CONDUCTING TWO-WAY ANALYSIS OF VARIANCE This section briefly introduces a simple method for using multiple regression as a means of conducting two-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 Analysis of Variance' before proceeding. The conduct of two-way ANOVA is largely an extension of the technique described for one-way ANOVA. The essential difference is that you now have one continuous dependent variable and two classification variables. You therefore must create new variables for each of the subclasses within each of the two classification variables. This will be illustrated with a simple example. Suppose you have a set of depression scores, as before, but you now have two classification variables. One represents 'Marital Status' and the other represents 'Social Class' as follows. Depression Marital Social Score Status Class 23 Single Lower 22 Single Middle 24 Single Lower 20 Single Middle 26 Single Upper 11 Married Lower 14 Married Middle 15 Married Upper 38 Divorced Lower 34 Divorced Middle 37 Divorced Upper 11 Other Lower 12 Other Middle 09 Other Middle 13 Other Upper As before, you must create one new variable for each subclass of each of the two classification variables. That is done as shown below. MS1 = Single SC1 = Lower MS2 = Married SC2 = Middle MS3 = Divorced SC3 = Upper MS4 = Other The next step is to code each person as 0 if they are NOT a member of the subclass and code the person as 1 to show that they are a member of the subclass. When that is done for each person in the sample we obtain the data shown as follows: Depression Marital Status Variables Social Class Score MS1 MS2 MS3 MS4 SC1 SC2 SC3 23 1 0 0 0 1 0 0 22 1 0 0 0 0 1 0 24 1 0 0 0 1 0 0 20 1 0 0 0 0 1 0 26 1 0 0 0 0 0 1 11 0 1 0 0 1 0 0 14 0 1 0 0 0 1 0 15 0 1 0 0 0 0 1 38 0 0 1 0 1 0 0 34 0 0 1 0 0 1 0 37 0 0 1 0 0 0 1 11 0 0 0 1 1 0 0 12 0 0 0 1 0 1 0 09 0 0 0 1 0 1 0 13 0 0 0 1 0 0 1 Now that we have created the basic data for the two classification variables, we must once again delete one of the subclass variables. However, that must be done for each of the two classification variables. As before, it does not matter which of the sublcass variables is deleted within each of the two classification variables. For purposes of illustration, we shall delete the last variable in each of the classification variables, and the results are shown below. Depression Score MS1 MS2 MS3 SC1 SC2 23 1 0 0 1 0 22 1 0 0 0 1 24 1 0 0 1 0 20 1 0 0 0 1 26 1 0 0 0 0 11 0 1 0 1 0 14 0 1 0 0 1 15 0 1 0 0 0 38 0 0 1 1 0 34 0 0 1 0 1 37 0 0 1 0 0 11 0 0 0 1 0 12 0 0 0 0 1 09 0 0 0 0 1 13 0 0 0 0 0 Now that we have constructed a suitable set of subclass variables for each of the two classification variables, we could set about to analyze the data as shown above. Unfortunately, the values of MS1 throuch MS3 represent only the 'main effect' of marital status and SC1 and SC2 represent only the 'main effect' of social class. We have not accounted for the possibility of an interation between marital status and social class. Such an interaction must be described with yet additional variables. Fortunately, it is easy enough to construct them. In order to construct the variables needed to represent the interaction between two classification variables, you need only follow a simple rule. Multiply each of the retained variables in the first 'factor' (marital status) by each of the retained variables in the second 'factor' (social class) to produce, in this example, M1xS1 = MS1 * SC1 M2xS1 = MS2 * SC1 M3xS1 = MS3 * SC1 M1xS2 = MS1 * SC2 M2xS2 = MS2 * SC2 M3xS2 = MS3 * SC2 If that is done for each case in the sample we obtain the data shown as follows: Score MS1 MS2 MS3 SC1 SC2 M1xS1 M2xS1 M3xS1 M1xS2 M2xS2 M3xS2 23 1 0 0 1 0 1 0 0 0 0 0 22 1 0 0 0 1 0 0 0 1 0 0 24 1 0 0 1 0 1 0 0 0 0 0 20 1 0 0 0 1 0 0 0 1 0 0 26 1 0 0 0 0 0 0 0 0 0 0 11 0 1 0 1 0 0 1 0 0 0 0 14 0 1 0 0 1 0 0 0 0 1 0 15 0 1 0 0 0 0 0 0 0 0 0 38 0 0 1 1 0 0 0 1 0 0 0 34 0 0 1 0 1 0 0 0 0 0 1 37 0 0 1 0 0 