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Review Sheet Analysis of Variance (ANOVA) ANOVA Steps and Notation Given the fo

Review Sheet Analysis of Variance (ANOVA) ANOVA Steps and Notation Given the following hypothetical data from an experiment examining learning performance under three temperature conditions: Temperature Conditions 1 2 3 50° 70° 90° 0 4 1 ΣX2 =106 1 3 2 G = 30 3 6 2 N = 15 1 3 0 k = 3 0 4 0 T1 = 5 T2 = 20 T3 = 5 SS1 = 6 SS2 = 6 SS3 = 4 n1 = 5 n2 = 5 n3 = 5 ___ X1 = 1 ___ X2 = 4 ___ X3 = 1 The letter k identifies the number of treatment conditions (the number of levels of the factor) The number of scores in each treatment is identified by a lowercase letter n The total number of scores in the entire study is specified by a capital letter N The total (ΣX) for each treatment condition is identified by the capital letter T The sum of all the scores in the research study (the grand total) is identified by G. You can compute G by adding up all of the N scores or by adding up the treatment totals G = ΣT The final calculation for ANOVA is the F-ratio which is composed of two variances F = variance between treatments variance within treatments sample variance = s2 = SS df Recall that SS = ΣX2 – (ΣX)2 N Total Sum of Squares, SStotal SStotal = ΣX2 – G 2 N In our example, we have: SStotal = 106 – 30 2 15 = 46 Within-Treatments Sum of Squares, SSwithin SSwithin = ΣSSinside each treatment SSwithin = 6 + 6 + 4 = 16 Between-Treatments Sum of Squares, SSbetween SSbetween = Σ(T2/n) – G2/N SSbetween = 52/5 + 202/5 + 52/5 - 302/15 = 5 + 80 + 5 – 60 = 90 – 60 = 30 To verify our analysis, check to see that the two components add up to the total. SStotal = SSbetween + SSwithin 46 = 30 + 16 Total Degrees of Freedom, dftotal dftotal = N – 1 dftotal = 15 – 1 = 14 Within-Treatments Degrees of Freedom, dfwithin dfwithin = Σ(n – 1) = Σdfin each treatment dfwithin = N – k dfwithin = 15 – 3 = 12 Between-Treatments Degrees of Freedom, dfbetween dfbetween = k – 1 dfbetween = 3 – 1 = 2 The two parts we obtained from this analysis of degrees of freedom add up to equal the total degrees of freedom dftotal = dfwithin + dfbetween 14 = 12 + 2 MS (variance) = s2 = SS df MSbetween = sbetween 2 = SSbetween = 30/2 = 15 dfbetween MSwithin = swithin 2 = SSwithin = 16/12 = 1.33 dfwithin F = MSbetween MSwithin F = 15/1.33 = 11.28 Then it is useful to organize the results in a table called ANOVA summary table Source SS df MS Between treatments 30 2 15 F = 11.28 Within treatments 16 12 1.33 Total 46 14 Hypothesis testing with ANOVA Given the data Placebo Drug A Drug B Drug C 0 1 3 7 N = 20 2 4 6 3 G = 60 0 1 4 5 ΣX2 = 262 0 1 3 6 3 3 4 4 T = 5 T = 10 T = 20 T = 25 SS = 8 SS = 8 SS = 6 SS = 10 H0: μ1 = μ2 = μ3 = μ4 Ha: At least one of the treatment means is different We will use α = .05 a. Analyze the SS to obtain SSbetween and SSwithin b. Use the SS values and the df values to calculate the two variances, MSbetween and MSwithin c. Finally, use the two MS values (variances) to compute the F-ratio The F-ratio for these data have df = 3, 16 Note that an F-ratio larger than 3.24 rejects the null (in this problem) uploads/Finance/ anova-steps-and-notation-total.pdf