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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.

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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: Factor B: Curriculum New Math Control Curriculum (B2) (B1) Factor A: Gender Females (A1) Males (A2) X 11 X 21

X .1

Overall Mean (marginal)

X 1.

X 2. X ..

X 12 X 22

X .2

Overall Mean (marginal)

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 SStotal 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 ) 2 In a simple one-way ANOVA the SSwithin = SSerror 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 SSerror 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 SSbetween 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, SSbetween is partitioned into SSA, SSB, and SSAB. SSA 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. SSB 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. SSAB 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 SSA and SSB. Basically, we compute the variability in the cell means and then subtract the variability due to the main effects.

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Example: Suppose we obtained the following data for the ANOVA design explained previously: Females - New 6 7 6 8 8 5 5 4 6 9 Females - Control 6 4 5 3 6 3 4 4 2 3 7 9 10 Males - New 9 10 8 8 6 7 6 5 4 5 Males - Control 3 3 4 2 3 4 1

3 Calculating the cell and marginal means we obtain the following: Factor B: Curriculum New Math Control Curriculum (B2) (B1) Factor A: Gender Females (A1) Males (A2) X 11 = 6.4 X 21 = 8.0

X .1 = 7.2

Overall Mean (marginal)

X 1. = 5.2 X 2. = 5.7 X .. = 5.45

X 12 = 4.0 X 22 = 3.4

X .2 =3.7

Overall Mean (marginal) The SSerror = ( X - X j ) =

(5 - 6.4) 2 + (8 - 6.4) 2 + ... + (6 - 4.0) 2 + (3 - 4.0) 2 + ... + (7 - 8.0) 2 + (9 - 8.0) 2 + ... + (5 - 3.4) 2 + (3 - 3.4) 2 = 72.8

The SSA = ni. ( X i. - X .. ) 2 = 20(5.2 - 5.45) 2 + 20(5.7 - 5.45) 2 = 1.25 + 1.25 = 2.5 The SSB = n. j ( X . j - X .. ) 2 = 20(7.2 - 5.45) 2 + 20(3.7 - 5.45) 2 = 61.25 + 61.25 = 122.5 The SSAB = [ nij ( X ij - X .. ) 2 ] SSA - SSB = [10(6.4 - 5.45) 2 + 10(4.0 - 5.45) 2 + 10(8.0 - 5.45) 2 + 10(3.4 - 5.45) ] 2.5 122.5 = [9.025 + 21.025 + 65.025 + 42.025] 126 = 137.1 125 = 12.1 SStotal = SSerror + SSA + SSB + SSAB = 72.8 + 2.5 + 122.5 + 12.1 = 209.9 · To obtain our F-ratios for each test we need to use the df associated with each main effect and interaction. dfA = Number of levels of Factor A 1 = 2 1 = 1, for our example dfB = Number of levels of Factor B 1 = 2 1 = 1, for our example

B

dfAB = (dfA)( dfB) = 1(1) = 1, for our example

B

dferror = N (number of cells) = 40 4 = 36, for our example dftotal = N 1 (checking this number is a good way to make sure you've entered your data correctly)

4 Using the appropriate df we can obtain the corresponding MS term needed to calculate our Fstatistic: MSA = SSA / df A = 2.5 / 1 = 2.5, for our example MSB = SSB / df B = 122.5 / 1 = 122.5 , for our example MSAB = SSAB / dfAB = 12.1 / 1 = 12.1, for our example MSerror = SSerror / dferror = 72.8 / 36 = 2.022, for our example The null hypothesis for each test is that there is no difference in the means. FA = MSA / MSerror = 2.5 / 2.022 1.24, (compare to a critical F with 1 and 36 df 4.125) FB = MSB / MSerror = 122.5 / 2.022 60.58, (compare to critical F with 1 and 36 df 4.125) FA = MSAB / MSerror = 12.1 / 2.022 5.98, (compare to critical F with 1 and 36 df 4.125) SPSS Output: Univariate Analysis of Variance - obtained using defaults under "Analyze" and "General Linear Model"and "Univariate"

Between-Subjects Factors

Value Label sex 1 2 curriculum 1 2 female male new program control N 20 20 20 20

Tests of Between-Subjects Effects

Dependent Variable: spatial Type III Sum of Squares 137.100a 1188.100 2.500 122.500 12.100 72.800 1398.000 209.900

Source Corrected Model Intercept sex curriculum sex * curriculum Error Total Corrected Total

df 3 1 1 1 1 36 40 39

Mean Square 45.700 1188.100 2.500 122.500 12.100 2.022

F 22.599 587.522 1.236 60.577 5.984

Sig. .000 .000 .274 .000 .019

a. R Squared = .653 (Adjusted R Squared = .624)

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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 · 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

Dependent Variable: score Type III Sum of Squares 399.111a 5292.089 51.511 235.244 112.356 230.800 5922.000 629.911

Source Corrected Model Intercept treatment severity treatment * severity Error Total Corrected Total

df 8 1 2 2 4 36 45 44

Mean Square 49.889 5292.089 25.756 117.622 28.089 6.411

F 7.782 825.456 4.017 18.347 4.381

Sig. .000 .000 .027 .000 .005

a. R Squared = .634 (Adjusted R Squared = .552)

