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Given observations of $\{y, x_1, x_2, \cdots, x_n\}$, we can always do a linear regression and get all the coefficients $\{c_i\}$ for the model

$$y = c_0 + c_1 x_1 + \cdots + c_n x_n.$$

However, this may not be the best answer. Let me explain it.

When we are doing a regression, we have estimates $\{d_i\}$ for the standard deviations of the coefficients $\{c_i\}$ and it may turn out, in my particular problem, that most of these coefficients have low $t$-values.

On the other hand, in my problem, I already know the underlying model is more like

$$y = \Sigma_i c_i (x_{m_i} - x_{n_i})$$

such as

$$y = c_1 (x_3 - x_4) + c_2 (x_1 - x_9)$$

and the problem is I don't know $\{m_i\}$ and $\{n_i\}$.

That is, in my strange case, if I already know it is of the form $y = c_1 (x_3 - x_4) + c_2 (x_1 - x_9)$, then when I find $c_1$ and $c_2$, I will find them to have high $t$-values.

Nevertheless I don't know $(x_3 - x_4)$ and $(x_1 - x_9)$ are the "special" combinations. And if I just solve $y = c_1 x_1 + \cdots + c_9 x_9$, I will find all $\{c_i\}$ to have low $t$-values.

(The reason for this strange phenomenon is, my $\{x_i\}$ have significant correlations with each other.)

It seems that I can solve the model

$$y = c_{1,2} (x_1 - x_2) + c_{1,3} (x_1 - x_3) + \cdots + c_{4,6} (x_4 - x_6) + \cdots$$

and find all $\{c_{i,j}\}$ with high $t$-values. But then there will be $36$ coefficients $\{c_{i,j}\}$ instead of $9$. I wonder if there are faster methods?

Thank you.

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  • $\begingroup$ Seems to me there might be some better way of formulating or parameterising the problem, but it's hard to tell without knowing a bit more.. could you give us some idea what the $x_i$s are and how you know the underlying model is of that form? $\endgroup$ – onestop Jan 6 '11 at 12:33
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(This response picks up where @AVB, who has provided useful comments, left off by suggesting we need to figure out which differences $X_i - X_j$ ought to be included among the independent variables.)

The big question here is what is an effective method to identify the model. Later we can worry about faster methods. (But regression is so fast that you could process dozens of variables for millions of records in a matter of seconds.)

To make sure I'm not going astray, and to illustrate the procedure, I simulated a dataset like yours, only a little simpler. It consists of 60 independent draws from a common multivariate normal distribution with five unit-variance variables $Z_1, Z_2, Z_3, Z_4,$ and $Y$. The first two variables are independent of the second two and have correlation coefficient 0.9. The second two variables have correlation coefficient -0.9. The correlations between $Z_i$ and $Y$ are 0.5, 0.5, 0.5, and -0.5. Then--this changes nothing essential but makes the data a little more interesting--I rescaled the variables, thus: $X_1 = Z_1, X_2 = 2 Z_2, X_3 = 3 Z_3, X_4 = 4 Z_4$.

Let's begin by establishing that this simulation emulates the stated problem. Here is a scatterplot matrix.

alt text

The full regression of $Y$ against the $X_i$ is highly significant ($F(4, 55) = 15.28,\ p < 0.0001$) but all four t-values equal 1.24 ($p = 0.222$), which is not significant at all. The estimated coefficients are 0.26, 0.13, 0.088, and -0.066 (rounded to two sig figs).

Here is my proposal: systematically combine variables in pairs (six pairs in this case, 36 pairs for nine variables), one pair at a time. Regress a pair along with all remaining variables, seeking highly significant results for the pairs.

What is a "pair"? It is the linear combination suggested by the estimated coefficients. In this case, they are

$$\eqalign{ X_{12} =& X_1 / 0.26 &+ X_2 / 0.13 \cr X_{13} =& X_1 / 0.26 &+ X_3 / 0.088 \cr X_{14} =& X_1 / 0.26 &- X_4 / 0.066 \cr X_{23} =& X_2 / 0.13 &+ X_3 / 0.088 \cr X_{24} =& X_2 / 0.13 &- X_4 / 0.066 \cr X_{34} =& X_3 / 0.088 &- X_4 / 0.066 \text{.} }$$

In general, with $\hat{\beta}_i$ representing the estimated coefficient of $X_i$ in this full regression, the pairs are defined by

$$X_{ij} = X_i / \hat{\beta}_i + X_j / \hat{\beta}_j\text{.}$$

This is so systematic that it's straightforward to script.

The "identification regressions" are the model

$$Y \sim X_{12} + X_3 + X_4$$

along with the five additional permutations thereof, one for each pair.

