ANALYSIS OF VARIANCE [ANOVA]

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ANALYSIS OF VARIANCE
Statistical methods are the techniques for studying variation in nature. Analysis of the variance is practical method of testing the signature. It has wider application as compared to t-test with the help of t-test, we can compare two means. When more tan two samples are to be compared, the technique used for the purpose is analysis of variance (ANOVA). This concept is known as F-ratio or F-test.  
The formula used to calculate the value of F-ratio is
F-ratio = MSB /MS W= Variance between groups/Variance with in Groups
To compare the value of F-ratio, the following steps to be done.
(i)                 sum of squares total (SST) or Total variance can be calculated. Formula is
SST = ∑ X2 – (∑X/N)2
(ii)              Sum of squares between groups (SSB) or Treatment Variance can be calculated. Formula is
SSB = (∑ X1) 2 /n 1+ (∑ X2) ) 2 /n 2 ‑‑ (∑ X ) 2 /N

(iii)            Sum of squares within groups (SSW) or Error Variance can be calculated formula is
SSW = SST _ SSB
(iv)             The degrees of freedom for SST,SSB and SSW are to be calculated with the use of the following formula.
SST = N-1 = Total subjects in all groups.
SSB = K-1 = number of groups.
SSW = N-K.
            Interpreting F-ratio
(i)                 If the obtained F-ratio value is less than the table value, we can say that there will be no significant difference between any of the paired mean difference. The problem is comes to end.
(ii)              If the obtained F-ratio value is equal or greater than the table value, we can say that there will be significant difference between the paired mean differences. But we can not say that all paired differences are having significant variance. To make this determination. Post hoc test must be applied for al differences between paired mean.
(From the post hoc test, we can calculate the value of confidence Interval. If the value of paired mean differences are more than the confidence interval, we can say that there is significant between two means and viceversa).
Note : If F-ratio found to be significant atleast anyone of the paired mean difference is having significant variance.
Post Hoc Tests
            As we discussed earlier, a significant F – ratio alone does not specify which groups differ from one another. It only indicates that there are differences somewhere among the groups. To identify the groups that differ significantly from one another, a post hoc test must be performed. There are several post hoc test may be applied to determine the location of group differences after a significant F has been found. The most commonly used post hoc test is scheffe’s confidence Interval (I). the formula is
                        I = S√ MSWXWg
Where I = Confidence interval
S = √(K-1) x F 0.05, K = Total number of groups
MSW = Mean Square within groups
Wg = 1/n1 + 1/n2 (Weighted mean)

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