A repeated measures analysis of variance (ANOVA)

  A repeated measures analysis of variance (ANOVA)




































A repeated measures analysis of variance (ANOVA) is a statistical procedure used to determine if there are significant differences between the means of two or more groups when the same subjects are measured multiple times. It is an extension of the one-way ANOVA that accounts for the correlation between the repeated measurements.


Assumptions


 The observations are normally distributed.

 The variances of the repeated measurements are equal.

 The repeated measurements are independent of each other.


Procedure


1. State the hypotheses:

   - Null hypothesis: There is no significant difference between the means of the groups.

   - Alternative hypothesis: There is a significant difference between the means of the groups.

2. Calculate the sum of squares:

   - Between groups: SSB

   - Within groups: SSW

   - Total: SST

3. Calculate the degrees of freedom:

   - Between groups: dfB = k - 1

   - Within groups: dfW = nk - k

   - Total: dfT = nk - 1

4. Calculate the mean square:

   - Between groups: MSB = SSB / dfB

   - Within groups: MSW = SSW / dfW

5. Calculate the F-test statistic:

   - F = MSB / MSW

6. Determine the p-value:

   - Obtain the p-value from the F-distribution with dfB and dfW degrees of freedom.

7. Make a decision:

   - If the p-value is less than the significance level, reject the null hypothesis and conclude that there is a significant difference between the means of the groups.

   - If the p-value is greater than the significance level, fail to reject the null hypothesis and conclude that there is no significant difference between the means of the groups.


Example


Suppose we want to compare the mean weight of rats in three different diets. We measure the weight of each rat at three different time points.


Data:


| Diet | Time 1 | Time 2 | Time 3 |

|---|---|---|---|

| Diet A | 100 | 110 | 120 |

| Diet B | 110 | 120 | 130 |

| Diet C | 120 | 130 | 140 |


ANOVA Table:


| Source | SS | df | MS | F | p-value |

|---|---|---|---|---|---|

| Diet | 600 | 2 | 300 | 10 | 0.001 |

| Time | 1200 | 2 | 600 | 20 | 0.0001 |

| Diet x Time | 200 | 4 | 50 | 1.67 | 0.15 |

| Within | 600 | 12 | 50 | | |

| Total | 2600 | 20 | | | |


Conclusion:


The F-test for the "Diet" effect is significant (p = 0.001), indicating that there is a significant difference between the mean weights of the rats in the three diets. The F-test for the "Time" effect is also significant (p = 0.0001), indicating that there is a significant difference in the mean weights of the rats over the three time points. However, the F-test for the "Diet x Time" interaction is not significant (p = 0.15), indicating that there is no significant interaction between diet and time.


Example 1:


Purpose: To test for differences in means across multiple measurements taken from the same subjects over time.


Example: A researcher conducts a study to investigate the effects of a new exercise program on the cardiovascular fitness of participants. Participants are assessed on their VO2 max (a measure of cardiovascular fitness) at three different time points: baseline, after 6 weeks of the exercise program, and after 12 weeks of the exercise program.


Hypotheses:

 H0: There is no difference in mean VO2 max across the three time points.

 Ha: There is a difference in mean VO2 max across the three time points.


Variables:

 Independent variable: Time (baseline, 6 weeks, 12 weeks)

 Dependent variable: VO2 max


Data Structure:

Each participant provides multiple measurements (three in this case, corresponding to the three time points). The data can be arranged in a rectangular data matrix, with participants as rows and time points as columns:


Participant | Baseline | 6 weeks | 12 weeks

-----------|---------|---------|---------

Participant 1 | 40 | 45 | 50

Participant 2 | 35 | 40 | 45

Participant 3 | 38 | 42 | 48



Assumptions:

 Normality of residuals

 Homogeneity of variances

 Sphericity (the assumption that the variances of the differences between each pair of means are equal)


Procedure:

1. Calculate the mean VO2 max for each participant at each time point.

2. Conduct a one-way repeated measures ANOVA, with Time as the independent variable and mean VO2 max as the dependent variable.

3. Test the sphericity assumption using Mauchly's test. If the assumption is violated, adjust the degrees of freedom using a correction factor.

4. Report the F-statistic, degrees of freedom, p-value, and effect size.


Results:

Suppose the ANOVA results indicate a significant F-statistic (p < 0.05). This suggests that there is a statistically significant difference in mean VO2 max across the three time points.


Post hoc Tests:

If the overall ANOVA is significant, the researcher may conduct post hoc tests to determine which specific time points differ from each other. Common post hoc tests for repeated measures ANOVAs include the Bonferroni test, the Tukey test, and the Dunnett test.


Interpretation:

The results of the repeated measures ANOVA indicate that the new exercise program has a positive effect on participants' cardiovascular fitness. VO2 max increases significantly from baseline to 6 weeks and from 6 weeks to 12 weeks.



Example 2: Repeated Measures ANOVA for Diabetes Trial

Data:

A clinical trial investigates the effects of two diabetes medications (Treatment A and Treatment B) on fasting blood glucose levels (mg/dL) over time. Patients are randomly assigned to one of the two treatment groups and have their blood glucose levels measured at baseline and at 3-month intervals for a total of 12 months.

Variables:

 Dependent variable: Fasting blood glucose levels (mg/dL)
 Independent variable (within-subjects): Time (baseline, 3 months, 6 months, 9 months, 12 months)
 Independent variable (between-subjects): Treatment group (Treatment A, Treatment B)

ANOVA Model:

Mixed-effects ANOVA with repeated measures on Time and Treatment group as a between-subjects factor


Assumptions:

 Sample size is sufficient (e.g., n ≥ 20 per group)
 Data is normally distributed
 Sphericity assumption is met (i.e., variances of the differences between repeated measures are equal)
 No significant outliers


Analysis:

The analysis is conducted using a mixed-effects ANOVA model with repeated measures on Time. The model fits the following parameters:

 Fixed effects: Treatment group, Time, Treatment x Time interaction
 Random effects: Intercept and Time (to account for within-subject variability)


Results:

Main Effect of Time:
 There is a significant main effect of Time (p < 0.05), indicating that blood glucose levels change over time.

Main Effect of Treatment:
 There is a significant main effect of Treatment (p < 0.05), indicating that Treatment A and Treatment B have different effects on blood glucose levels.

Treatment x Time Interaction:
 There is a significant Treatment x Time interaction (p < 0.05), indicating that the effect of Treatment on blood glucose levels varies over time.

Pairwise Comparisons:

To further explore the Treatment x Time interaction, pairwise comparisons are conducted between the two treatment groups at each time point. The results show that:

 At baseline, there is no significant difference in blood glucose levels between the two groups.
 At 3 months, Treatment A has a significantly lower blood glucose level than Treatment B (p < 0.05).
 At 6 months, 9 months, and 12 months, the difference in blood glucose levels between the two groups is no longer significant.

Conclusion:

The repeated measures ANOVA results indicate that:

 Both Treatment A and Treatment B lead to a decrease in blood glucose levels over time.
 Treatment A is more effective than Treatment B in lowering blood glucose levels at 3 months, but this difference disappears over time.

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