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Bivariate Analysis in SPSS: ANOVA (Textbook Resources: Chapter 22, pp. 541-546, and Session 13) Please use the SPSS data file GSS2004.sav for the following examples. I. Overview of the ANOVA Used to determine whether the mean of a given variable is statistically different for more than two different groups within the population Used for interval or ratio scale data. (At least one of the variables must be interval/ratio.) The groups must be independent. Groups are independent if there is no relationship between the objects or people in each of the different groups. This test can also be used to test for statistical differences in proportions (π) if you have a variable measured using a nominal scale with only two response categories, but the variable must be coded using “0” and “1”. (See handout on Univariate Analysis for an example of this.) II. Performing the ANOVA in SPSS Example: Is there a relationship between an individual’s level of education and their feelings about abortion? That is, are one’s feelings about abortion a function of education? Please use the variables ABANY and DEGREE for this example. In this example which variable is the dependent variable? ____________ What type of measurement scale is used to measure the dependent variable? Which variable is the independent or grouping variable? ____________ 1 What type of measurement scale is used to measure the independent variable? How many groups do we have under the independent variable? ____________ Are these groups independent? Even though both variables are nominal/ordinal scale variables we can use the ANOVA as a test of proportions. That is, we can focus on the proportion of individuals who say yes. However, the data is not coded correctly so we must fix this first. Please recode ABANY so that we can use a t-test of proportions. Don’t forget to recode missing values, too, and to specify the characteristics of your new variable in the variable view after you have finished recoding. 1. For this research problem, what are the null and alternative hypotheses? H0: Ha: The alternative hypothesis does not imply that all five mean values or proportions are statistically different from one another but rather that at least one of the five is statistically different from the others. 2. What is the appropriate test to test the null hypothesis? ___________________ 3. Calculate the test statistic Analyze → Compare Means → One-Way ANOVA → Select the “Dependent List” (the variable for which the mean will be calculated-- must be measured using an interval or ratio scale or nominal scale if a test of proportions) → Select “Factor” (the grouping/independent variable--may be measured using a nominal, ordinal, interval or ratio scale) 2 Under the Options tab you can: Generate descriptive statistics Perform the Levene test for “homogeneity of variances test” Specify the treatment of missing values Under the Options tab select “Descriptive”. 4. Do we accept or reject the null hypothesis? 5. Based on this what can we say? What can we conclude? Further Interpretation of the Results: While the ANOVA test allows us to conclude that the five proportions are not statistically the same, or rather that at least one of the proportions is different from the other four, it does not tell us which specific groups are statistically different from the others. For example, is the proportion of individuals who support abortion and have a high school diploma statistically different from the proportion of individuals who support abortion but have a college degree? Are all five statistically different from one another? The ANOVA test alone cannot answer this question. To get this information you need to use the Post Hoc tab under the ANOVA test. The appropriate Post Hoc test, however, depends upon whether or not the variances of the various groups are assumed to be equal or not. How do you know if the variances are equal? Just like the t-test for two independent means you use the test for homogeneity of variances. Under the Options tab you will find a box labeled “statistics”. In this box is the option “Homogeneity of variance test”. 3 The null and alternative hypotheses under this test are: H0: 21 = 22 = 23 = 24 = 25 Ha: 21 22 23 24 25 where the numbers refer to each of the five groups, respectively. If the p-value is greater than the significance level then accept the null hypothesis of equal variances. If the p-value is smaller than the significance level, then reject the null hypothesis that the population variances are equal. Given the results of the test for homogeneity of variances you then go back to: Analyze → Compare Means → One-Way ANOVA → Select the Post Hoc tab. Under this tab you will find many different tests. Which one should you choose? The actual choice of the post hoc test does not matter as long as you interpret the results of the test for homogeneity of variances correctly and pick your test from the appropriate category: “Equal variances assumed” or “Equal variances not assumed”. In this example should we assume equal variances? Given this, you can select any of the tests under equal variances not assumed. Run the Post Hoc test using Tamhane’s T2. How do you interpret the results of the Post Hoc test? The starred values are statistically different at the 5 percent significance level. 4 Based on this what can we say? What can we conclude? Example: Is the happiness that a person feels in their marriage a function of their level of education? That is, are people more or less likely to be happy in their marriage depending upon the amount of education that they have? Please use the variables HAPMAR and EDUC for this example. 1. Which variable in this example is the: Independent variable: ____________________ Dependent variable: _____________________ 2. What is the scale of measurement for the: Independent variable: ____________________ Dependent variable: _____________________ 3. What are the null and alternative hypotheses? 4. Calculate the test statistic. Be sure to check “Descriptive” and “Homogeneity of variance test” under Options. 5 5. Do we accept or reject the null hypothesis? 6. Should we assume equal variances? 7. What conclusions can we draw? 6