Quantitative assessment of brand promotion effect of agricultural products based on multiple regression analysis

General form of multiple linear regression model: 6yi=α0+α1x1i+α2x2i+⋯+αkxki+εii=1,2,⋯,n$${y_i} = {\alpha_0} + {\alpha_1}{x_{1i}} + {\alpha_2}{x_{2i}} + \cdots + {\alpha_k}{x_{ki}} + {\varepsilon_i}i = 1,2, \cdots ,n$$

In the model y_i is referred to as the dependent variable while x_1i, x_2i, ⋯, x_ki is referred to as the independent variable. At k = 1, equation (6) is known as a univariate linear regression model and at k > 2, equation (6) is known as a multiple linear regression model [32]. The dependent variable y_i consists of an error term random variable ε_i and a linear function α₀ + α₁x_1i + α₂x_2i + ⋯ + α_kx_ki of the k independent variables.

The parameter estimation of the multiple linear regression model is much more complicated than the univariate linear regression model, and the tool of matrix is introduced to simplify the computation and analysis in order to facilitate the computation and analysis, and to facilitate the generalization of the results from univariate totals to general multivariate totals.

A set of random samples in the overall set of (yi,x1i,x2i,⋯,xki)$$\left( {{y_i},{x_{1i}},{x_{2i}}, \cdots ,{x_{ki}}} \right)$$, i = 1, 2, ⋯, n. In then (6) can be written as the following matrix representation: 7y1=α0+α1x11+α2x21+⋯+αkxk1+ε1 y2=α0+α1x12+α2x22+⋯+αkxk2+ε2 ⋮ yn=α0+α1x1n+α2x2n+⋯+αkxkn+εn$$\begin{array}{*{20}{c}} {{y_1} = {\alpha_0} + {\alpha_1}{x_{11}} + {\alpha_2}{x_{21}} + \cdots + {\alpha_k}{x_{k1}} + {\varepsilon_1}} \\ {{y_2} = {\alpha_0} + {\alpha_1}{x_{12}} + {\alpha_2}{x_{22}} + \cdots + {\alpha_k}{x_{k2}} + {\varepsilon_2}} \\ \vdots \\ {{y_n} = {\alpha_0} + {\alpha_1}{x_{1n}} + {\alpha_2}{x_{2n}} + \cdots + {\alpha_k}{x_{kn}} + {\varepsilon_n}} \end{array}$$

Where k - the number of independent variables.

α₀, α₁, ⋯, α_k - parameters to be determined.

ε_i - random variable.

Expressed using matrix operations as: 8[ y1 y2 ⋮ yn]=[ 1 x11 ⋯ xk1 1 x12 ⋯ xk2 ⋮ ⋮ ⋮ 1 x1n ⋯ xkn][ α0 α1 ⋮ αk]+[ ε1 ε2 ⋮ εn]$$\left[ {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ \vdots \\ {{y_n}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{ {x_{11}}}& \cdots &{ {x_{k1}}} \\ 1&{ {x_{12}}}& \cdots &{ {x_{k2}}} \\ \vdots & \vdots &{ }& \vdots \\ 1&{ {x_{1n}}}& \cdots &{ {x_{kn}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\alpha_0}} \\ {{\alpha_1}} \\ \vdots \\ {{\alpha_k}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{\varepsilon_1}} \\ {{\varepsilon_2}} \\ \vdots \\ {{\varepsilon_n}} \end{array}} \right]$$

Denote by y=[ y1 y2 ⋮ yn]$$y = \left[ {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ \vdots \\ {{y_n}} \end{array}} \right]$$, x=[ 1 x11 ⋯ xk1 1 x12 ⋯ xk2 ⋮ ⋮ ⋮ 1 x1n ⋯ xkn]$$x = \left[ {\begin{array}{*{20}{c}} 1&{ {x_{11}}}& \cdots &{ {x_{k1}}} \\ 1&{ {x_{12}}}& \cdots &{ {x_{k2}}} \\ \vdots & \vdots &{ }& \vdots \\ 1&{ {x_{1n}}}& \cdots &{ {x_{kn}}} \end{array}} \right]$$, α=[ α0 α1 ⋮ αk]$$\alpha = \left[ {\begin{array}{*{20}{c}} {{\alpha_0}} \\ {{\alpha_1}} \\ \vdots \\ {{\alpha_k}} \end{array}} \right]$$, ε=[ ε1 ε2 ⋮ εn]$$\varepsilon = \left[ {\begin{array}{*{20}{c}} {{\varepsilon_1}} \\ {{\varepsilon_2}} \\ \vdots \\ {{\varepsilon_n}} \end{array}} \right]$$. This can be expressed as a matrix: 9y=Xα+ε$$y = X\alpha + \varepsilon$$

