Lecture 11ExperimentANOVAANOVA and research designANOVA and Research DesignExampleSlide 7Slide 8Test Statistic For ANOVA: F RatioBreaking it DownANOVA: RationaleA Closer Look…Between Treatments Variance (MS) NumeratorWithin Treatments Variance (MS) DenominatorSlide 15Putting it Together: F - RatioPutting it TogetherANOVA: Terms and NotationLet’s look with our ExampleANOVA FormulasAnalysis of SSReview: Breaking down SSAnalysis of Degrees of FreedomReview: Breaking down dfAnalysis of MS and F - ratioYou Try It!!FormulasSo, Now What? Hypothesis Testing!Hypothesis Testing: Setting the critical regionDistribution of F-ratiosHypothesis Testing: Setting the Critical RegionHypothesis Testing: Calculate the Test StatisticHypothesis Testing: Make a DecisionLet’s Do One: Hypothesis Testing with ANOVASlide 35Slide 36Effect Size in ANOVAIn the LiteraturePost Hoc TestsSlide 40Tukey’s Honestly Significant Difference (HSD) TestLet’s Try HSDYou Try OneScheffe TestCalculating the SheffeAssumptions of Independent Measures ANOVAF - Max RevisitedDo F-Max with our Flight DataHomeworkLecture 11Introduction to ANOVAExperiment First-born children tend to develop language skills faster than their younger siblings. One possible explanation for this phenomenon is that first-born have undivided attention from their parents. A researcher wanted to test this theory by comparing the language development of language across only children, twins and triplets. Davis (1973) predicted that the multiple birth children should have slower language development.ANOVAANOVA or analysis of variance: a hypothesis testing procedure that is used to evaluate differences between 2 or more samples.It is an omnibus test: permits the analysis of several variables at the same time–Nice because we have greater flexibility in designing our experiments. We can make multiple comparisons with one test.•single child vs. twins•single child vs. triplets•twins vs. tripletANOVA and research designIndependent Measures - 2 or more different samplesRepeated Measures - 2 or more measurements from the same sample. The same sample is tested across all the different treatment conditions.Mixed DesignANOVA and Research DesignIndependent variable - Now we can test more than one independent variable–Say we wanted to look at language development across 2 independent variables (# of births at one time) and SES–Note: SES and births are quasi-independent variables: we are differentiating our groups by them, but can’t manipulate them.Factors - The independent variables are called factors in ANOVA.–A study with only 1 IV is a single-factor design.–A study with more than 1 IV is a factorial design.Levels - individual groups w/in a factorsingle twins tripletsHi SESsample 1 sample 2 sample 3Lo SESsample 4 sample 5 sample 6ExampleMeditation Talk DrugHi Support sample 1 sample 2 sample 3Lo Support sample 4 sample 5 sample 6TherapyHow many factors are in this design? What are they?What are the levels of each factor?0 mg - placebo 25 mg 50 mg 100 mgsample 1 sample 2 sample 3 sample 4Drug DosageHow many factors are in this design? What?What are the levels of each factor?ANOVAToday we’ll just introduce ANOVA (although remember repeated measures and multiple factor designs are based on the statistics covered today) Goal of ANOVA is to help us decide (1) There are no differences between the populations (null hypothesis).(2) The differences represent real differences between populations (alternative hypothesis).Single-Factor Independent Measures ANOVAPopulation 1Treatment 1Population 2Treatment 2Population 3Treatment 3μ = ? μ = ? μ = ?Sample 13045404235Sample 25052545660Population 4Treatment 4μ = ?Sample 34243454752Sample 43142453433Test Statistic For ANOVA: F RatioReminder: t = obtained difference between samplesdifference expected by chance (error)Structure for ANOVA is the same; test statistic is call the F-ratio:F = variance (differences) between sample means variance (differences) expected by chanceF - ratio is based on VARIANCE instead of sample mean DIFFERENCES–With more than 2 samples how would we calculate differencesBreaking it DownNumerator - Can’t calculate mean difference score what would the diff. score bwtn 20, 30 and 35 be?–Variance simply gives us info about differences. The scores are different.•Set 1 has large differences btwn means and large variance•Set 2 has small differences btwn means and small varianceDenominator of t measures standard error or standard deviation expected by chance. Denominator of F simply squares that to variance.Set 1 Set 2M1 = 20 M1 = 28M2 = 30 M2 = 30M3 = 35 M3 = 31s2 = 58.33 s2 = 2.33ANOVA: RationaleANOVA = analysis of variance–Analysis means break into parts–We are going to break the total variance into 2 parts.Between - Treatments Variance = really measuring the differences between sample meansWithin - Treatments Variance = measuring differences inside each treatment conditionA Closer Look…Exp. 240 42 44s2 = MS = 4Exp. 140 55 70s2 = MS = 225Sample 1 Sample 2 Sample 3 Sample 1 Sample 2 Sample 3• Numerator = variance between samples.•Variance between samples is big for experiment 1 and small for experiment 2.•Denominator = variance within samples.•Measure variance within the group, so how much variability was there in the set of scores that made up Exp 1, sample 1 and so on.F = variance (differences) between sample means variance (differences) expected by chanceBetween Treatments Variance (MS)NumeratorVariance = DifferencesBetween treatments variance = how much difference between groups(1) Differences between treatments due to chance.•Chance = unpredictable or unplanned differences–(1) Individual Differences–(2) Experimental Errors (measurement)(2) Differences caused by the treatmentsWithin Treatments Variance (MS)DenominatorWithin - treatments variance = inside each treatment condition each participant is treated the same. Researcher doesn’t do anything to cause individual differences.–But there is still variability between individuals•ChanceTotal VariabilityBetween-treatments VarianceWithin-treatments VarianceMeasures difference due to:1. Chance2. Treatment EffectsMeasures difference due to:1. ChancePutting it Together: F - Ratio Total Variability (F) = MS Between MS WithinOR F = treatment + chance chance(1) Treatment has no effect = the numerator and denominator