Oneway ANOVA - Overview
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1. Students were randomly assigned to one of three groups and given a math test. The teacher then provided different levels of praise to each group: high praise to Group A, moderate praise to Group B, and no praise to Group C. 2. The next day, all students took another math test, and their scores were analyzed using ANOVA to determine if the different levels of praise had an effect. 3. The ANOVA analysis found a significant difference between the group means, with Group A scoring highest and Group C scoring lowest. This led to a rejection of the null hypothesis that the praise had no impact on test scores.
Oneway ANOVA - Overview
- 1. Analysis of Variance: Example
- 2. Learning ANOVA through an example • All students were given a math test. • Ahead of time, the students were randomly assigned to one of three experimental groups (but they did not know about it). • After the first math test, the teacher behaved differently with members of the three different experimental groups. • Data from Section 42 of Success at Statistics by Pyrczak
- 3. Creating different conditions in the groups • Regardless of their actual performance on the test, the teacher … • Gave massive amounts of praise for any correct answers to students in Group A. • Gave moderate amounts of praise for any correct answers to students in Group B. • Gave no praise for correct answers, just their score, to students in Group C.
- 4. Then the variable of interest was measured • The next day, at the end of the math lesson, the teacher gave another test. • Scores for all the students were recorded, as well as the amount of praise they had received for correct answers the day before. • The researchers thought that the earlier praise might have an effect on their scores on the second test. • ANOVA’s F-ratio will tell us if that’s true.
- 5. F test is a ratio of variance BETWEEN groups and variance WITHIN groups • On the top: difference between groups, which includes systematic and random components. • On the bottom: difference within groups, which includes only the random component. • When the systematic component is large, the groups differ from each other, and F > 1.00 difference s including any treatment effects difference s with no treatment effects F
- 6. Stating Hypotheses • H0: The amount of praise given has no impact on the math post-test. • HA: Groups who receive different amounts of praise will have different mean scores.
- 7. Scores on Test 2 for 18 students Group X A 7 A 6 A 5 A 8 A 3 A 7 B 4 B 6 B 4 B 7 B 5 B 7 C 3 C 2 C 1 C 3 C 4 C 1 ΣX 83 Mean 4.6111
- 8. Compare scores to M= 4.61
- 9. Part II: Variability: distance from the mean of all the scores on the test
- 10. Just as in Chapter 3, the differences are squared Σ(X-M)2 = Sum of Squares = SSTOTAL Group X Mtotal X-M (X-M)2 A 7 4.6111 2.3889 5.7068 A 6 4.6111 1.3889 1.9290 A 5 4.6111 0.3889 0.1512 A 8 4.6111 3.3889 11.4846 A 3 4.6111 -1.6111 2.5957 A 7 4.6111 2.3889 5.7068 B 4 4.6111 -0.6111 0.3735 B 6 4.6111 1.3889 1.9290 B 4 4.6111 -0.6111 0.3735 B 7 4.6111 2.3889 5.7068 B 5 4.6111 0.3889 0.1512 B 7 4.6111 2.3889 5.7068 C 3 4.6111 -1.6111 2.5957 C 2 4.6111 -2.6111 6.8179 C 1 4.6111 -3.6111 13.0401 C 3 4.6111 -1.6111 2.5957 C 4 4.6111 -0.6111 0.3735 C 1 4.6111 -3.6111 13.0401 Sum of squares: all scores = SSTOTAL = 80.278 G 83 80.27778 SSTOTAL Mean 4.6111 4.459877 Variance
- 11. What about the impact of praise? • The mean for all the students is 4.61. • Do all three groups of students have similar means? • H0: The amount of praise given has no impact on the math post-test. 0 1 2 3 H : • HA: Groups who receive different amounts of praise will have different mean scores.
