Mixed Model/Design ANOVA
A mixed design ANOVA is a statistical technique used to analyze data when two or more independent variables are involved, one of which is categorical (between-subjects) and the other is continuous (within-subjects).
Characteristics:
Two or more independent variables:
Between-subjects variable: Categorical variable with discrete levels.
Within-subjects variable: Continuous variable with multiple measurements for each participant.
Dependent variable: Measures the outcome or response of interest.
Types of Mixed Design ANOVAs:
Simple: Two-way mixed design with one between-subjects variable and one within-subjects variable.
Factorial: Complex mixed design with multiple between-subjects variables and multiple within-subjects variables.
Assumptions:
Independence: Data points are independent of each other.
Normality: Dependent variable is normally distributed within each cell.
Homogeneity of variance: Variances are equal across cells.
Sphericity: Covariances between within-subjects variables are equal.
Procedure:
1. Conduct a between-subjects ANOVA to assess the effect of the categorical variable.
2. Conduct a within-subjects ANOVA to assess the effect of the continuous variable.
3. Conduct interaction analyses to determine if the effects of the two variables interact.
Interpretation:
The main effect of the between-subjects variable indicates whether there are differences between the groups on the dependent variable.
The main effect of the within-subjects variable indicates whether there are differences between the levels of the continuous variable on the dependent variable.
The interaction effect indicates whether the relationship between the between-subjects variable and the within-subjects variable is different across levels of the between-subjects variable.
Advantages:
Can analyze the effects of both categorical and continuous variables.
Provides insight into how the two variables interact.
More powerful than separate between-subjects and within-subjects analyses.
Disadvantages:
Requires a large sample size.
Assumptions can be difficult to meet.
Can be complex to analyze.
Example 1:Mixed Design ANOVA in Diabetes Medicine
Objective: To investigate the effects of medication (drug vs. placebo) and lifestyle intervention (diet and exercise vs. usual care) on glycemic control in individuals with type 2 diabetes.
Design:
Between-subjects factor: Medication (2 levels: drug, placebo)
Within-subjects factor: Lifestyle intervention (2 levels: diet and exercise, usual care)
Repeated measures: Glycemic control assessed at baseline and follow-up (e.g., HbA1c, fasting blood glucose)
Data Collection:
Individuals with type 2 diabetes are randomly assigned to one of the four groups:
Drug + diet and exercise
Drug + usual care
Placebo + diet and exercise
Placebo + usual care
Glycemic control is measured at baseline and after a 6-month intervention period.
Analysis:
A mixed design ANOVA is conducted to analyze the data. The between-subjects effect of medication and the within-subjects effect of lifestyle intervention are tested. The interaction between medication and lifestyle intervention is also examined.
Results:
Medication: Significant main effect of medication, with individuals receiving the drug having lower glycemic control (e.g., lower HbA1c) compared to those receiving placebo.
Lifestyle intervention: Significant main effect of lifestyle intervention, with individuals in the diet and exercise group having lower glycemic control compared to those in the usual care group.
Interaction: Significant interaction between medication and lifestyle intervention. The beneficial effect of lifestyle intervention was greater for individuals receiving the drug compared to those receiving placebo.
Interpretation:
Medication (drug) is effective in improving glycemic control in individuals with type 2 diabetes.
Lifestyle intervention (diet and exercise) is also effective in improving glycemic control.
The combination of medication and lifestyle intervention is more beneficial than either intervention alone.
Conclusions:
Both medication and lifestyle intervention are important in the management of type 2 diabetes.
The most effective approach involves combining medication with a comprehensive lifestyle intervention program.
Example 2: Mixed Design ANOVA for Hypertension Medicine
Objective: To investigate the effects of two independent variables (medication and dosage) on blood pressure measurements in hypertensive patients.
Design: Mixed design ANOVA with:
Between-subjects factor: Medication (two levels: drug A vs. drug B)
Within-subjects factor: Dosage (three levels: low, medium, high)
Participants: A sample of hypertensive patients is randomly assigned to receive one of the two medications at three different dosages.
Procedure:
Patients are measured for baseline blood pressure.
They receive their assigned medication and dosage for a specified period.
After the treatment period, their blood pressure is measured again.
Data Analysis:
Model:
Y = μ + α + β + γ + αγ + βγ + ε
where:
Y = blood pressure measurement
μ = overall mean
α = effect of medication
β = effect of dosage
γ = within-subjects effect of dosage
αγ = interaction between medication and dosage
βγ = interaction between dosage and medication
ε = error term
Assumptions:
Normality of residuals
Homogeneity of variances
Independence of observations
Sphericity (for the within-subjects factor)
Hypothesis Testing:
Main effects: Test the effects of medication and dosage on blood pressure.
Interaction effect: Test the interaction between medication and dosage on blood pressure. This interaction would indicate that the effect of dosage differs depending on the medication used.
Post-hoc Tests:
If significant main effects or interactions are found, conduct post-hoc tests (e.g., Tukey's HSD) to determine which specific group comparisons are significant.
Expected Results:
If the medication has a significant effect, one medication may be more effective than the other in lowering blood pressure.
If the dosage has a significant effect, different dosages may have different effects on blood pressure.
If there is a significant interaction, the effects of dosage may differ depending on which medication is used.
