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When the scatter points in a correlation diagram show a linear trend, applying a straight line will clarify the relationship between the items on the y-axis and the x-axis. Fitting a straight line is called “fitting a relational equation.” The statistical method to find the relational expression is called a “simple regression analysis” or “linear regression analysis.”
The straight line function equation (y=ax+b) found by simple regression analysis is called the “simple regression equation.” We know that the simple regression equation can be found using the following formula:
[Formula]
*Sxy and Sxx are the sum of the products and the sum of the squared deviations, respectively, as shown in vol. 47.
Specifically, we explain this using an example in which a straight line determined by simple regression analysis is drawn on the correlation diagram between the circumference of the trunk of a male student (cm) and body fat percentage (%), as shown in the table below. Let us find this simple linear regression equation.
[Table 1]
[Figure 1] Fitting a straight line
Using the procedure in vol. 47, you can obtain the values as shown in the table below.
[Table 2]
Thus, a and b for y=ax+b can be obtained as follows:
b=74-1.76×17=44.08
Therefore, the linear simple regression equation is “y=1.76x+44.08.”
What are “strength” and “magnitude” in correlation?
Let the simple correlation coefficient be r and the simple regression equation be y=ax+b. r is an index to understand “strength,” and a is an index to grasp “size.” In that case, what are “strength” and “size” in the relationship between trunk circumference and body fat percentage?
r is a tool to understand whether “as the body fat percentage increases, the circumference of the trunk tends to increase” or, in other words, “does the body fat percentage affect the circumference of the trunk?”
r is a numerical value that expresses the degree of tendency (influence) and indicates the “strength” of body fat percentage relative to the circumference of the trunk.
The coefficient of the simple regression equation (a) is a tool to understand whether “if the body fat percentage increases by △%, will the circumference of the trunk increase by 0 cm?” In other words, “what is the influence of body fat percentage on the trunk circumference?”
a is a numerical representation of this effect and indicates the “size” of body fat percentage relative to the circumference of the trunk.
Now, let us compare the results of the study on the female students in the case above and look at the “strength” and “size” of each body fat percentage relative to the circumference diameter of the trunk.
[Table 3]
As shown in the table and figure below, the “strength” of body fat percentage relative to the circumference of the trunk is 0.81 for male students and 0.99 for female students, meaning that male students have a weaker degree of strength than female students.
In addition, the “size” of body fat percentage relative to the circumference of the trunk is 1.76 cm for male students and 1.21 cm for female students, indicating that male students are larger than female students.
In other words, if the value of r is large (strong), it does not necessarily mean that the circumference of the trunk will increase in response to an increase in body fat percentage.
[Table 4]
[Figure 2]
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