# Mixed-effects ANOVA

## Assess group differences across time or within-subjects

The

For example, let's say researchers are interested in the change of number of hours of reality TV watched (

The

**mixed-effects ANOVA**compares how a continuous outcome changes across time**(random effects)**between independent groups or levels**(fixed effects)**of a categorical predictor variable.For example, let's say researchers are interested in the change of number of hours of reality TV watched (

**continuous outcome)**between men and women**(fixed effect)**as the college football season leads into the college basketball season**(random effect)**. Gender is a "fixed" effect in that each participant is represented in one of the independent groups or levels of the "factor." Observations of number of hours of reality TV watched (let's say at the beginning of the college football season, then, at the beginning of the basketball season, and finally, at the end of March) is the "random" effect. Therefore, you can assess how the number of hours watched changes across time AND between different groups. You will be able to show evidence of how men change in number of hours watched from late August all the way until March of the next year and compare that level of change to how women change in number of hours watched along that same time frame.The

**marginal means and errors**for each level of the interaction should be presented in a mixed-effects ANOVA.**Significant main effects**must be further tested in a**post hoc**fashion to assess where among the levels of the interaction the significance exists and when the "**fixed"**or "**random**" effects are polychotomous (more than two "fixed" levels or observation of a variable) in the mixed-effects ANOVA analysis.The figure below depicts the use of mixed-effects ANOVA. There are two independent groups being compared on how they change across time in terms of an outcome taken at three time points.

### The steps for conducting a mixed-effects ANOVA in SPSS

1. The data is entered using a mixed method.

2. Click

3. Drag the cursor over the

4. Click on

5. In the

6. In the

7. Click the

8. Click

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

10. Click on the

11. Repeat Steps 8 and 9 until all the observations of the outcome are in the

12. Click on the "fixed" effect variable (groups, categorical variable) to highlight it.

13. Click on the

14. Click on the

15. Click on the "fixed" effect variable to highlight it.

16. Click on the

17. Click on the "random" effect variable to highlight it.

18. Click on the

19. Click the

20. Click

21. Click the

22. Look in the

23. Click on the

24. Repeat Steps 21 and 22 until all of the "fixed" and "random" effects are in the

25. Click on the

26. Look in the

27. Click

28. Click

2. Click

**.**__A__nalyze3. Drag the cursor over the

**drop-down menu.**__G__eneral Linear Model4. Click on

**.**__R__epeated Measures5. In the

**box, type the name of the outcome that is being observed multiple times or within-subjects.**__W__ithin-Subject Factor Name:6. In the

**Number of**box, type the number of observations of the outcome are being assessed. (Pretest -- Posttest = 2, Pretest -- Posttest -- Maintenance = 3, and so on)__L__evels:7. Click the

**button.**__A__dd8. Click

**Define**.9. Click on the first observation of the continuous outcome to highlight it.

10. Click on the

**arrow**to move the variable into the**box.**__W__ithin-Subjects Variables (Outcome name):11. Repeat Steps 8 and 9 until all the observations of the outcome are in the

**box.**__W__ithin-Subjects Variables (Outcome name):12. Click on the "fixed" effect variable (groups, categorical variable) to highlight it.

13. Click on the

**arrow**to move the variable into the**box.**__B__etween-Subjects Factor(s):14. Click on the

**Plo**button.__t__s15. Click on the "fixed" effect variable to highlight it.

16. Click on the

**arrow**to move the variable into the**box.**__S__eparate Lines:17. Click on the "random" effect variable to highlight it.

18. Click on the

**arrow**to move the variable into the**box.**__H__orizontal Axis:19. Click the

**button.**__A__dd20. Click

**Continue**.21. Click the

**button.**__O__ptions22. Look in the

**Estimated Marginal Means**table, in the**box. Click on the "fixed" effect variable to highlight it.**__F__actor(s) and Factor Interactions:23. Click on the

**arrow**to move the variable into the**box.**__D__isplay Means for:24. Repeat Steps 21 and 22 until all of the "fixed" and "random" effects are in the

**box.**__D__isplay Means for:25. Click on the

**C**box to select it.__o__mpare main effects26. Look in the

**Display**table, click on the**,**__D__escriptive statistics**,**__E__stimates of effect size**O**, and__b__served power**boxes to select them.**__H__omogeneity tests27. Click

**Continue**.28. Click

**OK**.### The steps for interpreting the SPSS output for a mixed-effects ANOVA

1. Look in the

2. Look in the

If this

If this

3. If the

If a

If a

4. If the

If the

If the

5. If researchers found a significant main effect, look in the

An example of this would be the half-life of a medication. As soon as the pill is ingested, the level of the medication in the bloodstream will significantly increase, but over time, the amount of the drug will dissipate as the body metabolizes the medicine.

