JP2021068456A - Calculation technique for eliminating "multicollinearity" or the like in regression analysis, and obtaining partial regression coefficient indicating contribution to proper objective variable of explanatory variable, as management material - Google Patents

Abstract

To solve the problem that a partial regression coefficient bi of an explanatory variable Xi of a cause indicating contribution to an effect Y is alienated from theory and experience, due to the presence of a large number of numerical values with a fairly high degree of correlation to an objective variable Y among selected explanatory variables of causes, in multiple regression analysis that is applied for relationship analysis between an industrial effect Y and its causes X.SOLUTION: Provided is a calculation technique for eliminating "multicollinearity" or the like. In order to eliminate this discrepancy, a determinant of correlation coefficients of an explanatory variable Xi and an effect Y of a normal equation in multiple regression analysis is created, and by using a diagonal element Xi that is a main force variable of a matrix since numeric balance of both right and left sides is not achieved, a portion overlapping with Xi is removed from Xj of a subordinate non-diagonal factor, and then, the remainder is used as a new non-diagonal factor of the matrix in a normal equation for which bi of a new correlation determinant is obtained.

Description

The present invention solves problems such as "multicollinearity" that occur when the relationship between sales, etc., which is an industrial result, and sales expenses, etc., which is the cause, is performed by multiple regression analysis, and sales expenses, etc. that contribute to sales. This is a calculation technique for the purpose of obtaining an appropriate partial regression coefficient bi indicating the contribution of the causative variable of.

When applying multiple regression analysis to elucidate the relationship between results and causes, problems such as "multicollinearity" occur in the explanatory variables, and appropriate management decisions cannot be made. In the following hypothetical analysis example, the distortion of the partial regression coefficient bi due to "multicollinearity" is shown, and the distortion of bi when there is a high correlation between the explanatory variables to some extent.

Multiple regression equation Y = b0 + b1X1 + b2X2 (1)
Multiple regression equations for standardized variables
Ay = bA1 + b2A2 (2)
Temporary analysis example Sales, labor costs, and sales floor area, which are the causes, are as follows.
Sales (million yen) Personnel costs (100,000 yen) Sales floor area (10m ² )
Y X1 X2
Store A 46 97 75
Store B 18 61 59
C store 36 83 72
Store D 22 37 51
E store 27 75 48
A multiple regression analysis is applied to this material when there are two explanatory variables.
The result of this calculation is as follows.
Correlation coefficient between sales and labor costs R1: 0.84
Correlation coefficient of sales to sales floor area R2: 0.79
Correlation coefficient between labor cost and sales floor area r12: 0.72
b1 = 0.56 b2 = 0.38
Although there is not much difference in the correlation coefficient between labor cost and sales floor area with respect to sales, there is a large difference in the numerical values of the partial regression coefficients b1 and b2, which indicate the degree of contribution of the cause to the result. This is due to the fact that there is a fairly high correlation between labor costs and sales floor area, and even managers with practical experience have doubts about the validity of this analysis result.

Analysis Example 2 (From Example 3, only the analysis results are shown.)
Case name JKL
For the objective variable
Correlation coefficient: R 0.51 0.62 0.54
Partial regression coefficient: bi 0.10 0,43 0.33
This is an analysis of a properly made case, but the numerical value of the partial regression coefficient with respect to the correlation coefficient of case J is small. This is because the value of the correlation coefficient between the explanatory variables of the examples is a little large, around 0.5, but it cannot be said that the correlation is particularly high if the value is about this level.

Regression analysis is an analysis of how much an explanatory variable of a set (X1, X2, ..., Xp) can explain the objective variable Y, and the numerical value of Xi that causes it is highly independent by physics, chemistry, etc. It is said that the contribution of Xi of each component constituting the set was not calculated. After that, when multiple regression analysis was used to analyze the relationship between corporate sales, which is the result of socio-economic phenomena, and sales expenses, which are the causes, the causes to be adopted have a high causal relationship with sales. Personnel costs, sales costs, sales floor area, etc. have come to be selected, the correlation coefficient between these causes is high through the sales figures, and bi, which indicates the degree of contribution to sales, is an abnormal value contrary to common sense and theory. Came to show. This is the problem of "multicollinearity".

