PSYC 612, FALL 2010Lecture 12: ANOVA IntroductionLecture Date: 11/17/2010Contents0.1 Preliminary Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Part I: ANOVA and t he GLM (70 minutes; 5 minute break) 21.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The t-test and correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Where does variance fit in ANOVA? . . . . . . . . . . . . . . . . . . . . . . 31.4 Parameter Estimation and Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . 31.4.1 The t-test: Comparing two levels at a time . . . . . . . . . . . . . . . . . . . 41.4.2 The rpb: Calculating the measure of association . . . . . . . . . . . . . . . . 51.4.3 The F-test: A simpler way to compare multiple level f actors . . . . . . . . . 52 Part II: AN OVA Details (50 minutes; 5 minute break) 62.1 Philosophy of ANOVA and MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 MRC and ANOVA - similarities and differences . . . . . . . . . . . . . . . . . . . . 72.2.1 Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 PART III: Module 3 demonstration (30 minutes) 80.1 Preliminary Questions•Have you read all the assigned reading for today?•Do you have any lingering questions about last week’s lecture?•Are you ready for Module 3?11 Part I: ANOVA and the GLM (70 minutes; 5 minutebreak)1.1 Purpose:To introduce you to the other side of the G LM1.2 Objectives:1. Provide a bridge between MRC and ANOVA via the t-test and rpb2. Discuss the primary differences between MRC and ANOVA3. Differentiate parameter estimation and hypothesis testing in ANOVA1.3 The t-test and correlation coefficientRemember when we discussed the correlation coefficient and I addressed a correlation between twovaria bles - o ne continuous and the other binary? That correlation was called the point biserial orrpband serves as an excellent bridge between MRC and ANOVA.rpb=st2t2+ dfRosenthal and Rosnow introduce the idea of significance tests as a function of both effect sizeand sample size and provide the following heuristic equation:Significance test = Size of effect × Size of studyand they then start by describing the r elationship between the correlation coefficient and the tstatistic.t =r√1 − r2×pdfThe equation above forms the ba sis for our bridge between MRC and ANOVA. Note that a testof mean differences - as expressed in the form of a t-test can be explained by using a measure ofassociation? This point is very important so we do not lose sight o f the fact that MRC and ANOVAboth come from the GLM. They must share some common terms if t hey come from the same family(think of genetics). The authors go on to express t as a function o f effect size and sample size byshowing that t can be computed using Cohen’s d (our favored effect size computation) and df withthe following equation:t = d ×√df2We will focus on this formulation of t as we move from two groups to more groups so pleaseremember this basic formulation. Before I move on, however, I want to stress that there are certainassumptions underlying the t-test that carry over to ANOVA. They are best summarized by thestatement “errors are IID normal” or...• Independent errors (residuals in MRC)• Identically distributed errors (homogeneity of error variance)2• Distribution of errors is normalAs we move from two g r oups to more t han two groups, we must change our way of thinking.Why? Because the t statistic only uses two groups. We can f ormulate a multiple t or rather an Fbased upon a similar logic as the t statistic. That is...F = t2=r21 − r2× dfWe may also express F as a function of effect sizes ( as measured by η2):F =η21 − η2×dferrordfmeanswhere dferroris the same df used in the t-test but now we have another df called the dfmeansthat represents the number of groups (k) being compared. So now we have a way of including morethan two groups but still test for mean differences. Pretty cool, huh?1.3.1 Where does variance fit in ANOVA?So why is ANOVA called the “analysis of variances” and not just mean difference testing? Varianceis a measure of our uncertainty of the mean; as varia nce increases, our certainty about the meanbeing the “best guess” decreases. You bega n your statistical training as an undergraduate learningthat a t-test was a test of mean differences. That test allows you t o assess whether two means aresimilar to one anot her but the test is really about whether the uncertainty around the means (i.e.,varia nce) is sufficient to treat them differently. If there is a great amount o f uncertainty - that isgreat variance - around each mean then we cannot judg e with any amount of certainty whether thetwo means differ. Hence, ANOVA is really not means testing but var iance testing. We can f urtherdefine F as a ratio of variances as in:F =σ2Bσ2W=MSBMSW=signalnoiseAs you can see, the F test is a test of variances that represent the signal to noise ratio and thattest can extend to multiple level factors, to between-subjects and to within-subjects designs. Theflexibility in the F ratio or F test makes it a useful statistical test for us and it fits well within theGLM.1.4 Parameter Estimation and Hypothesis TestingConsider the following example before I delve into parameter estimation and hypothesis testing. Ihave a simple dataset with the following observa t ions:The variables are:• F1: a treatment f actor with two levels• F2: another factor that is related to F1 but has four levels• F3: a crossed factor with F1 with two levels• Y: our dependent …