You are looking for results where $X_{ij}$ becomes significant: ignore the significance of the remaining $X_k$. To see what's going on, I will list the results of all six identification regressions for the simulation. As a shorthand, I list the variables followed by a vector of their t-values only:

$$\eqalign{ X_{12}, X_3, X_4:&\ (5.50, 1.24, -1.24) \cr X_{13}, X_2, X_4:&\ (1.36, 4.94, -1.13) \cr X_{14}, X_2, X_3:&\ (1.31, 5.16, 1.17) \cr X_{23}, X_1, X_4:&\ (1.64, 3.10, -1.09) \cr X_{24}, X_1, X_3:&\ (1.50, 4.15, 1.07) \cr X_{34}, X_1, X_2:&\ (5.56, 1.25, 1.25) } $$

As you can see from the first component of each vector (the t-value for the pair), precisely two disjoint pairs exhibit significant t-statistics: $X_{12}$, with $t = 5.50\ (p \lt 0.001)$, and $X_{34}$, with $t = 5.56\ (p \lt 0.001)$. The model thus identified is

$$Y \sim X_{12} + X_{34}\text{.}$$

(In general, we would also include--provisionally--any remaining $X_i$ not participating in any of the pairs. There aren't any in this case.)

The regression results are

$$\eqalign{ \hat{\beta_{12}} &= 0.027\ (t = 5.54,\ p \lt 0.001) \cr \hat{\beta_{34}} &= 0.0055\ (t = 5.58,\ p \lt 0.001), \cr F(2, 57) &= 30.92\ (p \lt 0.0001). }$$

Translating back to the original $X_i$, the model is

$$\eqalign{ Y &= 0.027(X_1 / 0.26 + X_2 / 0.13) + 0.0055(X_3 / 0.088 - X_4 / 0.066) \cr &= 0.103 X_1 + 0.206 X_2 + 0.0629 X_3 - 0.0839 X_4 \cr &= 0.103 (Z_1 + Z_2) + 0.021 (Z_3 - Z_4) \text{.} }$$

(The last line shows how this all relates to form of the original question.) That's exactly the form used in the simulation: $Z_1$ and $Z_2$ enter with the same coefficient and $Z_3$ and $Z_4$ enter with opposite coefficients. This method got the right answer.

I want to share a cool observation in this regard. First, here's the scatterplot matrix for the model.

alt text

Notice how $X_{12}$ and $X_{34}$ look uncorrelated. Furthermore, $Y$ is only weakly correlated with these variables. Doesn't look like much of a relationship, does it? Now consider an alternative set of pairs, $X_{13}$ and $X_{24}$. The regression of $Y$ on these is still highly significant ($F(2, 57) = 16.61\ (p \lt 0.0001).$ Moreover, the coefficient of $X_{24}$ is significant ($t = 2.39,\ p = 0.020$) even though that of $X_{13}$ is not ($t = 0.24,\ p = 0.812$). But look at the scatterplot matrix!

alt text

Clearly $X_{13}$ and $X_{24}$ are strongly correlated. But, even though this is the wrong model, $Y$ is also visibly correlated with these two variables, much more so than in the preceding scatterplot matrix!

The lesson here is that mere bivariate plots can be deceiving in a multiple regression setting: to analyze the relationship between any candidate independent variable (such as $X_{12}$) and the independent variable ($Y$), we must make sure to "factor out" all other independent variables. (This is done by regressing $Y$ on all other independent variables and, separately, regressing $X_{12}$ on all the others. Then one looks at a scatterplot of the residuals of the first regression against the residuals of the second regression. It's a theorem that the slope in this bivariate regression equals the coefficient of $X_{12}$ in the full multivariate regression of $Y$ against all the variables.)

This insight shows why we might want to systematically perform the "identification regressions" I have proposed, rather than using graphical methods or attempting to combine many of the pairs in one model. Each identification regression assesses the strength of the contribution of a proposed linear combination of variables (a "pair") in the context of all the remaining independent variables.

Note that although correlated variables were involved, correlation is not an essential feature of the problem or of the solution. Even where you don't expect the original variables $X_i$ to be strongly correlated, you could expect a model to have (unknown) linear constraints among the variables. That is the important issue to cope with. The presence of correlation only means that it can be problematic to identify such pairs solely by inspecting the original regression results.

Following the procedure I have proposed does not guarantee you will find a unique solution. It's conceivable, for instance, that you will find so many highly significant pairs that they are linearly dependent, forcing you to select among them by some other criterion. Nevertheless, the results you get ought to limit the sets of pairs you need to examine; they can be obtained with a straightforward procedure without intervention; and--if this simulation is any guide--they have a good chance of producing effective results.

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You can not use the 36-coefficient model, and not because it's going to be slow. Speed is the least of your worries here.

The real trouble is that you've taken an already under-determined problem (because of the correlations), and converted it into a problem which is severely under-determined for any data, because of linear dependencies. Simply put, $x_1-x_2=(x_1-x_3)-(x_2-x_3)$, so you can only determine 2 out of the 3 coefficients for these terms in the best case. The only way to fix this will be to prescribe some artificial regularization condition, like having minimal $\sum c_i^2$, or whatever may be right in your case.

It seems to me that you might be better off if you start by analyzing the correlation matrix and first figuring out which terms of the form $c_i-c_j$ should really appear in your problem.

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