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ANALYSIS OF VARIANCE

ANALYSIS OF VARIANCE. Multigroup experimental design. PURPOSES: COMPARE 3 OR MORE GROUPS SIMULTANEOUSLY TAKE ADVANTAGE OF POWER OF LARGER TOTAL SAMPLE SIZE CONSTRUCT MORE COMPLEX HYPOTHESES THAT BETTER REPRESENT OUR PREDICTIONS. Multigroup experimental design. PROCEDURES

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ANALYSIS OF VARIANCE

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  1. ANALYSIS OF VARIANCE

  2. Multigroup experimental design • PURPOSES: • COMPARE 3 OR MORE GROUPS SIMULTANEOUSLY • TAKE ADVANTAGE OF POWER OF LARGER TOTAL SAMPLE SIZE • CONSTRUCT MORE COMPLEX HYPOTHESES THAT BETTER REPRESENT OUR PREDICTIONS

  3. Multigroup experimental design • PROCEDURES • DEFINE GROUPS TO BE STUDIES: • Experimental Assignment VS • Intact or Existing Groups • OPERATIONALIZE NOMINAL, ORDINAL, OR INTERVAL/RATIO MEASUREMENT OF GROUPS • eg. Nominal: SPECIAL ED, LD, AND NON-LABELED • Ordinal: Warned, Acceptable, Exemplary Schools • Interval: 0 years’, 1 years’, 2 years’ experience

  4. Multigroup experimental design • PATH REPRESENTATION Ry.T e Treat y

  5. Multigroup experimental design • VENN DIAGRAM REPRESENTATION Treat SS SSy R2=SStreat/SSy SSerror SStreat

  6. Multigroup experimental design • dummy coding. Since the values are arbitrary we can use any two numerical values, much as we can name things arbitrarily- 0 or 1 A or B • Another nominal assignment of values is 1 and –1, called contrast coding: -1 = control, 1=experimental group Compares exp. with control: 1(E) -1(C)

  7. Multigroup experimental design • NOMINAL: If the three are simply different treatments or conditions then there is no preferred labeling, and we can give them values 1, 2, and 3 • Forms: • arbitrary (A,B,C) • interval (1,2,3) assumes interval quality to groups such as amount of treatment • Contrast (-2, 1, 1) compares groups • Dummy (1, 0, 0), different for each group

  8. Dummy Coding Regression Vars • Subject Treatment x1 x2 y • 01 A 1 0 17 • 02 A 1 0 19 • 03 B 0 1 22 • 04 B 0 1 27 • 05 C 0 0 33 • 06 C 0 0 21

  9. Contrast Coding Regression Vars • Subject Treatment x1 x2 y • 01 A 1 0 17 • 02 A 1 0 19 • 03 B 0 1 22 • 04 B 0 1 27 • 05 C -1 -1 33 • 06 C -1 -1 21

  10. Hypotheses about Means • The usual null hypothesis about three group means is that they are all equal: • H0 : 1 = 2 = 3 • while the alternative hypothesis is typically represented as • H1 : ij for some i,j .

  11. ANOVA TABLE • SOURCE df Sum of Mean F Squares Square • Treatment… k-1 SStreat SStreatSStreat/ k (k-1) SSe /k(n-1) • error k(n-1) SSe SSe / k(n-1) • total kn-1 SSy SSy / (n-1) • Table 9.2: Analysis of variance table for Sums of Squares

  12. F-DISTRIBUTION Central F-distribution power alpha Fig. 9.5: Central and noncentral F-distributions

  13. POWER for ANOVA • Power nomographs- available from some texts on statistics • Simulations- tryouts using SPSS • requires creating a known set of differences among groups • best understanding using means and SDs comparable to those to be used in the study • post hoc results from previous studies are useful; summary data can be used

  14. ANOVA TABLE QUIZ SOURCE DF SS MS F PROB GROUP 2 ___ 50 __ .05 ERROR __ ___ ___ TOTAL 20 R2 = ____

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