Review on Univariate Analysis of Variance (ANOVA) Pekka Malo 30E00500 Quantitative Empirical Research Spring 2016

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1 Review on Univariate Analysis of Variance (ANOVA) Pekka Malo 30E00500 Quantitative Empirical Research Spring 2016

2 Analysis of Variance Analysis of Variance (ANOVA) ~ special case of multiple regression, where the objective is to compare differences between two or more groups for single metric dependent variable. Consumers shown different advertising messages: Which message is more likely to lead to purchase? A company has several customer segments: Do the segments differ in terms of customer satisfaction? ANOVA 2

3 Univariate One-Way ANOVA ANOVA 3

4 Univariate One-Way ANOVA: model Do the means between the different groups 1 to k differ? H 0 : µ 1 = µ 2 = = µ k H 1 : one or more of the groups has a different mean ANOVA 4

5 Assumptions Independence of observations Equal variance (homoskedasticity) Normal distribution ANOVA 5

6 Example 1: Beverages and Reaction Times Sample 1: reaction times Sample 2: reaction times Sample 3: reaction times ANOVA 6

7 REACTION TIMES ANOVA 7

8 Does beverage have effect on reactions? H 0 : reaction times of all groups are the same (i.e. variation is due to people and not the drink) H 1 : reaction times of at least two groups are different (i.e., variation is due to drinks, not just people) ANOVA 8

9 Does beverage have effect on reactions? Compare the amounts of variance that come from The variance between the groups The variance within the groups Examine their ratio: F= Between groups variance/within groups variance ANOVA 9

10 F-statistic for One-Way ANOVA Based on the comparison of between groups and within groups variance When H 0 is true, we can estimate variance σ 2 in two different ways: Using pooled within-sample estimator: Using the variance of the sample means around overall mean: s 2 y = ki=1 (y i. y.. ) 2 k 1, MANOVA 10

11 Variation within the groups Sample 1: reaction times Sample 2: reaction times Sample 3: reaction times pooled within-sample estimate for variation within groups ANOVA 11

12 Variation between the groups Sample 1: reaction times Sample 2: reaction times Sample 3: reaction times ANOVA 12

13 F-statistic for One-Way ANOVA (cont.) Then the F-statistic is given by F = ns2 y s 2 e = ( i y2 i. /n y2.. /kn)/ (k 1) ( ij y2 ij i y2 i. /n ) /[k(n 1)] Where SSH is between group SS, and SSE is within group SS MANOVA 13

14 ANOVA and Regression ANOVA can be viewed as a form of multiple regression model, where independent variables are levels of discrete variables rather than the more usual continuous regression variables For example, a one-way between subjects ANOVA with two levels requires only one variable X to separate levels For more details, see Tabachnic and Fidell (2007) MANOVA 14

15 ANOVA and Regression (cont d) MANOVA 15

16 Identifying Differences Between Individual Groups Once you are confident that there are differences between groups, it is possible to perform a variety of tests to understand which groups differ Contrasts (~ a priori tests): Planned comparisons based on scientific goals Post-hoc tests (~ decide after experiment): Performed for each dependent variable separately Used for situations where you can decide which comparisons you want to make after looking at the data MANOVA 16

17 Contrasts Contrasts are used to test whether the levels of an effect (i.e. groups) are significantly different from one another Planned comparisons are more powerful due to their smaller number but difficult to specify correctly Appropriate when conceptual base is well defined where C = contrast value G i = group means W i = weights MANOVA 17

18 Commonly used contrasts Deviation: Compares the mean of each level (except a reference category) to the mean of all of the levels (grand mean). The levels of the variable can be in any order. Simple: Compares the mean of each level to the mean of a specified level. This type of contrast is useful when there is a control group. You can choose the first or last category as the reference. Difference: Compares the mean of each level (except the first) to the mean of previous levels. (Sometimes called reverse Helmert contrasts.) MANOVA 18

19 Commonly used contrasts (cont d) Helmert: Compares the mean of each level of the factor (except the last) to the mean of subsequent levels. Repeated: Compares the mean of each level (except the last) to the mean of the subsequent level. Polynomial: Compares the linear effect, quadratic effect, cubic effect, and so on. The first degree of freedom contains the linear effect across all categories; the second degree of freedom, the quadratic effect; and so on. These contrasts are often used to estimate polynomial trends. MANOVA 19

20 Post-hoc Multiple Comparisons Commonly used tests for multiple comparisons: Bonferroni s test: based on Student s t-test, adjusts the significance level for multiple comparisons Tukey s honestly significant difference test: uses Studentized range statistic to make all pairwise comparisons between groups Choose a test based on the number of pairwise comparisons: Use Tukey for large number of pairs: More powerful than Bonferroni when several pairs of means considered Use Bonferroni for small number of pairs MANOVA 20

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