CONDUCTING ONE-WAY ANALYSIS OF VARIANCE The multiple regression program can be used to rather easily analyze data using the technique of 'analysis of variance' or ANOVA. This section briefly explains how to conduct a one-way ANOVA and more complex ANOVA designs are described in subsequent sections. If you are not familiar with the conduct of ANOVA using a multiple regression program, you may wish to consult Chapter 5 in the text by Cohen, J. & Cohen, P. 'Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences', (2nd ed). Hillsdale, NJ: Lawrence Erlbaum, 1983. In order to illustrate the conduct of a one-way ANOVA, suppose you have a set of data consisting of scores on a 'depression' scale and 'marital status'. The data are shown in the next screen. Depression Marital Marital Status Score Status Code 24 Single 1 20 Single 1 26 Single 1 11 Married 2 14 Married 2 15 Married 2 38 Divorced 3 34 Divorced 3 37 Divorced 3 11 Other 4 12 Other 4 09 Other 4 13 Other 4 As you can see from the data, the depression scores represent a continuous variable and marital status represents a classification variable that has four subclasses. Each subclass can be represented by a verbal descriptor, such as 'single' or it can be represented by a numeral as shown in the column labeled as 'Marital Status Code'. In order to conduct a one-way ANOVA, you will have to create a separate variable for each of the subclasses of marital status. Then, each of these new variables will be scored as either 0 or 1. A score of 1 will mean that the individual is a member of the particular subclass, and a score of 0 means the individual is not a member of the subclass. This procedure can be used for virtually any classification variable. That is, create one new variable for each subclass and then score each of them as 0 or 1. For the above example, we shall create four variables for marital status (one for each subclass) and we shall give them the following names. MS1 = Single MS2 = Married MS3 = Divorced MS4 = Other Now, the first person in the sample was single and had a depression score of 24. Thus, we will represent that person using the scores of Depression Score MS1 MS2 MS3 MS4 24 1 0 0 0 By so scoring everyone in the sample, we obtain the following data which will be used to conduct the one-way ANOVA. Depression Marital Status Variables Score MS1 MS2 MS3 MS4 24 1 0 0 0 20 1 0 0 0 26 1 0 0 0 11 0 1 0 0 14 0 1 0 0 15 0 1 0 0 38 0 0 1 0 34 0 0 1 0 37 0 0 1 0 11 0 0 0 1 12 0 0 0 1 09 0 0 0 1 13 0 0 0 1 You are now ready to enter the above five variables and conduct a one-way ANOVA. The depression scores will be treated as the dependent variable and the four marital status variables will be treated as the independent variables. THERE IS ONE VERY IMPORTANT PROBLEM THAT MUST BE DEALT WITH!! When you conduct the regression analysis, it is ESSENTIAl that one of the above marital status variables be deleted. It does not matter, mathematically, which one you delete -- but one of them MUST be deleted. You can handle that problem by either entering only three of them or you can enter all four and then delete the one you prefer. There are good reasons for entering all of the subclass variables and then deleting one of them. We shall explain why after analyzing the data. If you wish to analyze these data, you can choose the raw data option and then use the file on the SPPC Disk 1 that has the name of ONEWAYR.DAT or you can choose the summary data option and use the file named ONEWAYS.DAT. In either case you must delete one of the 'marital status' (MS) variables. A partial analysis of the data is shown in below. In the following analysis the decision was to delete the fourth marital status variable, MS4. Make a note of the b-coefficients obtained from this solution. They are important and useful in the interpretation of the analysis. SIMULTANEOUS MULTIPLE REGRESSION RESULTS Raw Score b- Standardized Variable Name Coefficients Beta t-ratio p <= ------------- ------------ ------------ ---------- -------- Intercept 11.25000 MS1 12.08330 0.50584 7.09501 0.0001 MS2 2.08333 0.08721 1.22328 0.2514 MS3 25.08330 1.05007 14.72830 0.0000 As you saw from the foregoing screen, the intercept of the regression model has a value of 11.25. If you will compute the mean depression score for the fourth marital subclass you will discover it to be exactly equal to 11.25. That is no accident. Whenever you use this coding scheme for the subclasses of a classification variable, the intercept of the regression model will ALWAYS be exactly equal to the mean score for the subclass variable that you delete. Now compute the mean depression score for the 'single' persons. You will find it to be 23.3333. Now subtract the mean of the deleted subclass from the mean for the single persons. The difference is 23.3333 - 11.25 = 12.0833 which is identical to the b-coefficient for the 'single' subclass variable, MS1. In short, the b-coefficients will ALWAYS be identically equal to the difference between the subclass means and the intercept. Stated differently, each b-coefficient is a simple contrast that is computed as the difference between the retained subclass mean and the deleted subclass mean. In this case we have b0 = MS4 (Because we deleted the fourth subclass) b1 = MS1 - MS4 b2 = MS2 - MS4 b3 = MS3 - MS4 As an exercise, re-analyze the data using ONEWAYS.DAT and delete MS3. If you do that you will see that the b-coefficients are defined (and computed) in terms of the depression score means where b0 = MS3 (Because we deleted the third subclass) b1 = MS1 - MS3 b2 = MS2 - MS3 b3 = MS4 - MS3 Below you will see the analysis of variance table for this example. The traditional omnibus F-ratio is the one having three degrees of freedom (for the set). The single degree of freedom tests can be ignored for this analysis since none of them were 'planned' contrasts. However, the production of such single degree of freedom tests is very important for other types of analysis. In these types of analysis you will see that the Multiple Correlation is identical to the traditional Eta statistic and represents the degree of 'correlation' between depression and marital status. ANALYSIS OF VARIANCE Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= ------------- --- ---------- ---------- ------------ ------ MS1 1 35.70260 35.70260 7.18040 0.0242 MS2 1 157.73300 157.73300 31.72290 0.0005 MS3 1 1078.58000 1078.58000 216.92200 0.0000 For this set: 3 1272.02000 424.00600 85.27500 0.0000 Error 9 44.75000 4.97222 Total 12 1316.77000 SUMMARY STATISTICS Dependent Variable = Depression Omnibus F-ratio = 85.275030 Significance Level, p <= 0.000020 Multiple R = 0.982860 Squared Multiple R = 0.966015 Shrunken R = 0.977080 Shrunken Squared R = 0.954687 Determinant of Rxx = 0.676000 Regression Sum of Squares = 1272.019000 Error Sum of Squares = 44.750000 Standard Deviation of y' = 11.888460 Standard Error of Regression = 2.229848 The major point of the foregoing is that you can conduct any one-way ANOVA using this multiple regression program. You may have up to 50 subclasses in the classification variable and you can have as many cases as you like. Unequal subclass sample sizes will have no effect on the solution. We recommend that you create one variable for every subclass even though it is then necessary to delete one of them in order to carry out the analysis. The advantage is that you can then easily define or re-define the contrasts that you wish to interpret. The omnibus F-ratio and Eta statistic will not be affected by these choices but they can aid your interpretation of the subclass means and their differences.