# Is there a concept called “contribution” in statistics?

I am reading a research report. In that report, they used some elementary statistical methods to analyze their data. There is one quantity they computed called the contribution (of a factor), but they have not provided the definition for that. I am confused. I never heard of the contribution of a factor. It seems that they use these factors to build a linear model, e.g. $y=ax+b$. Does anyone know the meaning in this context?

• as in... x contributed to the model? – russellpierce Jan 22 '13 at 6:38
• @drknexus, kind of, they simply say it is the contributions of some xi to y (the outcome they tried to model). – user55647 Jan 22 '13 at 6:47
• Can you link the article or provide a citation or a snippet of the text where they use this term? – AdamO Jan 22 '13 at 6:54
• Well, it is an internal tech report, the so-called contrubtion are listed in one of the tables there, one "contribution" for each xi that used to develop the linear model (y=Ax+b), and the numeric value of these "contributions" varies from 0.0% to some <20%, almost look like some correlation factors or the co-efficient of the linear model, but get a percentage-representation. – user55647 Jan 22 '13 at 7:02
• It might be the contribution to sum of squares (the term is used there, but not defined). There are other possible uses, but given the extra details (which should be added to the question), that's my guess. – Glen_b -Reinstate Monica Jan 22 '13 at 10:34

It has no rigorous meaning. If I had to guess the context in which it was used, it may refer to the incremental reduction in sum of squared error attributed to adjusting for that factor, relative to a nested model which does not adjust for that factor. If that's the case, it may be demonstrating some seriously shortcomings in the understanding of statistics, since the goal (neither in prediction or inference) is not to make the sum of squared error vanish to zero (so "contribution" in this context means very little).

• Well, thats maybe the answer, actually it can be seen from the report that the authors's knowledge in statistcs are not very deep, anyway if there is not some technique term that i happen to be unwared of, then thats OK. – user55647 Jan 22 '13 at 7:04

I'm not aware of a formal definition, but I've been using the term "contribution" with a different meaning from AdamO's in this paper:

This is what I use it for and why:

Problem: I'm looking at spectroscopic data, i.e some kind of intensity at different wavelengths: each variate covers a different wavelength. As a spectroscopist, I can interprete the physical/chemical/biological meaning of the different wavelengths.
For such data, each variate will span a different range, and it usually doesn't make sense to scale variates to common range or variance (you'd blow up the noise in variates with low signal, and I also need these differences for my interpretation).
When interpreting the meaning of linear models, I can look at the coefficients. However, coefficients alone are not the whole truth, I need to take into account the range of intensities for the respective variate.

If I say the linear model produces some kind of scores $\mathbf s$ by matrix-multiplying $n$ spectra of $p$ wavelengths $\mathbf X$ with coefficients $\mathbf b$:

$\mathbf s^{(n \times 1)} = \mathbf X^{(n \times p)} \mathbf B^{(p \times 1)}$
with
$s_{i} = \sum_{j = 1}^p x_{i, j} \cdot b_{j}$

Now, I can put the the $x_{i, j} \cdot b_{j}$ into a matrix of the same size as $\mathbf X$. The columns correspond to the wavelengths. And each entry tells me how much of the final score of that spectrum is picked up at which wavelength, or which spectral band contributes how much and in which direction to the final score.
This allows me to do a spectroscopic interpretation of the model. These contributions are not real spectra, but they behave mostly like difference spectra with some additional possibilities. But I can relate them e.g. to known differences in the composition of my specimen. For details, please look into the paper.

Edit: If this already has a name, or you could suggest a better one, I'd be happy to hear it.