# Predicting the Aqueductal Cerebrospinal Fluid Pulse: A Statistical Approach
## Appendix A Relative weights method  
When multiple regression is used for explanatory purposes, it is often the case that investigators are interested in the extent to which each predictor variable contributes to prediction of the response variable. In short, investigators often want to quantify the relative importance of the predictor variables as the proportionate contribution that each makes to the overall \(r^{2}\) value of the model.  
In order to draw inference about the relative importance of predictors, researchers often examine the standardized regression coefficients. While this is acceptable when the predictor variables are uncorrelated, this method can be problematic when the predictors are correlated and may lead to erroneous conclusions being drawn. To overcome this problem, [PERSON] proposed a relative weights methodology [31], which involved mean centering and standardizing to unit variance the predictor and response variables, and then performing singular value decomposition (SVD) of the standardized predictor variables [32].  
\[X\ =\ U.S.V^{T} \tag{1}\]  
where \(X\) is a \([n\ xj]\) matrix of the standardized predictor variables; \(U\) is the left singular vector matrix; \(S\) is the matrix of singular values; and \(V\) is the eigenvector matrix of \(X^{\mathrm{T}}X\).  
Using SVD it is possible to obtain a new set of variables (\(Z_{X}\)) that relate to the original set of correlated predictor variables (\(X\)), but which are orthogonal to each another.  
\[Z_{x}\ =\ U.V^{T} \tag{2}\]where \(Z_{X}\) is a best fitting approximation of \(X\), which minimizes the residual sum of squares between original variables and the new orthogonal variables [32].  
Using the Moore-Penrose pseudoinverse algorithm [49], it is possible to regress the standardized response variable (\(Y\)) onto the new set of orthogonal predictor variables (\(Z_{X}\)) to obtain a new set of orthogonal regression coefficients (\(B\)), as follows:  
\[B\ =\ \left(Z_{x}^{T}.Z_{x}\right)^{-1}Z_{x}^{T}.Y \tag{11}\]  
Likewise, the original standardized correlated variables (\(X\)) can be regressed onto their orthogonal counterparts (\(Z_{X}\)), as follows:  
\[Q\ =\ \left(Z_{x}^{T}.Z_{x}\right)^{-1}Z_{x}^{T}.X \tag{12}\]  
where \(Q\) is a \([j\ xj]\) matrix of regression coefficients of \(X\) on \(Z_{X}\).  
In order to estimate the proportionate contribution that any particular variable \(X_{j}\) makes to \(Y\), we simply multiply the squares of \(Q\) and \(B\) together, and divide by the sum of the squares of \(B\), as follows:  
\[W_{abs}\ =\ Q^{2}.B^{2} \tag{13}\]  
This produces a vector of absolute weights for the original variables (\(W_{abs}\)), which can be expressed as relative weights (\(W_{rd}\)) (i.e., as a percentage contribution to the overall \(r^{2}\) value of the model), as follows:  
\[W_{rel}\ =\ \left(\frac{W_{abs}}{\sum B^{2}}\right)\times\left(\frac{100}{ \sum W_{abs}}\right) \tag{14}\]  
In summary, the relative weight approach enables the proportionate contribution that each predictor variable \(X_{j}\) makes to the prediction of Y to be determined, by multiplying the proportion of variance in each orthogonal \(Z_{xj}\) accounted for by \(X_{j}\) by the proportion of variance in Y accounted for by \(Z_{xj}\) and then summing these products [31; 32].