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TWO WAY ANOVA (G) In two-way ANOVA there is more than one experimental factor and one or more control factors. In case of the above one way ANOVA the three attitude scales for the test of performance are experimental factor where as the examiner is the control factor. (G) If the tests are conducted by two examiners so the over all factors will be 3 X 2 = 6 . It is also called double classication ANOVA. This procedure can be used for the application of three-way or larger class of ANOVA. .- .-i\\ a"! _ H Illustration : A farmer applied three types of fertilizers in 4 separate plots. The figure on yield per acre are tabulated below: Yield Fertilizers A B C D Total Plots Nitrogen 6 4 8 6 24 Potash 7 6 6 9 28 Phosphate 8 5 10 9 32 S Total 21 15 24 24 84Yield Square of Data Plots A B c B Row (X1)2 (X2)2 (X3)2 (X4): Fertilizer x1 x2 x3 X4 Total -36 Nitrogen Potash Column Total 6 Phosphates n Find out if the plots are materially different in fertility, as also, if the three fertilizers make any material difference in yields. Solution: O Step 1 Null Hypothesis:Let us take the hypothesis that: 1. All plots are not significantly differ in fertility (Column wise analysis) 2. All the fertilizers are not significantly differ in yields.(Row wise analysis) OStep 2 O Step 3 Correlation Factor (c) Sum of Square Total = EX? - C C = ( [ x) 2 EX2 = 149 + 77 + 200 + 198 = 624 N SSt = 624 -588 (84)2 12 SSt = 36 C =588O Step 4 Sum of Square Between Columns: SSc = ([x1)2 + ( [x2)2 + ([x3)2 + ([x4)2 _ C n1 n2 73 n4 (21)2 (15)2 SSc + (24) 2 + (24)2 -558 3 3 3 3 SSc = 147 + 75 + 192 + 192 - 588 O Step 5 SSc = 18 Sum of Square Between Rows: SSr ( [x1)2 ([ x2) 2 ([x3)2 = + + C n1 n2 n3 SSr = = (24) 2 (28)2 + + (32) 2 558 4 4 4 SSr =144 + 196 + 256 - 588 SSr = 80O Step 6 Step 7 Degree of Freedom: Sum of Square With in Groups: D.O.F for total sum of square = N - 1 D.O.F for SSt = 12- 1 = 11 SSw = SSt -( SSc + SSr ) D.O.F for sum of square between colums = c - 1 SSw = 36 - (18 +8 ) D.O.F for SSc = 4-1 =3 SSw =36 -26 D.O.F for sum of square between rows = r - 1 D.O.F for SSr = 3 - 1 = 2 SSw = 10 D.O.F for sum of square with in groups = (r - 1 ) ( c - 1 D.O.F for SSw = 2 x 3 = 6 Here : r = no. Of rows O Cor no. Of columnsANOVA TABLE Source of Sum of Degree of Freedom Mean Sum of F - Ratio Table Value Variation Square Square Between SSc = 18 C - 1 4 = 4 = 1.5 F Columns 4-1 =3 2.67 18/3 = 6 (3,6) = 8.94 HO = Accepted Between SSr = 8 r- 1 24 9 = 2.67 SSr / F (2,6) = 19.33 Rows 3-1 =2 SSw 8/2 = 4 HO = Accepted 4/1.667 = 2.4 Within SSw = 10 (r-1)(c-1) 24 Groups 2 x3 = 6 9 10/6 = 1.667 Total SSt = 36 N-1 12- 1 = 11 30Step 9 Interpretation 1. Columns wise analysis: The computed value of F = 3.6 is less than table value 8.94, hence the null hypothesis is accepted, it means the plots are not significantly differ in fertility. 2. Row wise analysis: The calculated value of F = 2.4 is less than table value 19.33, hence the null hypothesis is accepted. It means the fertilizers are alike so far as productivity concern. OUses of ANOVA O To test the significance between variance of two samples. O Used to study the homogeneity in case of Twoway Classification. O It is used in testing of correlation & regression. O ANOVA is used to test the significance of multiple correlation coefficient. The linearity of regression is also tested with the help of Analysis of Variance. Interpretation of significance of means & their interactions. OAdvantages of ANOVA O It is improved technique over t-test & z-test. O Suitable for multi-dimensional variables. O Analysis various factors at a time. O Can be used in three and more than three groups. OAdvantages of ANOVA Economical and good method of Parametric testing. Q It involve more than one independent variables in studying the main impact & interaction effect. The experimental design ( simple random design & level treatment design) are based on one way ANOVA technique. ' Disadvantages of ANOVA O It is much complex to analyze ANOVA under the strict assumptions regarding the nature of data. O This is not so much helpful in comparison with t-test that there is no special interpretation of the significance of two means.Limitations of ANOVA O It is difficult to analyze ANOVA under strict assumptions regarding the nature of data. It is not so helpful in comparison with t-test that there is no special interpretation of the significance of two means. O It is not always easy to interpret the cases of multiple interactions and their significance level. OLimitations of ANOVA O It has a fixed and difficult set for designing experiments for the researcher. O Requirement of post ANOVA t-test for further testing. O Sometimes, time consuming & also time requires knowledge & skills for solving numerical problems. O It provides no additional information as compared to t-test