# One-Factor Analysis of Variance (ANOVA)

Description | Usage | Output | Method | See Also

## Description

One-factor Analysis of Variance (ANOVA) provides a method for determining if the means of a given set of variables are statistically the same. The null hypothesis in the ANOVA calculation is that the variables all share the same mean. The calculation is performed using an F-test on specific parameters, as shown below. Draco allows any number of variables to be included in a one-factor ANOVA.

## Usage

The ANOVA option is available in the Data Menu when the Data Window is selected. Choosing the One-Factor ANOVA... option opens the ANOVA variable selection window. The variable selection window is shown below:
Each variable may be selected for inclusion in the ANOVA calculation. To begin the calculation, select Compute Analysis... from the Analysis Menu:

## Output

Running the ANOVA calculation will produce a results windows similar to the following:

## Results of the One-Factor Analysis of Variance (ANOVA) 3 Variables Considered: ser0 100 Observations ser1 100 Observations ser2 100 Observations Resulting F-Statistic: 1.11501 Degrees of Freedom:
Numerator: | 2
Denominator: | 297
Confidence Interval: 32.7912% Sum-of-Squares Partioning
Source | df | Ssq | Msq
Groups | 2 | 0.19737 | 0.09868
Error | 297 | 26.28573 | 0.0885
Total | 299 | 26.4831

The results above outline all parameters generated during the ANOVA calculation. The F-statistic is the F-test value resulting from the calculation, as explained in the Method section. The degrees of freedom outline the degrees of freedom for the numerator and denominator of the resultant F-test. The Confidence Interval specifies the confidence level of the null hypothesis (the variables share the same mean).
In the Sum-of-Squares partitioning table, the groups sum-of-squares (Ssq) is the sum of squares of the differences between each variables mean and the grand mean (the mean of all observations in all variables) scaled by the number of degrees of freedom. The groups mean squares is the groups' sum-of-squares divided by the number of degrees of freedom (number of variables - 1). The error sum-of-squares is represents the total of the individual group sum-of-squares of deviations from their corresponding group mean. The total sum-of-squares is the sum-of-squares of deviations of each observation from the overall grand mean value.

## Method

The ANOVA analysis relies on the calculation of the F-statistic for the null hypothesis. The F-statistic in this case is computed from the ratio of the of the estimated variance between groups and the estimated variance within groups. These variances can be estimated from the sum-of-squares in each case. The sum-of-squares between groups is defined as:
The total sum-of-squares can be rapidly calculated via the following formula:
In the above formula, a represents the number of variables being considered in the analysis, ni represents the number of observations in variable i, N represents the total number of observations, y bar represents the mean of variable i, and y bar bar represents the grand mean of all observations. The sum-of-squares due to errors can be calculated from the difference between the sum-of-squares between groups and the total sum-of-squares.
The mean square between groups is equal to the sum-of-squares between groups divided by the degrees of freedom in the sum-of-squares between groups (a - 1). Likewise, the mean square of errors is the sum-of-squares of errors divided by the degrees of freedom in the sum-of-squares of errors (N-a). The resulting F statistic is the ratio of the two mean squares:

## See Also

Data Window

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