1. Statistics in Applied Science and Technology Chapter 10. Analysis of Variance

2. Key Concepts in this Chapter Assumption involved in ANOVA. Sum of the square deviation between groups Sum of the square deviation within groups Sum of squared deviation of each x from the overall mean F statistics and its distribution Critical F value

3. Function of ANOVA What if I want to compare means of more than two groups? (Research Question) - Analysis of variance (ANOVA) Hypotheses/Research Question tested in ANOVA are usually like this: –H 0 :  1 =  2 =  3 …=  n –H 1 : not all the means are equal

4. Assumptions involved in ANOVA The observations are independent. Populations are normally distributed and population variances are equal (homogeneous). Level of measurement in interval-ratio. ANOVA is a robust technique, insensitive to departures from normality and homogeneity.

5. Rational for ANOVA The ratio of S b 2 /S w 2 follows F statistic (S b 2 measures between-group variance and S w 2 measures within- group variance) The test statistic of ANOVA is F statistic and its distribution is F distribution (Pg. 147, Figure 10.1). There is a family of F distribution. The one to use depends on the degree of freedom (k-1,  n i -k). k-1 is called the numerator degree of freedom.  n i -k is called the denominator degree of freedom. H 0 will be rejected when calculated F is more than critical F value, i.e., F fall into rejection/critical region (Decision Rule).

6. Calculation of test statistic F First some important notation: –k is the number of groups –n i is the number of observations in each group –SS b is the sum of the squared deviations between groups –SS w is the sum of the squared deviations within groups –SS t is the sum of squared deviation of each x ij from the overall mean. It measures the amount of variation about the overall mean

7. Calculation of test Statistic F Some of the most important equations include:

8. How would I find out Critical F? You need a Table called “Percentiles of the F distribution”, (You can download from “Schedule”). In-class demonstration of how to use the table.

9. Application of ANOVA 1. Is there a significant difference in birthweight among three groups of infants classified by the smoking status of the mothers? (textbook Pg. 150) Please pay attention to Table 10.3 (called as ANOVA table) and Table 10.4. SPSS can be used to calculated ANOVA table very easily.