4.4 Multiple comparison procedures
4 min read•august 13, 2024
Multiple comparison procedures are crucial when conducting several hypothesis tests simultaneously. They help control the , preventing an inflated risk of false positives that could compromise your study's validity.
and are two common methods for managing multiple comparisons. Bonferroni is simple but conservative, while Tukey's HSD is more powerful for pairwise comparisons after . Choose wisely based on your specific research needs.
Multiple Comparison Procedures
Recognizing the Need for Multiple Comparison Procedures
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- Multiple hypothesis tests increase the probability of making at least one (rejecting a true null hypothesis), known as the familywise error rate
- As the number of hypothesis tests increases, the likelihood of observing a rare event (Type I error) by chance alone also increases, leading to false positives
- Multiple comparison procedures are statistical methods designed to control the familywise error rate and maintain the overall significance level (α = 0.05) across all hypothesis tests
- Without multiple comparison procedures, the probability of making at least one Type I error can be much higher than the desired significance level, compromising the validity of the study's conclusions
Implications of Multiple Testing
- Failing to control for familywise error rate can lead to overestimating the significance of results, drawing incorrect conclusions, and compromising the reproducibility of the study
- Multiple comparison procedures aim to control familywise error rate by adjusting the significance level for each individual test or the critical values for rejecting the null hypothesis
Familywise Error Rate
Definition and Concept
- Familywise error rate () represents the probability of making at least one Type I error (false positive) among a family of hypothesis tests
- FWER increases as the number of hypothesis tests increases, even if each individual test is conducted at the desired significance level (α = 0.05)
Relationship with Number of Hypothesis Tests
- The relationship between the FWER and the number of hypothesis tests (m) can be approximated by: , assuming independence among tests
- For example, with 10 independent hypothesis tests and α = 0.05, the FWER is approximately 0.40, meaning there is a 40% chance of making at least one Type I error
Bonferroni vs Tukey's HSD
Bonferroni Correction
- Bonferroni correction divides the desired familywise significance level (α) by the number of hypothesis tests (m) to obtain an adjusted significance level (α_adjusted) for each individual test
- The Bonferroni-adjusted significance level is calculated as:
- Each individual hypothesis test is then conducted using the adjusted significance level, ensuring that the FWER is controlled at the desired level (α = 0.05)
- Bonferroni correction is simple to apply but can be conservative, especially when the number of tests is large or the tests are not independent
Tukey's Honestly Significant Difference (HSD)
- Tukey's HSD is a multiple comparison procedure specifically designed for pairwise comparisons following a significant ANOVA result
- Tukey's HSD calculates a critical value (q_critical) based on the number of groups, degrees of freedom, and the desired familywise significance level
- The absolute differences between group means are compared to the critical value multiplied by the standard error of the mean to determine statistical significance
- Tukey's HSD is more powerful than Bonferroni correction for pairwise comparisons but is limited to post-hoc analyses following ANOVA
Interpreting Multiple Comparisons
Adjusted Significance Levels and Critical Values
- When using a multiple comparison procedure, interpret the results based on the adjusted significance levels or critical values specific to the chosen method
- Bonferroni correction: Compare the p-value of each individual test to the adjusted significance level (α_adjusted). If the p-value is less than α_adjusted, reject the null hypothesis for that specific test
- Tukey's HSD: Compare the absolute difference between group means to the critical value multiplied by the standard error of the mean. If the absolute difference exceeds this threshold, conclude that the two groups are significantly different
Reporting Results and Drawing Conclusions
- Clearly state the multiple comparison procedure used, the adjusted significance levels or critical values, and the specific hypothesis tests that were found to be significant or non-significant based on the procedure
- Draw conclusions that are supported by the results of the multiple comparison procedure, considering the context of the study and the limitations of the chosen method
- Be cautious when interpreting non-significant results, as multiple comparison procedures may increase the risk of Type II errors (failing to reject a false null hypothesis) due to their conservative nature
Key Terms to Review (16)
ANOVA: ANOVA, or Analysis of Variance, is a statistical method used to compare the means of three or more groups to determine if at least one group mean is statistically different from the others. This technique helps in understanding the impact of one or more factors on a dependent variable and connects deeply to various statistical principles, including hypothesis testing, parametric assumptions, and the necessity for multiple comparison adjustments when significant differences are found.
Bonferroni Correction: The Bonferroni Correction is a statistical method used to address the problem of multiple comparisons by adjusting the significance level to reduce the chances of obtaining false-positive results. This technique involves dividing the desired alpha level (typically 0.05) by the number of tests being conducted, which helps to control the overall Type I error rate. By doing so, it ensures that findings from parametric or non-parametric tests remain reliable, especially when multiple comparison procedures are involved, such as in one-way ANOVA and repeated measures ANOVA scenarios.