Noting α^=[ α^0 α^1 ⋮ α^k]$$\widehat \alpha = \left[ {\begin{array}{*{20}{c}} {{{\hat \alpha }_0}} \\ {{{\hat \alpha }_1}} \\ \vdots \\ {{{\hat \alpha }_k}} \end{array}} \right]$$, e=[ e1 e2 ⋮ en]$$e = \left[ {\begin{array}{*{20}{c}} {{e_1}} \\ {{e_2}} \\ \vdots \\ {{e_n}} \end{array}} \right]$$, the matrix representation of the sample is: 10y=Xα^+e$$y = X\widehat \alpha + e$$

where α^0,α^1,⋯,α^k$${\hat \alpha_0},{\hat \alpha_1}, \cdots ,{\hat \alpha_k}$$ is the fitted value of α₀, α₁, ⋯, α_k respectively, the regression equation is: 11y^i=α^0+α^1x1i+α^2x2i+⋯+α^kxki$${\hat y_i} = {\hat \alpha_0} + {\hat \alpha_1}{x_{1i}} + {\hat \alpha_2}{x_{2i}} + \cdots + {\hat \alpha_k}{x_{ki}}$$

Where α^0$${\hat \alpha_0}$$ - constant.

α^1,⋯,α^k$${\hat \alpha_1}, \cdots ,{\hat \alpha_k}$$ - partial regression coefficient.

The purpose of the underlying assumptions is to facilitate the estimation of the parameters of the model, and the following assumptions are made about the equation:

Assumption 1 The model is assumed to be parametrically linear.

Assumption 2 The independent variables are independent of the random error term, i.e.: 12cov(xji,εi)=0,j=1,2,⋯,k$$\operatorname{cov} \left( {{x_{ji}},{\varepsilon_i}} \right) = 0,j = 1,2, \cdots ,k$$

Assumption 3 Zero mean assumption, i.e.: 13E(εi)=0,i=1,2,⋯,n$$E\left( {{\varepsilon_i}} \right) = 0,i = 1,2, \cdots ,n$$

Assumption 4 Homoskedasticity assumption, i.e.: 14var(εi)=σ2,i=1,2,⋯,n$$\operatorname{var} \left( {{\varepsilon_i}} \right) = {\sigma^2},i = 1,2, \cdots ,n$$

Assumption 5 No autocorrelation assumption: 15cov(εi,εj)=0,i≠j,i=1,2,⋯,n,j=1,2,⋯,n$$\operatorname{cov} \left( {{\varepsilon_i},{\varepsilon_j}} \right) = 0,i \ne j,i = 1,2, \cdots ,n,j = 1,2, \cdots ,n$$

Assumption 6 Normality is assumed, viz: 16εi~N(0,σ2),i=1,2,⋯,n$${\varepsilon_i}\sim N\left( {0,{\sigma^2}} \right),i = 1,2, \cdots ,n$$

After specifying the regression theoretical model, the unknown parameters of the model are estimated. Ordinary least squares is the most commonly used method for estimating the unknown parameters and is also the classical estimation method. When there are regression problems that do not meet the model assumptions, some new methods can be used, which are also based on the ordinary least squares method, such as partial least squares estimation, principal component regression, ridge regression and so on. Due to the fact that the computation is very large when there are many parameter variables, it is necessary to use computer software for the operation.

Based on the principle of least squares, the estimate α^i$${\hat \alpha_i}$$ of α_i is to be satisfied: 17Q=∑i=1n(yi−y^i)2→min$$Q = \sum\limits_{i = 1}^n {{{\left( {{y_i} - {{\hat y}_i}} \right)}^2}} \to \min$$

