The probability distribution used is cumulative logistic distribution which is applied to cumulated income, expenditure, or wealth on one hand and also to cumulated probabilities on the other hand. Logistic function or sigmoid function is defined as
𝑓(𝑥) = 𝐿 1+𝑒𝑥𝑝−𝑘(𝑥−𝑥0)
(2.1)
where L is the curve's maximum value, x0 is the x-value of the sigmoid's midpoint, and k = the steepness of the curve[85]. Logistic map, which is the basis for logistic function, is used to show how complex, chaotic behaviour can arise from very simple non-linear dynamical equations [86].
Pearson correlation coefficient (r)
R2 equals the squared Pearson correlation coefficient
Pearson correlation coefficient (r) or simply the correlation between two sets of data X and Y shows how strong is the relationship between them and is defined as
𝑟 =
∑(𝑋−𝑋 ̅ ̅)(𝑌−𝑌 ̅)
√∑(𝑋−𝑋 ̅ ̅)2√∑(𝑌−𝑌 ̅)2
(1.1)
In the Figure 1.4 we could see how the data are distributed for different values of r.
Figure 1.4 Variance according to different values [79]
Thus, r lies between 1 and -1. A value for r=0 indicates no relationship between the two variables. While for r=-1 and for r=1 there is a (perfect) strong relationship (negative in the first case and positive for the second)[80].
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Coefficient of determination (R2) is used to measure the goodness of the fit for two variables. It is actually the square of the correlation coefficient (r). Since the correlation coefficient lies between -1 and 1, the coefficient of determination is between 0 and 1.
R2= (variation attributed to independent variable)/ (total variation of dependent variable). If R2=1, this implies that 100% of the variation in the dependent variable can be explained by the variation in the independent variable [80].
T-test can be used for comparing two sets of data in order to test their significance (if they have different means) and the significance of the parameters of a regression function.
When we use it in order to assess the significance of two sets of data, we use the following formula [81]
𝑡 =
𝑥̅1−𝑥̅2
√𝑠1 2 𝑛1
+
𝑠2 2 𝑛1
(1.2)
where 𝑥̅1and 𝑥̅2 are the means of the two sets of values, s1 and s2 are the standard deviations of the two sets of data, and n1 and n2 are the number of values in the two sets of data. Standard deviation is calculated as follows
𝑠 = √∑(𝑥−𝑥̅)2 𝑛−1
(1.3)
The values for t test when comparing two sets of data can be determined according to the table of values calculated according to the number of degrees of freedom and a the confidence interval chosen. If the result is higher than the values from the table, we can conclude that the relationship between the two sets of data is significant. If the value from the t-test is below the value described in the table, the relationship is not significant..
In the case of regression, the t test statistic is t = (observed - expected) / (standard error). The expected value for the coefficient is 0 (the assumption is that the null hypothesis is true and the null hypothesis is that the β is 0), the test statistic is found by dividing the coefficient by the standard error of the coefficient [82].
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