Standardized variable Regarding the above equation (1), if the numerical unit of the explanatory variable Xi is the number of people or the area m ² , the meaning of the required numerical value of bi becomes ambiguous, so it is often used in the subsequent analysis. Since it is necessary to standardize the variables to create the correlation matrix, all variables use the standardized numerical values.
Hereinafter, A1 and A2 are standardized variables of X1 and X2.
Standardization: Ai = ( Xi-Xi ) / σi (3)
Xi : Average of Xi σi: Standard deviation of Xi By standardizing Y, X1 and X2, the numerical value of Ai becomes a numerical group having an average of 0 and a standard deviation of 1.

Correlation matrix of normal equations In the following analysis, the correlation matrix equation of the normal equations of multiple regression analysis is mainly used, so this is introduced here.
Correlation coefficient ΣA1A1 / n = 1 ΣA1A2 / n = r12 n: Number of samples The normal equation can now be displayed as follows.

Patent Document 1
Patent No. 5120820 "A device that evaluates an appropriate standard residential land by regression analysis that eliminates" multicollinearity ""
The solution to "multicollinearity" is how to satisfy the equation equal sign with the imbalance of the item values on both the left and right sides in the matrix display of the normal equation. Simply put, the two items on the left side have twice the correlation value (twice) of the one item on the right side, and the existence of overlapping parts is expected, and this part is removed.
In the case of the following two variables, the relatively high correlation between the explanatory variables mediated by the land price, which is the objective variable, is removed as follows to form the off-diagonal element 2 term of the first row of the matrix.
r12 = r12 (1-R1R2)
This is a method of collectively removing the numerical values of R1R2 for a series of cases, not for individual cases. This removal can be expected to have the effect of eliminating "multicollinearity" as it is, but it is a rough calculation.
Patent Document 2
Patent No. 5585921 "A device that applies multiple regression analysis to eliminate" multicollinearity "to analyze the relationship between sales, etc., which is an industrial achievement, and sales expenses, etc., which is the cause."
Compared to the removal of "Patent Document 1", the "multicollinearity" elimination of this patent is a more polite removal for individual cases, but since it is a removal for the standardized explanatory variable itself, the case Is classified into positive and negative sign tuning cases of the standardized variable, and is performed for each case. It is inconsistent and the obtained correlation matrix is also an asymmetric matrix.

これは、非対角要素の相関係数ｒ１２の数値を小さくする効果があり、その分だけ“多重共線性”が解消されるが、相関行列の対角要素に加える加算数値の選択に確たる基準がなく、恣意的であまり利用されていないと思う。
Ａｒｔｈｕｒ Ｅ．Ｈｏｅｒ ａｎｄ Ｒｏｂｅｒｔ，Ｗ．Ｋｅｎｎａｒｄ “Ｒｉｄｇｅ ｒｅｇｒｅｓｓｉｏｎ”１９７０年 ２月 Multicollinearity "Countermeasures for avoidance" Multicollinearity In order to avoid the problem, the following methods are generally considered.
Delete one of the highly correlated explanatory variables Principal component analysis Ridge regression Deletion of the explanatory variables is the number of employees and sales expenses that are significantly selected for the resulting sales and is omitted. Is not appropriate.
Principal component analysis is performed on P variables X1, X2. .. The coefficient bi of the explanatory variable is not obtained by the method of summarizing Xp into the integrated variable Xj (j = 1 to m) of p> m under a certain condition.
Ridge regression obtains the partial regression coefficient by adding a certain number of k to the diagonal elements of the correlation matrix, and the outline is shown below.

This has the effect of reducing the value of the correlation coefficient r12 of the off-diagonal element, and the "multicollinearity" is eliminated by that amount, but it is a reliable criterion for selecting the additional value to be added to the diagonal element of the correlation matrix. I don't think it's used, it's arbitrary and it's not used much.
Arthur E. Hoer and Robert, W. et al. Kennard “Ridge regression” February 1970

While the "multicollinearity" elimination of ridge regression is an indirect method of reducing the off-diagonal element values for diagonal elements, the present invention directly analyzes the off-diagonal elements of the correlation matrix. And.

Looking at the first line of equation (4), the left side is the sum of the correlation coefficients of the causative variables X1 and X2, and (1 + r12) is because r12 is considered to be a positive number greater than 0 from the background of variable selection. The value is larger than R1 (1>R1> 0) on the right side. Therefore, the following methods can be considered to satisfy the equal signs on the left and right sides.
The first method is the current regression analysis method, which can be said to be a method of forcibly aiming for equal signs on both the left and right sides. Multicollinearity "causes problems.
The second method is to remove the duplication when the formula (4) has two or more causal variables and the causal variable values of each term on the left side are expected to be duplicated. Even in this case, the values on both the left and right sides remain, but they are greatly relaxed. The suitability of this operation depends on whether the removal method is theoretically and practically rational. Since the first term of the first row on the left side is the synergy of X1 and the right side is also the correlation of only the objective variables Y and X1, X2 is a variable in the dependent position with respect to this row. Therefore, if we want to remove the duplication between variables, we think that it is appropriate to remove it from the subordinate X2 based on the main Xi.

