Factor B: Curriculum New Math Control Curriculum (B (B 1 ) Overall Mean (marginal) Females (A 1 ) Factor A: Gender Males (A 2) X 21

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1 1 Factorial ANOVA The ANOVA designs we have dealt with up to this point, known as simple ANOVA or oneway ANOVA, had only one independent grouping variable or factor. However, oftentimes a researcher has more than one independent grouping variable, or factor of interest. Factorial ANOVA is used when we want to consider the effect of more than one factor on differences in the dependent variable. A factorial design is an experimental design in which each level of each factor is paired up or crossed with each level of every other factor. In other words each combination of the levels of the factors is included in the design. This type of design is often depicted in a table. We typically refer to ANOVA designs by the number of factors and/or by the number of levels within a factor. A one-way ANOVA refers to a design with one factor, two-way ANOVA has two factors, three-way ANOVA has three factors, etc. A two-by- three ANOVA is a two-way ANOVA with two levels of the first factor and three levels of the second factor. A three-by-four-by-two ANOVA is a three-way ANOVA with three levels of the first factor, four of the second, and two of the third. Factorial designs allow us to determine if there are interactions between the independent variables or factors considered. An interaction implies that differences in one of the factors depend on differences in another factor. Example Consider a researcher who is interested in determining whether a new mathematics curriculum is better at helping students develop spatial visualization skills. Furthermore, he wonders whether there is a difference between boys and girls, because it is known that males tend to be better at spatial visualization than females. The researcher has the following two-way (two-by-two) factorial design: Females (A 1 ) Factor B: Curriculum New Math Control Curriculum (B (B 1 ) ) Overall Mean (marginal) X X 1 1. Factor A: Gender Males (A ) X 1 Overall Mean (marginal). 1 X X X. X X. X.. Suppose the new curriculum was found to improve spatial visualization scores equally as well for both males and females. Then there would be main effect differences only. Main effect differences reflect differences in the means of one of the factors, ignoring other factors. However, if, for example, the new curriculum worked better for females then there would be an interaction effect. Typically we graph each of the cell means to depict differences obtained in factorial ANOVA. The assumptions underlying the statistical tests associated with factorial ANOVA are the same as those associated with a simple one-way ANOVA. Specifically, it is assumed the dependent variable is normally distributed within each cell, that the population variances are

2 identical within each cell, and that the observations and groups are independent of each other. Conceptually, the way we calculate the statistics associated with factorial ANOVA designs is comparable to what we did for simple one-way ANOVA designs. Basically, we determine the variability associated with different means; there are just more means to deal with now. The SS total in a factorial design is exactly the same as it was in simple ANOVA. It represents the total variability among all observations around the grand mean or ( X X ) In a simple one-way ANOVA the SS within = SS error represented the variability of observations within a particular group. However, now we are partitioning the groups even further so each group is represented by a cell in our table. In other words, the SS error represents the variability of observations within a particular cell of the table. It is the variability that is expected among individuals and can be thought of as an estimate of variability that is common to all cells. In a factorial ANOVA the SS between still represents the variability of the group means from the overall mean. However, now we have to determine which of the variability is due to main effects and which is due to interaction effects. For a two-way ANOVA design, as depicted in the example above, SS between is partitioned into SS A, SS B, and SS AB. SS A represents the variability in the marginal means associated with the different levels of factor A, when compared to the overall mean. In our example, it would represent the variability in the means obtained for boys and girls, ignoring curriculum. It is computed by using the row marginal means and the grand mean. SS B represents the variability in the marginal means associated with the different levels of factor B, when compared to the overall mean. In our example, it would represent the variability in the different curriculum programs, ignoring gender. It is computed by using the column marginal means and the grand mean. SS AB represents the variability in the cell means, after controlling for main effect differences, when compared to the overall mean. It is computed by using the cell means and the overall means, as well as SS A and SS B. Basically, we compute the variability in the cell means and then subtract the variability due to the main effects. Example: Suppose we obtained the following data for the ANOVA design explained previously: Females - New Females - Control Males - New Males - Control

3 3 Calculating the cell and marginal means we obtain the following: Factor B: Curriculum New Math Control Curriculum (B (B 1 ) ) Overall Mean (marginal) Females (A 1 ) X = 6.4 X 1 = 4.0 X 1. = 5. Factor A: Gender Males (A ) X 1 = 8.0 X = 3.4 X. = 5.7 Overall Mean (marginal) X. 1= 7. X. =3.7 X.. = 5.45 The SS error = ( X X j ) = (5 6.4) + (8 6.4) (6 4.0) + (3 4.0) (7 8.0) + (9 8.0) (5 3.4) + (3 3.4) = 7.8 The SS A = n ( X X ) = 0( ) + 0( ) = i. i... = The SS B = n ( X X ) = 0( ) + 0( ) = j. j.. = The SS AB = [ n ij ( X ij X.. ) ] SS A - SS B = [10( ) + 10( ) + 10( ) + 10( ) ] = [ ] 16 = = 1.1 SS total = SS error + SS A + SS B + SS AB = = 09.9 To obtain our F-ratios for each test we need to use the df associated with each main effect and interaction. df A = Number of levels of Factor A 1 = 1 = 1, for our example df B B = Number of levels of Factor B 1 = 1 = 1, for our example df AB = (df A )( df B ) B = 1(1) = 1, for our example df error = N (number of cells) = 40 4 = 36, for our example df total = N 1 (checking this number is a good way to make sure you ve entered your data correctly)

