7.3 Repeated measures ANOVA
5 min read•august 21, 2024
is a powerful statistical tool used in biostatistics to analyze data from experiments where subjects are measured multiple times. It extends one-way ANOVA to account for within-subject variability, making it ideal for longitudinal studies and treatment effect assessments over time.
This method offers increased statistical power by reducing individual differences, as each participant serves as their own control. It allows for smaller sample sizes while maintaining robust analysis, making it particularly valuable in clinical trials with limited patient populations.
Overview of repeated measures ANOVA
- Analyzes data from experimental designs where subjects are measured multiple times
- Extends one-way ANOVA to account for within-subject variability in longitudinal studies
- Crucial in biostatistics for assessing treatment effects over time or under different conditions
Within-subjects vs between-subjects designs
- Within-subjects designs measure each participant under all conditions, reducing individual differences
- Between-subjects designs assign different groups to each condition, controlling for order effects
- Repeated measures ANOVA primarily used for within-subjects designs, offering increased statistical power
- Allows for smaller sample sizes while maintaining robust statistical analysis
Assumptions of repeated measures ANOVA
Sphericity assumption
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- Requires equality of variances of the differences between all possible pairs of groups
- Assessed using Mauchly's test of
- Violation leads to increased
- Corrections (, ) applied when sphericity is violated
Normality assumption
- Assumes normally distributed data within each level of the within-subjects factor
- Checked using visual inspection (Q-Q plots) or statistical tests ()
- Robust to minor violations with sufficiently large sample sizes
Homogeneity of variance
- Assumes equal variances across all levels of the within-subjects factor
- Tested using or
- Less critical in repeated measures designs due to within-subjects nature
Conducting repeated measures ANOVA
Data organization
- Requires long format data structure for most statistical software
- Each row represents a single observation for a participant at a specific time point
- Includes columns for participant ID, time point or condition, and dependent variable
Calculation of F-statistic
- Compares the variance between conditions to the variance within conditions
- calculated as:
- MS represents Mean Square, obtained by dividing Sum of Squares by
Degrees of freedom
- Between-subjects df = k - 1 (k = number of conditions)
- Within-subjects df = N - k (N = total number of observations)
- Error df = (n - 1)(k - 1) (n = number of participants)
Post-hoc tests for repeated measures
Pairwise comparisons
- Conducted to determine which specific conditions differ significantly
- Common methods include t-tests or Tukey's Honestly Significant Difference (HSD)
- Helps identify patterns or trends in the data across different time points or conditions
Bonferroni correction
- Adjusts p-values to control for Type I error rate in multiple comparisons
- Calculated by dividing the desired alpha level by the number of comparisons
- Conservative approach, may increase
- Alternative methods include Holm's sequential Bonferroni or False Discovery Rate (FDR)
Effect size in repeated measures ANOVA
Partial eta squared
- Measures the proportion of variance explained by the factor, excluding other factors
- Calculated as:
- Values typically interpreted as small (0.01), medium (0.06), or large (0.14)
Cohen's f
- Alternative measure, particularly useful for power analysis
- Calculated as:
- Interpreted as small (0.1), medium (0.25), or large (0.4)
Advantages of repeated measures design
Increased statistical power
- Requires fewer participants to detect significant effects compared to between-subjects designs
- Controls for individual differences by using each participant as their own control
- Particularly beneficial in clinical trials with limited patient populations
Reduced subject variability
- Eliminates between-subject variability from the error term
- Increases sensitivity to detect treatment effects
- Allows for more precise estimation of within-subject changes over time
Limitations and considerations
Carryover effects
- Occur when the effect of one condition influences subsequent conditions
- Mitigated through counterbalancing or randomization of condition order
- May require washout periods in certain experimental designs (pharmacological studies)
Practice effects
- Improvement in performance due to repeated exposure to tasks or measures
- Can confound treatment effects, especially in cognitive or skill-based assessments
- Addressed through careful experimental design and statistical control methods
Repeated measures ANOVA vs other methods
One-way ANOVA vs repeated measures
- One-way ANOVA used for between-subjects designs, repeated measures for within-subjects
- Repeated measures offers higher statistical power and requires fewer participants
