MORE COMPLEX ANALYSIS OF VARIANCE This section briefly introduces a simple method for using multiple regression as a means of conducting N-way analysis of variance. If you are not familiar with the use of regression to conduct ANOVA you should review the instructional materials for 'One-Way' and 'Two-Way Analysis of Variance' before proceeding. The conduct of N-way ANOVA is largely an extension of the technique described for two-way ANOVA. The essential difference is that you now have one continuous dependent variable and three or more classification variables. You therefore must create new variables for each of the subclasses within each of the classification variables. This will be illustrated with a simple example. Suppose you have a set of depression scores (DEP), as before, but you now have four classification variables. The first three have two subclasses and the fourth has three subclasses. In traditional parlance this is often referred to as a 2x2x2x3 crossed factorial design. In this example, the first variable is gender, the second is marital status, the third is social status, and the fourth is age. As before, we create one special variable for each subclass within each of the classification variables. We could therefore define these special subclass variables as shown below. Depression DEP = Dependent Variable Gender G1 = Males, G2 = Females Marital Status M1 = Single, M2 = Other Social Status S1 = Lower, S2 = Upper Age A1 = Young, A2 = Middle, A3 = Old As before, each subclass variable is scored as 1 to indicate that the person is a member of the subclass or as 0 to indicate that the person is NOT a member of the subclass. Once you have defined the subclass variables and then coded each person, you must then eliminate one subclass variable for each of the classification variables. For this example we shall choose to eliminate G2, M2, S2, and A3. We therefore retain the coded variables DEP G1 M1 S1 A1 A2 Now suppose we have someone with a depression score of 37 who is male, married, upper class and middle aged. Such a person would then be coded as DEP G1 M1 S1 A1 A2 37 1 0 0 0 1 The complete coded data for 48 individuals are shown as follows: DEP G1 M1 S1 A1 A2 DEP G1 M1 S1 A1 A2 DEP G1 M1 S1 A1 A2 36 1 1 1 1 0 11 1 0 1 0 0 44 0 1 0 0 1 37 1 1 1 1 0 13 1 0 1 0 0 43 0 1 0 0 1 41 1 1 1 0 1 21 1 0 0 1 0 56 0 1 0 0 0 44 1 1 1 0 1 23 1 0 0 1 0 59 0 1 0 0 0 46 1 1 1 0 0 78 1 0 0 0 1 42 0 0 1 1 0 48 1 1 1 0 0 76 1 0 0 0 1 44 0 0 1 1 0 23 1 1 0 1 0 28 1 0 0 0 0 68 0 0 1 0 1 26 1 1 0 1 0 31 1 0 0 0 0 71 0 0 1 0 1 39 1 1 0 0 1 41 0 1 1 1 0 16 0 0 1 0 0 38 1 1 0 0 1 42 0 1 1 1 0 18 0 0 1 0 0 51 1 1 0 0 0 46 0 1 1 0 1 26 0 0 0 1 0 54 1 1 0 0 0 49 0 1 1 0 1 28 0 0 0 1 0 37 1 0 1 1 0 51 0 1 1 0 0 83 0 0 0 0 1 39 1 0 1 1 0 53 0 1 1 0 0 81 0 0 0 0 1 63 1 0 1 0 1 28 0 1 0 1 0 33 0 0 0 0 0 66 1 0 1 0 1 31 0 1 0 1 0 36 0 0 0 0 0 As was the case with the two-way ANOVA, the foregoing values for the subclass variables only represent the so-called main effects of the 2x2x2x3 ANOVA model. Thus, in order to compute the subclass variables that will represent the interactions, we must form product terms. Since gender could interact with the remaining three variables, we must form the subclass variables for the two-way interactions with gender by simply multiplying the G1 value with the remainder of the subclass values to produce G1xM1 G1xS1 G1xA1 G1xA2 However, to simplify the notation and conserve space, we shall denote these as GM GS GA1 GA2 Of course, marital status can also interact with social status and age, so we form those products as M1xS1 M1xA1 M1xA2 or, more simply for notation purposes, MS MA1 MA2 Finally, social status can interact with age so we form S1xA1 S1xA2 or, for notation convenience, SA1 SA2 We have now accomodated the potential for all possible two-way interactions. However, there is also the possibility for three different three-way interactions and one four-way interaction. These are: gender x marital x social gender x marital x age marital x social x age gender x marital x social x age As before, these interaction variables are created by simply producting the appropriate subclass variables. These are shown as follows without comment. G1xM1xS1 or GMS G1xM1xA1 or GMA1 G1xM1xA2 or GMA2 M1xS1xA1 or MSA1 M1xS1xA2 or MSA2 G1xM1xS1xA1 or GMSA1 G1xM1xS1xA2 or GMSA2 Thus, all the needed subclass variables for this analysis are: DEP G1 M1 S1 A1 A2 GM GS GA1 GA2 MS MA1 MA2 SA1 SA2 GMS GMA1 GMA2 GSA1 GSA2 MSA1 MSA2 GMSA1 GMSA2 Space makes it difficult to display all of