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Computing our example

Computing our example. Step 1: compute sums of squares Recall our data…. 1. 2. N = 15. Computing our example. Step 1: compute sums of squares SS total = [10 2 + 13 2 + 5 2 + 9 2 + 8 2 + 6 2 + 8 2 + 10 2 + 4 2 +12 2 + 1 2 + 3 2 + 4 2 + 5 2 + 2 2 ] -

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Computing our example

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  1. Computing our example • Step 1: compute sums of squares • Recall our data… 1 2 N = 15

  2. Computing our example • Step 1: compute sums of squares • SStotal = [102 + 132 + 52 + 92 + 82 + 62 + 82 + 102 + 42 +122 + 12 + 32 + 42 + 52 + 22] - = 854 – 666.67 = 187.33 1

  3. Computing our example • Step 1: compute sums of squares • SSgroup = 27.14 + 8.84 + 67.34= 103.32 1 2

  4. Computing our example • Step 1: compute sums of squares • SSerror • =SStotal-SSgroup = 187.33 – 103.32 = 84.01 • So… • SSgroup = 103.32 • SSerror = 84.01 • Sstotal = 187.33 1

  5. Computing our example • Step 2: Compute df • df group = k – 1 = 3 – 1 = 2 • dferror = N – k = 15 – 3 = 12 • df total = N – 1 = 15 – 1 = 14 1

  6. Computing our example • Step 3: Compute Mean Squares (MS) 1

  7. Computing our example • Step 4: Put all the info in the ANOVA table: 1

  8. Computing our example • Step 5: Compare Fobs to Fcritical: • Fobs = 7.38 • Fcritical = …need to obtain Fcrit from tables for F • df will be (numerator, denominator) in F-ratio • df = 2, 12 • F (2,12, α = .05) = 3.89 • Reject H0 (Fobs > Fcritical) 1 2

  9. 1-way ANOVA in SPSS Data: 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

  10. 1-way ANOVA in SPSS Procedure: Choose the appropriate procedure, and… 1

  11. 1-way ANOVA in SPSS Dialog box: slide the variables… 1 …into the appropriate places

  12. n – k = 15 - 3 = 12 k-1 = 3-1 = 2 n-1 = 15-1 = 14 Here we see the between and within sources of variance Here are the SD’s (here expressed as the “mean square” – that’s the average sum of squares, which is after all a ‘standardized’ deviation) 1-way ANOVA in SPSS 1 Result!

  13. Significant result…now what? 1 • Follow-up tests • ONLY compute after a significant ANOVA • Like a collection of little t-tests • But they control overall type 1 error comparatively well • They do not have as much power as the omnibus test (the ANOVA) – so you might get a significant ANOVA & no sig. Follow-up • Purpose is to identify the locus of the effect (what means are different, exactly?) 2

  14. Significant result…now what? • Follow-up tests – most common… • Tukey’s HSD (honestly sig. diff.) • Formula: • But it’s easier to use SPSS… 1

  15. Follow-ups to ANOVA in SPSS 1 2 Choose “post-hoc” test (meaning ‘after this’) Check the appropriate box for the HSD (Tukey, not Tukey’s b)

  16. Sig. levels between pairs of groups Groups that do not differ Follow-ups to ANOVA in SPSS 2 And one that does (from the other 2) 1 3

  17. Follow-ups to ANOVA in SPSS So “TV Movie” differs from both “Soap Opera” and “infomercial” , significantly 1 “Soap Operas” and “infomercials” do not differ significantly

  18. Assumptions to test in One-Way • Samples 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 small…if there’s a problem, can use Kruskal-Wallis test) • The k samples should have equal variances (this is the homogeneity of variance assumption, and we’ll look at it shortly…violations are important mostly with small samples and unequal n’s) 1

  19. Homogeneity of variance - SPSS 1. Click on the ‘options’ button 2. Choose homogeneity of variance 3. Click continue

  20. Homogeneity of variance - SPSS Homogeneity test output As you can see, no problems here. The test has to be significant for there to be a violation

  21. Interpret output • “The amount of aggression arising from watching TV changed according to the type of program watched, F(2,12) = 7.38, p .05. Tukey’s 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.” 1

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