PSY 307 – Statistics for the Behavioral SciencesFisher’s F-Test (ANOVA)Testing Yields in AgricultureANOVATwo Sources of VariabilityF-TestDifference vs ErrorF-RatioSlide 9Testing HypothesesFormulas for FFormula for SStotalHours of Sleep DeprivationFormula for SSbetweenFormula for SSwithinDegrees of FreedomChecking Your WorkCalculating F (Computational)F-DistributionANOVA AssumptionsCautionsPSY 307 – Statistics for the Behavioral SciencesChapter 16 – One-Factor Analysis of Variance (ANOVA)Fisher’s F-Test (ANOVA)Ronald FisherTesting Yields in AgricultureX1X2X1= =ANOVAAnalysis of Variance (ANOVA) – a test of more than two population means.One-Way ANOVA – only one factor or independent variable is manipulated.ANOVA compares two sources of variability.Two Sources of VariabilityTreatment effect – the existence of at least one difference between the population means defined by IV.Between groups variability – variability among subjects receiving different treatments (alternative hypothesis).Within groups variability – variability among subjects who receive the same treatment (null hypothesis).F-TestIf the null hypothesis is true, the numerator and denominator of the F-ratio will be the same.F = random error / random errorIf the null hypothesis is false, the numerator will be greater than the denominator and F > 1.F = random error + treatment effectrandom errorDifference vs ErrorDifference on the top and the error on the bottom:Difference is the variability between the groups, expressed as the sum of the squares for the groups.Error is the variability within all of the subjects treated as one large group.When the difference exceeds the variability, the F-ratio will be large.F-RatioF = MSbetween MSwithinMS = SS dfSS is the sum of the squared differences from the mean.F-RatioF = MSbetween MSwithinMSbetween treats the values of the group means as a data set and calculates the sum of squares for it.MSwithin combines the groups into one large group and calculates the sum of squares for the whole group.Testing HypothesesIf there is a true difference between the groups, the numerator will be larger than the denominator.F will be greater than 1Writing hypotheses:H0: 1 = 2 = 3H1: H0 is falseFormulas for FDescription in words of what is being computed.Definitional formula – uses the SS, described in the Witte textComputational formula – used by Aleks and in examples in class.Formula for SStotalSStotal is the total Sum of the SquaresIt is the sum of the squared deviations of scores around the grand mean.SStotal = ∑(X – Xgrand)2SStotal = ∑(X2 – G2/N)Where G is the grand total and N is its sample sizeHours of Sleep Deprivation0 24 480 3 64 6 82 6 10 Grand Total6 15 24 G = 455492025357632253369)45(3)24(3)15(3)6(2222betweenSS  2210064363694160)10()8()6()6()3()2()4()0(22222222ab oveaboveSSwithinFormula for SSbetweenSSbetween is the between Sum of the SquaresIt is the sum of the squared deviations for group means around the grand mean.SSbetween = n∑(X – Xgrand)2SSbetween = ∑(T2/n – G2/N)Where T is each group’s total and n is each group’s sample sizedefinitioncomputationFormula for SSwithinSSwithin is the within Sum of the SquaresIt is the sum of the squared deviations for scores around the group mean.SSwithin = ∑(X – Xgroup)2SSwithin = ∑X2 – ∑T2/n)Where T is each group’s total and n is each group’s sample sizedefinitioncomputationDegrees of Freedomdftotal = N-1The number of all scores minus 1dfbetween = k-1The number of groups (k) minus 1dfwithin = N-kThe number of all scores minus the number of groups (k)Checking Your WorkThe SStotal = SSbetween + SSwithin.The same is true for the degrees of freedom:dftotal = dfbetween + dfwithinCalculating F (Computational)SSbetween =  T2 – G2 n NWhere T is the total for each group and G is the grand totalSSwithin =  X2 -  T2NSStotal =  X2 – G2/NF-DistributionCritical valueCommon – retain nullRare – reject nullLook up F critical value in the F table using df for numerator and denominatorANOVA AssumptionsAssumptions for the F-test are the same as for the t-testUnderlying populations are assumed to be normal with equal variances.Results are still valid with violations of normality if:All sample sizes are close to equalSamples are > 10 per groupOtherwise use a different testCautionsThe ANOVA presented in the text assumes independent samples.With matched samples or repeated measures use a different form of ANOVA.The sample sizes shown in the text are small in order to simplify calculations.Small samples should not be