ESTABLISHING THE RELATIVE CONTRIBUTIONS OF VARIABLES

Defining the Model

Ŷ = a + b₁X₁ + b₂X₂ + … + b_pX_p for a total of ‘p’ terms

• only the contributions of individual variables are of interest; therefore there can be only one variable per term and one term per variable

• variables can be transformed (i.e., ‘linearized’) if they have a nonlinear relationship with Y

• there can be no ‘redundancy’ among the predictor variables in the model; i.e., none can have a ‘high’ intercorrelation, otherwise ‘multi-collinearity’ is said to exist

Testing for Multi-Collinearity

1. Bivarite Correlation Matrix (the quick-and-dirty approach)

Step 1: create a matrix of bivariate correlations, where every variable (Y, X₁,…X_p) is paired with every other variable

Note: ‘total’ linear correlations are standard but it would be better to use either ‘total’ quadratic correlations or ‘partial’ linear correlations

Step 2: predictor variables with correlations > 0.80 are highly intercorrelated; select for the model the one predictor variable with the highest correlation with Y

2. Multiple Linear Regressions (the more statistically sound approach)

Step 1: regress each predictor variable on the linear combination of all other predictor variables

Given Ŷ = f(X₁, X₂, X₃, X₄)

Find X₁ = f(X₂, X₃, X₄) and R₁

X₂ = f(X₁, X₃, X₄) and R₂

X₃ = f(X₁, X₂, X₄) and R₃

X₄ = f(X₁, X₂, X₃) and R₄

Step 2: if any R > 0.8 then remove the one variable corresponding to the highest R (i.e., if R_k is highest, remove X_k)

Step 3: repeat step 2 recursively until no R > 0.8

Determining Variable Contributions

Step 1: perform multivariate regression analysis using the reduced set of predictor variables (i.e., those for which no multi-collinearity exists)

Step 2: calculate the ‘Standardized Partial Regression Coefficients’ (a.k.a. ‘Beta Weights, B_k)

for b_k•X_k, B_k = │b_k│• (S_Xk / S_Y)

the B_k is proportional to the contribution of predictor variable X_k to the regression of the response variable

Note 1: if the predictor variables were standardized prior to Step 1 (i.e., X_k is replaced by X_k’ where X_k’ = (X_k – X_mean) / S_X) then

│b_k│ = B_k

Note 2: if all intercorrelations among predictor variables equal zero then

B_k = r_YXk