One-Way Analysis of Variance

Computing our exampleStep 1: compute sums of squaresRecall our dataKNR 445StatisticsANOVA (1w)Slide 1TV MovieSoap OperaInfomercial1610381341055492128n = 5n = 5n = 5 = 3= 8= 9

N = 15

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Computing our exampleStep 1: compute sums of squaresSStotal

= [102 + 132 + 52 + 92 + 82 + 62 + 82 + 102 + 42 +122 + 12 + 32 + 42 + 52 + 22] -

= 854 666.67 = 187.33 KNR 445StatisticsANOVA (1w)Slide 2

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Computing our exampleStep 1: compute sums of squaresSSgroup

= 27.14 + 8.84 + 67.34= 103.32

KNR 445StatisticsANOVA (1w)Slide 3

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Computing our exampleStep 1: compute sums of squaresSSerror=SStotal-SSgroup= 187.33 103.32 = 84.01SoSSgroup = 103.32SSerror = 84.01Sstotal = 187.33KNR 445StatisticsANOVA (1w)Slide 41

Computing our exampleStep 2: Compute dfdf group = k 1 = 3 1 = 2dferror = N k = 15 3 = 12df total = N 1 = 15 1 = 14KNR 445StatisticsANOVA (1w)Slide 51

Computing our exampleStep 3: Compute Mean Squares (MS)KNR 445StatisticsANOVA (1w)Slide 6

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Computing our exampleStep 4: Put all the info in the ANOVA table:KNR 445StatisticsANOVA (1w)Slide 7SourceSum of SquaresDFMSFsig.Between Groups103.32251.66MSB/MSW=51.66/7=7.38p-valueWithin Groups84.01127Total187.33141

Computing our exampleStep 5: Compare Fobs to Fcritical:Fobs = 7.38Fcritical = need to obtain Fcrit from tables for Fdf will be (numerator, denominator) in F-ratiodf = 2, 12F (2,12, = .05) = 3.89Reject H0 (Fobs > Fcritical)KNR 445StatisticsANOVA (1w)Slide 81

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KNR 445StatisticsANOVA (1w)Slide 91-way ANOVA in SPSSData: One column for the grouping variable (IV: group in this case), one for the measure (DV: fitness in this case)Data: Note grouping variable has 3 levels (goes from 1 to 3)1

KNR 445StatisticsANOVA (1w)Slide 101-way ANOVA in SPSSProcedure: Choose the appropriate procedure, and1

KNR 445StatisticsANOVA (1w)Slide 111-way ANOVA in SPSSDialog box: slide the variablesinto the appropriate places1

KNR 445StatisticsANOVA (1w)Slide 121-way ANOVA in SPSSHere we see the between and within sources of varianceHere are the SDs (here expressed as the mean square thats the average sum of squares, which is after all a standardized deviation)k-1 = 3-1 = 2n k = 15 - 3 = 12n-1 = 15-1 = 14Result!1

KNR 445StatisticsANOVA (1w)Slide 13Significant resultnow what?Follow-up testsONLY compute after a significant ANOVALike a collection of little t-testsBut they control overall type 1 error comparatively wellThey do not have as much power as the omnibus test (the ANOVA) so you might get a significant ANOVA & no sig. Follow-upPurpose is to identify the locus of the effect (what means are different, exactly?)12

KNR 445StatisticsANOVA (1w)Slide 14Significant resultnow what?Follow-up tests most commonTukeys HSD (honestly sig. diff.)Formula:

But its easier to use SPSS

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KNR 445StatisticsANOVA (1w)Slide 15Follow-ups to ANOVA in SPSSChoose post-hoc test (meaning after this) 12Check the appropriate box for the HSD (Tukey, not Tukeys b)

KNR 445StatisticsANOVA (1w)Slide 16Follow-ups to ANOVA in SPSSSig. levels between pairs of groupsGroups that do not differAnd one that does (from the other 2)123

KNR 445StatisticsANOVA (1w)Slide 17Follow-ups to ANOVA in SPSS1So TV Movie differs from both Soap Opera and infomercial , significantlySoap Operas and infomercials do not differ significantly

KNR 445StatisticsANOVA (1w)Slide 18Assumptions to test in One-WaySamples should be independent (as with independent t-test does not mean perfectly uncorrelated)Each of the k populations should be normal (important only when samples are smallif theres a problem, can use Kruskal-Wallis test)The k samples should have equal variances (this is the homogeneity of variance assumption, and well look at it shortlyviolations are important mostly with small samples and unequal ns)1

KNR 445StatisticsANOVA (1w)Slide 19Homogeneity of variance - SPSS1. Click on the options button2. Choose homogeneity of variance3. Click continue

KNR 445StatisticsANOVA (1w)Slide 20Homogeneity of variance - SPSSHomogeneity test outputAs you can see, no problems here. The test has to be significant for there to be a violation

Interpret outputThe amount of aggression arising from watching TV changed according to the type of program watched, F(2,12) = 7.38, p .05. Tukeys HSD follow-up tests showed that those watching violent movies (M = 3) experienced less aggression than those watching soap operas (M = 8) or infomercials (M = 9). There was no difference in aggression level between those who watched soap operas and those who watched infomercials.KNR 445StatisticsANOVA (1w)Slide 211