Implications:
The results of this ANOVA can provide valuable information regarding the effectiveness of different hypertension medications and the optimal dosage for individual patients. This information can aid medical practitioners in making informed treatment decisions for hypertensive patients.
Example 3: Mixed Design ANOVA for Hypertension Medicine
Independent Variable: Medicine (Between-Subjects)
Levels: Drug A, Drug B, Placebo
Dependent Variable: Blood Pressure Reduction (Within-Subjects)
Levels: Baseline, Month 1, Month 3
Mixed Design ANOVA:
Mixed Design ANOVA for Hypertension Medicine
------------------------------------------
Source of Variation | df | MS | F | p |
------------------------------------------
Medicine (Between) | 2 | 150 | 10.0 | 0.05
Time (Within) | 2 | 100 | 5.0 | 0.01
Medicine x Time (Interaction) | 4 | 50 | 2.5 | 0.05
Error | 60 | 20 |
------------------------------------------
Results:
Medicine Effect: There is a significant difference in blood pressure reduction between the different medicine groups (F(2,60) = 10.0, p = 0.05).
Time Effect: There is a significant improvement in blood pressure over time (F(2,120) = 5.0, p = 0.01).
Medicine x Time Interaction: There is a significant interaction between medicine and time, indicating that the rate of blood pressure reduction varies across the different medicine groups (F(4,120) = 2.5, p = 0.05).
Conclusion:
The mixed design ANOVA suggests that both medicine type and time have significant effects on blood pressure reduction. Additionally, the significant interaction indicates that the effectiveness of the medicine varies over time.
Example 4:
Example of a Mixed Design ANOVA Related to Pain Score
Design:
Independent variable: Impact group (impact vs. non-impact)
Dependent variable: Pain score
Time: Repeated measure with 3 time points (e.g., baseline, 1 week, 2 weeks)
Data:
| Subject | Group | Time | Pain Score |
|---|---|---|---|
| 1 | Impact | Baseline | 5 |
| 1 | Impact | 1 Week | 3 |
| 1 | Impact | 2 Weeks | 2 |
| 2 | Non-Impact | Baseline | 7 |
| 2 | Non-Impact | 1 Week | 6 |
| 2 | Non-Impact | 2 Weeks | 5 |
Hypothesis:
The impact group will have lower pain scores compared to the non-impact group over time.
Statistical Analysis:
Mixed Design ANOVA:
Time as a within-subjects factor
Impact group as a between-subjects factor
Results:
F(1, 18) = 10.23, p = 0.004
Interpretation:
There is a significant main effect of impact group, indicating that the impact group had lower pain scores compared to the non-impact group.
Post-Hoc Tests (e.g., Tukey's HSD):
t(18) = 2.56, p = 0.019
t(18) = 2.89, p = 0.009
Follow-Up Analysis:
The impact group had significantly lower pain scores than the non-impact group at both 1 week (p = 0.019) and 2 weeks (p = 0.009).
There was no significant difference in pain scores between the two groups at baseline.
Conclusion:
The results support the hypothesis that the impact group had lower pain scores compared to the non-impact group over time. This suggests that the impact intervention was effective in reducing pain levels.
Example 5:
Mixed Design ANOVA Example: Pain Score in IPACT vs. Non-IPACT Groups
Research Question: Does participation in an IPACT (Interpersonal Psychotherapy for Chronic Pain) program reduce pain scores over time compared to a non-IPACT control group?
Design:
• Independent Variable 1: Group (between-subjects factor):
* IPACT group
* Non-IPACT group
• Independent Variable 2: Time (within-subjects factor):
* Time point 1 (baseline)
* Time point 2 (mid-treatment)
* Time point 3 (post-treatment)
• Dependent Variable: Pain score (measured using a validated pain scale)
Hypotheses:
• Main Effect of Group: There is a significant difference in pain scores between the IPACT and non-IPACT groups.
• Main Effect of Time: There is a significant change in pain scores over time.
• Interaction Effect: There is a significant interaction between group and time, meaning the change in pain scores over time differs between the IPACT and non-IPACT groups.
Example Data:
| Group | Time Point | Pain Score |
|---|---|---|
| IPACT | Baseline | 7 |
| IPACT | Mid-treatment | 5 |
| IPACT | Post-treatment | 3 |
| Non-IPACT | Baseline | 8 |
| Non-IPACT | Mid-treatment | 7 |
| Non-IPACT | Post-treatment | 6 |
| ... | ... | ... |
Analysis:
A mixed design ANOVA would be conducted to analyze the data. The analysis would test for:
• Main effects: Whether there is a significant difference in pain scores between the IPACT and non-IPACT groups (Group main effect), and whether there is a significant change in pain scores over time (Time main effect).
• Interaction effect: Whether the change in pain scores over time differs between the IPACT and non-IPACT groups.
Interpretation:
• Significant main effect of Group: Indicates that the IPACT group had significantly different pain scores than the non-IPACT group, regardless of time.
• Significant main effect of Time: Indicates that pain scores changed significantly over time, regardless of group.
• Significant interaction effect: Indicates that the change in pain scores over time was different between the IPACT and non-IPACT groups. This suggests that the IPACT program had a different effect on pain scores compared to the control group.
Conclusion:
The results of the mixed design ANOVA would provide evidence for or against the hypothesis that participation in an IPACT program reduces pain scores over time compared to a non-IPACT control group.
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