UNDER CONSTRUCTION

Purpose of Two-Way ANOVA

One-way ANOVA examines whether the means of two or more groups differ
Two-way ANOVA allows us to add a second factor (independent variable)
Two-way mixed ANOVA allows us have two factors, one between groups (independent) and another within groups (related).

Fundamentals of 2-way ANOVA

• Independent variables are categorical (e.g., year in school, experimental condition)
• The independent variable is often called a "factor".
• The values of the independent variable are called the "levels".
• Dependent variable is continuous
  
• Kind of Effects:
  Main effect
  • The effects of each individual factor
    
  • Ignores the level of the other factor
    
  • Tests the null hypothesis that the means of all groups of Factor 1 are equal (The main effect of factor 1)
    
  • Tests the null hypothesis that the means of all groups of Factor 2 are equal (The main effect of factor 2)
    

  Interaction

  • The cumulative effect of two variables
    
  • Present when the relationship between one factor and the dependent variable changes for different levels of the other factor
    
  • Tests the null hypothesis that the relationship between each factor and the dependent variable is the same for all levels of the other factor (The interaction of factors 1 and 2)
    
• Assumptions for the two-way ANOVA are the same as for the one-way ANOVA
  
  • Observations are independent of one another
    
  • The continuous dependent variable is distributed close to normal
    
  • The variance of the continuous dependent variable is roughly equal in all groups
    
• If the interaction is significant it does not make sense to interpret (only the) main effects
  

Brief Discussion of How ANOVA works

• ANOVA analyzes variance by separating it into two parts
  • Within-groups variability
    
  • Between-groups variability
• F statistic indicates whether the between-groups variability is significantly greater than the within-groups variability
  
• If the F statistic is significant (p < .05), at least one group mean is significantly different from one or more of the others
  
• A significant F statistic suggests that we reject the null hypothesis
  

Using SPSS to do 2 factor mixed measures ANOVA

To set up a mixed measure ANOVA is very much like a repeated measures (related or within groups) one-way ANOVA. You will need at least three columns of data, one for each group (one for each level of your independent variable). The data in these columns correspond to values of the dependent variable (the thing you measured). You will also need a column that that specifies the between groups independent variable (containing categorical values for the different levels of this variable).

For this example, consider whether they attendend a review session as the between groups variable and their quiz score on quizes 1 through 5 as the repeated groups independent variable.

Go to the Analyze menu and select the General Linear Model. In this submenu you'll see several tests. The one that we're interested in today is Repeated Measures.
After selecting Repeated Measures you'll get a window that looks like this. Here you should select the variables that you are testing. Your "Within-Subjects Variables" are the columns that correspond to the different levels of your Independent variable. In the "betwee-subjects variables" field you specify your between groups independent variable.
You may also want to examine some of the buttons at the bottom of this window to get your descriptive statistics and a graph of the group means.
Your ANOVA output Here is what the output will look like (actually this is just some of it).
Notice that the output is more complex than we saw with independent groups ANOVAs. This is because we need to worry about the assumption of "spericity". For now, let's assume that this assumption is satisfied and only look at the top lines of each of the two rows (the ones labeled "spericity assumed"). Also notice that the within groups main effects (and the interaction) are presented separately from the between groups main effect. As usual, SPSS doesn't tell you to reject or fail to reject the H0, nor does it give you the Fcrit. To make your decision about the H0 you must compare the p-value with your a-level. If the p-value is equal to or smaller than the your a-level, then you should reject the H0, otherwise you should fail to reject H0.

Your graph

I recommend that you change the line graph (which is the default) to a clustered bar graph in the Chart editor (use the Gallery menu). This is because bars are more appropriate than lines for these cagegorical variables.

UNDER CONSTRUCTION

Follow-Up Analyses

UNDER CONSTRUCTION

Recall that the ANOVA only tells us if there is any difference between the groups. If we want to know where the differences are, then we need to do some additional analyses. There are generally two kinds of additional analyses, planned means contrasts and post hoc tests. The "Planned" kind of additional tests are done if, in advance of doing your ANOVA, you know specifically what groups you expect to be different are (note: you should limit yourself to a fairly small set of comparisons). The "post hoc" tests are those that you do after you've done an ANOVA and found a difference and now want to konw what groups differ from one another.

Post Hoc Multiple Comparisons: Post hoc means "after the fact." "Multiple comparisons" means that all possible pairs of factors are compared. There are many options regarding post hoc tests on SPSS. However, some are more commonly used than others.

• LSD: the "least significant difference." This is the most liberal of the tests, since you are most likely to show significant differences in comparisons. This procedure is simply a series of t tests.
  • Scheffe and Bonferroni: most conservative of the tests.
  • Tukey: (HSD-Honestly Significant Difference). This calculates a number that represents the minimum difference between mean values in order to identify a significant difference. SPSS also has additional statistical information for this post hoc test.
  • Bonferroni procedure is a series of t-tests with an adjusted significance level
• After entering variables you wish to test, *Options
• *Descriptives, *Homogeneity of variances, *Continue
• * Post hoc test(s) you desire, *Continue, * OK

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• To do planned means comparisons you need to set up orthogonal comparisions. Notice that the SPSS box makes you enter some "coefficients". This is where the "orthogonal" part will come in. Basically it means that the coefficients that you enter have to sum up to zero. The coeffiecients are how you tell SPSS which groups to compare. You assign positive numbers for one (or more, but we won't worry about these more advanced comparisions) group and negative numbers to the group you want to compare it to. Then you assign zeros to the groups that you want to ignore. Consider the following set of coefficents.

  Coefficents Comparison 1, -1, 0 group 1 vs. group 2 1, 0, -1 group 1 vs. group 3 0, 1, -1 group 2 vs. group 3 To the right is what the output from SPSS looks like. The results of Comparision 1 (group 1 vs 2) is significant (p = 0.001) Comparison 2 (group 1 vs 3) is not significant (p > 0.05) Comparison 3 (group 2 vs 3) is significant (p = 0.001)

• To do post hoc tests is even easier. All you need to do is to click on the box of the kind of post hoc test that you want to do. Some of the most common are Tukey's HSD, Fisher's LSD, and Scheffe (a very conservative post hoc test). Notice that to do these tests you need to specify what level of a you want to use.

  The output for these three tests is presented below. For each, you will see the results of each pairwise comparision. For example, the Tukey HSD test, book alone vs. notes alone, is significant (p = 0.004), while book alone vs. borrowed notes is not significant (p > 0.05). The next set is notes alone vs book alone and notes alone vs borrowed, and the set after that is borrowed against book alone and borrowed against notes alone. The results for the other post hoc tests are aranged in the same way.

  

  Based on these results (either the planned comparisons, if we had some reason in advance to test the groups against one another, or the post hocs, if we found rejected the H₀ based on our ANOVA first) we can reject several of the alternative hypotheses:

  m1 not equal to m2 not equal to m3	REJECT
  m1 not equal to m2 = m3	REJECT
  m1 = m2 not equal to m3	REJECT
  m1 = m3 not equal to m2	FAIL TO REJECT