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A one-way analysis of variance produces SSTotal = 80 and SSWithin = 30. For this analysis, what is SSBetween?
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- 1. Chapter 12 Introduction to Analysis of Variance PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
- 2. Chapter 12 Learning Outcomes • Explain purpose and logic of Analysis of Variance1 • Perform Analysis of Variance on data from single-factor study2 • Know when and why to use post hoc tests (posttests)3 • Compute Tukey’s HSD and Scheffé test post hoc tests4 • Compute η2 to measure effect size5
- 3. Tools You Will Need • Variability (Chapter 4) – Sum of squares – Sample variance – Degrees of freedom • Introduction to hypothesis testing (Chapter 8) – The logic of hypothesis testing • Independent-measures t statistic (Chapter 10)
- 4. 12.1 Introduction to Analysis of Variance • Analysis of variance – Used to evaluate mean differences between two or more treatments – Uses sample data as basis for drawing general conclusions about populations • Clear advantage over a t test: it can be used to compare more than two treatments at the same time
- 5. Figure 12.1 Typical Situation for Using ANOVA
- 6. Terminology • Factor – The independent (or quasi-independent) variable that designates the groups being compared • Levels – Individual conditions or values that make up a factor • Factorial design – A study that combines two or more factors
- 7. Figure 12.2 Two-Factor Research Design
- 8. Statistical Hypotheses for ANOVA • Null hypothesis: the level or value on the factor does not affect the dependent variable – In the population, this is equivalent to saying that the means of the groups do not differ from each other • 3210 : H
- 9. Alternate Hypothesis for ANOVA • H1: There is at least one mean difference among the populations (Acceptable shorthand is “Not H0”) • Issue: how many ways can H0 be wrong? – All means are different from every other mean – Some means are not different from some others, but other means do differ from some means
- 10. Test statistic for ANOVA • F-ratio is based on variance instead of sample mean differences effecttreatmentnowithexpectedes)(differencvariance meanssamplebetweenes)(differencvariance F
- 11. Test statistic for ANOVA • Not possible to compute a sample mean difference between more than two samples • F-ratio based on variance instead of sample mean difference – Variance used to define and measure the size of differences among sample means (numerator) – Variance in the denominator measures the mean differences that would be expected if there is no treatment effect
- 12. Type I Errors and Multiple-Hypothesis tests • Why ANOVA (if t can compare two means)? – Experiments often require multiple hypothesis tests—each with Type I error (testwise alpha) – Type I error for a set of tests accumulates testwise alpha experimentwise alpha > testwise alpha • ANOVA evaluates all mean differences simultaneously with one test—regardless of the number of means—and thereby avoids the problem of inflated experimentwise alpha
- 13. 12.2 Analysis of Variance Logic • Between-treatments variance – Variability results from general differences between the treatment conditions – Variance between treatments measures differences among sample means • Within-treatments variance – Variability within each sample – Individual scores are not the same within each sample
- 14. Sources of Variability Between Treatments • Systematic differences caused by treatments • Random, unsystematic differences – Individual differences – Experimental (measurement) error
- 15. Sources of Variability Within Treatments • No systematic differences related to treatment groups occur within each group • Random, unsystematic differences – Individual differences – Experimental (measurement) error effectstreatmentnowithsdifference effectstreatmentanyincludingsdifference F
- 16. Figure 12.3 Total Variability Partitioned into Two Components
- 17. F-ratio • If H0 is true: – Size of treatment effect is near zero – F is near 1.00 • If H1 is true: – Size of treatment effect is more than 0. – F is noticeably larger than 1.00 • Denominator of the F-ratio is called the error term
- 18. Learning Check • Decide if each of the following statements is True or False • ANOVA allows researchers to compare several treatment conditions without conducting several hypothesis tests T/F • If the null hypothesis is true, the F-ratio for ANOVA is expected (on average) to have a value of 0 T/F
- 19. Learning Check - Answers • Several conditions can be compared in one testTrue • If the null hypothesis is true, the F-ratio will have a value near 1.00 False
- 20. 12.3 ANOVA Notation and Formulas • Number of treatment conditions: k • Number of scores in each treatment: n1, n2… • Total number of scores: N – When all samples are same size, N = kn • Sum of scores (ΣX) for each treatment: T • Grand total of all scores in study: G = ΣT • No universally accepted notation for ANOVA; Other sources may use other symbols
- 21. Figure 12.4 ANOVA Calculation Structure and Sequence
- 22. Figure 12.5 Partitioning SS for Independent-measures ANOVA
- 23. ANOVA equations N G XSStotal 2 2 treatmenteachinsidetreatmentswithin SSSS N G n T SS treatmentsbetween 22
- 24. Degrees of Freedom Analysis • Total degrees of freedom dftotal= N – 1 • Within-treatments degrees of freedom dfwithin= N – k • Between-treatments degrees of freedom dfbetween= k – 1
- 25. Figure 12.6 Partitioning Degrees of Freedom
- 26. Mean Squares and F-ratio within within withinwithin df SS sMS 2 between between betweenbetween df SS sMS 2 within between within between MS MS s s F 2 2
