# Normality and repeated-measures ANOVA

## Assess normality of difference scores with three observations of a continuous outcome

The assumption of normality of difference scores is a statistical assumption that needs to be tested for when comparing three or more observations of a continuous outcome with repeated-measures ANOVA. Normality of difference scores for three or more observations is assessed using skewness and kurtosis statistics. In order to meet the statistical assumption of normality, skewness and kurtosis statistics should be below an absolute value of 2.0. If either skewness or a kurtosis statistic is above an absolute value of 2.0, then the continuous distribution is assumed to not be normal. Oftentimes, if the distributions for each observation of the outcome are normally distributed, the difference scores between the multiple observations will be normally distributed. Repeated-measures ANOVA should not be conducted when the assumption of normality of difference scores is violated. Repeated-measures ANOVA should only be conducted on normally distributed continuous outcomes.

### The steps for conducting skewness and kurtosis statistics on difference scores in SPSS

1. The data is entered in a within-subjects fashion.

2. Click

3. Click

4. In the

5. Click on the first observation of the continuous outcome to highlight it.

6. Click on the

7. Click on the "

8. Click on the second observation of the continuous outcome to highlight it.

9. Click on the

10. Click

11. Go to

12. Click

13. Click

14. In the

15. Click on the second observation of the continuous outcome to highlight it.

16. Click on the

17. Click on the "

18. Click on the third observation of the continuous outcome to highlight it.

19. Click on the

20. Click

21. Go to

22. Click

23. Drag the mouse pointer over the

24. Select

25. Click on the first difference score variable to highlight it. Example: "

26.

27. Click on the second difference score variable to highlight it. Example: "

28.

29. Click the

30. Deselect

31. Select the

32. Click

33. Select the

34. Click

2. Click

**.**__T__ransform3. Click

**.**__C__ompute Variable4. In the

**box, give the outcome variable a name with a "**__T__arget Variable:**D**" in front of it. Example: "**DOutcome**"5. Click on the first observation of the continuous outcome to highlight it.

6. Click on the

**arrow**to bring it into the**Num**box.__e__ric Expression:7. Click on the "

**-**" button or simply type "-" in the**Num**box.__e__ric Expression:8. Click on the second observation of the continuous outcome to highlight it.

9. Click on the

**arrow**to bring it into the**Num**box.__e__ric Expression:10. Click

**OK**.11. Go to

**Data View**, there is a new variable that contains the difference scores between the two observations of the continuous outcome with the variable name.12. Click

**.**__T__ransform13. Click

**.**__C__ompute Variable14. In the

**box, give the outcome variable a name with a "**__T__arget Variable:**D**" in front of it and a "**2**" at the end. Example: "**DOutcome2**"15. Click on the second observation of the continuous outcome to highlight it.

16. Click on the

**arrow**to bring it into the**Num**box.__e__ric Expression:17. Click on the "

**-**" button or simply type "-" in the**Num**box.__e__ric Expression:18. Click on the third observation of the continuous outcome to highlight it.

19. Click on the

**arrow**to bring it into the**Num**box.__e__ric Expression:20. Click

**OK**.21. Go to

**Data View**, there is a new variable that contains the difference scores between the two observations of the continuous outcome with the variable name.22. Click

**.**__A__nalyze23. Drag the mouse pointer over the

**D**drop-down menu.__e__scriptive Statistics24. Select

**.**__D__escriptives25. Click on the first difference score variable to highlight it. Example: "

**Doutcome**"26.

**Click the arrow button**to bring the variable over to the**box.**__V__ariable(s):27. Click on the second difference score variable to highlight it. Example: "

**Doutcome2**"28.

**Click the arrow button**to bring the variable over to the**box.**__V__ariable(s):29. Click the

**tab.**__O__ptions30. Deselect

**Mi**and__n__imum**Ma**boxes under the__x__imum**Dispersion**section.31. Select the

**and**__K__urtosis**Ske**boxes under the__w__ness**Distribution**section.32. Click

**Continue**.33. Select the

**Save standardi**box.__z__ed values as variables34. Click

**OK**.### The steps for interpreting the SPSS output for skewness and kurtosis of difference scores

1. Under the

**skewness**and**kurtosis**columns of the**Descriptive Statistics**table, if the**Statistic is less than an absolute value of 2.0**, then researchers can assume**normality**of the difference scores.### Was the assumption of normality of difference scores met for the repeated-mesures ANOVA?

## Hire A Statistician

**DO YOU NEED TO HIRE A STATISTICIAN?**

Eric Heidel, Ph.D., PStat** **will provide the following statistical consulting services for undergraduate and graduate students at $100/hour. Secure checkout is available with Stripe, Venmo, Zelle, or PayPal.

- Statistical Analysis
- Research Design
- Sample Size Calculations
- Diagnostic Testing and Epidemiological Calculations
- Survey Design and Psychometrics