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### ANALYSIS OF VARIANCE (ANOVA) PT-III

One-way within-subjects ANOVA
A one-way within-subjects ANOVA allows you to determine if there is a relationship between a categorical IV and a continuous DV, where each subject is measured at every level of the IV. Within-subject ANOVA should be used whenever want to compare 3 or more groups where the same subjects are in all of the groups. To perform a within-subject ANOVA in SPSS you must
have your data set organized so that the subject is the unit of analysis and you have different variables containing the value of the DV at each level of your within-subjects factor.

To perform a within-subject ANOVA in SPSS:
Choose Analyze thenGoto General linear model thenGoto Repeated measures.
Type the name of the factor in the Within-Subjects Factor Name box.
Type the number of groups the factor represents in the Number of Levels box.
Click the Define button.
Move the variables representing the different levels of the within-subjects factor to the Within-Subjects Variables box.
Click the OK button.

The output of this analysis will contain the following sections.
Within-Subjects Factors. Tells you what variables represent the different levels of your factor.
Multivariate Tests. Contains the first test of your within-subject factor, making use of multivariate analysis. Multivariate analysis actually provides a matrix of results, which would naturally be very difficult to interpret on its own. Statisticians have therefore developed four common methods of converting the results of a multivariate test to an Ftest. The most commonly used and accepted statistic is Wilk’s Lambda. More recently statisticians have used the Pillai-Bartlett Trace, since research has indicated that this statistic is somewhat more robust to violations of the model assumptions than Wilk.s lambda. It is therefore recommended that you base your conclusions on one of these two statistics. The decision is often moot, however, since the different statistics almost always produce similar F conversions. When you report your results in a paper, you typically state the method you used, provide the resulting F-statistic, its degrees of freedom, and its p-value.
Mauchly’s Test of Sphericity. The second way of testing your within-subjects factor is called repeated measures. This method more powerful than multivariate analysis but makes the additional assumption that the correlations between your within-subjects levels are all the same. This table provides a test of this, also called the assumption of sphericity. A significant test means that sphericity has been violated, indicating that you should not use the uncorrected results of a repeated-measures analysis. You can either use the multivariate results, or you can apply a correction for the violation to the repeated-measures results. If your within-subjects factor only has two levels you will always have perfect sphericity.
Tests of Within-Subjects Effects. Contains the results of a repeated-measures test of your within-subjects factor. If the assumption of sphericity is satisfied, you examine the test provided in the row labeled Sphericity Assumed. If the sphericity assumption is violated, you should examine the tests provided in either the row titled Greenhouse-Geisser or Huynh-Feldt, which provide tests corrected for your assumption violations. If you observe a significant effect, this indicates (like in between-subjects ANOVA) that there is a difference somewhere among the means. As with between-subjects ANOVA, post-hoc comparisons are required to pinpoint which means are different from each other.
Tests of Within-Subjects Contrasts. Provides the results of polynomial contrasts among your within-subject conditions. These can provide you with some information about the specific nature of the relation between your factor and the DV. However, the results will be meaningless unless your groups have some type of ordinal relation with each other.
Tests of Between-Subjects Effects. This section is not typically examined when performing a one-way within-subjects ANOVA.

You can ask SPSS to provide you with the means within each level of your within-subjects factor by clicking the Options button in the variable selection window and moving your within-subjects variable to the Display Means For box. This will add a section to your output titled Estimated Marginal Means containing a table with a row for each level of your factor. The values within each row provide the mean, standard error of the mean, and the boundaries for a 95% confidence interval around the mean for observations within that cell.

Multifactor within-subjects ANOVA
Just as you can use ANOVA to examine multiple between-subjects factors, so can you use it to examine multiple within-subjects factors. Multifactor ANOVA can determine the independent influence of each of your IVs on the DV (main effects) as well as the extent to which the effect of an IV on your DV depends on the level of other IVs in your model (interactions).

To perform an ANOVA with two or more within-subject factors in SPSS
Choose Analyze thenGoto General linear model thenGoto Repeated measures.
Next you define the within-subject factor(s). For each factor
o Enter the name of the factor in the Within-Subject Factor Name box.
o Enter the number of levels the factor has in the Number of Levels box.
Click the Define button.
The next thing you will need to do is identify which variables correspond to the particular combinations of your within-subject factors. The Within-Subject Variables box in the next window will contain something that looks like the following.
__?__ (1,1)
__?__ (1,2)
__?__ (2,1)
__?__ (2,2)
__?__ (3,1)
__?__ (3,2)
The numbers on the right-hand side correspond to the levels of your factors. The first number corresponds to the level of your first factor, while the second number corresponds to the level of your second factor. The order of your factors will be listed above the Within-Subjects Variables box. For each combination represented in this box you should select the corresponding variable in the box on the left-hand side and then press the arrow button next to the Within-Subjects Variables box. It doesn.t matter what level of a variable you decide to associate with a number on this screen, but you must be sure that you are consistent. For example, the variable you put in the (3,1) slot should have the same level of the first factor as the variable in the (3,2) slot, and the variable you put in the (1,2) slot should have the same level of the second factor as the variable you put in the (2,2) slot.
Click the OK button.