- One-way ANOVA assumes independence of observations, not suitable for longitudinal data
Paired t-test vs repeated measures
- Paired t-test limited to two time points or conditions
- Repeated measures ANOVA extends to three or more time points or conditions
- Repeated measures more efficient for multiple comparisons, reducing Type I error rate
Reporting results
Tables and figures
- Present descriptive statistics (means, standard deviations) for each condition in a table
- Use line graphs or box plots to visualize changes across time points or conditions
- Include error bars (standard error or confidence intervals) to represent variability
Interpretation of findings
- Report F-statistic, degrees of freedom, , and effect size
- Describe the nature and direction of significant effects
- Discuss practical implications of findings in the context of the research question
- Address any violations of assumptions and their potential impact on results
Software implementation
SPSS for repeated measures ANOVA
- Accessed through the General Linear Model > Repeated Measures menu
- Requires specification of within-subjects factor(s) and dependent variable
- Offers options for post-hoc tests, effect sizes, and assumption checks
- Provides syntax for reproducibility and customization of analyses
R for repeated measures ANOVA
- Conducted using packages like
,ez
, orafexlme4
function inezANOVA()
package specifically designed for repeated measuresez- Allows for flexible modeling of complex designs and custom contrasts
- Provides tools for visualization (ggplot2) and advanced post-hoc analyses
Real-world applications
Clinical trials
- Assessing drug efficacy over multiple time points (baseline, 1 month, 3 months, 6 months)
- Evaluating changes in physiological measures (blood pressure, cholesterol) pre- and post-intervention
- Comparing different treatment regimens in crossover designs
Longitudinal studies
- Tracking developmental changes in cognitive abilities across childhood and adolescence
- Monitoring disease progression or recovery in patients over extended periods
- Investigating the long-term effects of public health interventions on population health metrics
Key Terms to Review (30)
ANOVA table: An ANOVA table is a structured display of the results from an Analysis of Variance (ANOVA) test, which helps determine if there are significant differences between the means of multiple groups. This table breaks down the sources of variance in the data, typically including the variation between groups and within groups, along with associated statistics such as sums of squares, degrees of freedom, mean squares, and F-statistics. It plays a crucial role in repeated measures ANOVA by showing how repeated measurements impact the variability among subjects over time or conditions.
Bartlett's Test: Bartlett's Test is a statistical test used to determine if there are significant differences in the variances among multiple groups. It is particularly important when comparing groups with repeated measures, as it helps assess the assumption of homogeneity of variances, which is crucial for the validity of various statistical analyses, including ANOVA.
Between-subjects design: A between-subjects design is an experimental setup where different participants are assigned to each condition of an experiment, ensuring that each participant experiences only one level of the independent variable. This type of design helps eliminate potential carryover effects from one condition to another, allowing researchers to assess the impact of the independent variable on distinct groups. It's crucial for comparing outcomes across groups without the influence of previous experiences from other conditions.
Bonferroni Correction: The Bonferroni correction is a statistical adjustment made to account for the increased risk of Type I errors when multiple comparisons are conducted. It involves dividing the significance level (alpha) by the number of tests being performed, thus making it more stringent and reducing the chances of incorrectly rejecting the null hypothesis. This method is particularly relevant in the context of various analysis techniques, where multiple groups or conditions are compared.
Cohen's f: Cohen's f is a measure of effect size used to quantify the strength of the relationship between variables in statistical analyses, particularly in the context of ANOVA. It helps researchers understand how much variance in the dependent variable can be explained by the independent variable(s). A larger Cohen's f indicates a more substantial effect and is useful for comparing effects across different studies or experiments.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. It is a crucial concept in statistics, influencing the calculation of variability, the performance of hypothesis tests, and the interpretation of data across various analyses. Understanding degrees of freedom helps in determining how much information is available to estimate parameters and influences the shape of probability distributions used in inferential statistics.