1 Even if a certain number of the standardized variable Ai is added or multiplied, the original numerical value of Ai does not change. See standardization formula (3) 2 ΣA1A2 / n = r12.
3 When removing duplicates with Ai values as they are, the calculation becomes complicated when there are negative numbers, so make sure that all Ai values are positive in the removal calculation, and that all Ai values are unified. Add a certain number (1.8 in this example, aiming at the maximum negative number of A).
4 Regarding the left side (diagonal element A1, off-diagonal element A2), in order to remove the part synchronized with A1 from A2 as follows, remove from A2 up to the lower of the absolute values of A2 and A1. Find the remainder.
Removal example A1 A2 Removal residual residue
0.3 0.2 0.2 0
0.2 0.5 0.2 0.3
It should be noted that the sum of the numerical values of A1 and A2 in the embodiment becomes equal to 18 by adding one constant of 3 above.
From this, as can be understood in the following simple example, in the normal equation, the numerical values of the off-diagonal elements at the symmetrical positions of the diagonal elements located diagonally in the square matrix are equal.
Let the sum of the numerical values of A1 and A2 be equal to 0.7.
When A1 is a diagonal element (removal standard) and A2 is an off-diagonal element to be removed A1 A2 Removal residual amount 0.3 0.5 0.3 0.2
0.4 0.2 0.2 0
In the opposite case A2 A1 Removal amount Residual 0.5 0.3 0.3 0
0.2 0.4 0.2 0.2
Therefore, the numerical values of the off-diagonal elements at symmetrical positions with respect to the diagonal diagonal elements of the correlation matrix of the normal equation are equal.
5 Since the removal residue is a part of A2, this is multiplied by A1 * A2 (this is the correlation coefficient r12 before removing the duplication. See formula (4)) to obtain r12 again and after removing the duplication. Create a new correlation matrix and find the bi (i = 1, 2) after removing the duplicates.

In order to confirm the validity of the obtained bi, the relative ratio of bi when the original maximum correlation coefficient of the explanatory variable with respect to the objective variable Y is set to 100 is close to the original correlation value, and "multicollinearity". It can be seen that such things have been resolved.

Implementation procedure

実施例２ ２変数、３変数、４変数の計算例
２変数の場合

３変数の場合

４変数の場合

実施例３
Ｙに対する相関係数が０．５レベルの事例ｘｊ、Ｘｋ，Ｘｌの３事例についてのｂｉを求める回帰分析である。

1 The objective variable Y and the explanatory variable Xi of the normal equation of the multiple regression analysis are standardized in order to suppress the fluctuation of the explanatory variable depending on the measurement unit and to create the correlation matrix of the normal equation.
2 Add a constant of standardized variables. Use the characteristic that the numerical value does not change by multiplication.
3 Add a certain number to the reference variable for the numerical value of each line of the normal equation and to prevent the negative number from being generated by the duplication removal.
4 The calculation of [0012] is applied to the deduplication of the dependent variable of the off-diagonal element based on the main variable which is diagonal.
5 Since the obtained residual numerical value is a part of the dependent variable that is the off-diagonal element of the matrix [0012], a new determinant is created again as the correlation coefficient of the off-diagonal element of the normal equation.
6 Find the bi indicating the degree of contribution to the objective variable Y from the new matrix of the normal equations obtained by the above operations.
7 In order to verify whether the calculation method for eliminating "multicollinearity" and the like up to this point is appropriate, the validity of the method when explanatory variables are sequentially added to the same case is verified. Looking at this, the result of the calculation method for eliminating "multicollinearity" and the like applied in the case of four explanatory variables is close to the relative ratio difference of the correlation coefficient of the explanatory variables with respect to the original objective variable and is generally valid.
Example 1
The following objective variables Y and explanatory variables Xh, Xi, Xj, Xk, and Xl are numerical values set appropriately. Find the correlation coefficient between each variable to simplify the calculation.

Example 2 Calculation example of 2 variables, 3 variables, and 4 variables In the case of 2 variables

In the case of 3 variables

In the case of 4 variables

Example 3
This is a regression analysis for obtaining bi for three cases xj, Xk, and Xl in which the correlation coefficient with respect to Y is 0.5 level.