Estimated Marginal Means - obtained under "options" button

3. treatment * severity

Dependent Variable: score 95% Confidence Interval treatment hypnosis severity mild moderate severe CBT mild moderate severe behavioral mild moderate severe Mean 13.200 11.400 10.400 16.800 12.000 5.800 11.000 9.000 8.000 Std. Error 1.132 1.132 1.132 1.132 1.132 1.132 1.132 1.132 1.132 Lower Bound 10.903 9.103 8.103 14.503 9.703 3.503 8.703 6.703 5.703 Upper Bound 15.497 13.697 12.697 19.097 14.297 8.097 13.297 11.297 10.297

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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:

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19 17 15 13 11 9 7 5 Hypnosis CBT Behavioral Mild Moderate Severe

19 17 15 13 11 9 7 5 Mild Moderate Severe Hypnosis CBT Behavioral

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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,

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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

Dependent Variable: score Type III Sum of Squares 187.917a 7252.083 89.542 56.333 42.042 234.000 7674.000 421.917

Source Corrected Model Intercept treatment severity treatment * severity Error Total Corrected Total

df 5 1 2 1 2 42 48 47

Mean Square 37.583 7252.083 44.771 56.333 21.021 5.571

F 6.746 1301.656 8.036 10.111 3.773

Sig. .000 .000 .001 .003 .031

a. R Squared = .445 (Adjusted R Squared = .379)

3. treatment * severity

Dependent Variable: score 95% Confidence Interval treatment hypnosis severity mild severe CBT mild severe behavioral mild severe Mean 13.250 11.875 14.000 13.625 12.875 8.125 Std. Error .835 .835 .835 .835 .835 .835 Lower Bound 11.566 10.191 12.316 11.941 11.191 6.441 Upper Bound 14.934 13.559 15.684 15.309 14.559 9.809

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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 mildly depressed patients so the researcher may be justified in interpreting the main effect:

15 14 13 12 11 10 9 8 7 Mild Severe Hypnosis CBT Behavioral

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15 14 13 12 11 10 9 8 7 Hypnosis CBT Behavior Mild Severe

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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 SStherapy for mildly depressed patients and SStherapy for severely depressed patients. We do this using the cell means for the different therapy treatments and the following marginal means for severity of depression.

2. severity

Dependent Variable: score 95% Confidence Interval severity mild severe Mean 13.375 11.208 Std. Error .482 .482 Lower Bound 12.403 10.236 Upper Bound 14.347 12.181

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SStherapy for mild depression = n ( X therapy,mild - X mild ) 2 = 8[(13.25 13.375)2 + (14 13.375)2 + (12.875 13.375)2] = 8[0.016 + 0.391 + 0.25] = 5.25 SStherapy for severe depression = n ( X therapy, severe - X severe ) 2 = 8[(11.875 11.208)2 + (13.625 11.208)2 + (8.125 11.208)2] = 126.333 Each of these SS have 2 df because they use 3 means in the calculation. The F ratio for testing main effect differences of therapy for patients that are mildly depressed is based on MStherapy = 5.25/2 = 2.625 and MSwithin = 5.571 so F = 2.625 / 5.571 = 0.471 which needs to be compared to a critical F with 2 and 42 df, which is approximately 3.23. There are obviously no main effect differences for therapy treatments for patients that with mild

9 depression. The F statistic for testing main effect difference of therapy for patients that are severely depressed is (126.33/2) / 5.571 = 63.167/5.571 = 11.338. So there is a difference in depression scores for the different therapy treatments for patients that are severely depressed. One could also test the main effect of depression severity within each of the treatment levels. This is accomplished by obtaining the SSseverity 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

Dependent Variable: score 95% Confidence Interval treatment hypnosis CBT behavioral Mean 12.563 13.813 10.500 Std. Error .590 .590 .590 Lower Bound 11.372 12.622 9.309 Upper Bound 13.753 15.003 11.691

SSseverity for hypnosis = n ( X severity,hypnosis - X hypnosis ) 2 = 8[(13.25 12.563)2 + (11.875 12.563)2 ] = 8[0.472 + 0.473] = 7.563 SSseverity for CBT = n ( X severity,CBT - X CBT ) 2 = 8[(14.0 13.813)2 + (13.625 13.813)2 ] = 0.563 SSseverity for behavioral therapy = n ( X severity,behaviroal - X behavioral ) 2 = 8[(12.875 10.5)2 + (8.125 10.5)2 ] = 90.25 Each of these SS have 1 df because they use 2 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 MSwithin = 5.571 so F = 7.563 / 5.571 = 1.357 which needs to be compared to a critical F with 1 and 42 df, which is approximately 4.08. 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 0.563 / 5.571 = 0.101. 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 90.25 / 5.571 = 16.199 so there is a main effect difference for severity of depression for patients that are treated with behavioral therapy

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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 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).

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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 SSeffect by SStotal. Although this is conceptually simple, estimates of SSeffect and SStotal 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-2 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; (2) 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.

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