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Variance

Variance gives a measure of how the data distributes itself about the mean or expected value.

Where,

σ2 = Variance

x = Values given in a set of data

x̄ = Mean of the data

n = Total number of values.

Procedure

To apply an ANOVA,you have to first sets up the null and alternative hypothesis. The null hypothesis assumes that there is no significant difference between the groups. The alternative hypothesis assumes that there is a significant difference between the groups. After cleaning the data, you must test the above assumptions and see if the data meets them. They must then do the necessary calculation and calculate the F-ratio. Simply look at the p value against the established alpha. Accordingly, we reject the null hypothesis or we fail to reject the null hypothesis. Rejecting the null hypothesis, we will conclude that the mean of the groups are not equal.

Logic

The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance. Calculating a treatment effect is then trivial, "the effect of any treatment is estimated by taking the difference between the mean of the observations which receive the treatment and the general mean"

Partitioning of the Sum of Squares

ANOVA uses traditional standardized terminology. The definition equation of sample variance is

, where the divisor is called the degrees of freedom (DF), the summation is called the sum of squares (SS), the result is called the mean square (MS) and the squared terms are deviations from the sample mean. ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model. For example, the model for a simplified ANOVA with one type of treatment at different levels.

The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect.

The F-test

The F-test is used for comparing the factors of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic

where MS is mean square, = number of treatments and = total number of cases

to the F-distribution with , degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution.

The expected value of F is which is 1 for no treatment effect. As values of F increase above 1, the evidence is increasingly inconsistent with the null hypothesis. Two apparent experimental methods of increasing F are increasing the sample size and reducing the error variance by tight experimental controls.

There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result:

  1. The textbook method is to compare the observed value of F with the critical value of F determined from tables. The critical value of F is a function of the degrees of freedom of the numerator and the denominator and the significance level (α). If F ≥ FCritical, the null hypothesis is rejected.
  2. The computer method calculates the probability (p-value) of a value of F greater than or equal to the observed value. The null hypothesis is rejected if this probability is less than or equal to the significance level (α).

The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level).