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 09 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 If you wish to analyze these data, choose the raw data option and then use the file on the SPPC Disk 1 diskette that has the name TWOWAYR.DAT. You can also choose the summary data option and use the file named TWOWAYS.DAT. Partial results for this analysis are shown below along with a brief interpretation of them. SIMULTANEOUS MULTIPLE REGRESSION RESULTS Raw Score b- Standardized Variable Name Coefficients Beta t-ratio p <= -------------- ------------ ------------ ---------- -------- Intercept 13.00000 MS1 13.00000 0.65189 6.01783 0.0078 MS2 2.00000 0.08510 0.92582 0.5749 MS3 24.00000 1.02120 11.10980 0.0012 SC1 -2.00000 -0.10029 -0.92582 0.5749 SC2 -2.50000 -0.13028 -1.33631 0.2738 M1xS1 -0.50000 -0.01808 -0.17496 0.8656 M2xS1 -2.00000 -0.05306 -0.65465 0.5613 M3xS1 3.00000 0.07960 0.98198 0.5996 M1xS2 -2.50000 -0.09040 -0.94491 0.5834 M2xS2 1.50000 0.03980 0.52489 0.6363 M3xS2 -0.50000 -0.01326 -0.17496 0.8656 The raw-score b-coefficients are interpreted in the same manner as in the one-way ANOVA but there are some differences. For example, since the 'other' marital status subclass variable was eliminated, that causes the b-coefficients for 'Marital Statis' to be defined as contrasts in terms of the mean depression score for MS4. That is, b1 = MS1 - MS4 b2 = MS2 - MS4, and b3 = MS3 - MS4 The coefficients for 'Social Class' are defined as contrasts with the mean score for SC3 since that was the subclass variable omitted from the model. Thus, b4 = SC1 - SC3, and b5 = SC2 - SC3 If you will compute the subclass mean depression scores for the two ways of classifying the individuals in this sample you will discover that the above b-coefficients, or contrasts, are not exactly equal to the subclass mean differences as was the case with the one-way ANOVA example. The reason for that arises from the fact that each cell in the design does not have the same number of cases. This may seem to be a disadvantage. Actually, what is seen here is an approach to factorial analysis of variance that provides a simple and powerful means of dealing with designs having unequal and disproportionate numbers of cases in the subclasses. For an important and extensive discussion of this topic see Chapter 5 of the Cohen & Cohen text. The ANOVA table for this example is presented below. It is important to note the hierarchical step-down F-ratios and their associated probabilities. In other words, it is necessary to interpret the model in an hierarchical manner whenever one has unequal and disproportionate subclass sample sizes. Again, it is urged that the unfamiliar user consult discussions such as those found in Chapter 5 of the Cohen & Cohen text. ANALYSIS OF VARIANCE Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= ------------- ---- ------------ ---------- ----------- ------ MS1 1 43.20000 43.20000 18.51430 0.0212 MS2 1 157.73300 157.73300 67.60000 0.0030 MS3 1 1078.58000 1078.58000 462.25000 0.0002 For this set: 3 1279.52000 426.50600 182.78800 0.0008 SC1 1 0.00211 0.00211 0.00090 0.9765 SC2 1 21.50360 21.50360 9.21584 0.0545 For this set: 2 21.50570 10.75290 4.60837 0.1221 M1xS1 1 0.69122 0.69122 0.29623 0.6248 M2xS1 1 8.14787 8.14787 3.49194 0.1580 M3xS1 1 3.90516 3.90516 1.67364 0.2865 M1xS2 1 3.73333 3.73333 1.60000 0.2955 M2xS2 1 1.02857 1.02857 0.44081 0.5562 M3xS2 1 0.07142 0.07142 0.03061 0.8656 For this set: 6 17.57760 2.92960 1.25554 0.4594 Error 3 7.00000 2.33333 Total 14 1325.60000 Space does not allow a more extended discussion of the interpretation of hierarchical analysis. However, any two-way crossed factorial ANOVA can be constructed by use of the above procedures. It does not matter how many subclasses are present within each of the two classification variables for such designs. The present limits of the program are such that you may have no more than 50 retained subclass and interaction variables in any design or regression analysis.