4 4 Using the appropriate df we can obtain the corresponding MS term needed to calculate our F- statistic: MS A = SS A / df A =.5 / 1 =.5, for our example MS B = SS B / df B = 1.5 / 1 = 1.5, for our example MS AB = SS AB / df AB = 1.1 / 1 = 1.1, for our example MS error = SS error / df error = 7.8 / 36 =.0, for our example The null hypothesis for each test is that there is no difference in the means. F A = MS A / MS error =.5 /.0 1.4, (compare to a critical F with 1 and 36 df 4.15) F B = MS B / MS error = 1.5 / , (compare to critical F with 1 and 36 df 4.15) F A = MS AB / MS error = 1.1 / , (compare to critical F with 1 and 36 df 4.15) SPSS Output: Univariate Analysis of Variance - obtained using defaults under "Analyze" and General Linear Model and "Univariate" Between-Subjects Factors sex curriculum 1 1 Value Label female 0 male 0 new program N 0 control 0 Tests of Between-Subjects Effects Dependent Variable: spatial Source Corrected Model Intercept sex curriculum sex * curriculum Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig a a. R Squared =.653 (Adjusted R Squared =.64) Under the model option in SPSS you can choose to use either Type II SS, Type III SS (default) or Type IV SS. It is recommended that you go with the default which adjusts the tests conducted when you have an unequal number of observations in each cell and conducts each test independently of other tests.

5 5 Typically when one finds an interaction they graph it to aid in the interpretation. However, our example wasn t very interesting so let s consider a more interesting example. Suppose a counseling psychologist conducted a study to determine the best type of therapy for various levels of depression and obtained the following data: Tests of Between-Subjects Effects Source Corrected Model Intercept treatment severity treatment * severity Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig a a. R Squared =.634 (Adjusted R Squared =.55) Estimated Marginal Means - obtained under "options" button 3. treatment * severity 95% Confidence Interval treatment hypnosis CBT behavioral severity moderate moderate moderate Mean Std. Error Lower Bound Upper Bound There is a significant interaction in this example and the best way to interpret it is to create separate line graphs for each level of one factor that depicts the cell means for the other factor. This can be done in two different ways as the following demonstrates:

6 Mild Moderate Severe 7 5 Hypnosis CBT Behavioral Hypnosis CBT Behavioral 7 5 Mild Moderate Severe If the interaction was not found to be significant than the lines in the above plots would be parallel. If, for example, we had only compared hypnosis to behavioral therapy then we would not have found a significant interaction. Once we find a significant interaction many methodologists would argue that any significant main effects that are found should not be interpreted. However, this is somewhat dependent on the type of interaction that is obtained. In the example above a disordinal interaction was obtained, because the interaction lines intersect (or move in opposite directions). In this case it is not appropriate to interpret any significant main effects because differences found in different levels of one factor depend on differences in the second factor. However, it is also possible to obtain an ordinal interaction. In this case, the lines would not be parallel, however the lines would not cross or move in different directions. For example,

7 7 suppose the following results had been obtained from 8 patients at each severity level in each of the 3 therapy groups: Tests of Between-Subjects Effects Source Corrected Model Intercept treatment severity treatment * severity Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig a a. R Squared =.445 (Adjusted R Squared =.379) 3. treatment * severity treatment hypnosis CBT behavioral severity 95% Confidence Interval Mean Std. Error Lower Bound Upper Bound In this case the interaction is significant, but the following interaction graphs would be obtained, which makes it clear that all treatments seemed to work better for ly depressed patients so the researcher may be justified in interpreting the main effect: Hypnosis CBT Behavioral Mild Severe

8 Mild Severe Hypnosis CBT Behavior Whenever an interaction is obtained, one might want to do a test of simple main effects. This test teases apart the interaction. A test of simple main effects is different from simply interpreting the main effects, which ignores different levels of the second factor. Rather a test of simple main effects is a test for main effect differences at each level of the other factor. For example, one might want to test the main effect of treatment within each of the two different levels of depression severity. This is accomplished by obtaining the SS therapy for ly depressed patients and SS therapy for ly depressed patients. We do this using the cell means for the different therapy treatments and the following marginal means for severity of depression.. severity 95% Confidence Interval severity Mean Std. Error Lower Bound Upper Bound SS therapy for depression = n X therapy X = (, ) 8[( ) + ( ) + ( ) ] = 8[ ] = 5.5 SS therapy for depression = n X therapy X = (, ) 8[( ) + ( ) + ( ) ] = Each of these SS have df because they use 3 means in the calculation. The F ratio for testing main effect differences of therapy for patients that are ly depressed is based on MS therapy = 5.5/ =.65 and MS within = so F =.65 / = which needs to be compared to a critical F with and 4 df, which is approximately 3.3. There are obviously no main effect differences for therapy treatments for patients that with

