Standardized Regression Coefficient

A sensitivity measure of a model can be obtained using a multiple regression to fit the input data to a theoretical equation that could produce the output data with as small error as possible. The most common technique of regression in sensitivity analysis is that of least squares linear regression. Thus the objective is to fit the input data to a linear equation ( Yhat = aX + b) approximating the output Y , with the criterion that the sum of the squared difference between the line and the data points in Y is minimized. A linear regression model of the N × k input sample X to the output Y takes the form:

Y_{i} = β_{0}+∑β_{j}X_{ij}+ε_{i}, j=1:k

where β_{j} are regression coefficients to be determined and ε_{i} is the error due to the approximation, i.e. ε_{i} = Y_{i}-Yhat_{i}.

The regression coefficients β_{j}, j=1,…, k, measures the linear relationship between the input factors and the output. Their sign indicates whether the output increases (positive coefficient) or decreases (negative coefficient) as the corresponding input factor increases. Since the coefficients are dependent on the units in which X and Y are expressed, the normalized form of the regression model is used in sensitivity analysis:

(Yhat_{i}-Ybar_{i})/shat = ∑((β_{j}*shat _{j})/shat* (X

where

shat = (∑((Y_{i}-Yhat)^{2})/(N-1))^{1/2},

shat_{j} = (∑((X_{ij}-Xbar_{j})^{2})/(N-1))^{1/2}

In sensitivity analysis, the standardized coefficients **β _{j}shat_{j}/shat** called standardized regression coefficients (SRCs), are used as a sensitivity measure.

- Francesca Campolongo, Andrea Saltelli, Tine Sørensen, and Stefano Tarantola. Hitchhiker’s guide to sensitivity analysis. In
*Sensitivity analysis*, Wiley Ser. Probab. Stat., pages 15–47. Wiley, Chichester, 2000.