1. Chapter 8 Analysis of METOC Variability

2. Contents 8.1. One-factor Analysis of Variance (ANOVA) 8.2. Partitioning of METOC Variability 8.3. Mathematical Model of One-way ANOVA 8.4. Multiple Comparisons 8.5. Two-Factor ANOVA

3. within-sample and between- sample variations. Sample-1Sample-3Sample-2

4. 8.1. One-factor (or One-way) ANOVA The one-factor ANOVA involves sampled data from two or more populations of a factor. The ANOVA is used to test if the true population means are all the same or at least two of them are different. The ANOVA test a hypothesis about means (of several populations) using the F- distribution as the test statistic.

5. Assumed Conditions (1) All samples are independent; (2) All populations are normally distributed;

6. Null Hypothesis

7. ANOVA Sample-1Sample-2Sample-3 Sample-p

8. sample

9. d.f. = p-1 d.f. = N-p sample

12. 8.2. Partitioning of Variability SST=SSW+SSB

19. Unequal Sample Sizes

20. 8.3. Mathematical Model of One-way ANOVA

21. 8.4. Multiple Comparisons When a hypothesis testing from ANOVA rejects the null hypothesis, we only know that not all means are equal or at least one mean differs from others. There is no information on which one (or ones) is (or are) different from others. To obtain further information, we have to conduct comparison for each pair of population means, i.e., pairwise comparison. For p populations, there are p(p – 1)/2 pairs of population means for further comparison. For three populations (A, B, C), the pairwise comparison includes [i.e., 3(3-1)/2 = 3]: (A-B), (B-C), (C-A). A procedure for making pairwise comparison among a set of p population means is called the multiple comparison. There are many multiple comparison procedures. Here, we introduce three most commonly used methods.

22. Fisher ‘s Least Significant Difference Method For the same sample size 

23. Fisher ‘s LSD Procedure 

25. Tukey ‘s Method

28. Scheffe ‘s Method

29. 8.5. Two-Factor (or Two-Way) ANOVA Under many situations, there are more than one factors that have effects and have to be considered in the analysis of variance. This type of analysis is called the multifactor analysis of variance. If two factors are considered in the analysis, it is called the two- factor analysis of variance (or two-factor ANOVA).

30. One Observation Per Cell

36. Multiple Observations Per Cell