Looking for name of statistical property (variance explained by regressor in addition to all other regressors)

Question

There is a statistical property - and I only know the name in German and cannot find any proper translation - that is defined as: variance explained of a specific regressor x1 in addition to the variance explained by all other regressors.

That is, the shared variance is not included in the explained variance of this statistical property - it is the minimum estimate of its shared variance with the dependent variable. It's a classical concept in psychology.

In German the term is "Nützlichkeit", which translates into "utility". But I'm afraid in English the name is something so different that my google-fu does not work.

Answer 1

I think you're talking of "usefulness".

In simplest terms it can be explained with the help of a cholesky-decomposition of the correlation/covariance-matrix, where the dependent variable is at the end, and the independent variable, whose "usefulness" is asked for, preceeds it directly. Say, we have $x_1,x_2,x_3$ as independent variables, and $y$ as dependent in a common correlation matrix R, then write its cholesky-factor-loadings matrix L as $$ L = \begin{array} {|r|rrrr|} x_1 & 1 & . & . & . \\ x_2 & a & b & . & . \\ x_3 & c & d & e & . \\ \hline y & f & g & h & i \\ \end{array} $$ where the sums of squares along the rows are the variances of the variables, the entries along the columns are "factor-loadings" or coordinates in an orthogonal euclidean space. Then $f^2 + g^2 + h^2 $ is the explained variance of the dependent variable (the multiple $r^2$) , $i^2$ the unexplained variance in the dependent variable, $ r_{y,x_3 \cdot x_2,x_1} = e \cdot h $ is the partial correlation between $x_3$ and $y$ and the "usefulness" $u^2$ of $x_3$ wrt the composition of $y$ by the independent variables $x_k$ is $u^2 = h^2$ .

Note, that if we want to see the "usefulness" of the variable $x_2$ we simply rearrange the order of the independent variables and get the different choleskly-factor L' as
$$ L' = \begin{array} {|r|rrrr|} x_1 & 1 & . & . & . \\ x_3 & c & d' & . & . \\ x_2 & a & b' & e' & . \\ \hline y & f & g' & h' & i \\ \end{array} $$ where still $f^2+g'^2+h'^2 = f^2+g^2+h^2$ but $u_2^2 = h'^2 \ne h^2 = u_3^2 $ the usefulness of $x_2$ determined by $h'^2$ is different from that of $x_3$ determined by $h^2$.

I assume, you mean this thing?

[update]: see this link ;-) Here the same question was asked (and answered). ...looks like I got a bit old meanwhile ...

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some sources from that linked earlier message:

Bortz, J.: Statistik für Sozialwissenschaftler, Pg 442 "usefulness" 5'th ed (1999); Springer-Verlag

(Following references by Bortz:)

Budescu, D.V.: Dominance Analysis: A new approach to the problem of relative importance of predictors in multi- ple regression. Psych.Bull. 114, 542-551 (1993)

Darlington, R.B.: Multiple Regression in psychological research and practice, Psych.Bull. 69 161-162 (1968)

Answer 2

This sounds like partial correlation, the wikipedia page on partial correlation gives specific detail so you can see if it matches what you are asking about. That page also mentions some other measures that might be what you are looking for if partial correlation is not.

Answer 3

In commercial market research your formula is sometimes the definition of:

1. "Importance". Importance has many other interpretations, so this is not a particularly helpful translation.

2. The "Shapely Value". For example: http://marketing.gfkamerica.com/website/articles/ShapelyValueRegression.pdf

As regards "utility", in English it is generally used only in the context of models of preference, or, when models commonly used to model preference are used for something else (e.g., if applying a multinomial logit model).