A

A

6. If researchers found a significant main effect, scroll down to the

7. Based on having a significant interaction effect, the "

8. Lastly, there is a graph that serves a visual aid for understanding the

**Box's Test of Equality of Covariance Matrices**table. If the*p*-value in the**Sig.**row is**MORE THAN .05**, continue with the analysis. If the*p*-value is**LESS THAN .05**, reassess the observations for outliers and rerun the analysis.2. Look in the

**Mauchly's Test of Sphericity**table, under the**Sig.**column.If this

*p*-value is**MORE THAN .05**, researchers will interpret the*p*-values in the**Multivariate Tests**table.If this

*p*-value is**LESS THAN .05**, researchers will interpret a**Greenhouse-Geisser**corrected analysis in the**Tests of Within-Subjects Effects**table.3. If the

*p*-value was**MORE THAN .05**in the table above, look in the**Multivariate Tests**table of the output, under the**Sig.**column. These are the*p*-values that are interpreted for the change across time for all study participants and the interaction between the "fixed" and "random" effects.If a

*p*-value is**LESS THAN .05**, then researchers have evidence of a significant main effect.If a

*p*-value is**MORE THAN .05**, then researchers do not have evidence of a significant main effect.4. If the

*p*-value was**LESS THAN .05**(which happens more times than not), look in the**Tests of Within-Subjects Effects**table, under the**Sig.**column. Interpret the*p*-values for the change across time for all study participants and the interaction between the "fixed" and "random" effects that are in the**Greenhouse-Geisser**row.If the

*p*-value is**LESS THAN .05**, then researchers have evidence of a significant main effect.If the

*p*-value is**MORE THAN .05**, then researchers do not have evidence of a significant main effect.5. If researchers found a significant main effect, look in the

**Tests of Within-Subjects Contrasts**table, under the**Sig.**column. The*p*-values in this column are focused on testing linear and quadratic effects. A linear effect travels in one direction, either "up" or "down." A quadratic effect is an effect that goes "up" and then goes "down" or it will go "down" and then go back "up."An example of this would be the half-life of a medication. As soon as the pill is ingested, the level of the medication in the bloodstream will significantly increase, but over time, the amount of the drug will dissipate as the body metabolizes the medicine.

A

*p*-value of**LESS THAN .05**denotes either a significant linear or quadratic effect.A

*p*-value of**MORE THAN .05**means there was not a significant linear or quadratic effect.6. If researchers found a significant main effect, scroll down to the

**Estimated Marginal Means**section of the output. The means for both "fixed" and "random" effects are presented first. These*p*-values are testing the entire sample, without taking the other variable into consideration. For the "fixed" effect, look in the**Pairwise Comparisons**table, under the**Sig.**column. These are the*p*-values associated with comparing the independent groups or levels of the categorical "fixed" effect. For the "random" effect, look at the**Pairwise Comparisons**table, under the**Sig.**column. These are the*p*-values associated with the "random" effects or observations of the outcome. Any*p*-value that is**LESS THAN .05**means there is evidence of a significant difference between-groups or within-subjects. A*p*-value that is**MORE THAN .05**means that there is not a significant difference between-groups or within-subjects.7. Based on having a significant interaction effect, the "

**fixed"*"random**" table presents the marginal means and standard errors associated with the interaction.8. Lastly, there is a graph that serves a visual aid for understanding the

*p*-values.Click on the

**Download Database**and**Download Data Dictionary**buttons for a configured database and data dictionary for mixed-effects ANOVA.**Click on the****Validation of Statistical Findings**button to learn more about bootstrap, split-group, and jack-knife validation methods.## Statistician Services for Professionals

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