10 Replies to “How to Perform a Repeated Measures ANOVA By Hand”

1. Great, thank you.

   Reply

2. Hello
   There seems to be an error in the formula step 3
   SSS =(Σ rk^2/c) – (N^2/rc) : N should be Grand Total (as in the calculation), not number of obs. as explained underneath the formula
   and in the following calculation:
   ((742 + 422 + 622 + 922 + 682)/3) – (3382/(6)(3)) = 441.1:
   should be:
   ((742 + 422 + 622 + 922 + 682)/3) – (3382/(5)(3)) = 441.1

   Reply

   1. Good catch, just fixed it!

      Reply

3. how did you get 64.26 for the variance? what were the steps?

   Reply

4. Is this different from a 1-way ANOVA without repeated measures? And how so?

   Reply

5. I couldn’t get the right result on the first step…I found it =sst=538.596

   Reply

6. what about two way ANOVA with repetitions?

   Reply

   1. Two-way ANOVA with repetitions (also known as **repeated measures ANOVA**) is a statistical method used to analyze the effect of two categorical independent variables on a continuous dependent variable, while accounting for **within-subject variability**. This is particularly useful when measurements are repeated for the same subjects under different conditions.

      ### When to Use Two-Way ANOVA with Repetitions
      – **Two Independent Variables:** You have two categorical independent variables (factors). For example:
      – Factor A: Treatment type (e.g., Drug A, Drug B)
      – Factor B: Time (e.g., Day 1, Day 2, Day 3)
      – **Repetition:** You measure the dependent variable (response) multiple times for the same subjects across all combinations of the two factors.

      —

      ### Objectives
      1. **Main Effects:**
      – Determine if each independent variable has a significant effect on the dependent variable.
      2. **Interaction Effect:**
      – Examine if there’s a combined effect of the two factors on the dependent variable.
      3. **Subject Variability:**
      – Account for variability due to repeated measurements within subjects.

      —

      ### Assumptions
      1. **Normality:** The dependent variable is approximately normally distributed for each group.
      2. **Sphericity:** Variances of the differences between conditions (repeated measures) are equal.
      – This is tested using **Mauchly’s Test of Sphericity.**
      – If violated, use corrections like **Greenhouse-Geisser** or **Huynh-Feldt.**
      3. **Homogeneity of Variances:** The variance of the dependent variable is equal across groups.
      4. **Independence:** Observations within each group are independent of one another.

      —

      ### Steps to Perform Two-Way ANOVA with Repetitions
      #### 1. **Define Hypotheses**
      – **Main Effect A:** \( H_0: \mu_A = \mu_B \) (No effect of Factor A)
      – **Main Effect B:** \( H_0: \mu_B = \mu_C \) (No effect of Factor B)
      – **Interaction Effect:** \( H_0: \text{No interaction between Factors A and B} \)

      #### 2. **Calculate the ANOVA Table**
      – The table includes:
      – **Sum of Squares (SS):** Total variance decomposed into components (main effects, interaction, error).
      – **Degrees of Freedom (df):** Associated with each effect.
      – **Mean Squares (MS):** Variance estimates (SS/df).
      – **F-statistics and p-values:** Used to test hypotheses.

      #### 3. **Post-Hoc Analysis**
      – If the main effects or interaction are significant, perform post-hoc tests (e.g., Tukey’s HSD) to pinpoint the specific groups with differences.

      #### 4. **Report Results**
      – Include:
      – Main effects (F, df, p-value).
      – Interaction effect (F, df, p-value).
      – Post-hoc results (if applicable).

      —

      ### Example in Python
      Here’s a basic Python implementation using **statsmodels**:

      “`python
      import pandas as pd
      import numpy as np
      import statsmodels.api as sm
      from statsmodels.stats.anova import AnovaRM

      # Example data: subjects measured under 2 treatments (A, B) and 3 time points
      data = {
      ‘Subject’: np.repeat(np.arange(1, 11), 6),
      ‘Treatment’: np.tile(np.repeat([‘A’, ‘B’], 3), 10),
      ‘Time’: np.tile([‘T1’, ‘T2’, ‘T3’], 20),
      ‘Response’: np.random.randn(60) # Replace with actual data
      }
      df = pd.DataFrame(data)

      # Perform Repeated Measures ANOVA
      anova = AnovaRM(df, ‘Response’, ‘Subject’, within=[‘Treatment’, ‘Time’])
      result = anova.fit()

      print(result)
      “`

      —

      ### Example in R
      For those who use R, the `aov()` function or `ezANOVA()` from the `ez` package is helpful:

      “`R
      library(ez)

      # Example data frame
      data <- data.frame( Subject = rep(1:10, each = 6), Treatment = rep(c("A", "B"), each = 3, times = 10), Time = rep(c("T1", "T2", "T3"), times = 20), Response = rnorm(60) # Replace with actual data ) # Run two-way ANOVA with repetitions anova_result <- ezANOVA( data = data, dv = .(Response), wid = .(Subject), within = .(Treatment, Time) ) print(anova_result) ``` --- ### Key Takeaways - **Interpret Results Carefully:** Focus on the significance of main effects and interaction. - **Check Assumptions:** Always validate assumptions before drawing conclusions. - **Visualization:** Use interaction plots to visualize the effects and interactions. If you have specific data or need help interpreting results, feel free to share more details!

      Reply

7. Is your SST a different formula then what others use? I’m seeing:

   SST = Sum(x – xbar_total)^2 on other websites.

   Reply

   1. Hi Sarah…Great question — and yes, **SST** (Total Sum of Squares) can be presented using slightly different **formulations**, but they all fundamentally calculate the same thing: **the total variability in the data**.

      Let’s break this down clearly:

      —

      ### ✅ What is SST?

      **SST** measures the total variation in all observations around the **grand mean** (𝑥̄_total).
      So regardless of how it’s written, it’s capturing:
      > *“How far is each data point from the overall mean?”*

      —

      ### ✅ Common Formula You Saw:

      \[
      SST = \sum (x_{ij} – \bar{x}_{..})^2
      \]

      Where:
      – \(x_{ij}\) is each individual observation
      – \(\bar{x}_{..}\) is the grand mean (mean of all values across all groups and subjects)

      This is the standard textbook way of defining SST — and **yes**, this is the correct and most common formula.

      —

      ### ❓Are There Other Equivalent Forms?

      Yes! Some authors or sources might **reorganize the SST** using other formulas — for instance, using:
      – Raw scores squared minus correction factor
      – Matrix notation (especially in stats software)
      – Partitions that add up to SST: SSB (between groups) + SSW (within groups), or more specific:
      – SSA (effect of treatment/time)
      – SSS (effect of subject)
      – SSE (error)

      So you might see SST = SSA + SSS + SSE
      — which is still correct, just broken into components.

      —

      ### 🔍 Why the Confusion?

      Sometimes people mislabel one of the **component sums of squares** as the total SST. For example:
      – They might call **SSA** (treatment sum of squares) the “SST”
      – Or calculate the **within-subject variation only**

      So it’s always good to **check the context** in which the formula is being used.

      —

      ### ✅ Final Word

      If you’re doing **repeated measures ANOVA by hand**, stick with:
      \[
      SST = \sum (x_{ij} – \bar{x}_{..})^2
      \]

      That’s the **true total sum of squares** — from which you can then partition the variability as needed.

      —

      Reply