Analysis of variance

ANOVA - STATISTICAL TOOL

1. ANALYSIS OF VARIANCE (ANOVA) Presented by, Neethu Asokan

2. Biostatistics is the science of collection , analysis and interpretation of facts and numbers connected with biology. It is also called biometrics .  STUDENT’S-t TEST  CHI-SQUARE TEST  FISHER’S TEST(F)  ANOVA BIOSTATISTICS

3. WHAT IS ANOVA? ANOVA refers to the examination of differences among the samples . It is an extremely useful technique concerning research in biology. The term ANOVA was first proposed by R.A.Fisher . It is a different way to summarize the differences between several means and comparing these means in one step. This method is called ANOVA or one-way ANOVA

4. WHY ANOVA? In real life things do not result in two groups being compared. Two-sample t-tests are problematic Increasing the risk of an error At .05 level of significance, with 100 comparisons, 5 will show a difference when none exists ( experiment wise error) So the more t-tests you run, the greater the risk of an error (rejecting the null when there is no difference) ANOVA allows us to see if there are differences between means with an OMNIBUS test A STATISTICAL TEST

5. HYPOTHESIS OF ANOVA H : The (population) means of all groups under consideration are equal. H a : The (pop.) means are not all equal.

6. 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.

7. 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 squares of the individual values are usually used. This method is more convenient.

8. TECHNIQUES OF ANOVA The analysis of variance has been classified into a. One-Way classification b. Two-Way classification

9. One-Way ANOVA:  Single independent variable is involved. Example: Effect of pesticide(independent variable) on the oxygen consumption (dependent variable) in a sample of insect. Two-Way ANOVA:  Two independent variable is involved. Example: Effects of different levels of combination of a pesticide(independent variable) and on insect hormone (dependent variable) on the oxygen consumption of a sample of insect.

10. ILLUSTRATION A certain manure was used on four plots of land A,B,C and D. Four beds were prepared in each plot and manure used. The output of the crop in the beds of plots A,B,C and D is given below . Using ANOVA find out whether the difference in the means of the production of crops of the plots is significant or not. A B C D 6 15 9 8 8 10 3 12 10 4 7 1 8 7 1 3 ONE-WAY ANOVA

11. FIND OUT THE MEAN OF EACH SAMPLE SUM OF THE SAMPLES ÷ NUMBER OF SAMPLE

12. FIND COMBINED MEAN

13. FIND THE SUM OF SQUARES BETWEEN THE SAMPLES or SS BETWEEN

14. FIND MS BETWEEN OR MEAN SQUARE BETWEEN THE SAMPLES How to find the degree of freedom??

15. FIND SUM OF SQUARE WITHIN THE SAMPLE OR SS WITHIN

16. SAMPLE X1 X1 X1- X1¯ (X1- X1¯) 2 ∑ (X1- X1¯) 2

17. FIND MS WITHIN DEGREES OF FREEDOM

18. SOURCE OF VARIATION SUM OF SQUARE DEGREES OF FREEDOM MEAN SQUARE BETWEEN SAMPLES WITHIN SAMPLES TOTAL

19. MAKING ANOVA TABLE

20. FIND F VALUE

21. INFERENCE

22. TWO-WAY ANOVA It is used when the data are classified on the basis of two factors. It is also called two factor analysis of variance. ILLUSTRATION: Set up two-way ANOVA table for the following results. Per acre production data for sorghum. NAME OF FERTILIZERS VARIETY OF SORGHUM SEEDS CO.1 CO.5 CO.9 UREA 6 5 5 AMMONIUM SULPHATE 7 5 4 ZINC SULPHATE 3 3 3 POTASH 8 7 4

23. STEP 8 DEGREE OF FREEDOM c =number of item column, r= number of item row

24. SETTING A TWO WAY ANOVA TABLE

25. Source of variation Sum of square Df Mean square F calculated value F table value at 5% Between columns Between rows Error total

26. WHEN ANOVA? Data must be experimental If you do not have access to statistical software, an ANOVA can be computed by hand With many experimental designs, the sample sizes must be equal for the various factor level combinations. ANOVA formulas change from one experimental design to another

27. 3 WAY ANOVA The three-way ANOVA is used to determine if there is an interaction effect between three independent variables on a continuous dependent variable. It is only appropriate to use a three-way ANOVA if your data "passes" six assumptions that are required for a three-way ANOVA to give you a valid result.

28. ASSUMPTIONS Assumption #1:  Your  dependent variable  should be measured at the  continuous  level (i.e., it is an  interval  or  ratio  variable). Assumption #2:  Your  three independent variables  should each consist of  two or more categorical ,  independent groups . Assumption #3:  You should have  independence of observations , which means that there is no relationship between the observations in each group or between the groups themselves.

29. Assumption #4:  There should be  no significant outliers . Assumption #5:  Your  dependent variable  should be  approximately normally distributed for each combination of the groups of the three independent variables . Assumption #6:  There needs to be  homogeneity of variances for each combination of the groups of the three independent variables . You can check assumptions #4, #5 and #6 using SPSS Statistics.

30. REFERENCE Fundamentals of Mathematical Statistics Paperback – 2014; S.C. Gupta Research Methodology And Statistical Techniques; Santhosh gupta,2002 Research methodology- tools and technique; C . R Kothari