- 27. ANOVA Summary Table Source SS df MS F Between Treatments 40 2 20 10 Within Treatments 20 10 2 Total 60 12 •Concise method for presenting ANOVA results •Helps organize and direct the analysis process •Convenient for checking computations •“Standard” statistical analysis program output
- 28. Learning Check • An analysis of variance produces SStotal = 80 and SSwithin = 30. For this analysis, what is SSbetween? • 50A • 110B • 2400C • More information is neededD
- 29. Learning Check - Answer • An analysis of variance produces SStotal = 80 and SSwithin = 30. For this analysis, what is SSbetween? • 50A • 110B • 2400C • More information is neededD
- 30. 12.4 Distribution of F-ratios • If the null hypothesis is true, the value of F will be around 1.00 • Because F-ratios are computed from two variances, they are always positive numbers • Table of F values is organized by two df – df numerator (between) shown in table columns – df denominator (within) shown in table rows
- 31. Figure 12.7 Distribution of F-ratios
- 32. 12.5 Examples of Hypothesis Testing and Effect Size • Hypothesis tests use the same four steps that have been used in earlier hypothesis tests. • Computation of the test statistic F is done in stages – Compute SStotal, SSbetween, SSwithin – Compute MStotal, MSbetween, MSwithin – Compute F
- 33. Figure 12.8 Critical region for α=.01 in Distribution of F-ratios
- 34. Measuring Effect size for ANOVA • Compute percentage of variance accounted for by the treatment conditions • In published reports of ANOVA, effect size is usually called η2 (“eta squared”) – r2 concept (proportion of variance explained) total treatmentsbetween SS SS 2
- 35. In the Literature • Treatment means and standard deviations are presented in text, table or graph • Results of ANOVA are summarized, including – F and df – p-value – η2 • E.g., F(3,20) = 6.45, p<.01, η2 = 0.492
- 36. Figure 12.9 Visual Representation of Between & Within Variability
- 37. MSwithin and Pooled Variance • In the t-statistic and in the F-ratio, the variances from the separate samples are pooled together to create one average value for the sample variance • Numerator of F-ratio measures how much difference exists between treatment means. • Denominator measures the variance of the scores inside each treatment
- 38. 12.6 post hoc Tests • ANOVA compares all individual mean differences simultaneously, in one test • A significant F-ratio indicates that at least one difference in means is statistically significant – Does not indicate which means differ significantly from each other! • post hoc tests are follow up tests done to determine exactly which mean differences are significant, and which are not
- 39. Experimentwise Alpha • post hoc tests compare two individual means at a time (pairwise comparison) – Each comparison includes risk of a Type I error – Risk of Type I error accumulates and is called the experimentwise alpha level. • Increasing the number of hypothesis tests increases the total probability of a Type I error • post hoc (“posttests”) use special methods to try to control experimentwise Type I error rate
- 40. Tukey’s Honestly Significant Difference • A single value that determines the minimum difference between treatment means that is necessary to claim statistical significance–a difference large enough that p < αexperimentwise – Honestly Significant Difference (HSD) n MS qHSD within
- 41. The Scheffé Test • The Scheffé test is one of the safest of all possible post hoc tests – Uses an F-ratio to evaluate significance of the difference between two treatment conditions groupstwoofSSwithcalculatedBA versus within between MS MS F
- 42. Learning Check • Which combination of factors is most likely to produce a large value for the F-ratio? • large mean differences and large sample variancesA • large mean differences and small sample variancesB • small mean differences and large sample variancesC • small mean differences and small sample variancesD
- 43. Learning Check - Answer • Which combination of factors is most likely to produce a large value for the F-ratio? • large mean differences and large sample variancesA • large mean differences and small sample variancesB • small mean differences and large sample variancesC • small mean differences and small sample variancesD
- 44. Learning Check • Decide if each of the following statements is True or False • Post tests are needed if the decision from an analysis of variance is “fail to reject the null hypothesis” T/F • A report shows ANOVA results: F(2, 27) = 5.36, p < .05. You can conclude that the study used a total of 30 participants T/F
- 45. Learning Check - Answers • post hoc tests are needed only if you reject H0 (indicating at least one mean difference is significant) False • Because dftotal = N-1 and • Because dftotal = dfbetween + dfwithin True
- 46. 12.7 Relationship between ANOVA and t tests • For two independent samples, either t or F can be used – Always result in same decision – F = t2 • For any value of α, (tcritical)2 = Fcritical
- 47. Figure 12.10 Distribution of t and F statistics
- 48. Independent Measures ANOVA Assumptions • The observations within each sample must be independent • The population from which the samples are selected must be normal • The populations from which the samples are selected must have equal variances (homogeneity of variance) • Violating the assumption of homogeneity of variance risks invalid test results
- 49. Figure 12.11 Formulas for ANOVA
- 50. Figure 12.12 Distribution of t and F statistics
- 51. Any Questions ? Concepts ? Equations?