One-Way Repeated Measures MANOVA
One-Way Repeated Measures MANOVA
Multivariate Analysis of Variance (MANOVA) is a statistical technique used to analyze the effects of independent variables on multiple dependent variables simultaneously. One-way repeated measures MANOVA is a specific type of MANOVA that is used when the dependent variables are measured on the same group of subjects over time or under different conditions.
Assumptions
Dependent variables are normally distributed.
Covariance matrices of the dependent variables are equal across all levels of the independent variable.
Observations are independent.
Procedure
1. Data Preparation: Collect data on multiple dependent variables measured on the same subjects over time or conditions.
2. Multivariate Test of Significance: Perform a multivariate test to determine if there is an overall effect of the independent variable on the set of dependent variables. This is done by comparing the within-subjects covariance matrix to the between-subjects covariance matrix.
3. Univariate Tests of Significance: If the multivariate test is significant, conduct univariate tests of significance on each individual dependent variable to identify which specific dependent variables are affected by the independent variable.
4. Effect Size Estimation: Calculate effect sizes to estimate the magnitude of the effect of the independent variable on the dependent variables.
Advantages
Allows for simultaneous analysis of multiple dependent variables related to a single construct.
Can identify the overarching effect of the independent variable on the set of dependent variables.
Reduces the risk of Type I error (false positives) by controlling multiple comparisons.
Disadvantages
Assumptions can be difficult to meet, especially the assumption of equal covariance matrices.
Interpretation can be complex due to the need to consider both multivariate and univariate results.
Requires a sufficient sample size to ensure statistical power.
Applications
One-way repeated measures MANOVA is commonly used in research settings where the effects of an independent variable (e.g., treatment, condition, time) are being investigated on multiple related dependent variables (e.g., anxiety, depression, stress). It is particularly useful when the dependent variables are correlated and the researcher wants to understand the overall effect of the independent variable on the construct being measured.
Example 1:
Objective: To investigate the effect of a new treatment on pain intensity in patients with chronic back pain.
Design:
One-way repeated measures MANOVA
Independent variable: Treatment group (new treatment vs. control)
Dependent variables: Pain intensity (measured at baseline, 1 week, 4 weeks, and 12 weeks)
Data:
The data consists of pain intensity scores for 50 patients randomly assigned to the new treatment (n=25) or control group (n=25).
Analysis:
1. Assumptions:
Normality of residuals
Sphericity (homogeneity of variances-covariances)
2. Hypothesis:
The null hypothesis is that there is no difference in pain intensity over time between the two treatment groups.
3. Results:
The MANOVA test results show:
Wilks' Lambda = 0.75, F(3, 46) = 6.05, p < 0.01
This indicates a significant overall effect of treatment group on pain intensity over time.
4. Post-hoc Analysis:
To determine specific differences between the groups at each time point, univariate F-tests are performed:
Baseline: p = 0.25 (no significant difference)
1 week: p = 0.02 (significant difference, new treatment has lower pain intensity)
4 weeks: p = 0.001 (significant difference, new treatment continues to have lower pain intensity)
12 weeks: p = 0.005 (significant difference, new treatment still has lower pain intensity)
5. Conclusion:
The study finds that the new treatment is effective in reducing pain intensity in patients with chronic back pain over a 12-week period compared to the control group.
Example 2:
A repeated measures-MANOVA was conducted to examine the effects of an 8-week diabetes management program on multiple outcome measures in diabetes patients.
Participants:
50 adults with type 2 diabetes were randomly assigned to two groups:
Intervention group (n=25) received the 8-week diabetes management program
Control group (n=25) received standard diabetes care
Repeated Measures Variables:
Hemoglobin A1c (HbA1c)
Fasting blood glucose (FBG)
Body mass index (BMI)
Systolic and diastolic blood pressure (SBP and DBP)
Baseline and Follow-Up Assessments:
All participants completed baseline assessments prior to the intervention. Follow-up assessments were conducted at 4 weeks, 8 weeks, and 12 weeks post-intervention.
Statistical Analysis:
The repeated measures-MANOVA was used to test for the overall effect of intervention group on the multiple outcome measures, while controlling for time (baseline, 4 weeks, 8 weeks, 12 weeks). Post hoc univariate ANOVAs with Bonferroni correction were conducted to examine significant group differences for each outcome variable.
Results:
The repeated measures-MANOVA revealed a significant overall effect of intervention group (Wilks' Lambda = 0.78, p < 0.001). The univariate ANOVAs indicated:
HbA1c: Significant reduction in HbA1c in the intervention group compared to the control group at 8 weeks (p < 0.01) and 12 weeks (p < 0.001).
FBG: Significant reduction in FBG in the intervention group compared to the control group at 4 weeks (p < 0.05), 8 weeks (p < 0.01), and 12 weeks (p < 0.001).
BMI: Significant reduction in BMI in the intervention group compared to the control group at 8 weeks (p < 0.05) and 12 weeks (p < 0.01).
SBP: Significant reduction in SBP in the intervention group compared to the control group at 8 weeks (p < 0.05) and 12 weeks (p < 0.01).
DBP: No significant difference in DBP between the intervention and control groups at any time point.
Conclusion:
The 8-week diabetes management program was effective in improving multiple outcome measures in diabetes patients, including HbA1c, FBG, BMI, and SBP. These findings suggest that the program may be a valuable tool for managing diabetes and improving patient health.
Example 3:
Purpose: To assess changes in glycemic control and lipid profiles over time in diabetes patients without a control group.
Design: Repeated measures-MANOVA.
Participants: 100 diabetes patients (40% male, mean age 55 years) were followed for 12 months.
Variables:
Glycemic control: HbA1c levels at baseline and 6-month follow-up.
Lipid profiles: Total cholesterol (TC), low-density lipoprotein cholesterol (LDL-C), high-density lipoprotein cholesterol (HDL-C), and triglycerides (TG) at baseline and 6-month follow-up.
Procedure: All participants underwent measurements of HbA1c and lipid profiles at baseline and after 6 months. Repeated measures-MANOVA was used to analyze the main effects of time (baseline vs. 6 months) and the interaction between time and subject.
Results:
Glycemic control: HbA1c levels significantly decreased from baseline (9.2 ± 1.2%) to 6 months (7.8 ± 1.0%; p < 0.001).
Lipid profiles: TC (192 ± 35 mg/dL vs. 185 ± 32 mg/dL), LDL-C (125 ± 28 mg/dL vs. 120 ± 26 mg/dL), and HDL-C (48 ± 12 mg/dL vs. 51 ± 13 mg/dL) showed improvements over time, while TG (152 ± 45 mg/dL vs. 155 ± 46 mg/dL) remained unchanged.
Interaction: No significant interaction was detected between time and subject, indicating that the observed changes were consistent across all participants.
Conclusion:
Repeated measures-MANOVA revealed significant improvements in glycemic control and lipid profiles over 6 months in diabetes patients without a control group. This suggests that lifestyle interventions or medication adjustments may have contributed to these positive outcomes. However, it is important to note that the absence of a control group limits the interpretation of these results.
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