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I'm not too good at math but I'm going to give this ago anyway and try and explain.

Say I have a dataset, where each object in the set (not too sure about my correct use of terminology!) has $4$ different characteristics, and an overall score. If I had $5$ different objects, with all the data for each one, is there a way to try and find a formula which when applied to the characteristics for each object will give the overall score?

Ivo Terek
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user166754
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  • Is there any link between characteristic $A$ and characteristic $B$ (or $B,C$, etc.), or are they guaranteed to be independent? – abiessu Jul 28 '14 at 21:04
  • @abiessu They are all independent from each other. They are all however between 0 and 10, if that makes any difference. – user166754 Jul 28 '14 at 21:05
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    Do you know if the overall score depends linearly on the characteristics? If the answer is yes, this may be of use: http://en.wikipedia.org/wiki/General_linear_model. – Andrew D Jul 28 '14 at 21:07
  • @AndrewD I'm not sure if the overall score does depend linearly, although I think it's likely. Is there any easy(ish) resource which helps explain General Linear Models? – user166754 Jul 28 '14 at 21:14
  • @user166754 Define easy(ish) - are you after something theoretical, or something which just describes how to use the model? To what extent have you studied mathematics? – Andrew D Jul 28 '14 at 21:30
  • @AndrewD AP Calc in HS. I just really need something to describe how to use the model, if it's too advanced then don't worry. – user166754 Jul 28 '14 at 21:33
  • @user166754 Eh, that should be enough - I'll write an answer out as I can't think of a good resource off the top of my head. – Andrew D Jul 28 '14 at 21:38
  • @AndrewD Thanks very much. – user166754 Jul 28 '14 at 21:39

1 Answers1

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One way of modelling what is going on here is via the General Linear Model. In your case, if we label the objects $1$ through to $5$, then supposing the $i$-th object $(i = 1, \ldots, 5)$

  • is given a score $a_i$, $b_i$, $c_i$ and $d_i$ in each of the four characteristics you have, and
  • the total/overall score of the object is $Y_i$,

we model the relationship between $Y_i$ (here the dependent variable) and the $a_i, \ldots, d_i$ (the independent variables) as $$ Y_i = \alpha a_i + \beta b_i + \gamma c_i + \delta d_i + \epsilon_i $$ where $\alpha, \beta, \gamma, \delta$ are fixed constants, and the $\epsilon_i$ are identically and independently distributed Normal random variables with mean $0$ and some unknown variance $\sigma^2$ - if you don't know what this means, then just think of this term accounting for any random deviation from the model. Note here that here I make two assumptions:

  • that if an object has score $0$ for every characteristic, it's overall score is $0$ also; and
  • the random deviation in each overall score is the same every time (this is known as the "homoscedastic" case, if you are curious).

Now, in this case, we have $4$ unknown parameters, and more than $4$ (specifically $5$) sets of data, so more than likely we should be able to proceed with no bother. (If you know any linear algebra, then I can expand on why it's "more than likely" rather than "certainly" - I'm not too aware of how the US system works, I'm afraid).

Assuming everything is well behaved, then we can find what are known as the "least squares estimators" for the values of $\alpha, \beta, \gamma$ and $\delta$ - I'll save the explanation as to how to arrive at these estimators. The "least squares estimators" are then given by (note the hat to denote that they are estimators)

$$ \begin{pmatrix} \hat\alpha \\ \hat\beta \\ \hat\gamma \\ \hat\delta \\ \end{pmatrix} = \left[ \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_5 \\ b_1 & b_2 & b_3 & b_4 & b_5 \\ c_1 & c_2 & c_3 & c_4 & c_5 \\ d_1 & d_2 & d_3 & d_4 & d_5 \\ \end{pmatrix} \begin{pmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ a_4 & b_4 & c_4 & d_4 \\ a_5 & b_5 & c_5 & d_5 \\ \end{pmatrix} \right]^{-1} \begin{pmatrix} \Sigma_{i=1}^5 a_iY_i \\ \Sigma_{i=1}^5 b_iY_i \\ \Sigma_{i=1}^5 c_iY_i \\ \Sigma_{i=1}^5 d_iY_i \\ \end{pmatrix}$$

where I've had to use matrices as otherwise writing out the answer would be a complete mess, and the notation that $\Sigma_{i=1}^5 a_iY_i = a_1Y_1 + a_2Y_2 + \cdots + a_5Y_5$ (and in a similar manner for the other uses of $\Sigma$). You can then get a computer to do these calculations, or some form of stats package (I'd recommend R as it's free, but I don't know how to use it - asking on the Statistics StackExchange may be a good idea, but I'd read their house rules first).

Andrew D
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