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ANOVA

ANOVA. Analysis of Variance or The F distribution. Example.

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ANOVA

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  1. ANOVA Analysis of Variance or The F distribution

  2. Example • A randomized clinical trial carried out to compare the effect of three treatments A, B and C for the reduction of serum cholesterol levels in obese patients. The reduction in serum cholesterol levels after the intervention is shown in the table 1.Can we conclude from these data there is a difference in effect of treatments used for the reduction of serum cholesterol level? 2. Which treatment is most effective?

  3. K = number of groups • n = Sample size of single groups(n1,n2.n3…..) • N = Total sample size • T = sum of scores in a single group(T1, T2, T3…..) • G = sum of all scores or Grant total

  4. Solution

  5. Now we will calculate: • Sum of Squares: SSWithin, SSBetween, SSTotal • Degrees of freedom: dfWithin, dfBetween,dfTotal • Mean Squares: MSWithin, MSBetween • F-Ratio: F = MSBetween/ MSWithin

  6. Sum of Squares: SSWithin (SSW), SSBetween (SSB), SSTotal (SST) • SSB= ∑( T²j/nj) – (T²)/N 19689.57 – 703²/26 = 681.53 • SSW = ∑ X²ij - ∑ (T²j/nj) 20471 – 19689.57 =781.43 • SST = 681.53 + 781.43 = 1462.96 Degrees of freedom: dfWithin, dfBetween, dfTotal • dfWithin = N-1=3-1 = 2 • dfBetween = K- N= 26 – 3 = 23 • dfTotal = K- 1 = 26 – 1= 25 Mean Squares: MSWithin (MSqW), MSBetween (MSqB) • MSqB =SSB/d.fbetween = 681.58/2= 340.77 • MSqW = SSW/d.fwithin = 781.43/23 =33.98 • F-Ratio: F = MSBetween/ MSWithin= 340.77/ 33.98= 10.03

  7. Analysis of ANOVA can be presented in a table

  8. Multiple comparisons between treatment • Since Ho : null hypothesis is rejected. We are interested to know which pair of treatments means significantly differ. • H01=µ1 =µ2 (X1 – X2) – (µ1- µ2) • Test statistic = • √Error mean square(1/n1+1/n2)

  9. Statistical decision : Student t distribution table value for α = 0.05 and d.f. 23 is 2.069. • There is no significantly different from the treatment A & C, where as treatment B is significantly different from the treatment A & C • Conclusion : since the average reduction in serum cholesterol levels is highest in treatment B and which is significantly different from the treatment A and C. Hence we conclude that treatment B is most effective for the reduction of serum cholesterol levels.

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