12.2 Multiple comparisons and post-hoc tests
3 min read•august 7, 2024
When conducting multiple statistical tests, the risk of false positives increases. help control this risk by adjusting significance levels. These methods ensure that overall error rates stay within acceptable limits, maintaining the integrity of research findings.
Post-hoc tests come into play after finding significant effects in analyses like . They allow for between group means, helping researchers pinpoint specific differences. Various post-hoc tests exist, each with unique strengths for different research scenarios.
Multiple Comparison Corrections
Controlling False Positives
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- (FWER) represents the probability of making at least one (false positive) among all hypotheses tested
- FWER increases as the number of hypotheses tested increases, leading to a higher chance of obtaining false positives
- Multiple comparison corrections aim to control the FWER by adjusting the significance level (α) for each individual hypothesis test
- Controlling FWER ensures that the overall Type I error rate is maintained at the desired level (usually 0.05) across all comparisons
Bonferroni and Holm-Bonferroni Corrections
- is a simple and conservative method for controlling FWER
- Divides the desired overall significance level (α) by the number of hypotheses tested (m) to obtain the adjusted significance level for each individual test:
- Ensures that the FWER is controlled at the desired level, but may be overly conservative, leading to reduced (increased )
- is a step-down procedure that improves upon the Bonferroni correction
- Orders the p-values from smallest to largest and compares each p-value to a sequentially adjusted significance level: , where is the rank of the p-value
- Offers more power than the Bonferroni correction while still controlling FWER
False Discovery Rate
- (FDR) is an alternative approach to multiple comparison corrections that controls the expected proportion of false positives among all significant results
- FDR is less conservative than FWER control methods and provides a better balance between Type I and Type II errors
- is a popular method for controlling FDR
- Orders the p-values from smallest to largest and compares each p-value to a sequentially adjusted threshold: , where is the rank of the p-value and is the total number of hypotheses tested
- Identifies the largest p-value that satisfies the condition and declares all hypotheses with smaller or equal p-values as significant
Post-hoc Tests
Pairwise Comparisons
- Post-hoc tests are used to make pairwise comparisons between group means after a significant overall effect has been found in an ANOVA
- Pairwise comparisons involve testing the differences between all possible pairs of group means
- Multiple comparison corrections are often applied to control the FWER or FDR when conducting pairwise comparisons
- Common post-hoc tests for pairwise comparisons include test, , and
Tukey's HSD and Scheffe's Tests
- Tukey's Honest Significant Difference (HSD) test is a widely used post-hoc test for pairwise comparisons
- Computes a critical value based on the studentized range distribution, which depends on the number of groups and the degrees of freedom for the error term
- Controls the FWER for all pairwise comparisons and is more powerful than the Bonferroni correction when the number of groups is large
- Scheffe's test is another post-hoc test that can be used for pairwise comparisons and complex contrasts
- Uses the F-distribution to compute a critical value and is more conservative than Tukey's HSD test
- Offers simultaneous confidence intervals for all possible contrasts, making it flexible for testing any linear combination of means
Dunnett's Test
- Dunnett's test is a specialized post-hoc test used when comparing several treatment groups to a single
- Computes a critical value based on the Dunnett's distribution, which accounts for the correlation between the comparisons to the control group
- Controls the FWER for the comparisons between each treatment group and the control group
- Useful in experiments where the main interest lies in comparing treatments to a control (e.g., drug trials comparing different doses to a placebo)
Key Terms to Review (21)
ANOVA: ANOVA, or Analysis of Variance, is a statistical method used to test differences between two or more group means. This technique helps determine if at least one of the group means is significantly different from the others, making it a powerful tool in experimental design for comparing multiple treatments or conditions.
Benjamini-Hochberg Procedure: The Benjamini-Hochberg procedure is a statistical method used to control the false discovery rate (FDR) when conducting multiple hypothesis tests. This technique allows researchers to identify which findings are statistically significant while reducing the chances of falsely declaring results as significant, especially in high-dimensional datasets common in big data contexts. By ranking p-values and comparing them to a threshold determined by the number of tests, it provides a more powerful approach to multiple comparisons compared to traditional methods.
Bonferroni correction: The Bonferroni correction is a statistical method used to counteract the problem of multiple comparisons by adjusting the significance level when conducting multiple tests. By dividing the desired alpha level (e.g., 0.05) by the number of comparisons being made, it helps to reduce the likelihood of Type I errors, which occur when a true null hypothesis is incorrectly rejected. This adjustment is particularly relevant in analyses involving multiple groups or factors, ensuring that findings remain statistically valid.
Control Group: A control group is a baseline group in an experiment that does not receive the experimental treatment or intervention, allowing researchers to compare it with the experimental group that does receive the treatment. This comparison helps to isolate the effects of the treatment and determine its effectiveness while accounting for other variables.
Dunnett's Test: Dunnett's Test is a statistical method used for making multiple comparisons between a control group and several treatment groups while controlling for Type I error. This test is particularly useful in scenarios where researchers want to determine if the means of various treatment groups differ significantly from the mean of a single control group, rather than comparing all treatment groups against each other. By focusing on the control group, Dunnett's Test minimizes the chances of false positives that can occur when conducting multiple comparisons.
