3. For example… Does class level influence amount of study time? Do gender and education level interact in determining one’s susceptibility to sexual harassment? How do gender and marital status contribute to one’s level of anxiety? Which of three therapeutic methods are most effective at battling depression?
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5. Basic terminology Factor An independent variable (or grouping variable) Level A particular value that a factor can possess Group mean The mean value of the DV across observations within a particular level of an IV Class Grand mean The mean value of the DV across observations in the experiment as a whole Freshman Sophomore Junior Senior
10. Group 1 Group 2 H0 vs. H1 - 3 Groups Group 3 Group Mean Grand Mean X X X X
11. Decomposing variance “ Natural variability” “ Variability across group means” The essence of an ANOVA is to determine how the variability across group means (treatment effect) relates to the natural variability (or error in measurement). Specifically, we want to know the relative amount of total variability that is attributable to each of these sources. F =
12. F = 1 Variability due to groups = Natural variability Decomposing variance F > 1 Variability due to groups > Natural variability F > 1 Variability due to groups > Natural variability
16. One-way ANOVA: Examples H 1 : Amount of study time varies by class level μ freshman , μ sophomore , μ junior , μ senior are not all equal H 0 : Amount of study time does not vary by class level μ freshman = μ sophomore = μ junior = μ senior H 1 : Three therapeutic methods have differing degrees of effectiveness in treating depression μ cognitive , μ psychodynamic , μ biomedical , are not all equal H 0 : Three therapeutic methods have the same degree of effectiveness in treating depression μ cognitive = μ psychodynamic = μ biomedical
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19. Example = 6.38 Control 1 Control 2 Experimental X 7.0 7.4 5.0 s 1.00 1.14 .89 n 5 5 6 X
20. 3 4 5 6 7 8 Control 2 Control 1 Experimental Example X
21. Decomposing variance “ Natural variability” “ Variability across group means” F = “ Estimate of population variance” “ Average deviation from grand mean”
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23. Decomposing variance F = “ Average deviation from grand mean” “ Estimate of population variance” General formula for variance of a set of numbers: SS df MS B MS W Σ ( X – X ) 2
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26. Variance within-groups Mean squared error (or within-groups), MS W SS df N - 1 N - k MS W = SS 1 + SS 2 + … + SS k Number of groups Σ ( X i,1 – X 1 ) 2 Σ ( X i,2 – X 2 ) 2 Σ ( X i,k – X k ) 2 … Σ ( X i,j – X j ) 2 s p 2 = SS 1 SS 2 + df 2 df 1 + + … + …
27. Variance within-groups Mean squared error (or within-groups), MS W 3 4 5 6 7 8 Control 2 Control 1 Experimental X
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29. Back to our Example… = 6.38 Control 1 Control 2 Experimental X 7.0 7.4 5.0 s 1.00 1.14 .89 n 5 5 6 X
30. Back to our Example… = 6.38 = 13.20 2 X j 7.0 7.4 5.0 s 1.00 1.14 .89 n 5 5 6 X Control 1 Control 2 Experimental X i 7,7,8,8,6 7,7,8,9,6 4,4,6,6,5,5 SS w = Σ ( X i,j – X j )
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32. Decomposing variance General formula for variance of a set of numbers: SS df MS B MS W MS W = SS w /df w MS W = 13.20/13 = 1.105 Next step… we need to find MS B (Mean Square Between)
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