The derivation of Eq: 18{ nα^0+(∑i=1nx1i)α^1+⋯+(∑i=1nxki)α^k=∑i=1nyi (∑i=1nx1i)α^0+(∑i=1nx1i2)α^1+(∑i=1nx1ix2i)α^2+⋯+(∑i=1nx1kxki)α^k=∑i=1nx1iyi …⋯ (∑i=1nxki)α^0+(∑i=1nx1ixki)α^1+(∑i=1nx2ixki)α^2+⋯+(∑i=1nxki2)α^k=∑i=1nxkiyi$$\left\{ {\begin{array}{*{20}{c}} {n{{\hat \alpha }_0} + \left( {\sum\limits_{i = 1}^n {{x_{1i}}} } \right){{\hat \alpha }_1} + \cdots + \left( {\sum\limits_{i = 1}^n {{x_{ki}}} } \right){{\hat \alpha }_k} = \sum\limits_{i = 1}^n {{y_i}} } \\ {\left( {\sum\limits_{i = 1}^n {{x_{1i}}} } \right){{\hat \alpha }_0} + \left( {\sum\limits_{i = 1}^n {x_{1i}^2} } \right){{\hat \alpha }_1} + \left( {\sum\limits_{i = 1}^n {{x_{1i}}} {x_{2i}}} \right){{\hat \alpha }_2} + \cdots + \left( {\sum\limits_{i = 1}^n {{x_{1k}}} {x_{ki}}} \right){{\hat \alpha }_k} = \sum\limits_{i = 1}^n {{x_{1i}}} {y_i}} \\ { \ldots \cdots } \\ {\left( {\sum\limits_{i = 1}^n {{x_{ki}}} } \right){{\hat \alpha }_0} + \left( {\sum\limits_{i = 1}^n {{x_{1i}}} {x_{ki}}} \right){{\hat \alpha }_1} + \left( {\sum\limits_{i = 1}^n {{x_{2i}}} {x_{ki}}} \right){{\hat \alpha }_2} + \cdots + \left( {\sum\limits_{i = 1}^n {x_{ki}^2} } \right){{\hat \alpha }_k} = \sum\limits_{i = 1}^n {{x_{ki}}} {y_i}} \end{array}} \right.$$

is further written in matrix form: 19Aa=B$$Aa = B$$

Assume that the coefficient matrix A is full rank, where: 20A = XTX,B=XTY,X=[ 1 x11 ⋯ xk1 1 x12 ⋯ xk2 ⋮ ⋮ ⋮ 1 x1n ⋯ xkn] Y = [ y1 y2 ⋮ yn],a=[ α^0 α^1 ⋮ α^k]$$\begin{array}{rcl} A &=& {X^T}X,B = {X^T}Y,X = \left[ {\begin{array}{*{20}{c}} 1&{ {x_{11}}}& \cdots &{ {x_{k1}}} \\ 1&{ {x_{12}}}& \cdots &{ {x_{k2}}} \\ \vdots & \vdots &{ }& \vdots \\ 1&{ {x_{1n}}}& \cdots &{ {x_{kn}}} \end{array}} \right] \\ Y &=& \left[ {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ \vdots \\ {{y_n}} \end{array}} \right],a = \left[ {\begin{array}{*{20}{c}} {{{\hat \alpha }_0}} \\ {{{\hat \alpha }_1}} \\ \vdots \\ {{{\hat \alpha }_k}} \end{array}} \right] \\ \end{array}$$

The measured least squares method is estimated as: 21a=A−1B=(XTX)−1XTY$$a = {A^{ - 1}}B = {\left( {{X^T}X} \right)^{ - 1}}{X^T}Y$$

After the unknown parameters of the model are estimated and the multiple linear regression equation is established, the multiple linear regression equation needs to be tested for significance. 1)

Goodness-of-fit test of regression equation the magnitude of the coefficient of determination R² can indicate how well the estimated equation fits the sample observations, i.e. SST = SSR + SSE.

Where SST=∑i=1n(yi−y¯)2$$SST = \sum\limits_{i = 1}^n {{{\left( {{y_i} - \bar y} \right)}^2}}$$ is the total sum of squared deviations, SSR=∑i=1n(y^i−y¯)2$$SSR = \sum\limits_{i = 1}^n {{{\left( {{{\hat y}_i} - \bar y} \right)}^2}}$$ is the regression sum of squares, which is a parameter that reflects the effect of regression, and SSE=∑i=1n(yi−y^i)2$$SSE = \sum\limits_{i = 1}^n {{{\left( {{y_i} - {{\hat y}_i}} \right)}^2}}$$ is the residual sum of squares. y_i is the regression value on the ith sample point (x1i,x2i,⋯,xki)$$\left( {{x_{1i}},{x_{2i}}, \cdots ,{x_{ki}}} \right)$$ and y¯$$\bar y$$ is the sample mean of y.

The correlation coefficient is: 22r=cov(X,Y)DX⋅DY,r^=∑i=1n(xi−x¯)(yi−y¯)∑i=1n(xi−x^)2⋅∑i=1n(yi−y¯)2$$r = \frac{{\operatorname{cov} (X,Y)}}{{\sqrt {DX \cdot DY} }},\hat r = \frac{{\sum\limits_{i = 1}^n {\left( {{x_i} - \bar x} \right)} \left( {{y_i} - \bar y} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} - \hat x} \right)}^2}} } \cdot \sqrt {\sum\limits_{i = 1}^n {{{\left( {{y_i} - \bar y} \right)}^2}} } }}$$

R² is the square of the correlation coefficient: 23R2=1−SSESST$${R^2} = 1 - \frac{{SSE}}{{SST}}$$

In one-way linear regression, if the value of R² is close to 1, then it indicates that the regression equation is a good fit to the actual observations, and vice versa, the worse the fit.