the raw data for this analysis. However, if you wish to examine the complete set of raw data for this example, use your favorite word processor and examine the file on your SPPC Disk 1 named FOURWAYR.DAT or use the 'type' command from DOS to examine that file. Better yet, run the program using the raw data option and use the FOURWAYR.DAT file. Of course, you can also choose the summary data option of the program and then use the FOURWAYS.DAT data file. A partial solution for this example is shown below along with a limited discussion of the analysis. For further excellent discussion of this approach to the solution of complex ANOVA designs, you might wish to consult Chapter 5 of the Cohen & Cohen text. In this example we show only the ANOVA Table and point out that it is very important to analyze the sources of variance by grouping the subclass variables into sets. The program will present an opportunity to do that. In defining sets of subclass variables it was decided to examine the main effects of the model. Thus, the first three sets each had one variable (for gender, marital status, and social class), and the fourth set had two variables to represent the main effect of age. This portion of the ANOVA Table is shown as follows: ANALYSIS OF VARIANCE MAIN EFFECTS: Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= --------------- -- ----------- ----------- ----------- ------ Gender G1 1 300.00000 300.00000 107.46300 0.0000 Marital M1 1 0.75000 0.75000 0.26865 0.6146 Social Class S1 1 4.08333 4.08333 1.46269 0.2367 Age A1 1 2460.37000 2460.37000 881.32800 0.0000 A2 1 3321.13000 3321.13000 1189.66000 0.0000 For this set: 2 5781.50000 2890.75000 1035.49000 0.0000 It was not known whether any of the two-way interactions would be significant and we had no a' priori hypotheses concerning them. Thus, the decision was to include all of the two-way interactions as a single set. Since there were nine such interactions we indicated that the fifth set would have nine variables. The portion of the ANOVA Table that deals with the two-way interactions is shown as follows: TWO-WAY INTERACTIONS: Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= --------------- -- ----------- ----------- ----------- ------ GM 1 0.00000 0.00000 0.00000 0.9955 GS 1 0.00000 0.00000 0.00000 0.9955 GA1 1 0.00000 0.00000 0.00000 0.9955 GA2 1 0.00000 0.00000 0.00000 0.9955 MS 1 200.08300 200.08300 71.67160 0.0000 MA1 1 3.37500 3.37500 1.20896 0.2822 MA2 1 7021.13000 7021.13000 2515.03000 0.0000 SA1 1 1276.04000 1276.04000 457.09000 0.0000 SA2 1 105.12500 105.12500 37.65670 0.0000 For this set: 9 8605.75000 956.19400 342.51700 0.0000 It was also not known whether any of the three-way interactions would be significant and we had no a' priori hypotheses concerning them. Thus, the decision was to include all of the three-way interactions as a single set. Since there were seven such interactions we indicated that the sixth set would have seven variables. The portion of the ANOVA Table that deals with the three-way interactions is shown as follows: THREE-WAY INTERACTIONS: Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= --------------- -- ----------- ----------- ----------- ------ GMS 1 0.00000 0.00000 0.00000 0.9955 GMA1 1 0.00000 0.00000 0.00000 0.9955 GMA2 1 0.00000 0.00000 0.00000 0.9955 GSA1 1 0.00000 0.00000 0.00000 0.9955 GSA2 1 0.00000 0.00000 0.00000 0.9955 MSA1 1 222.04200 222.04200 79.53730 0.0000 MSA2 1 10.12500 10.12500 3.62687 0.0658 For this set: 7 232.16700 33.16670 11.88060 0.0000 The following shows that neither of the two four-way interactions were significant (they were isolated by indicating that the seventh set would have two variables). Also shown is the error term for the ANOVA model. Again, the reader is reminded that an hierarchical analysis is very powerful in dealing with unequal and disproportionate subclass sample sizes. In this example the subclass sample sizes were all equal so that an ordinary orthogonal partition would produce the same results as the hierarchical analysis shown here. Such would not be the case, however, if subclass sample sizes were unequal and disproportionate. In such an event, the ordinary orthogonal partitioning would produce misleading results. FOUR-WAY INTERACTION: Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= --------------- -- ----------- ----------- ----------- ------ GMSA1 1 0.00000 0.00000 0.00000 0.9955 GMSA2 1 0.00000 0.00000 0.00000 0.9955 For this set: 2 0.00000 0.00000 0.00000 0.9999 Error 24 67.00000 2.79167 Total 47 14991.20000