OneWay ANOVA Analysis of Variance By Ahlam Lee

One-Way ANOVA (Analysis of Variance) By Ahlam Lee

Contents • Independent Samples T-test vs. One-Way ANOVA • <SPSS> One-Way ANOVA • HW#10 Hints

T-test vs. ANOVA Definition Assumption Independent Samples T-TEST One-Way ANOVA Is used when comparing Is used when you have to compare means of two different groups means of several groups (i. e. , more than two different groups) Normally distributed outcome Equal variances between the groups Groups are independent Independent Samples T-Test Variance Test Levene’s Test for Equality of Variance One-Way ANOVA Test of Homogeneity of Variances

One Way ANOVA • The purpose of a one-way analysis of variance (one-way ANOVA) is to compare the means of two or more groups (the independent variable) on one dependent variable to see if the group means are significantly different from each other.

One Way ANOVA Group A vs. Group B vs. Group D, Group E…. . VS. etc. Group C

One Way ANOVA • A one-way ANOVA tests the hull hypothesis (Ho) that states that all the groups represent populations that have the same means. Where there are three means, the null hypothesis is: Ho : µ 1 = µ 2 = µ 3 Ha: µ 1 ≠ µ 2 and/or µ 1 ≠ µ 3 and/or µ 2 ≠ µ 3

Research Question • Is there a significant difference in math scores among general, academic, and vocation programs?

<SPSS> One-Way ANOVA

<SPSS> One-Way ANOVA Math Scores should be moved into “Dependent List. ” Type of Program should be moved into “Factor. ”

<SPSS> One-Way ANOVA Click on “Options. ”

<SPSS> One-Way ANOVA After clicking on “Options, you will see the following SPSS dialogue box named “One. Way ANOVA Options. ”

<SPSS> One-Way ANOVA In this SPSS dialogue box, check off “Descriptive” and “Homogeneity of variance test. ”

<SPSS> One-Way ANOVA Then, Click on “OK. ”

<SPSS> One-Way ANOVA

Equal variance among the groups should be met in order to use ANOVA. Non-Significant result on this test mean equal variance has been met. Before seeing the results of our One-Way ANOVA, Test of Homogeneity of Variances is reported, which examines whether our assumption (i. e. , equal variances among the groups) has been met. In this test, the null hypothesis is “there is no significant difference in the variances among the groups” and the alternative hypothesis is “there is a significant difference in the variances among the group”. So, if the Levene’s Test is significant (the value under “Sig” is less than. 05), the variances between the groups are different. However, if it is not significant (“Sig” is greater than. 05), the variances between the groups are not significantly different; that is, the variances between the groups are approximately equal. Here, we see that the significance is. 350, which is greater than. 05. We assume that the variances between the groups are approximately equal. There is a significant difference in math scores among the three groups, F(2, 197) = 29. 279, p <. 001. HOWEVER, This ANOVA Result DOES NOT SHOW the DIFFERENCE Between the TWO GROUPS (i. e. , General vs. Academic; Academic vs. Vocation; General vs. Vocation).

There is a significant difference in math scores among the three groups, F(2, 197) = 29. 279, p <. 001. HOWEVER, This ANOVA Result DOES NOT SHOW the DIFFERENCE Between the TWO GROUPS (i. e. , General vs. Academic; Academic vs. Vocation; General vs. Vocation). So when running one-way ANOVA, we should conduct a post-hoc test in order to see the difference between the TWO GROUPS. General Program Academic Program vs. vs. Academic Program Vocation Program

<SPSS> Post Hoc Test Click on “Post Hoc. ” After clicking on “Post Hoc, ” you will see the following SPSS dialog box.

<SPSS> Post Hoc Test 1. Check off “Bonferroni” in this dialog box. 2. Then, Click on “Continue. ”

<SPSS> Post Hoc Test Click on “OK. ”

Post Hoc Tests The post-hoc test showed that there is a significant difference between general and academic programs in the math score ( p <. 001). Also, there is a significant difference between academic and vocation programs in the math score ( p <. 001). However, there is no significant difference between general and vocation program in the math score ( p >. 1). Note: In terms of types of post-hoc tests, there are many different kinds of post-hoc tests in the option box. Among them, we (researchers) often consider Bonferroni’s and Tukey’s tests. Both tests control the Type I error rate very well but are conservative tests. Of the two, Bonferroni has more power when the number of comparisons is small, whereas Tukey is more powerful when testing large numbers of means. In our example, we have only three comparisons, so we chose Bonferroni.

<SPSS> HW#10 hints • Using the dataset (hsb 2. sav), please compare the social study score (socst) among the three program types (prog) using One-Way ANOVA. Attach the SPSS outputs and interpret the results.

<SPSS> HW#10

<SPSS> HW#10 Social Science Scores should be moved into “Dependent List. ” Type of Program should be moved into “Factor. ”

<SPSS> HW#10 Click on “Options. ”

<SPSS> HW#10 After clicking on “Options, you will see the following SPSS dialogue box named “One. Way ANOVA Options. ”

<SPSS> HW#10 In this SPSS dialogue box, check off “Descriptive” and “Homogeneity of variance test. ”

<SPSS> Post-Hoc Test Click on “Post Hoc. ”

<SPSS> Post-Hoc Test After clicking on “Post Hoc, ” you will see the following SPSS dialog box.

<SPSS> Post-Hoc Test 1. Check off “Bonferroni” for Post-Hoc Test. 2. Click on “Continue”.

<SPSS> HW#10 Then, Click on “OK. ”

<SPSS> Post-Hoc Test (HW#10)

Test of Homogeneity of Variances examines whether or not our assumption (i. e. , equal variances among the groups) has been met. In this test, Ho: “there is no significant difference in the variances among the groups. ” Ha: “there is a significant difference in the variances among the groups. ” The result shows that there is no significant difference in the variances among the group ( p =. 288). Thus, we fail to reject Ho and conclude that the variances among the groups are not significant different, suggesting that equal variance among the groups has been met. There is a significant difference in social studies scores among the three programs, F(2, 197) = 26. 112, p <. 001.

Post Hoc Tests The post-hoc test revealed the following findings: (1) there is a significant difference in the social studies scores between general and academic programs, p <. 01; (2) there is a significant difference in the social studies scores between general and vocation programs, p <. 05; and (3) there is a significant difference in the social studies scores between academic and vocation programs, p <. 001.