means differ significantly, limiting the interpretability and practical implications of the findings
- Provide a more detailed understanding of the nature of the significant effects found in the two-way ANOVA, allowing researchers to draw more precise conclusions and make more targeted recommendations based on the results
Controlling Type I Error
- Post-hoc tests are designed to control the familywise error rate, which increases with the number of pairwise comparisons conducted
- Familywise error rate is the probability of making at least one Type I error (false positive) across all pairwise comparisons
- Without controlling for the familywise error rate, the likelihood of finding a significant difference by chance alone increases as more comparisons are made
- Post-hoc tests adjust the significance level for each comparison to maintain the overall Type I error rate at the desired level (usually 0.05)
- Examples of post-hoc tests that control the familywise error rate include , , and Scheffe's test
Choosing Post-Hoc Tests
Factors to Consider
- Number of pairwise comparisons: Some post-hoc tests (Bonferroni) are more conservative and appropriate when the number of comparisons is relatively small, while others (Tukey's HSD) are suitable for a larger number of comparisons
- Sample size: Post-hoc tests may have different performance depending on the sample size; some tests (Tukey's HSD) are more robust to unequal sample sizes than others
- Assumption of homogeneity of variances: Some post-hoc tests (Tukey's HSD) assume equal variances across groups, while others (Games-Howell) are more robust to violations of this assumption
- Research question and specific comparisons of interest: Some post-hoc tests (Dunnett's test) are designed for comparing treatment groups to a control group, while others (Tukey's HSD) compare all possible pairs of means
Commonly Used Post-Hoc Tests
- Tukey's Honestly Significant Difference (HSD) test: Compares all possible pairs of group means while controlling the familywise error rate; appropriate when sample sizes are equal and the assumption of homogeneity of variances is met
- Bonferroni correction: Adjusts the significance level for each pairwise comparison to control the familywise error rate; more conservative than Tukey's HSD and appropriate when the number of comparisons is small
- Scheffe's test: A more conservative post-hoc test that is robust to violations of the assumption of homogeneity of variances; suitable when the number of comparisons is large
- Dunnett's test: Compares each treatment group to a control group while controlling the familywise error rate; appropriate when the research question specifically involves comparisons to a control condition
Interpreting Post-Hoc Results
Presenting Results
- Post-hoc test results are typically presented as a matrix or table showing the pairwise comparisons between group means and their corresponding p-values
- A significant (p < .05) indicates that the difference between the two group means is statistically significant, while a non-significant p-value suggests that the difference is not significant
- In addition to p-values, post-hoc test results may include mean differences, standard errors, confidence intervals, and effect sizes to provide a more comprehensive understanding of the findings
Drawing Conclusions
- When interpreting post-hoc test results, researchers should focus on the specific pairwise comparisons that are relevant to their research question and hypotheses
- Consider the magnitude of the differences between group means, in addition to their statistical significance, to assess the practical importance of the findings
- Be cautious not to overinterpret non-significant differences or to make causal inferences without proper experimental design
- Discuss the implications of the post-hoc test results in the context of the research question, previous literature, and the limitations of the study
- Clear and concise reporting of post-hoc test results, along with effect sizes and confidence intervals, can enhance the interpretability and replicability of the findings
Key Terms to Review (16)
Bonferroni Correction: The Bonferroni correction is a statistical adjustment made to account for the increased risk of Type I errors when performing multiple comparisons. This method involves dividing the desired alpha level (significance level) by the number of comparisons being made, which helps to control the overall error rate. By adjusting the significance threshold, the Bonferroni correction ensures that findings remain reliable, particularly in contexts where multiple hypotheses are tested simultaneously.
Clinical trials: Clinical trials are research studies conducted with human participants to evaluate the effectiveness and safety of new medical interventions, treatments, or devices. They are essential in determining how well a drug or treatment works in people, and the results can influence medical guidelines and policies.
