Numerical Example of One-Way ANOVA
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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.
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Contributed by: Olexandr Eugene Prokopchenko and Pylyp Prokopchenko (September 2012)
Open content licensed under CC BY-NC-SA
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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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