The present invention solves problems such as "multicollinearity" that occur when the relationship between sales, etc., which is an industrial result, and sales expenses, etc., which is the cause, is performed by multiple regression analysis, and sales expenses, etc. that contribute to sales. This is a calculation technique for the purpose of obtaining an appropriate partial regression coefficient bi indicating the contribution of the causative variable of.

When applying multiple regression analysis to elucidate the relationship between results and causes, problems such as "multicollinearity" occur in the explanatory variables, and appropriate management decisions cannot be made. In the following hypothetical analysis example, the distortion of the partial regression coefficient bi due to "multicollinearity" is shown, and the distortion of bi when there is a high correlation between the explanatory variables to some extent.

Multiple regression equation Y = b0 + b1X1 + b2X2 (1)
Multiple regression equation of standardized variables Ay = bA1 + b2A2 (2)
Temporary analysis example Sales, labor costs, and sales floor area, which are the causes, are as follows.
Sales (million yen) Personnel costs (100,000 yen) Sales floor area (10m ² )
Y X1 X2
Store A 46 97 75
Store B 18 61 59
C store 36 83 72
Store D 22 37 51
E store 27 75 48
A multiple regression analysis is applied to this material when there are two explanatory variables.
The result of this calculation is as follows.
Correlation coefficient between sales and labor costs R1: 0.84
Correlation coefficient of sales to sales floor area R2: 0.79
Correlation coefficient between labor cost and sales floor area r12: 0.72
b1 = 0.56 b2 = 0.38
Although there is not much difference in the correlation coefficient between labor cost and sales floor area with respect to sales, there is a large difference in the numerical values of the partial regression coefficients b1 and b2, which indicate the degree of contribution to the result of the cause. This is due to the fact that there is a fairly high correlation between labor costs and sales floor area, and even managers with practical experience have doubts about the validity of this analysis result.

Analysis Example 2 (From Example 3, only the analysis results are shown.)
Case name JKL
For the objective variable
Correlation coefficient: R 0.51 0.62 0.54
Partial regression coefficient: bi 0.10 0.43 0.33
This is an analysis of a properly made case, but the numerical value of the partial regression coefficient with respect to the correlation coefficient of case J is small. This is because the value of the correlation coefficient between the explanatory variables in the examples is a little large, around 0.5, but it cannot be said that the correlation is particularly high if the value is about this level.

Regression analysis is an analysis of how much an explanatory variable of a set (X1, X2, ..., Xp) can explain the objective variable Y, and the numerical value of Xi that causes it is highly independent by physics, chemistry, etc. It is said that the contribution of Xi of each component constituting the set was not calculated. After that, when multiple regression analysis was used to analyze the relationship between corporate sales, which is the result of socio-economic phenomena, and sales expenses, which are the causes, the causes to be adopted have a high causal relationship with sales. Personnel costs, sales costs, sales floor area, etc. have come to be selected, the correlation coefficient between these causes is high through the sales figures, and bi, which indicates the degree of contribution to sales, is an abnormal value contrary to common sense and theory. Came to show. This is the problem of "multicollinearity".

Standardized variable Regarding the above equation (1), if the numerical unit of the explanatory variable Xi is the number of people or the area m ² , the meaning of the required numerical value of bi becomes ambiguous, so it is often used in the subsequent analysis. Since variables need to be standardized to create a correlation matrix, all variables use standardized numbers.
Hereinafter, A1 and A2 are standardized variables of X1 and X2.
Standardization: Ai = ( Xi-Xi ) / σi (3)
Average σi: Standard deviation of Xi By standardizing Y, X1 and X2, the numerical value of Ai becomes a numerical group with an average of 0 and a standard deviation of 1.

Correlation matrix of normal equations In the following analysis, the correlation matrix equation of the normal equations of multiple regression analysis is mainly used, so this is introduced here.

