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Numerical Example of One-Way ANOVA

This Demonstration illustrates some basic principles of one-way ANOVA (factor analysis of variance) and shows how it works so you can analyze the statistical variability of a statistical complex.
You can vary the sample sizes of the three groups , , separately from 3 to 18, and observe how the ANOVA table changes. The total variation depends not only on the sizes but also on the variability within each sample.
An ANOVA table helps in understanding the overall relationship of random variables and in learning the principle of calculating the variations.
This Demonstration considers the calculation of the ANOVA table manually and illustrates the repeated-measures design. Repeated measures analysis of variance (rANOVA) is one of the most commonly used statistical approaches to repeated measures designs.

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In ANOVA logic, partitioning the sources of variance and hypothesis testing can be done individually. According to the ANOVA method, the total variation is decomposed into two parts: a source of variation due to the group or factor effect (expressed with ) and a source of variation due to the measurement error (expressed with ).
This Demonstration uses the ANOVA table algorithm based on deviations.
By definition, the variance , where is the sum of squared deviations; is the total deviation, based on the differences between variants and the mean of statistical complex; is the intergroup or between-group deviation, based on differences between each group (or sample) and the complex means; and is the intragroup or within-group deviation, based on differences between the variants and the sample (group) mean.
This Demonstration illustrates some basic principles of one-way ANOVA only. We know that the Fisher -test is used for comparisons of the components of the total deviation. The -value is the ratio of variance between and variance within samples (groups).
Consider an experiment to study the effect of three different levels of a factor on a response , , . With (here we use but say there are groups) observations for each level, we write the outcome of the experiment in a work table.
To calculate the -ratio:
Calculate the size of the statistical complex (first row in the work table) .
Calculate the sum within each sample (group), and the complex total sum, adding values in the row.
Calculate , the square of the sum.
Find the value and the total sum in the row:
.
Calculate the (sum of square).
Find the total sum in the row .
Find the value :
and the deviations:
,
,
.
Calculate total variance , between-group variance , and within-group variance .
The statistical ANOVA complex degrees of freedom is .
The between-group degrees of freedom is one less than the number of groups (samples): .
The within-group degrees of freedom is .
And finally we find the -ratio:
and the relative factor effect:
.
It is well known that the -test is used to compare the components of the total deviation. In this Demonstration, we are using the textbook method of concluding the hypothesis test: compare the observed -value with the critical value of determined from tables. The critical value of is a function of the numerator degrees of freedom, the denominator degrees of freedom, and the significance level alpha. If the experimental -value is more than critical -value, then reject the null hypothesis.
The Demonstration has thus illustrated how we can get and apply the experimental -value.
References
[1] A. Gelman, "Analysis of Variance—Why It Is More Important Than Ever," Annals of Statistics, 33(1), 2005. pp. 1–53. doi:10.1214/009053604000001048.
[2] D. C. Montgomery, Design and Analysis of Experiments, 5th ed., New York: Wiley, 2001.
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