Effect Size: Effect size is a quantitative measure that reflects the magnitude of a phenomenon or the strength of the relationship between variables. It helps researchers understand how meaningful a statistically significant result is, bridging the gap between statistical significance and practical significance in research findings.
F-ratio: The f-ratio is a statistical measure used in ANOVA (Analysis of Variance) that compares the variance between group means to the variance within groups. This ratio helps determine if the means of different groups are statistically significantly different from each other. A higher f-ratio indicates that the group means are more spread out compared to the variation within the groups, suggesting that at least one group mean is significantly different.
F-statistic: The f-statistic is a ratio used in statistical tests to compare the variances between two or more groups. It helps determine if the group means are significantly different from one another, and it is a key component in various analyses including multiple linear regression, ANOVA, and other hypothesis testing methods. This statistic plays an essential role in assessing the overall significance of the model being tested.
Factorial design: Factorial design is a type of experimental setup that allows researchers to evaluate the effects of multiple independent variables simultaneously, as well as their interactions. This design is crucial for understanding how different factors influence an outcome and provides a more comprehensive view than one-variable-at-a-time approaches. By incorporating two or more factors into the experiment, factorial design helps reveal complex relationships and interactions that might otherwise be overlooked.
Georgy Snedecor: Georgy Snedecor was a prominent statistician known for his work in agricultural statistics and the development of statistical methods. He is best recognized for creating the analysis of variance (ANOVA) framework, which is crucial in understanding the variation among group means and the significance of differences among groups, particularly in repeated measures contexts.
Greenhouse-Geisser: Greenhouse-Geisser is a correction applied in repeated measures ANOVA that adjusts the degrees of freedom to account for violations of sphericity. This correction is crucial because it helps to provide more accurate statistical results when the assumption of sphericity—which assumes that the variances of the differences between all combinations of related groups are equal—is not met. By adjusting the degrees of freedom, the Greenhouse-Geisser method aims to minimize Type I error rates in hypothesis testing.
Homogeneity of Variance: Homogeneity of variance refers to the assumption that different samples have the same variance. This concept is crucial when conducting various statistical tests, as violations of this assumption can lead to incorrect conclusions. Inconsistent variances can affect the results of hypothesis testing, particularly in comparing groups or conditions.
Huynh-Feldt: The Huynh-Feldt correction is a method used in repeated measures ANOVA to adjust the degrees of freedom when the sphericity assumption is violated. This correction helps provide more accurate F-tests, ensuring that the results are reliable even when the variances of the differences between the levels of the within-subjects factors are not equal. It is especially important in studies with multiple measurements on the same subjects over time or under different conditions.
Interaction effect: An interaction effect occurs when the relationship between one independent variable and the dependent variable changes depending on the level of another independent variable. This means that the effect of one variable is not constant across all levels of the other variable, revealing a more complex relationship among the factors being studied. Understanding interaction effects helps in accurately interpreting results, particularly in analyses that involve multiple variables.
Levene's Test: Levene's Test is a statistical procedure used to assess the equality of variances across different groups. This test is particularly important in the context of repeated measures ANOVA, as it checks the assumption that the variances of the different groups being compared are equal, which is crucial for ensuring the validity of the analysis. If the assumption of equal variances is violated, it can lead to inaccurate conclusions about the relationships between the groups being studied.
Main effect: A main effect is the primary influence of one independent variable on a dependent variable in an experimental study. It highlights how changes in a single factor impact outcomes, while ignoring interactions with other factors. Understanding main effects is crucial for analyzing results in complex designs, allowing researchers to draw conclusions about individual variables without confounding influences from others.
Normality: Normality refers to the condition where data is symmetrically distributed around the mean, forming a bell-shaped curve known as the normal distribution. This concept is crucial because many statistical tests and methods assume that the data follow a normal distribution, which influences the validity of the results and conclusions drawn from analyses.
