>>Return to Tell Me About Statistics!
In Vol.27, we explained the analysis of variance (ANOVA) and the decomposition of variation.
Here, we explain how to decompose the variation.
About global variation
First, find the mean and variance across subjects for drugs X, Y, and Z. Variation is expressed as the “sum of squared deviations.” The calculated sum of squared deviations is called “total variation” in ANOVA.
About within-group variation
“Intra-group variation” is a numerical representation of the extent to which individual data vary from the average effect of each drug.
Consider the group of effects for each drug as a “group” and calculate the deviation of the data minus the mean for each group. Square the resulting deviations (deviation squared) and add these deviation squares for each group (deviation sum of squares). This sum of squared deviations is called “fluctuation.”
The total of the sum of squared deviations for each group (in this example, drugs X, Y, and Z) is the “within-group variation” (Figure 1).
[Figure 1] Within-group variations of the three drugs
About variation between groups
The variation between groups” is a numerical representation of the extent to which the average effect of each drug varies from the overall average. Figure 2 shows the mean effect of each drug.
Consider the cluster of effects for each drug as a “group,” and calculate the deviation of each group’s mean minus the overall mean. We square the resulting deviation (deviation squared) and multiply the deviation squared by the number of subjects per group (in this case, the number of subjects taking drugs X, Y, and Z). The sum of the three values obtained is the variation between groups.
[Figure 2] Variation between groups of three drugs.
The sum of the within- and between-group variations always agrees with the overall variation (Figure 3).
[Figure 3] Total value of overall variation
Test method for 3 or more groups
Whether the population means of three or more groups are equal can be determined by comparing the magnitude of the between-group and within-group variation. To compare these two variations, it is common to calculate the variance ratio and perform an F-test. An ANOVA can elucidate whether there are differences in the effectiveness of drugs if the test is conducted using variance ratios.
A large F-value (p < 0.05) indicates a significant difference in the mean effect of the drugs in the population.
Where are the significant differences?
If the results of the F-test show a significant difference, there is a difference in the mean value of the effect of each drug. However, the ANOVA only indicates that there are some differences between the groups but does not know where the differences are.
Next, we explain the types of tests that need to be performed after implementing ANOVA.
>>Return to Tell Me About Statistics!
Comments are closed