F test and ANOVA
- 1. F – TEST & ANALYSIS OF VARIANCE (ANOVA)
- 2. F - Test • F -test is a test of hypothesis concerning two variances derived from two samples. • F-statistic is the ratio of two independent unbiased estimators of population variances and expressed as: • • F= 1 2 /2 2 • n1 – 1 the degrees of freedom for numerator and n2- 1 the degrees of freedom for denominator. • F –table gives variance ratio values at different levels of significance at d= (n1-1) given horizontally and degrees of freedom (d) = (n2-1) given vertically. • Generally 1 2 is greater than 2 2 but if 2 2 is greater than 1 2, in such cases the two variances should be interchanged so that the value of ‘F’ is always greater than 1.
- 3. • If the F – ratio value is smaller than the table value, the null hypothesis (H0) is accepted. It indicates that the samples are drawn from the same population. • If the calculated F value is greater than the table value, null hypothesis (H0) is rejected and conclude that the standard deviations in the two populations are not equal.
- 4. • Working Procedure • Set up the null hypothesis (H0) 1 2 = 2 2 and alternative hypothesis (H1) = 1 2 2 2 • Calculate the variances of two samples and then calculate the F statistic i.e., • F= 1 2 /2 2 if 1 2 2 2 • Or F= 2 2 /1 2 if 2 2 1 2 • Take level of significance at 0.05 • Compare the compound F- value with the table value and degrees of freedom (n1 – 1) horizontally and degrees of freedom (n2 -1) vertically.
- 5. • Assumptions of F test: • Normality: The values in each group should be normally distributed. • Independence of Error: Variation of each value around its own group mean i.e., error should be independent of each value. • Homogenity: The variances within each group should be equal for all groups i.e. 1 2 =2 2= 3 2 =…….n 2
- 6. • Uses • F test to check - equality of population variances. • To test the two independent samples (x and y) have been drawn from the normal populations with the same variances (2). • Whether the two independent estimates of the populations variances are homogenous or not.
- 7. ANOVA • The ‘Analysis of Variance’ (ANOVA) is the appropriate statistical technique to be used in situations where we have to compare more than two groups OR • It is a powerful statistical procedure for determining if differences in means are significant and for dividing the variance into components.
- 8. • Variance (2) is an absolute measure of dispersion of raw scores around the sample (group) mean, the dispersion of the scores resulting from their varying differences (error terms) from the means. • Mean square – The measure of variability used in the analysis of variance is called a mean square • Sum of squared deviation from mean divided by degrees of freedom. • • Mean square = Sum of squared deviation from mean • ---------------------------------------------- • Degrees of freedom
- 9. • Assumptions in analysis of variance • The samples are independently drawn • The population are normally distributed, with common variance • They occur at random and independent of each other in the groups • The effects of various components are additive.
- 10. • Technique for analysis of variance • One - way ANOVA : Here a single independent variable is involved • Eg: Effect of pesticide (independent variable) on the oxygen consumption (dependent variable) in a sample of insect. • Two -way ANOVA: Here two independent variables are involved. • Eg: Effects of different levels of combination of a pesticide (independent variable) and an insect hormone (independent variable) on the oxygen consumption of a sample of insect.
- 11. • Working Procedure • The procedure of calculation in direct method are lengthy as well as time consuming and this is not popular in practice for all purposes. • Therefore a short cut method based on the sum of the squares of the individual values are usually used. • This method is more convenient.
- 23. Critical Value = 3.89 F = 22.59