One-Way Analysis of Variance

One-Way Analysis of VarianceComparing means of more than 2 independent samples1

KNR 445StatisticsANOVA (1w)Slide 2Why not multiple t-tests?# comparisons = k(k-1)/2, where k is the # of groups. In other words, when there are many more than 2 groups, the number of comparisons is largeIn each separate comparison, you only use information within that pair but information within other pairs may increase powerIf you do multiple tests, you have multiple answers, not just oneWith many tests, type 1 error rates increase radically 1234

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KNR 445StatisticsANOVA (1w)Slide 3ANalysis Of VAriance1-way ANOVAGrouping variable = factor = independent variableThe variable will consist of a number of levelsIf 1-way ANOVA is being used, the number will be >2.E.G. What type of program has the greatest impact on aggression?Violent movies, soap operas, or infomercials?Type of program is the independent variable or factor, while soap operas is one level of the factor, & aggression is the dependent variable123

KNR 445StatisticsANOVA (1w)Slide 41-way ANOVA: HypothesesNull:

Alternative (experimental):

Note: no directional hypothesis; Null may be false in many different ways

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KNR 445StatisticsANOVA (1w)Slide 51-way ANOVA: LogicChange in test statisticRecall with t-tests, test statistic is

But with ANOVA, test statistic is

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1-way ANOVA: LogicReturn to our example

KNR 445StatisticsANOVA (1w)Slide 61234TV MovieSoap OperaInfomercial1610381341055492128 = 3= 8= 9

61-way ANOVA: LogicVariance between sampling means

Known as between group varianceThink of what could cause these means to differ (vary) from each otherTreatment effect: differences due to the different ways the groups were treated (systematic variation)Chance: Individual differences, experimental error (unsystematic, unexplained variation)KNR 445StatisticsANOVA (1w)Slide 71 = 3= 8= 9

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71-way ANOVA: LogicVariance expected by chance (error)

Known as within group varianceThink of what could cause the scores within the groups to differ from each otherChance: Individual differences, experimental error (unsystematic, unexplained variation)KNR 445StatisticsANOVA (1w)Slide 81TV MovieSoap OperaInfomercial1610381341055492128

81-way ANOVA: LogicPartitioning the varianceKNR 445StatisticsANOVA (1w)Slide 9123Total varianceWithin group varianceChanceBetween group variance:TreatmentChance

+=91-way ANOVA: LogicThe test statistic for the 1-way ANOVAThe F-ratio

If null is true

If null is false

KNR 445StatisticsANOVA (1w)Slide 10123

101-way ANOVA: LogicMore on the F-statisticF is a statistic that represents ratio of two variance estimatesDenominator of F is called error termWhen no treatment effect, F 1If treatment effect, observed F will be > 1 How large does F have to be to conclude there is a treatment effect (to reject H0)? Compare observed F to critical values based on sampling distribution of FA family of distributions, each with a pair of degrees of freedom KNR 445StatisticsANOVA (1w)Slide 111

11As with t-distribution, a family of curvesShape of distribution changes with dfIf null is true, F1p largest at 1p tapers as F > 1F-values always positive (variance cant be negative)KNR 445StatisticsANOVA (1w)Slide 12The F-distribution12012345678F.05 = 5.14F valuesProbabilityRegion of rejection

Hypothesis Testing with ANOVAResearch questionDoes the type of programming affect levels of aggression? Statistical hypothesesH0: 1 = 2 = ... = kH1: At least 2 means are significantly differentDecision rule (critical value)Compute observed F-ratio from dataMake decision to reject or fail to reject H0If H0 rejected, conduct multiple comparisons as neededKNR 445StatisticsANOVA (1w)Slide 131

Computing ANOVAFor those interested (?!) well cover the steps and do an example (not necessary if you understand the conceptbut it may help)If youre really confident, skip to slide 21Steps to completionCompute SS (sums of squares)Compute dfCompute MS (mean squares)Compute FKNR 445StatisticsANOVA (1w)Slide 14123

Computing ANOVAVocabulary/Symbols: k = Number of groupsnj = Sample size of the jth group (n1, n2,nj,nk)N = Total sample size = Mean of the jth group = Grand (overall) meanSS (Sum of squares) = Sum of squared deviations around a meanKNR 445StatisticsANOVA (1w)Slide 15

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Computing ANOVAStep 1: Compute Sums of Squares (SS)Need total, group, and error sums of squaresThese are combined with appropriate df to give variance calculations, & then generate F-ratioKNR 445StatisticsANOVA (1w)Slide 1612

Computing ANOVAStep 1: Compute Sums of Squares (SS)Total sum of squares (looks a little daunting, but its really not)KNR 445StatisticsANOVA (1w)Slide 17

1. This is just the sum of the squared deviations of each observation from the overall mean 2. The second part of the formula is the one well use in the calculations, as its easier to work with. The first one is easier to conceptualize.1234

Computing ANOVAStep 1: Compute Sums of Squares (SS)Group sum of squaresKNR 445StatisticsANOVA (1w)Slide 18

11. This is the sum of the squared deviations of each group mean from the overall mean (multiplied by each samples size)

Computing ANOVAStep 1: Compute Sums of Squares (SS)Error sum of squaresTotal variance is composed of SSgroup & SSerrorSStotal = SSgroup + SSerrorRearrange this formula to get:SSerror= SStotal SSgroup

KNR 445StatisticsANOVA (1w)Slide 19121. To calculate, take the sum of squares of each observation within a group from its group mean, for all groups

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Computing ANOVAStep 2: Compute degrees of freedom(Used to adjust SS to variance estimates)df group:

df total:

df error (or whats left over):

KNR 445StatisticsANOVA (1w)Slide 20123

Computing ANOVAStep 3: Compute Mean Squares (MS) & F-ratioMean Square = variance KNR 445StatisticsANOVA (1w)Slide 21123

Computing ANOVAThe ANOVA summary table (may help you understand where it comes from in SPSS) all the processes are alluded to aboveKNR 445StatisticsANOVA (1w)Slide 221SourceSum of SquaresDFMSFsig.Between GroupsSSgroupdfgroupSSgp/dfgp=MSBMSB/MSWp-valueWithin GroupsSSerrordferrorSse/dfe=MSWTotalSStotaldftotal