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:
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: