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I have an index that is formulated as follows, for party $j$, group $s$, where $w$ indicates weight of party or group as share of population :

$$ \lambda =\sqrt{\sum^J\sum^S w_j w_s(\alpha_j+\beta_{js}-w_j)^2)} $$

There are regressions for each party $j$ predicted by group $s$: $$ \text{party}_j \sim \alpha_j + \beta_{js} \text{group}_s + \epsilon $$

I want to extract the confidence intervals (95%) for this index based on the standard errors for each $\beta$ coefficient. I have seen here that this is the formula to calculated sums of coefficients:

$$ \Delta \text{SE} = \sqrt{\sum{w^2_i \text{SE}^2_i}} $$

My impression is that whichever transformations you apply to the $beta$ coefficient before summing it up, you have to apply to the standard error and then apply this formula. Is this correct?

$$ \text{For} \sum{f(\beta)} \\ \Delta \text{SE} = \sqrt{\sum{w^2_i f(\text{SE})^2_i}} $$

So in the case of my index, the standard errors should be calculated as follows: $$ \text{SE}_\lambda= \sqrt{ \sqrt{ \sum^J{ w_j^2{( \sqrt{ \sum^{S}{ w_s^2(\alpha_j + \text{SE}_{js} - w_j)^2 } } )}^2 } } } $$

Is this th proper way to apply transformations to confidence intervals for the sum of regression coefficients? Thanks.

The Pointer
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Marco Pastor Mayo
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  • You never define or describe the $\beta_{js}:$ did you perhaps omit something in a formula? Perhaps they are the coefficients of "$\text{group}_s$"? Could you explain the point of squaring a square root in your formula and then taking *two* more square roots? I suspect searching our site for [delta method](https://stats.stackexchange.com/search?tab=votes&q=%22delta%20method%22) might clear a few things up for you quickly. In particular, you will want to pay attention to advice about working with correlated random variables (the coefficient estimates often are correlated). – whuber Feb 02 '21 at 15:54
  • @whuber yes, thanks for the heads up. Beta is the coefficient for a social group predicting a party choice. I edited the formula to fix it. – Marco Pastor Mayo Feb 02 '21 at 17:32
  • @whuber On the squring of a square root. Yes, it is redundant becuase they cancel each other out, but I left it so that it’s clear how it follows the method outlined. I presume this is called the delta method, correct? I’ll read more about it. In the meantime, I wanted to know if these assumptions are correct or if there’s anything glaringly wrong. – Marco Pastor Mayo Feb 02 '21 at 17:42
  • I'm afraid this is not a correct application, which is why I referred you to other posts about the method. You can tell it won't work out by applying the units calculus. – whuber Feb 02 '21 at 18:00
  • Ok, are there any specific posts on the Delta Method applied to transformations of regression coefficients? – Marco Pastor Mayo Feb 02 '21 at 19:02
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    Since that requires the covariance matrix of the estimates and those are typically extracted in `R` using the function `vcov`, I [added this term to the search](https://stats.stackexchange.com/search?q=delta+method+vcov) and found at least three examples (some of them buried in code, but they are explicit): https://stats.stackexchange.com/questions/322604, https://stats.stackexchange.com/questions/231074, and https://stats.stackexchange.com/questions/66861. – whuber Feb 02 '21 at 19:17

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