9 9 depression. The F statistic for testing main effect difference of therapy for patients that are ly depressed is (16.33/) / = /5.571 =.338. So there is a difference in depression scores for the different therapy treatments for patients that are ly depressed. One could also test the main effect of depression severity within each of the treatment levels. This is accomplished by obtaining the SS severity within each treatment, using the cell means for the different levels of depression severity and the following marginal means for the different therapy treatments. 1. treatment 95% Confidence Interval treatment hypnosis CBT behavioral Mean Std. Error Lower Bound Upper Bound SS severity for hypnosis = n X severity X = (, hypnosis hypnosis ) 8[( ) + ( ) ] = 8[ ] = SS severity for CBT = n X severity X = (, CBT CBT ) 8[( ) + ( ) ] = SS severity for behavioral therapy = n X severity X = 8[( ) + ( ) ] = 90.5 (, behaviroal behavioral ) Each of these SS have 1 df because they use means in the calculation. The F ratio for testing main effect differences of severity of depression for patients that treated using hypnosis is based on MS = 7.563/1 and MS within = so F = / = which needs to be compared to a critical F with 1 and 4 df, which is approximately There are obviously no main effect differences for severity of depression for patients that are treated with hypnosis therapy. The F statistic for testing main effect difference for severity of depression for patients that are treated with CBT is / = So there are no main effect differences for severity of depression for patients that are treated with CBT. The F statistic for testing main effect differences for severity of depression for patients treated with behavioral therapy is 90.5 / = so there is a main effect difference for severity of depression for patients that are treated with behavioral therapy It should be noted that there is no way to test simple main effects in SPSS without using the syntax window. Rather the syntax window in SPSS must be used. The following SPSS syntax can be used to obtain a test of simple main effects, as well as an ANOVA. The majority of the syntax is what is run when one clicks General Linear Model under the Analyze menu option and then chooses Univariate. The middle lines beginning with /EMMEANS are added when one chooses to get an estimate of the means under the options button. All of this syntax can be obtained by choosing the paste button when

10 10 running an ANOVA from the point and click menu. The last two lines provide a test of the simple main effects (as well as some extraneous output) and must be typed in by the user. UNIANOVA score BY treatment severity /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /EMMEANS = TABLES(treatment) /EMMEANS = TABLES(severity) /EMMEANS = TABLES(treatment*severity) /CRITERIA = ALPHA(.05) /DESIGN = treatment severity treatment*severity / EMMEANS = tables (treatment * severity) comp (treatment) / EMMEANS = tables (treatment * severity) comp (severity). All of the multiple comparison procedures discussed in terms of simple one-way ANOVA can be generalized to higher way ANOVA designs and these are easily obtained using the point and click menu options in SPSS. However, it should be noted that these are tests of the main effects that ignore other factors. Therefore, I would not recommend interpreting pair-wise comparisons if a significant interaction is obtained. Power analyses for factorial ANOVA designs can also be conducted, similar to how they were conducted for simple one-way ANOVA designs. For factorial ANOVA designs we simply conduct separate power analyses for each factor individually, ignoring any additional factors that may exist. Once again, statistical significance does not imply differences that are important from a practical perspective. An effect size measure can be estimated by dividing the SS effect by SS total. Although this is conceptually simple, estimates of SS effect and SS total are dependent on knowing how to determine the expected mean squares, which is technically difficult. However, estimates of effect size can be obtained under the options button when running a factorial ANOVA in SPSS. Effect size measures will be printed out in the ANOVA table, next to each of the F-statistics for the main effects and the interaction terms. Having unequal cell sizes in a factorial ANOVA is a complex issue, from a technical perspective, because it results in a dependency among the main effect and interaction estimates of variability. Using the Type III SS, which is the default in SPSS, will provide you with a test of unweighted means, which is usually the appropriate test to conduct with unequal cell sizes. It should be noted that higher-order factorial designs are typical in Social Science research and all of the procedures that relate to a two-way ANOVA can easily be applied to higherorder factorial designs. However, with higher order designs there are more interaction terms to deal with and considering anything above a three-way ANOVA makes interpreting the results extremely difficult. Suppose one had a 3-by-4-by- factorial design. In other words, a three-way factorial design with three levels of factor A, four levels of factor B, and two levels of factor C. The corresponding ANOVA would be a test of the following: (1) Three tests of the Main Effects of Factor A, Factor B, and Factor C; () Three tests of the Two-way Interaction Effects of AB, AC, and BC, and (3) One test of the Three-way Interaction Effect of ABC.