In practical problems, if R² increases by adding an independent variable, it is assumed that the model can be made to fit better by adding an independent variable [33]. However, the reality is that an increase in R² due to an increase in the number of independent variables does not correlate well with a good fit, and then R² needs to be adjusted. This brings us to the adjusted goodness of fit:

In a real problem, if R² increases by adding an independent variable, this would lead one to believe that the model would be fitted better by adding an independent variable. However, the reality is that an increase in R² due to an increase in the number of independent variables does not correlate well with goodness of fit, and then R² needs to be adjusted. This brings us to the adjusted goodness of fit: 24R12=1−SSEn−k−1/SSTn−1$$R_1^2 = 1 - {{\frac{{SSE}}{{n - k - 1}}} \mathord{\left/ {\vphantom {{\frac{{SSE}}{{n - k - 1}}} {\frac{{SST}}{{n - 1}}}}} \right. } {\frac{{SST}}{{n - 1}}}}$$

Where n − k − 1 - Degrees of freedom (residual sum of squares).

n − 1 - degrees of freedom (overall sum of squares).

In multiple linear regression, the closer R12$$R_1^2$$ is to 1, the better the fit. 2)

The test of significance of a regression equation can be illustrated by the F-test for significance:

F - test of the sum of squares decomposition formula: 25∑i=1n(yi−y¯)2=∑i=1n(yi−y^i)2+∑i=1n(y^i−y¯)2$$\sum\limits_{i = 1}^n {{{\left( {{y_i} - \bar y} \right)}^2}} = \sum\limits_{i = 1}^n {{{\left( {{y_i} - {{\hat y}_i}} \right)}^2}} + \sum\limits_{i = 1}^n {{{\left( {{{\hat y}_i} - \bar y} \right)}^2}}$$

Denoted as L_yy = Q + U. Can be obtained: 26F=UQ/(n−2)~F(1,n−2)$$F = \frac{U}{{Q/(n - 2)}}\sim F(1,n - 2)$$

Rejecting domain χ0={F>F1−α(1,n−2)}$${\chi_0} = \left\{ {F > {F_{1 - \alpha }}(1,n - 2)} \right\}$$ indicates that the regression is good.

China’s agricultural product promotion data mainly come from several major websites and platforms, such as Taobao, Shake, Xiaohongshu, WeChat Video No. These websites and platforms regularly publish relevant transaction data, which are objective, true and fair. The experiments in this section take the promotional sales data of an agricultural product on the Taobao platform on December 1, 2023 as the research object. Descriptive statistics are shown in Table 3.

The study of the process of explaining changes in a dependent variable by multiple influences as independent variables is multiple regression analysis. For multiple linear regression models the estimation of structural parameters and variance is done mainly by least squares. The collected data were substituted into the model and the regression and statistical tests obtained. The regression results are shown in Table 4. According to the relevant theory in statistics, the closer the P-value of the results of the statistical analysis software in the table is to 1, it means that the influence of the factor on the effect of agricultural brand promotion is more insignificant, and the closer the P-value is to 0, it means that the influence of the factor on the effect of agricultural brand promotion is more significant, and the better the results are. As can be seen from the table, the correlation between financing effect, information exhaustiveness and project reputation is relatively high, and its P-value is 0, 0.0761 and 0.0842 respectively.

After parametric regression analysis and interpretation of the economic significance of the model, the assessment model needs to be quality tested to verify the consistency (or accuracy) of its evaluation results. The assessment consistency test is usually conducted through ratio analysis. The calculation results of assessment consistency indicators are summarized in Table 5. As can be seen from the table, the median assessment ratio is 1.00261, the arithmetic mean of the ratios is 1.0352, and the weighted mean of the ratios is 0.9522, which are all within the range of 0.9-1.1 required by IAAO. The coefficient of dispersion is 4.02311, which meets the requirement of ≦15 and can be considered reasonable. The correlation difference of 0.9877 meets the IAAO’s requirement of 0.98-1, so the assessment model meets the test criteria, indicating that the model is usable in batch assessment. This shows that it is feasible to use the multiple regression analysis method in the brand promotion effect of agricultural products.