- 12. Compute mean score in the groups • Mean of Group A: MA=6.00 • Mean of Group B: MB=5.50 • Mean of Group C: MC=2.33 Group X A 7 A 6 A 5 A 8 A 3 A 7 Mean 6 Group X B 4 B 6 B 4 B 7 B 5 B 7 Mean 5.5 Group X C 3 C 2 C 1 C 3 C 4 C 1 Mean 2.333
- 13. Compare each student’s score to the mean score for his or her own group MA=6.00 MB=5.50 MC=2.33
- 14. Variability within each group is random: all within group had same amount of praise SSA=16.00 SSB=9.50 SSC=7.333 Group X MA (X-MA)2 A 7 6 1 A 6 6 0 A 5 6 1 A 8 6 4 A 3 6 9 A 7 6 1 Mean 6 SSA 16.000 Group X MB (X-MB)2 B 4 5.5 2.25 B 6 5.5 0.25 B 4 5.5 2.25 B 7 5.5 2.25 B 5 5.5 0.25 B 7 5.5 2.25 Mean 5.5 SSB 9.500 Group X MC (X-MC)2 C 3 2.333 0.445 C 2 2.333 0.111 C 1 2.333 1.777 C 3 2.333 0.445 C 4 2.333 2.779 C 1 2.333 1.777 Mean 2.333 SSC 7.333
- 15. Variability within each group is random: all within group had same amount of praise To find the amount of random variability, add the SS from all the groups together. Within Sum of squares SSwithin=16+9.5+7.33 SSwithin=32.833 SSA=16.00 SSB=9.50 SSC=7.333
- 16. Part V: Analysis of Variance: Partitioning variability into components • SSTOTAL is all the variability in the Sample • Some of it is systematic variability between groups related to the treatment, level of praise by the teacher • Some of it is random within groups, due to the many differences among students besides the praise level • SSTOTAL = SSWITHIN + SSBETWEEN
- 17. Variability between groups is due to the teacher’s level of praise • The means of the groups are not the same • MA=6.00 MB=5.50 MC=2.33 • SSBETWEEN represents the variability due to the different praise level treatments • SSTOTAL and SSWITHIN have been computed • SSTOTAL = 80.28 and SSWITHIN = 32.83 • SSBETWEEN = SSTOTAL – SSWITHIN • SSBETWEEN = 80.28 – 32.83 • SSBETWEEN = 47.45
- 18. Part VI: Asking the research question a new way: as a ratio between variances • Is the SSBETWEEN large relative to SSWITHIN ? • If SSBETWEEN is large relative to the SSWITHIN then the treatment (teacher praise) had an effect. • If SSBETWEEN is large, REJECT the null hypothesis. • The F-statistic is a ratio of those two components of variability, adjusted for sample size. variabilit y including any treatment effects variabilit y with no treatment effects F
- 19. Compute degrees of freedom for Sstotal Sswithin and SSbetween
- 20. Each kind of SS has its own df • Total degrees of freedom for SSTOTAL dftotal= N – 1 (N is the total number of cases) dftotal = 18 – 1 = 17 • Between-treatments degrees of freedom for SSBETWEEN dfbetween= k – 1 (k is the number of groups) dfbetween= 3 – 1 = 2 • Within-groups degrees of freedom for SSWITHIN dfwithin= N – k dfwithin= 18 – 3 = 15
- 21. “Average” the SSwithin and SSbetween over their df These are called “Mean Squares”
- 22. Equations for Mean Squares & F • The between and within sums of squares are divided by their df to create the appropriate variance • These are called the Mean Squares • The SS is averaged (mean) across df • The F-ratio test statistic is the ratio of MSbetween to MSwithin between within SS SS within MS MS within df between between df between within MS MS F
- 23. Computing the Mean Squares • SSBETWEEN = 47.45 dfbetween= 3 – 1 = 2 • SSWITHIN = 32.833 dfwithin = 18 – 3 = 15 23.75 47.45 between SS 2 between between df MS 2.189 32.833 within SS 15 within within df MS
- 24. Computing F for the example F = 23.725 / 2.189 10.849 F = 10.849 between 23.735 MS between MS 2.189 within MS F within MS F
- 25. Testing hypotheses with F 10.849 23.75 between MS 2.189 within MS F • When the p-value for F is less than the alpha you chose for your test, then you can Reject H0 • There are critical values for F that define a rejection region – but they vary by both types of df and (outside of intro statistics courses) no one knows any of them by heart. • In this class: we use p-value only, from the F Distribution calculator.
- 26. Testing hypotheses with F 10.849 23.75 between MS 2.189 within MS F • The p-value of F = 10.849 for df = 2, 15 is p=.0012 • Using = .05 • Since the p-value is less than (<) the alpha level, we Reject the null hypothesis. • Some groups had different levels of performance on the test due to the level of the teacher’s praise.
- 27. ANOVA table – a tool for computing F Source SS df MS F Between 47.45 2 23.725 10.84 Within 32.83 15 2.189 Total 80.28 17 • The SS and df columns add up to the total • In each row, SS divided by df equals MS • In the final column, F is MSB divided by MSW
- 28. 1. Fill in the blanks. 2. How many subjects were in the study? 3. How many groups were in the study?
- 29. Review of the ANOVA test • Hypotheses and significance level are stated • Sum of Squared differences from the mean of all the scores is computed = SStotal • Sum of Squared differences from the mean of each group is computed = SSwithin • Sum of Squared differences between groups is computed by subtraction = SSbetween • Degrees of Freedom df are computed for each SS • Mean Squares MSbetween and MSwithin are computed. • F ratio is computed and its p –value determined. • Decision is made regarding the null hypothesis.