Box Plots: Box plots, also known as whisker plots, are graphical representations used to summarize and visualize the distribution of a dataset through its quartiles. They provide insights into the central tendency, variability, and presence of outliers within the data, making them particularly useful for comparing multiple groups side by side, which is essential in multiple comparison procedures.
Cohen's d: Cohen's d is a statistical measure used to quantify the effect size, or the magnitude of difference, between two groups. It expresses the difference in means between the groups in terms of standard deviations, making it a useful tool for comparing results across different studies and tests, whether parametric or non-parametric. By providing a standardized measure of effect size, Cohen's d can help interpret results in multiple comparison situations, as well as within more complex analyses such as ANCOVA and MANOVA, while also fitting into the framework of robust estimation and hypothesis testing.
Eta-squared: Eta-squared is a measure of effect size that indicates the proportion of variance in a dependent variable that can be attributed to the effects of an independent variable. It is particularly useful in the context of analysis of variance (ANOVA) and helps researchers understand how much of the total variability in their data is explained by their experimental conditions, making it essential for assessing the practical significance of findings, especially during multiple comparisons.
Familywise error rate: The familywise error rate (FWER) is the probability of making one or more Type I errors when performing multiple hypothesis tests simultaneously. It reflects the risk of incorrectly rejecting at least one null hypothesis across a family of tests, which can inflate the overall error rate as the number of comparisons increases. Controlling for the FWER is essential in statistical analyses involving multiple comparisons to maintain the integrity and validity of the findings.
FWER: FWER, or Family-Wise Error Rate, is the probability of making one or more Type I errors when conducting multiple hypothesis tests. It is a crucial concept in multiple comparison procedures, as it addresses the increased likelihood of incorrectly rejecting at least one null hypothesis as the number of tests increases. Keeping FWER in check helps maintain the integrity of statistical conclusions by controlling for these errors during hypothesis testing.
Homogeneity of variance: Homogeneity of variance refers to the assumption that different groups in a statistical test have similar variances or spread in their data. This concept is crucial because many statistical tests, particularly parametric ones, rely on this assumption to ensure that results are valid and reliable. When this assumption is met, it supports the integrity of comparisons made between groups, influencing the interpretation of various analyses, such as comparisons among group means or in more complex models.
Interaction plots: Interaction plots are graphical representations that help visualize how two or more independent variables affect a dependent variable, especially when their effects are not additive. They reveal the nature of interactions between factors, showing how the levels of one factor influence the relationship between the levels of another factor on the outcome being measured. These plots are crucial for understanding complex relationships in data, particularly when conducting multiple comparison procedures, as they can highlight significant interactions that may affect the results.
John Tukey: John Tukey was an influential American statistician known for his pioneering work in data analysis, particularly in developing methods that made statistical analysis more accessible. His contributions to multiple comparison procedures, logistic regression, and repeated measures ANOVA transformed how statisticians interpret and analyze data, helping to shape modern statistical practice.
Linear models: Linear models are statistical tools used to describe the relationship between one dependent variable and one or more independent variables through a linear equation. They assume that changes in the independent variable(s) will result in proportional changes in the dependent variable, making them useful for prediction and analysis. In the context of multiple comparison procedures, linear models help to evaluate the differences among groups when assessing multiple hypotheses simultaneously.
Mixed models: Mixed models are statistical models that incorporate both fixed effects and random effects, allowing for the analysis of data that have multiple levels of variability. They are particularly useful in situations where observations are not independent, such as when data is collected from groups or clusters, making them ideal for hierarchical or longitudinal data. Mixed models can handle complexities in data structures, enabling researchers to make more accurate inferences about the effects of predictors.
Normality: Normality refers to a statistical concept where data is distributed in a symmetrical, bell-shaped pattern known as a normal distribution. This property is crucial for many statistical methods, as it underpins the assumptions made for parametric tests and confidence intervals, ensuring that results are valid and reliable.
P-value adjustment: P-value adjustment refers to the statistical techniques used to modify p-values when multiple comparisons are made in order to control for the increased risk of Type I errors. When conducting multiple tests, the chance of obtaining at least one statistically significant result just by chance increases, so adjustments are necessary to maintain the integrity of the results. Common methods for adjusting p-values include the Bonferroni correction and the Benjamini-Hochberg procedure.
Tukey's HSD: Tukey's HSD (Honestly Significant Difference) is a statistical test used to identify which specific group means are significantly different from each other after performing an ANOVA. This test is particularly useful for making multiple comparisons between group means while controlling the overall Type I error rate, ensuring that the likelihood of incorrectly rejecting a true null hypothesis remains at a desired level.
Type I Error: A Type I error occurs when a null hypothesis is incorrectly rejected, indicating that there is a significant effect or difference when, in reality, none exists. This error is crucial in understanding the reliability of hypothesis testing, as it directly relates to the alpha level, which sets the threshold for determining significance.