False Discovery Rate: The false discovery rate (FDR) is the expected proportion of false positives among all the significant findings in a statistical analysis. It plays a crucial role in hypothesis testing, especially when dealing with large datasets and multiple comparisons, where the chances of incorrectly rejecting a null hypothesis increase. Managing FDR is essential in high-dimensional experiments and when applying multiple comparisons adjustments to control the number of false discoveries.
Family-wise error rate: The family-wise error rate (FWER) is the probability of making one or more type I errors when performing multiple hypothesis tests. This concept is crucial when analyzing data from studies involving multiple comparisons, as the risk of falsely identifying a significant effect increases with the number of tests conducted. Therefore, controlling the FWER is essential to ensure the validity of conclusions drawn from statistical analyses.
Holm-Bonferroni Method: The Holm-Bonferroni method is a statistical procedure used to control the family-wise error rate when conducting multiple hypothesis tests. It is a stepwise approach that adjusts the significance levels for individual tests based on their rank order, making it a more powerful alternative to the traditional Bonferroni correction. This method helps researchers determine which hypotheses can be considered statistically significant while reducing the risk of Type I errors.
Homogeneity of variance: Homogeneity of variance refers to the assumption that different groups in a statistical test have the same variance or spread in their data. This concept is crucial when performing analyses like ANOVA, as violating this assumption can lead to incorrect conclusions about the differences between groups. Ensuring homogeneity of variance helps validate the results and interpretations derived from statistical tests, making it a fundamental consideration when comparing multiple groups.
Interaction effect: An interaction effect occurs when the relationship between one independent variable and a dependent variable changes depending on the level of another independent variable. This concept highlights how different factors can work together to produce unique outcomes, demonstrating that the combined influence of multiple variables may not simply be additive, but can actually modify each other's effects in significant ways.
Main Effect: A main effect refers to the direct influence of an independent variable on a dependent variable in an experimental design. This concept is crucial in understanding how different levels of a factor affect outcomes, separate from any interaction effects that may occur between factors. Recognizing main effects helps researchers interpret the results of complex experiments and evaluate the significance of individual variables in various designs.
Multiple comparison corrections: Multiple comparison corrections are statistical techniques used to reduce the chances of obtaining false-positive results when conducting multiple hypothesis tests. When multiple comparisons are made, the likelihood of incorrectly rejecting the null hypothesis increases, which can lead to misleading conclusions. These corrections adjust the significance levels or p-values to account for the increased risk of Type I errors, ensuring that findings are more reliable and valid.
Normality: Normality refers to the assumption that the data being analyzed follows a normal distribution, which is a bell-shaped curve where most of the observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions. This concept is crucial in many statistical methods, as violations of this assumption can lead to misleading results, especially when comparing means across groups or examining relationships between variables.
P-value adjustment: P-value adjustment is a statistical technique used to modify the significance levels of p-values to control for Type I error rates when multiple comparisons are made. When performing multiple tests, the chance of incorrectly rejecting a null hypothesis increases, so adjusting p-values helps to maintain the overall error rate at an acceptable level. This concept is particularly important in contexts where multiple hypotheses are tested simultaneously, such as in sequential analyses and post-hoc testing.
Pairwise comparisons: Pairwise comparisons refer to a statistical method used to compare the means of different groups against one another. This technique helps determine which specific groups differ from each other after conducting an overall test, like ANOVA. It is particularly useful in understanding the relationships and differences among multiple treatments or conditions within a study.
Randomization: Randomization is the process of assigning participants or experimental units to different groups using random methods, which helps eliminate bias and ensures that each participant has an equal chance of being placed in any group. This technique is crucial in experimental design, as it enhances the validity of results by reducing the influence of confounding variables and allowing for fair comparisons between treatments.
Scheffe's test: Scheffe's test is a statistical method used for making multiple comparisons among group means after conducting an Analysis of Variance (ANOVA). It provides a way to control the family-wise error rate, allowing researchers to determine if there are significant differences between specific group pairs or combinations, while remaining conservative in the findings. This test is particularly useful when the number of groups is large or when the researcher wants to conduct a variety of comparisons without inflating the risk of Type I error.
Statistical Power: Statistical power is the probability that a statistical test will correctly reject a false null hypothesis, which means detecting an effect if there is one. Understanding statistical power is crucial for designing experiments as it helps researchers determine the likelihood of finding significant results, influences the choice of sample sizes, and informs about the effectiveness of different experimental designs.
Tukey's HSD: Tukey's HSD (Honestly Significant Difference) is a post-hoc test used to determine which specific group means are different after conducting an ANOVA. It helps in comparing all possible pairs of means while controlling the overall error rate, making it particularly useful in situations with multiple comparisons. This test provides a straightforward way to identify significant differences between groups when the initial analysis indicates that at least one group mean is significantly different from others.
Type I Error: A Type I error occurs when a null hypothesis is incorrectly rejected, leading to the conclusion that there is an effect or difference when none actually exists. This mistake can have serious implications in various statistical contexts, affecting the reliability of results and decision-making processes.
Type II Error Rate: The type II error rate is the probability of failing to reject a null hypothesis that is actually false, often denoted as $$\beta$$. This error rate is significant in understanding the sensitivity of a statistical test, as it directly relates to the power of the test, which is defined as 1 minus the type II error rate. A high type II error rate indicates that a test might miss a true effect or difference when it exists, especially when multiple comparisons are being conducted.