The output of this analysis will include the following sections.
Within-Subjects Factors. Tells you what variables represent each combination of your factors.
Multivariate Tests. Contains the first test of your main effects and interactions, making use of multivariate analysis. The test statistics provided here can be interpreted in the same way as described in the One-way within-subjects ANOVA section. A significant main effect indicates that at least two of the groups composing that factor have significantly different means. A significant interaction between a set of factors indicates that the influence of any one factor involved in the interaction significantly changes under different levels of the other factors in the interaction.
Mauchly’s Test of Sphericity. Provides a test of the sphericity assumption for each of your within-subject terms (including both main effects and interactions).
Tests of Within-Subjects Effects. Contains the repeated-measures tests of your withinsubjects terms. If the assumption of sphericity is satisfied, you examine the test provided in the row labeled Sphericity Assumed. If the sphericity assumption is violated, you should examine the tests provided in either the row titled Greenhouse-Geisser or Huynh-
Feldt, which provide tests corrected for your assumption violations.
Tests of Within-Subjects Contrasts. Provides the results of polynomial contrasts among your within-subject conditions. These will have no meaning unless your levels have an ordinal relationship.
Tests of Between-Subjects Effects. This section is not typically examined when performing a multifactor within-subjects ANOVA. You can ask SPSS to provide you with the means within the levels of your main effects or your interactions by clicking the Options button in the variable selection window and moving the appropriate term to the Display Means For box. This will add a section to your output titled Estimated Marginal Means containing a table for each main effect or interaction in your model. The table will contain a row for each cell within the effect. The values within each row provide the mean, standard error of the mean, and the boundaries for a 95% confidence interval around the mean for observations within that cell.

Post-hoc comparisons involving within-subject factors. SPSS does not provide any options that allow you to easily compare the different levels of a within-subject factor. However, there is a relatively easy way to do this. Recall that the different levels of your within-subject factor will be stored in different variables in SPSS. If you want to see if there is a difference between two particular levels of a within-subject factor, you can create a new variable that is a difference between the two variables corresponding to the different levels you want to compare. To see if there is a significant difference, all you need to do is test whether the mean of the difference variable is significantly different from zero. This method is completely valid, and is called a modern within-subject contrast. You can apply a Bonferroni correction to prevent inflation of your Type I error by dividing the alpha of each contrast by the total number of post-hoc contrasts you perform from the same analysis.

Mixed ANOVA
Mixed ANOVA allow you to simultaneously examine the effect of within-subjects and betweensubjects factors within the same experiment. It allows you to detect all of the following types of effects.
Main effects of between-subjects factors
Main effects of within-subjects factors
Interactions involving between-subjects factors
Interactions involving within-subjects factors
Interactions involving both between-subjects and within-subjects factors To perform a mixed ANOVA in SPSS
Choose Analyze thenGoto General linear model thenGoto Repeated measures.
Next you define the within-subject factor(s). For each factor
o Enter the name of the factor in the Within-Subject Factor Name box.
o Enter the number of levels the factor has in the Number of Levels box.
Click the Define button.
Identify which variables are associated with each combination of your within-subjects conditions as described above in the Multifactor within-subjects ANOVA section.
Move any between-subjects IVs to the Between-subjects factor(s) box.
Click the OK button.

The output from this analysis will contain the following sections.
Within-Subjects Factors. Tells you what variables represent each combination of your within-subjects factors.
Between-Subjects Factors. Lists how many subjects are in the combination of each of your between-subjects factors.
Multivariate Tests. Contains multivariate tests of the main effects of your within-subjects factors as well as interactions that contain at least one within-subjects factor. The test statistics provided here can be interpreted in the same way as described in the One-way within-subjects ANOVA section. A significant main effect indicates that at least two of the groups composing that factor have significantly different means. A significant interaction between a set of factors indicates that the influence of any one factor involved in the interaction significantly changes under different levels of the other factors in the interaction.
Mauchly’s Test of Sphericity. Provides a test of the sphericity assumption for each of your within-subject terms (including the main effects of your within-subjects factors as well as interactions involving at least one within-subjects factor).
Tests of Within-Subjects Effects. Contains repeated-measures tests of the main effects of your within-subjects factors as well as interactions that contain at least one withinsubjects factor. If the assumption of sphericity is satisfied, you examine the test provided in the row labeled Sphericity Assumed. If the sphericity assumption is violated, you should examine the tests provided in either the row titled Greenhouse-Geisser or Huynh-Feldt, which provide tests corrected for your assumption violations.
Tests of Within-Subjects Contrasts. Provides the results of polynomial contrasts among your within-subject main effects and interactions. These will have no meaning unless your levels have an ordinal relationship.
Tests of Between-Subjects Effects. Contains tests of the main effects of your between-subjects factors as well as tests of any interactions that only involve between-subjects factors.

You can ask SPSS to provide you with the means within the levels of your main effects or your interactions (whether they involve within-subjects factors, between-subjects factors, or both) by clicking the Options button in the variable selection window and moving the appropriate term to the Display Means For box. This will add a section to your output titled Estimated Marginal Means containing a table for each main effect or interaction in your model. The table will contain a row for each cell within the effect. The values within each row provide the mean, standard error of the mean, and the boundaries for a 95% confidence interval around the mean for observations within that cell.