Patent Document 1
Patent No. 5120820 "A device that evaluates an appropriate standard residential land by regression analysis that eliminates" multicollinearity ""
The solution of "multicollinearity" depends on how to reasonably suppress the imbalance of the item values on both the left and right sides in the matrix display of the normal equations. In this patent, when the explanatory variables are two variables, the relatively high correlation between the explanatory variables mediated by the land price, which is the objective variable, is removed as follows, and the off-diagonal element 2 term in the first row of the matrix is removed. It is configured.
r12 = r12 (1-R1R2)
This is a method of collectively removing the numerical values of R1R2 for a series of cases, not for individual cases. This removal can be expected to have the effect of eliminating "multicollinearity" as it is, but it is a rough calculation.
Patent Document 2
Patent No. 5585921 "A device that applies multiple regression analysis to eliminate" multicollinearity "to analyze the relationship between sales, etc., which is an industrial achievement, and sales expenses, etc., which is the cause."
The elimination of "multicollinearity" in this patent is a more polite removal for individual cases than the removal in [Patent Document 1], but it is a removal for the standardized explanatory variable itself, so it is a case. Is classified into positive and negative sign tuning cases of the standardized variable, and is performed for each case. It is inconsistent and the obtained correlation matrix is also an asymmetric matrix.

これは、非対角要素の相関係数ｒ１２の数値を小さくする効果があり、その分だけ“多重共線性”が解消されるが、相関行列の対角要素に加える加算数値の選択に確たる基準がなく、恣意的であまり利用されていないと思う。
Ａｒｔｈｕｒ Ｅ．Ｈｏｅｒ ａｎｄ Ｒｏｂｅｒｔ，Ｗ．Ｋｅｎｎａｒｄ“Ｒｉｄｇｅ ｒｅｇｒｅｓｓｉｏｎ”１９７０年 ２月 The following methods are generally considered to avoid the "multicollinearity" problem.
Deletion of one of the highly correlated explanatory variables Principal component analysis Ridge regression Deletion of explanatory variables is not appropriate to omit the number of employees and sales expenses that are the significantly selected causes for the resulting sales.
Principal component analysis is performed on P variables X1, X2. .. The coefficient bi of the explanatory variable is not obtained by the method of summarizing Xp into the integrated variable j (j = 1 to m) of p> m under a certain condition.
Ridge regression obtains the partial regression coefficient by adding a certain number of k to the diagonal elements of the correlation matrix, and the outline is shown below.

This has the effect of reducing the value of the correlation coefficient r12 of the off-diagonal element, and the "multicollinearity" is eliminated by that amount, but it is a reliable criterion for selecting the additional value to be added to the diagonal element of the correlation matrix. I don't think it's used, it's arbitrary and it's not used much.
Arthur E. Hoer and Robert, W. et al. Kennard "Ridge regression" February 1970

While the "multicollinearity" elimination of ridge regression is an indirect method of reducing the off-diagonal element values for diagonal elements, the present invention uses the off-diagonal elements of the correlation matrix as the direct analysis target. To do.

Looking at the first line of equation (4), the left side is the sum of the correlation coefficients of the causative variables X1 and X2, and (1 + r12) is because r12 is considered to be a positive number greater than 0 from the background of variable selection. The value is larger than R1 (1>R1> 0) on the right side. Therefore, the following methods can be considered to satisfy the equal signs on the left and right sides.
The first method is the current regression analysis method, which can be said to be a method of forcibly aiming for equal signs on both the left and right sides.
Multicollinearity "causes problems.
The second method is to remove the duplication when the formula (4) has two or more causal variables and the causal variable values of each term on the left side are expected to be duplicated. Even in this case, the values on both the left and right sides remain, but they are greatly relaxed. The suitability of this operation depends on whether the removal method is theoretically and practically rational. Since the first term of the first row on the left side is the synergy of X1 and the right side is also the correlation of only the objective variables Y and X1, X2 is a variable in the dependent position with respect to this row. Therefore, if we want to remove the duplication between variables, we think that it is appropriate to remove it from the subordinate X2 based on the main Xi.

1 Even if a certain number of the standardized variable Ai is added or multiplied, the original numerical value of Ai does not change.
See standardization formula (3) 2 ΣA1A2 / n = r12.
3 When removing duplicates with Ai values as they are, the calculation becomes complicated when there are negative numbers, so make sure that all Ai values are positive in the removal calculation, and that all Ai values are unified. Add a certain number (1.8 in this example, aiming at the maximum negative number of A).
4 Regarding the left side (diagonal element A1, off-diagonal element A2), in order to remove the part synchronized with A1 from A2 as follows, remove from A2 up to the lower of the absolute values of A2 and A1. Find the remainder.
Removal example A1 A2 Removal residual residue
0.3 0.2 0.2 0
0.2 0.5 0.2 0.3
It should be noted that the sum of the numerical values of A1 and A2 in the embodiment becomes equal to 18 by adding one constant of 3 above.
From this, as can be understood in the following simple example, in the normal equation, the numerical values of the off-diagonal elements at the symmetrical positions of the diagonal elements located diagonally in the square matrix are equal.
Let the sum of the numerical values of A1 and A2 be equal to 0.7.
When A1 is a diagonal element (removal standard) and A2 is an off-diagonal element to be removed A1 A2 Removal residual amount 0.3 0.5 0.3 0.2
0.4 0.2 0.2 0
In the opposite case A2 A1 Removal amount Residual 0.5 0.3 0.3 0
0.2 0.4 0.2 0.2
Therefore, the numerical values of the off-diagonal elements at symmetrical positions with respect to the diagonal diagonal elements of the correlation matrix of the normal equation are equal.
5 Since the removal residue is a part of A2, this is multiplied by A1 * A2 (this is the correlation coefficient r12 before removing the duplicates. See formula (4)), and r12 is calculated again to obtain r12 after the duplicates are removed. Create a new correlation matrix and find bi (i = 1, 2) after removing the duplicates.

