An Illustrative Example

In order to illustrate different cases related to overlapping, we include an example where we observe a variable у which assumes values from 1 to 10. The overall n = 90 observations are divided in two subgroups, A and B, with n_A = 60 and «g = 30.

Four different situations, representing the four main overlapping typologies, are reported. In the first case, the subgroups A and В are not overlapping, in the second there is low overlap between A and B, in the third we observe an high level of overlap, while, in the fourth case the overlapping is complete.

The у values for the four cases are presented in Table 4.1 and in Figure 4.1.

Table 4.2 shows the mean, the median, the coefficient of variation cvj = (Tj/i/j, the probability, and the intensity of transvariation for the four cases illustrated in Table 4.1 and in Figure 4.1.

Given плпв = 60 * 30 = 1800 possible differences (yg, - улг), in the first case, where y,_m.g = 7, y_meA = 2 and (y„„g - y_meA) = (7 - 2) = 5, all 1800 differences are positive and, since both the null and the negative differences are zero, 2«gл = 0 and зпва = 0, also the probability of transvariation is zero:

TABLE 4.1

The intensity of transvariation is based on the difference between the subgroup's means, that is, (i/g - уд) = (7.70 - 3.05) = 4.65, and on Tba, which is the sum of the negative differences (yg,■ - уд,). In this case there are no negative differences, since _Зивд = 0, also Т_вл = 0 and consequently the intensity of transvariation is zero:

In the second case, still with y_n,_eB = 7, у,„,д = 2 and (y_meB - у,_ж.д) = 5, the presence of some overlapping units leads to iп_Вл = 61 null and to 3пвл = 43 negative differences (yg, - у a?)', the corresponding probability of transvariation results:

The difference between the subgroup's means is (yg - у a) = (7.40 - 3.20) = 4.20 and Твл, that is, the sum of the _Зивд = 43 negative differences, is equal to 63; the related intensity of transvariation is calculated as:

In the third case, a higher level of overlapping between the subgroups A and В leads to 2ивд = 136 null and to 3пвл = 226 negative differences (yg, - уд,),

FIGURE 4.1

Example of overlapping: 90 observations are divided in two non-overlapping subgroups (I), two weakly overlapping subgroups (II), two highly overlapping subgroups (III) and two perfectly overlapping subgroups (IV)

with a probability of transvariation given by:

The difference between the subgroup's means is (yg - ул) = (6.60 - 3.60) = 3 and the sum of the pi в a = 226 negative differences is Tba = 515; the intensity of transvariation results:

The fourth case refers to the situation of maximum overlapping, where the subgroup's medians have the same value y,_m.g = y_meA = 3 as holds for the subgroup's means yg = у a = 4.60.

In this extreme case, the number of positive differences (yg, - улг), пвл, is 788, while the null differences are 2^ba = 224 and the negative differences з«вл = 788; when the overlapping is maximum, with y_meg = y_meA, it follows

TABLE 4.2

Sample Size, Mean, Median, Coefficient of Variation, Probability of Transvariation and Intensity of Transvariation for the Subgroups A and В in the Four Cases (Zero, Low, High, Maximum Overlapping) Reported in Table 4.1

that the number of positive differences is equal to the number of negative differences: пвл = з«вл- The property ща = з«вл represents the basis for assigning, in the case of maximum overlapping, the difference between the subgroups in equal parts to the inequality between and to the overlapping component.

Starting from гщл = 224, зпва = 788, па = 60, and «в = 30, the probability of transvariation results:

The sum of the _Зивл = 788 negative differences is T= 2540, which is the same value as the sum of the П_Вл = 788 positive differences, thus leading to the maximum of the intensity of transvariation:

As expected, the probability of transvariation Нвл is always lower than the intensity of transvariation Uba, except in case IV, when the two indicators are equal to one.