Introduction to Analysis of Variance
- 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
Editor's Notes
- #6: FIGURE 12.1 A typical situation in which ANOVA would be used. Three separate samples are obtained to evaluate the mean differences among three populations (or treatments) with unknown means.
- #8: FIGURE 12.2 A research design with two factors. The research study uses two factors: One factor uses two levels of therapy technique (I versus II), and the second factor uses three levels of time (before, after and 6 months after). Also notice that the therapy factor uses two separate groups (independent measures) and the time factor uses the same group for all three levels (repeated measures).
- #10: Some instructors may want to spend some time listing some of the ways that the null hypothesis cold be wrong that do NOT include μ1 ≠ μ2 ≠ μ3 because many students make the logic error of assuming that if null is wrong then the ONLY way it can be wrong is if all the groups are significantly different from each other.
- #13: Testwise Type I error rate and Experimentwise Type I error rate are sufficiently complex concepts that some instructors may want to expand the lecture materials on this topic.
- #17: FIGURE 12.3 The independent-measures ANOVA partitions, or analyzes, the total variability into two components: variance between treatments and variance within treatments.
- #22: FIGURE 12.4 The structure and sequence of calculations for the ANOVA.
- #23: FIGURE 12.5 Partitioning the sum of squares (SS) for the independent-measures ANOVA.
- #26: FIGURE 12.6 Partitioning degrees of freedom (df) for the independent-measures ANOVA.
- #28: Some instructors may want to begin the discussion of the computational procedures with this slide. Students sometimes get lost in all the formulas, but if they can see that the summary table is the “goal” of the process, they can break it down into the small steps required to fill in each part of the table.
- #32: FIGURE 12.7 The distribution of F-ratios with df = 2, 12. Of all the values in the distribution, only 5% are larger than F = 3.88, and only 1% are larger than F = 6.93.
- #34: FIGURE 12.8 The distribution of F-ratios with df = 3, 20. The critical values for α = .01 is 4.94.
- #37: A visual representation of the between-treatments variability and the within0-treatments variability that form the numerator and denominator, respectively, of the F-ratio. In (a), the difference between treatments is relatively large and easy to see. In (b), the same 4-point difference between treatments is relatively small and is overwhelmed by the within-treatments variability.
- #48: FIGURE 12.10 The distribution of t statistics with df = 18 and the corresponding distribution of F-ratios with df = 1, 18. Notice that the critical values for α = .05 are t = ±2.101 and F = 2.1012 = 4.41.
- #50: FIGURE 12.10 Formulas for ANOVA.
- #51: FIGURE 12.12 SPSS Output of the ANOVA for the studying strategy experiment in Example 12.1.