In order to confirm the validity of the obtained bi, the relative ratio of bi when the original maximum correlation coefficient of the explanatory variable with respect to the objective variable Y is set to 100 is close to the original correlation value, and "multicollinearity". It can be seen that such things have been resolved.

Implementation procedure

1 The objective variable Y and the explanatory variable Xi of the normal equation of the multiple regression analysis are standardized in order to suppress the fluctuation of the explanatory variable due to the measurement unit and to create the correlation matrix of the normal equation. I do.
2 Add a constant of standardized variables. Use the characteristic that the numerical value does not change by multiplication.
3 Add a certain number to the reference variable for the numerical value of each line of the normal equation and to prevent the negative number from being generated by the duplication removal.
4 Deduplication of dependent variables of off-diagonal elements based on diagonal main variables is [0012].
Apply the calculation of.
5 Since the obtained residual numerical value is a part of the dependent variable which is an off-diagonal element of the matrix, [0012]
, Create a new determinant that is the correlation coefficient of the off-diagonal elements of the normal equation.
6 Find the bi indicating the degree of contribution to the objective variable Y from the new matrix of the normal equations obtained by the above operations.
7 In order to verify whether the calculation method for eliminating "multicollinearity" etc. up to this point is appropriate, the validity of the method when explanatory variables are sequentially added to the same case is verified. Looking at this, the result of the calculation method for eliminating "multicollinearity" applied in the case of four explanatory variables is close to the relative ratio difference of the correlation coefficient of the explanatory variable with respect to the original objective variable, and is generally valid.

実施例２ ２変数、３変数、４変数の場合

計算結果は採用変数２変数、追加しｉた３変数、４変数の全ての場合で、現行分析方法で求めたｂｉ数値は乱れているが、重複分除去の分析ではいずれの場合もＲ値の相対比に比較的近い結果得ている。
実施例３
Ｙに対する相関係数が０．５レベルの事例ｘｊ，Ｘｋ，Ｘｌの３事例についてのｂｉを求める回帰分析である。

計算結果は現行分析方法のｂ値は乱れているが、重複分除去の計算でほぼＲ値に近い。Example 1
The following objective variables Y and explanatory variables Xh, Xi, Xj, Xk, and Xl are numerical values set appropriately. Find the correlation coefficient between each variable to simplify the calculation.

Example 2 In the case of 2 variables, 3 variables and 4 variables

The calculation results are in all cases of 2 variables adopted, 3 variables added, and 4 variables, and the bi value obtained by the current analysis method is distorted, but in the analysis of duplication removal, the R value is the R value in each case. The results are relatively close to the relative ratio.
Example 3
This is a regression analysis for obtaining bi for three cases xj, Xk, and Xl in which the correlation coefficient with respect to Y is 0.5 level.

Although the b value of the current analysis method is disturbed in the calculation result, it is close to the R value in the calculation of duplication removal.

Claims (1)

A normal equation for multiple regression analysis with sales as the objective variable Y and sales expenses as the explanatory variable Xi, and standardized variables Y and Xi (A = ( Xi-Xi ) / σi, Xi: mean, σi: standard deviation , Y is also the same), and the reference variable of the normal equation is the reference variable, the diagonal element of the matrix in which a certain number is added so as not to generate a negative number by unifying the numerical values and removing the duplication. Partial regression showing the contribution to Y from the new matrix expression with the diagonal element as the dependent variable, removing the overlap with the reference variable from the dependent variable, and again using the remainder as the off-diagonal element of the correlation matrix expression of the normal equation. This is a calculation technique that eliminates "multiple co-linearity" and the like for obtaining the coefficient bi.