Cohen's d: Cohen's d is a statistical measure used to quantify the effect size, or the strength of a difference, between two groups. It provides a standardized way to interpret the magnitude of an effect in research studies, often used in the context of comparing means from different groups, such as after conducting analyses like Two-Way ANOVA. This measure helps researchers understand not just whether an effect exists, but how large that effect is in practical terms.
Experimental Psychology: Experimental psychology is a branch of psychology that focuses on the scientific study of behavior and mental processes through controlled experiments. It seeks to understand how various factors influence psychological phenomena, such as perception, cognition, emotion, and learning, often employing statistical methods to analyze data and draw conclusions.
F-statistic: The f-statistic is a ratio used in statistical hypothesis testing to compare the variances of two populations or groups. It plays a crucial role in determining the overall significance of a regression model, where it assesses whether the explained variance in the model is significantly greater than the unexplained variance, thereby informing decisions on model adequacy and variable inclusion.
Factorial design: Factorial design is a statistical experiment design that investigates the effects of two or more factors by considering all possible combinations of the factor levels. This design allows researchers to study not only the main effects of each factor but also the interaction effects between factors, which can provide deeper insights into the data.
Homogeneity of variance: Homogeneity of variance refers to the assumption that different samples in a statistical test have similar variances. This concept is crucial for ensuring the validity of various statistical analyses, as violating this assumption can lead to inaccurate results and interpretations. When applying methods such as ANOVA, it's essential to check this assumption to ensure that any differences found among group means are not influenced by unequal variances.
Interaction Effects: Interaction effects occur when the relationship between one predictor variable and the response variable changes depending on the level of another predictor variable. This concept is crucial in understanding complex relationships within regression and ANOVA models, revealing how multiple factors can simultaneously influence outcomes.
Levels of Factors: Levels of factors refer to the specific conditions or values that independent variables can take in an experiment or statistical analysis. In the context of a two-way ANOVA, these levels represent the different groups or categories being compared, allowing researchers to understand how changes in one factor affect the response variable across the different levels of another factor.
Main effects: Main effects refer to the individual impact of each predictor variable on the outcome variable in a statistical model. They help to understand how different factors independently influence the response, without considering the interaction between them. Analyzing main effects is crucial when evaluating the contributions of various predictors and can guide decisions regarding model specifications and interpretations.
Normality: Normality refers to the assumption that data follows a normal distribution, which is a bell-shaped curve that is symmetric around the mean. This concept is crucial because many statistical methods, including regression and ANOVA, rely on this assumption to yield valid results and interpretations.
P-value: A p-value is a statistical measure that helps to determine the significance of results in hypothesis testing. It indicates the probability of obtaining results at least as extreme as the observed results, assuming that the null hypothesis is true. A smaller p-value suggests stronger evidence against the null hypothesis, often leading to its rejection.
Partial eta squared: Partial eta squared is a measure of effect size that indicates the proportion of total variance in a dependent variable that is attributable to a specific independent variable, while controlling for other variables. It helps in understanding the strength of the relationship between an independent variable and a dependent variable in the context of ANOVA and ANCOVA analyses, providing a clearer picture of how much variance is explained by each factor after accounting for others.
R: In statistics, 'r' is the Pearson correlation coefficient, a measure that expresses the strength and direction of a linear relationship between two continuous variables. It ranges from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation. This measure is crucial in understanding relationships between variables in various contexts, including prediction, regression analysis, and the evaluation of model assumptions.
SPSS: SPSS, which stands for Statistical Package for the Social Sciences, is a software tool widely used for statistical analysis and data management in social science research. It provides users with a user-friendly interface to perform various statistical tests, including regression, ANOVA, and post-hoc analyses, making it essential for researchers to interpret complex data efficiently.
Tukey's HSD: Tukey's HSD (Honestly Significant Difference) is a statistical test used for multiple comparisons following an ANOVA, specifically designed to determine which group means are significantly different from each other. It is particularly useful because it controls the overall Type I error rate when comparing multiple groups, ensuring that the probability of falsely identifying at least one significant difference remains low. This test provides a straightforward way to identify significant pairwise differences while maintaining robustness against the assumptions of equal variances and normality.