One Wa ANOVA (Analsis of Variance) The one-wa analsis of variance (ANOVA) is used to determine whether there are an significant differences between the means of two or more independent (unrelated) groups (although we tend to onl see it used when there are a minimum of three, rather than two groups) For example, we could use a one-wa ANOVA to understand whether exam performance differed based on test anxiet levels amongst students, dividing students into three independent groups (eg, low, medium and high-stressed students) It is important to realize that the one-wa ANOVA is an omnibus test statistic and cannot tell ou which specific groups were significantl different from each other; it onl tells us that at least two groups were different Since we ma have three, four, five or more groups in our stud design, determining which of these groups differ from each other is important We can do this using a post-hoc test Assumptions: 1 Our dependent variable should be measured at the interval or ratio level (ie, the are continuous) Examples of variables that meet this criterion include revision time (measured in hours), intelligence (measured using IQ score), exam performance (measured from 0 to 100), weight (measured in kg), and so forth 2 Our independent variable should consist of two or more categorical, independent groups Tpicall, a one-wa ANOVA is used when ou have three or more categorical, independent groups, but it can be used for just two groups (but an independent-samples t-test is more commonl used for two groups) Example independent variables that meet this criterion include ethnicit (eg, 3 groups: Caucasian, African American and Hispanic), phsical activit level (eg, 4 groups: sedentar, low, moderate and high), profession (eg, 5 groups: surgeon, doctor, nurse, dentist, therapist), and so forth 1
3 We should have independence of observations, which means that there is no relationship between the observations in each group or between the groups themselves For example, there must be different participants in each group with no participant being in more than one group 4 There should be no significant outliers Outliers are simpl single data points within our data that do not follow the usual pattern (eg, in a stud of 100 students' IQ scores, where the mean score was 108 with onl a small variation between students, one student had a score of 156, which is ver unusual, and ma even put her in the top 1% of IQ scores globall) The problem with outliers is that the can have a negative effect on the one-wa ANOVA, reducing the validit of our results Fortunatel, when using SPSS to run a one-wa ANOVA on our data, ou can easil detect possible outliers 5 Your dependent variable should be approximatel normall distributed for each categor of the independent variable 6 There needs to be homogeneit of variances You can test this assumption in SPSS using Levene's test for homogeneit of variances Scenario example: A manager wants to raise the productivit at his compan b increasing the speed at which his emploees can use a particular spreadsheet program As he does not have the skills in-house, he emplos an external agenc which provides training in this spreadsheet program The offer 3 courses: a beginner, intermediate and advanced course He is unsure which course is needed for the tpe of work the do at his compan, so he sends 10 emploees on the beginner course, 10 on the intermediate and 10 on the advanced course When the all return from the training, he gives them a problem to solve using the spreadsheet program, and times how long it takes them to complete the problem He then compares the three courses (beginner, intermediate, advanced) to see if there are an differences in the average time it took to complete the problem 2
Data laout: Treatments 1 2 3 k 11 21 31 k1 12 22 32 k 2 1r 1 2 r 2 3r 3 k r k T 1 T 2 T 3 T k 1 2 3 k Hpothesis to be tested: H 0 H 1 : All the treatment means are equal Or, H 0 : 1 = 2 = 3 = = k : The are not equal ANOVA Table Source of Variation DF SS MS F c P Value Treatments k-1 Treatment Treatment SS MST SS MST F c k 1 MSE Error n-k Error SS Error SS MSE n k Total n-1 k= total no of groups n= total no of observations Multiple Comparisons: ANOVA is primaril concentrated on testing the hpothesis H 0 : 1 = 2 = 3 = = k b means of a single F-test If this hpothesis of equalit of treatment means is rejected we ma conclude that there are differences among the treatment means But this is not enough since this 3
test indicates ver little about the nature of differences among the means Hence the researcher ma often desire to decide which pairs of treatments are different and he ma want to compare one treatment effect with average of some other treatment effects The procedures adopted for deciding which pairs of means are different and for examining wide variet of possible differences among the means are called multiple comparison procedures Important multiple comparison procedures are i) Fisher s LSD method, ii) Tuke s test, iii) Scheffe s method etc The Tuke post-hoc test is generall the preferred test for conducting post-hoc tests on a onewa ANOVA Perform ANOVA and test the significance of the difference between fertilizer means Hpothesis: Fertilizer 1 Fertilizer 2 Fertilizer 3 77 72 76 81 58 85 71 74 82 76 66 80 80 70 77 H 0 H 1 : All the fertilizer means are equal : The are not equal Procedure: Analze Compare Means One-Wa ANOVA Dependent List Factor : Yield/Response : Fertilizer Click Post Hoc: Click Tuke Continue Oka 4
Comment: From ANOVA table it has been found that the P-value corresponding to the fertilizer is 0005 That is fertilizer is highl significant at 5% level of significance That is the fertilizer means differ significantl Comment: Fertilizer 1 significantl differs from fertilizer 2 (p<005) Fertilizer 2 significantl differs from fertilizer 1 and from fertilizer 3 (p<005) Fertilizer 3 significantl differs from fertilizer 2 (p<005) 5