One-way Analysis of Variance and Post Hoc TestsOne-way Analysis of Variance: PurposeOne-way Analysis of Variance: Hypotheses - 1One-way Analysis of Variance: Hypotheses - 2One-way Analysis of Variance: Hypotheses - 3One-way Analysis of Variance: Assumptions and RequirementsOne-way Analysis of Variance: Effect Size - 1One-way Analysis of Variance: Effect Size - 2One-way Analysis of Variance: APA Style - 1One-way Analysis of Variance: APA Style - 2Homework problems: One-way Analysis of VarianceHomework problems: One-way Analysis of Variance - Data set, variables, and sampleHomework problems: One-way Analysis of Variance – Specific Relationship TestedHomework problems: One-way Analysis of Variance – Choosing an answerSolving the problem with SPSS: Level of measurement - 1Solving the problem with SPSS: Level of measurement - 2Solving the problem with SPSS: Evaluating normality - 1Solving the problem with SPSS: Evaluating normality - 2Solving the problem with SPSS: Evaluating normality - 3Solving the problem with SPSS: Evaluating normality - 4Solving the problem with SPSS: Evaluating normality - 5Solving the problem with SPSS: One-way analysis of variance - 1Solving the problem with SPSS: One-way analysis of variance - 2Solving the problem with SPSS: One-way analysis of variance - 3Solving the problem with SPSS: One-way analysis of variance - 4Solving the problem with SPSS: One-way analysis of variance - 5Solving the problem with SPSS: One-way analysis of variance - 6Solving the problem with SPSS: Evaluating normality with the central limit theorem - 7Solving the problem with SPSS: Evaluating equality of group variancesSolving the problem with SPSS: Answering the question - 1Solving the problem with SPSS: Answering the question - 2Solving the problem with SPSS: Answering the question - 3Logic for one-way analysis of variance: Level of measurementLogic for one-way analysis of variance: Assumption of normalityLogic for One-way ANOVA: Assumption of equality of variancesLogic for one-way analysis of variance: Decision about null hypothesisLogic for one-way analysis of variance: Decision about post hoc testPower Analysis: One-way ANOVA Problem that was FalsePower Analysis: Statistical Results for False One-way ANOVA - 1Power Analysis: Statistical Results for False One-way ANOVA - 2Access to SPSS’s SamplePower ProgramPower Analysis for One-way ANOVA - 1Power Analysis for One-way ANOVA - 2Power Analysis for One-way ANOVA - 3Power Analysis for One-way ANOVA - 4Power Analysis for One-way ANOVA - 5Power Analysis for One-way ANOVA - 6Power Analysis for One-way ANOVA - 7Power Analysis for One-way ANOVA - 8Power Analysis for One-way ANOVA – 9Power Analysis for One-way ANOVA – 10Power Analysis for One-way ANOVA – 11Power Analysis for One-way ANOVA - 12Power Analysis for One-way ANOVA - 13Power Analysis for One-way ANOVA - 14SW388R6Data Analysis and Computers ISlide 1One-way Analysis of Variance and Post Hoc TestsKey Points about Statistical TestSample Homework ProblemSolving the Problem with SPSSLogic for One-way Analysis of Variance with Post Hoc TestsPower AnalysisSW388R6Data Analysis and Computers ISlide 2One-way Analysis of Variance: PurposePurpose: test whether or not the populations represented by the samples have a different meanExamples: Social work students have higher GPA’s than nursing students and education majorsSocial work students volunteer for more hours per week than education majors, liberal arts majors, and communications majorsUT social work students score higher on licensing exams than graduates of Texas State University, Baylor University, and University of HoustonSW388R6Data Analysis and Computers ISlide 3One-way Analysis of Variance: Hypotheses - 1Hypotheses:Null: mean of population 1 = mean of population 2 = mean of population 3 = mean of population 4 ….VersusResearch: at least one population mean is different from the othersDecision:Reject null hypothesis if pSPSS ≤ alpha (≠ relationship)SW388R6Data Analysis and Computers ISlide 4One-way Analysis of Variance: Hypotheses - 2A significant finding tells us that at least one mean is different from the other, but it does not tell us which one or ones differ.To answer that question, we might try to use t-tests to compare each of the pairs of means to identify which are significantly different. However, the multiple tests would increase the probability of making a Type I error increases above the alpha that we set, i.e. greater than the risk of drawing a wrong conclusion.One strategy for controlling the error rate is the Bonferroni inequality by which we divide alpha by the number of tests we need and only report those that meet this reduced level of significance.SW388R6Data Analysis and Computers ISlide 5One-way Analysis of Variance: Hypotheses - 3As an alternative, statisticians have developed post hoc tests which control the error rate for the number of comparisons that are made. Thus, for a one-way analysis of variance, we only report findings for combinations of differences if the main test (often referred to as the omnibus test) shows statistical significance.We will use the Tukey HSD (Honestly Significant Difference) test to identify statistically significant differences between pairs of groups.SW388R6Data Analysis and Computers ISlide 6One-way Analysis of Variance: Assumptions and RequirementsVariable is interval level (ordinal with caution)Variable is normally distributedAcceptable degree of skewness and kurtosisorUsing the Central Limit Theorem (10+ in each group)The variance of the groups is not differentSW388R6Data Analysis and Computers ISlide 7One-way Analysis of Variance: Effect Size - 1For analysis of variance, the effect size statistic, there are multiple measures of effect size:eta squared (η²), partial Eta squared (ηp²), omega squared (ω²), and the Intraclass correlation (ρI) We are particularly interested in eta squared which is the percent of the total variance in the dependent variable accounted for by the variance between categories (groups) defined by the independent variable, i.e. the ratio of the between-groups sum of squares to the total sum of squares.SW388R6Data Analysis and Computers ISlide 8One-way Analysis of Variance: Effect Size - 2The effect size statistic used in power analysis is f, which is computed as the square root of η² ÷ 1 + η²Cohen’s interpretation of f: small: f