1. ONE WAY ANALYSIS OF VARIANCE ANOVA o It is used to investigate the effect of one factor which occurs at h levels (≥3). Example: Suppose that we wish to test the effect of temperature at levels (20, 30, 35, 40 o C) on the serum total proteins. Biostatistics and Data analysis 3 rd Lecture

2. RANDOM MODEL HYPOTHESIS

3. The summary statistics for each row are shown in the table below 20 o C25 o C30 o C Sample size (n)798 Mean2.28572.4445.625 S.D.0.4870.8821.922 Variance (S 2 )0.2370.7783.694 Temperature ( o C) Serum Total Proteins (g/dL) 20 2, 3, 2, 2, 3, 2, 2 25 4, 3, 2, 3, 1, 2, 2, 3, 2 30 5, 6, 7, 4, 2, 6, 7, 8

4. o The sum of the squares of the deviations between a value and the mean of the value SS between groups SS(B) SS within groups SS(W) o The average squared deviation from the mean and are found by dividing the variation by the degrees of freedom MS = SS / df MS between groups MS(B) MS within groups MS(W) Variances (Mean of Squares) = MS Variation (Sum of Squares) = SS

5. Are all of the values identical? – There are variations among the data called the total variation SS(T). Variation (Sum of Squares) = SS Temperature ( o C) Serum Total Proteins (g/dL)Means 20 2, 3, 2, 2, 3, 2, 22.2857 25 4, 3, 2, 3, 1, 2, 2, 3, 22.444 30 5, 6, 7, 4, 2, 6, 7, 85.625

6. Are all of the sample means identical? – There variation called between group SS(B)variation or variation due to Factor. Temperature ( o C) Serum Total Proteins (g/dL)Means 20 2, 3, 2, 2, 3, 2, 22.2857 25 4, 3, 2, 3, 1, 2, 2, 3, 22.444 30 5, 6, 7, 4, 2, 6, 7, 85.625

7. Are each of the values within each group identical? – There is variation within group SS(W) (error variation). Temperature ( o C) Serum Total Proteins (g/dL)Means 20 2, 3, 2, 2, 3, 2, 22.2857 25 4, 3, 2, 3, 1, 2, 2, 3, 22.444 30 5, 6, 7, 4, 2, 6, 7, 85.625

8. – The variation between groups, SS(B), or the variation due to the factor – The variation within groups, SS(W), or the error variation There are two sources of variation

9. Here is the basic one-way ANOVA table SourceSSdfMSFP Between (Factor) Within (Error) Total

10. The summary statistics for the grades of each row are shown in the table below 20 o C25 o C30 o C Sample size (n)798 Mean2.28572.4445.625 S.D.0.4870.8821.922 Variance (S 2 )0.2370.7783.694 Temperature ( o C) Serum Total Proteins (g/dL) 20 2, 3, 2, 2, 3, 2, 2 25 4, 3, 2, 3, 1, 2, 2, 3, 2 30 5, 6, 7, 4, 2, 6, 7, 8

11. Grand Mean – The grand mean is the average of all the values – It is a weighted average of the individual sample means

12. Between Group Variation, SS(B)

13. Within Group Variation, SS(W)

14. After filling in the sum of squares, we have … SourceSSdfMSFp Between56.4412 Within33.50421 Total89.94523

15. – MS = SS / df MS(B)= 56.441 / 2= 28.221 MS(W)= 33.504 / 21= 1.595 Variances

16. After filling in the sum of squares, we have … SourceSSdfMSFp Between56.441228.221 Within33.504211.595 Total89.94523

17. – An F test statistic is the ratio of two sample variances – The MS(B) and MS(W) are two sample variances and that’s what we divide to find F. – F = MS(B) / MS(W) F = 28.2 / 1.595 = 17.69 F test

18. After filling in the sum of squares, we have … SourceSSdfMSF cal P Between56.441228.22117.69 Within33.504211.595 Total89.94523 Tabulated F 2,21(5%) = 3.47, F 2,21(1%) = 5.78, F 2,21(0.1%) = 5.78 Thus calculated F at df 2,21 > Tabulated at F 2,21(0.1%) = 5.78 Thus reject null hypothesis