2. Computing our example  Step 1: compute sums of squares  Recall our data… KNR 445 Statistics ANOVA (1w) Slide 1 TV MovieSoap OperaInfomercial 1610 3813 4105 549 2128 n = 5 = 3= 8= 9 N = 15 1 2

3. 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 ] - = 854 – 666.67 = 187.33 KNR 445 Statistics ANOVA (1w) Slide 2 1

4. Computing our example  Step 1: compute sums of squares  SS group = 27.14 + 8.84 + 67.34= 103.32 KNR 445 Statistics ANOVA (1w) Slide 3 1 2

5. Computing our example  Step 1: compute sums of squares  SS error  =SS total -SS group = 187.33 – 103.32 = 84.01  So…  SS group = 103.32  SS error = 84.01  Ss total = 187.33 KNR 445 Statistics ANOVA (1w) Slide 4 1

6. Computing our example  Step 2: Compute df  df group = k – 1 = 3 – 1 = 2  df error = N – k = 15 – 3 = 12  df total = N – 1 = 15 – 1 = 14 KNR 445 Statistics ANOVA (1w) Slide 5 1

7. Computing our example  Step 3: Compute Mean Squares (MS) KNR 445 Statistics ANOVA (1w) Slide 6 1

8. Computing our example  Step 4: Put all the info in the ANOVA table: KNR 445 Statistics ANOVA (1w) Slide 7 Source Sum of Squares DFMSFsig. Between Groups 103.32251.66 MS B /MS W =51.66/7 =7.38 p-value Within Groups 84.01127 Total187.3314 1

9. Computing our example  Step 5: Compare F obs to F critical :  F obs = 7.38  F critical = …need to obtain F crit from tables for F  df will be (numerator, denominator) in F-ratio  df = 2, 12  F (2,12, α =.05) = 3.89  Reject H 0 (F obs > F critical ) KNR 445 Statistics ANOVA (1w) Slide 8 1 2

10. KNR 445 Statistics ANOVA (1w) Slide 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

11. KNR 445 Statistics ANOVA (1w) Slide 10 1-way ANOVA in SPSS Procedure: Choose the appropriate procedure, and… 1

12. KNR 445 Statistics ANOVA (1w) Slide 11 1-way ANOVA in SPSS Dialog box: slide the variables… …into the appropriate places 1

13. KNR 445 Statistics ANOVA (1w) Slide 12 1-way ANOVA in SPSS 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) k-1 = 3-1 = 2n – k = 15 - 3 = 12n-1 = 15-1 = 14 Result! 1

14. KNR 445 Statistics ANOVA (1w) Slide 13 Significant result…now what?  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?) 1 2

15. KNR 445 Statistics ANOVA (1w) Slide 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

16. KNR 445 Statistics ANOVA (1w) Slide 15 Follow-ups to ANOVA in SPSS Choose “post-hoc” test (meaning ‘after this’) 1 2 Check the appropriate box for the HSD (Tukey, not Tukey’s b)

17. KNR 445 Statistics ANOVA (1w) Slide 16 Follow-ups to ANOVA in SPSS Sig. levels between pairs of groups Groups that do not differ And one that does (from the other 2) 1 2 3

18. KNR 445 Statistics ANOVA (1w) Slide 17 Follow-ups to ANOVA in SPSS 1 So “TV Movie” differs from both “Soap Opera” and “infomercial”, significantly “Soap Operas” and “infomercials” do not differ significantly

19. KNR 445 Statistics ANOVA (1w) Slide 18 Assumptions to test in One-Way 1. Samples should be independent (as with independent t- test – does not mean perfectly uncorrelated) 2. Each of the k populations should be normal (important only when samples are small…if there’s a problem, can use Kruskal-Wallis test) 3. 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

20. KNR 445 Statistics ANOVA (1w) Slide 19 Homogeneity of variance - SPSS 1. Click on the ‘options’ button 2. Choose homogeneity of variance 3. Click continue

21. KNR 445 Statistics ANOVA (1w) Slide 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

22. 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.” KNR 445 Statistics ANOVA (1w) Slide 21 1