Independence Weight

Independence Weight Calculation is a method used in multi-criteria decision analysis, primarily designed to compute weights by considering the strength of collinearity among indicators. It is located in SPSSAU -> Comprehensive Evaluation -> Independence Weight

SPSSAU Operations

Simply drag the analysis items to the right panel and click 'Start'. In SPSSAU, a key parameter is the composite score:

Composite Score: When this parameter is selected, SPSSAU saves the composite score as a new column with a title like "CompScore_****."

SPSSAU Data Format

The independence weight calculation is used to compute indicator weights, with each indicator occupying a single column of data. The sample numbers in the figure serve only as identifiers and have no actual significance in the analysis.

Algorithm

1.Data Preparation

Gather the indicators to be analyzed and construct a data matrix X where rows represent decision objects, and columns represent various indicators.

2.Standardize the Data Matrix

Perform standardization to eliminate the influence of different units. A common standardization formula is:

$$x_{ij}^{\prime }=\frac{x_{ij}-min(x_j)}{max(x_j)-min(x_j)}$$

x_ij represents the original data, and x_ij' represents the standardized data.

3.Calculate Multiple Correlation Coefficients for Each Indicator

Treat X₁ as Y and the remaining X variables as independent variables, then perform linear regression to calculate the multiple correlation coefficient. A higher coefficient indicates a stronger relationship between X₁ and other variables, implying weaker independence and a lower assigned weight.

Repeat this process for each X variable to obtain multiple correlation coefficient values.

4.Calculate Independence Weights

The formula is:

$$w_j=\frac{1}{R_j}$$

w_j represents the weight of indicator j, and R_i is the multiple correlation coefficient.

5.Normalize Weights

Normalize the computed weights to ensure their sum equals 1:

$${w_j}^{\prime }=\frac{w_j}{\sum _{j=1}^n{w}_j}$$

6.Calculate Composite Score

The formula is:

$$CompScor{e}_i=\sum _{j=1}^n{w}_j\times x_{ij}$$

• CompScore_i represents the composite score of sample i.
• w_j is the weight of indicator j.
• x_ij is the value of item i in indicator j.
• n is the number of evaluation indicators.