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I am interested in understanding the different options for gauging relative importance of variables in the results of a linear model. One way I've done this is multiplying the raw variable coefficients by the standard deviation of the variable. I realize that there is no reason why a 1 SE change in one variable should be comparable to a 1 SE change in another... however, for similar variables, this is a decent approximation.

My question is, do any manipulations with the standard error (as opposed to the standard deviation) ever figure into determining the relative importance of variables? I think my question reveals a misunderstanding of the true meaning / importance of standard errors.

Thanks.

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  • $\begingroup$ I don't see how the question or an answer reveals anything about the importance of the standard error. $\endgroup$ – Michael R. Chernick Apr 4 '17 at 15:11
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    $\begingroup$ What do you mean by "relative importance"? Your characterization of this (vague) term will indicate how one might go about answering this question. $\endgroup$ – whuber Apr 4 '17 at 15:41
  • $\begingroup$ Relative importance - if we put all the variables on the same scale (whichever way), we want to understand which variable has a bigger effect on the response variable. If we could only select 5 variables out of 40, what would they be? What about 10 variables? $\endgroup$ – matsuo_basho Apr 4 '17 at 16:11
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Multiplying the coefficient by the std error doesn't make any sense to me but dividing the coefficient by the standard error does. This produces a t-value or a chi-square depending on the model. The absolute value of the resulting test statistic is a quite reasonable, quick and dirty proxy or heuristic for relative variable importance.

For a fairly comprehensive review of the many metrics (both good and bad) used for relative importance, read Ulrike Groemping's papers. She recommends a computationally intensive metric she calls RELAIMPO and has an R module for its implementation.

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  • $\begingroup$ DJohnson, what I want to understand is: how does the t-value heuristic compare with the heuristic that I outline. Namely, multiplying the coefficient by the standard deviation of the variable. $\endgroup$ – matsuo_basho Apr 4 '17 at 16:12
  • $\begingroup$ As already stated, your heuristic doesn't make any sense to me. Therefore, any comparative evaluation by me is moot. Read Groemping's reviews...if you find your approach discussed in her comprehensive listing of relative importance metrics, then she would be also be likely to provide a comparison of its merits. If it's not there, then consider it a closed case. $\endgroup$ – Mike Hunter Apr 4 '17 at 16:19
  • $\begingroup$ DJohnson, you wrote "multiplying the coefficient by the std error doesn't make any sense to me". This is different than what I wrote. I am multiplying the coefficient by the standard deviation of the actual variable. That way, if the coefficient is 10 but the standard deviation is 0.1, the effect of a st. dev. increase on the response is +1. This allows us to compare the effects of standard deviation increases across the various variables. $\endgroup$ – matsuo_basho Apr 4 '17 at 17:09
  • $\begingroup$ Your heuristic still doesn't make sense to me as you are combining apples and oranges by adjusting a conditional metric (the coefficient) with an unconditional one (variable s.d.). The std error associated with each coefficient is a more appropriate standardizing metric...hence, the recommendation to use the associated t- or chi-square value as a proxy for relative importance. Again, read Groemping. Her review of the many metrics for relative importance is comprehensive. If you find yours among them, great! I'll defer to her wrt any further questions. $\endgroup$ – Mike Hunter Apr 4 '17 at 17:19
  • $\begingroup$ Let's say the coefficients for variables x1 and x2 are 10 and 50, respectively. We are interested in interpreting the relative effect of the change in these variables on the response variable. We can compare the effect from a unit change in x1 and x2, but it turns out that x1 and x2 never change by a unit. X1 usually changes by 0.1 and x2 usually changes by 0.3. So actually, from this, we know that a usual change in x1 and x2 is associated with changes of 1 and 15, respectively in the response. Please explain in simple terms why such an approach is incorrect. $\endgroup$ – matsuo_basho Apr 5 '17 at 19:33

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