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How to compare the difference on >2 groups on one or more variables If it is only one variable, we could compare three groups with multiple ttests: M1 vs. M2, M1 vs. M3, M2 vs. M3 >2 variables? For example, how two teaching methods are different for three different sizes of classes. ANOVA allows you to see if there is any difference between groups on some variables. 3

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“Analysis of Variance” A hypothesis-testing procedure used to evaluate mean differences between two or more treatments (or populations) on different variables. ANOVA is available for both parametric (score data) and non-parametric (ranking) data. Advantages: 1) Can work with more than two samples. 2) Can work with more than one independent variable 4

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Assume that you have data on student performance in non-assessed tutorial exercises as well as their final grading. You are interested in seeing if tutorial performance is related to final grade. ANOVA allows you to break up the group according to the grade and then see if performance is different across these grades. 5

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One-way between groups Differences between the groups The groups are categorized in one way, such as groups were divided by age, or grade. This is the simplest version of ANOVA It allows us to compare variable between different groups, for example, to compare tutorial performance from different students grouped by grade. 6

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One-way repeated measures A single group has been measured by a variable for a few times Example 1: one group of patients were tested by a new drug in different times: before taking the drug, after taking the drug Example 2: student performance on the tutorial over time. 7

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Two-way between groups For example: the grades by tutorial analysis could be extended to see if overseas students performed differently to local students. What you would have from this form of ANOVA is: The effect of final grade The effect of overseas versus local The interaction between final grade and overseas/local Each of the main effects are one-way tests. The interaction effect is simply asking "is there any significant difference in performance when you take final grade and overseas/local acting together". 8

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Two-way repeated measures Use the repeated measures Include an interaction effect For example, we want to see the performance of tutorial about gender and time of testing. We have the same two groups (male, and female groups) and test them in different times to compare the difference. 9

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In ANOVA an independent or quasi- independent variable is called a factor. Factor = independent (or quasi-independent) variable. Levels = number of values used for the independent variable. One factor → “single-factor design” More than one factor → “factorial design” 10

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An example of a single-factor design A example of a two-factor design 11

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ANOVA calculates the mean for each of the final grading groups on the tutorial exercise figure - the Group Means. It calculates the mean for all the groups combined - the Overall Mean. Then it calculates, within each group, the total deviation of each individual's score from the Group Mean - Within Group Variation. Next, it calculates the deviation of each Group Mean from the Overall Mean - Between Group Variation. Finally, ANOVA produces the F statistic which is the ratio Between Group Variation to the Within Group Variation. If the Between Group Variation is significantly greater than the Within Group Variation, then it is likely that there is a statistically significant difference between the groups. The statistical package will tell you if the F ratio is significant or not. All versions of ANOVA follow these basic principles but the sources of Variation get more complex as the number of groups and the interaction effects increase. 12

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Variance between treatments can have two interpretations: Variance is due to differences between treatments. Variance is due to chance alone. This may be due to individual differences or experimental error. 13

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Data Analysis—Analysis Tools— three different ANOVA: Anova: Single Factor (one-way between groups) Anova: Two-factors With Replication Anova: Two-Factors Without Replication 14

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Three groups of preschoolers and their language scores, whether they are overall different? Group 1 ScoresGroup 2 ScoresGroup 3 Scores 87 89 868591 769996 568587 787989 988190 778289 667896 758596 679193 15

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Step1: a statement of the null and research hypothesis One-tailed or two-tailed (there is no such thing in ANOVA) 16

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Step2: Setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis 0.05 17

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Step3: Selection of the appropriate test statistics ANOVA: Single factor 18

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Group 1 Scoresx squareGroup 2 Scoresx squareGroup 3 Scoresx square 877569877569897921 867396857225918281 765776999801969216 563136857225877569 786084796241897921 989604816561908100 775929826724897921 664356786084969216 755625857225969216 674489918281938649 n10 N30 ∑x 766 852 916 ∑∑X2534 76.6 85.2 91.6 214038.5333 59964 72936 84010 216910 58675.6 72590.4 83905.6 215171.6

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Between sum of squares 215171.6-214038.531133.07 within sum of squares 216910-215171.601738.40 total sum of squares 216910-214038.532871.47

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Between-group degree of freedom=k-1 k: number of groups Within-group degree of freedom=N-k N: total sample size source sums of squaresdf mean sums of squaresF Between groups1133.072566.538.799 Within gruops1738.402764.39 Total2871.4729

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Between-group degree of freedom=k-1 k: number of groups Within-group degree of freedom=N-k N: total sample size 22

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Step4: (cont.) df for the denominator = n-k=30-3=27 df for the numerator = k-1=3-1=2 23

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Step4: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic Table-Distribution of F (http://www.socr.ucla.edu/applets.dir/f_table.html) 24

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Step5: comparison of the obtained value and the critical value If obtained value > the critical value, reject the null hypothesis If obtained value < the critical value, accept the null hypothesis 8.80 and 3.36 25

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Step6 and 7: decision time What is your conclusion? Why? How do you interpret F (2, 27) =8.80, p<0.05 26

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