Chapter 8 Analysis of METOC Variability. Contents 8.1. One-factor Analysis of Variance (ANOVA) 8.2.…

Chapter 8

Analysis of METOC Variability

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

within-sample and between-sample variations.

Sample-1 Sample-3Sample-2

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.

Assumed Conditions

• (1) All samples are independent;

• (2) All populations are normally distributed;

ANOVA

Sample-1 Sample-2 Sample-3Sample-3 Sample-p

sample

sample

d.f. = p-1

d.f. = N-p

sample

8.2. Partitioning of Variability

SST=SSW+SSB

Unequal Sample Sizes

8.3. Mathematical Model of One-way ANOVA

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.

Fisher ‘s Least Significant Difference Method

For the same sample size

Fisher ‘s LSD Procedure

Tukey ‘s Method

Scheffe ‘s Method

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).

One Observation Per Cell

Multiple Observations Per Cell