P-value: A p-value is a statistical measure that helps to determine the significance of results in hypothesis testing. It represents the probability of observing the obtained results, or more extreme results, assuming that the null hypothesis is true. This value provides insight into the strength of the evidence against the null hypothesis and is critical for making decisions about the validity of claims in various statistical tests.
Partial eta squared: Partial eta squared is a measure of effect size that quantifies the proportion of the total variance in a dependent variable that is attributed to an independent variable, while controlling for other variables in the analysis. This statistic helps to understand the strength of the relationship between variables in various analyses, particularly in designs like two-way ANOVA and repeated measures ANOVA, where multiple factors or repeated observations can complicate interpretation. It gives researchers insight into the significance of effects when comparing groups or conditions, making it a valuable tool for interpreting results.
R: In statistics, 'r' typically refers to the correlation coefficient, which quantifies the strength and direction of the linear relationship between two variables. Understanding 'r' is essential for assessing relationships in various statistical analyses, such as determining how changes in one variable may predict changes in another across multiple contexts.
Repeated measures ANOVA: Repeated measures ANOVA is a statistical technique used to analyze data where the same subjects are measured multiple times under different conditions. This method allows researchers to assess differences in means while controlling for variability among subjects, making it ideal for experiments where measurements are taken at various time points or conditions.
Ronald Fisher: Ronald Fisher was a pioneering statistician and geneticist, known for his significant contributions to the field of statistics, particularly in experimental design and analysis of variance. His work laid the groundwork for modern statistical methods, including the development of the ANOVA framework, which is essential for understanding the relationships between multiple groups in repeated measures designs.
Shapiro-Wilk: The Shapiro-Wilk test is a statistical test used to determine whether a dataset follows a normal distribution. This test is particularly useful for small sample sizes and provides a quantitative measure of normality, which is crucial when applying various statistical techniques that assume normality, like repeated measures ANOVA.
Sphericity: Sphericity refers to the assumption in repeated measures ANOVA that the variances of the differences between all combinations of related groups are equal. This concept is crucial because it affects the validity of the statistical tests used in repeated measures designs, ensuring that the results are reliable and meaningful. When sphericity is violated, it can lead to inaccurate conclusions about the relationships between groups.
SPSS: SPSS (Statistical Package for the Social Sciences) is a powerful software tool widely used for statistical analysis, data management, and data visualization in various fields such as social sciences, health, and market research. Its user-friendly interface allows researchers to perform complex statistical tests and analyses, making it essential for interpreting data results related to various statistical methods.
Tukey's HSD: Tukey's Honestly Significant Difference (HSD) is a statistical test used to determine if there are significant differences between the means of multiple groups after conducting an ANOVA. It helps identify which specific groups' means are different when a significant effect is found, making it a post-hoc analysis method. This test controls the family-wise error rate and is commonly applied in various contexts, including one-way ANOVA, two-way ANOVA, and repeated measures designs.
Type I Error Rate: The Type I error rate is the probability of rejecting a true null hypothesis, often denoted by the Greek letter alpha (α). It reflects the risk of finding a statistically significant result when there is no actual effect or difference. In the context of repeated measures ANOVA, managing the Type I error rate is crucial since multiple comparisons can inflate this risk, leading to misleading conclusions.
Type II Error Rate: The Type II error rate, often denoted as \(\beta\), refers to the probability of failing to reject a null hypothesis when it is actually false. This concept is crucial in statistical hypothesis testing, particularly in contexts where repeated measures are involved, as it can affect the interpretation of results and the power of the tests used. Understanding Type II error rate helps researchers assess the risks of overlooking a true effect or difference, especially when analyzing data that involves multiple measurements from the same subjects.
Within-subjects design: Within-subjects design is a type of experimental design where the same participants are used in all conditions of an experiment. This approach allows researchers to control for individual differences by comparing participants to themselves across different conditions, which can enhance the sensitivity of the statistical tests used to analyze the data. The design is particularly useful when studying changes over time or differences in responses to various treatments or conditions.