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Re: st: Comparing coefficients across sub-samples

From   David Hoaglin <>
Subject   Re: st: Comparing coefficients across sub-samples
Date   Thu, 2 Aug 2012 21:25:27 -0400


The formula that Lisa gave is not Welch's t-test formula.  Welch's
formula produces a t-test on the difference between the two underlying
means, and the expression inside the square-root sign is (s1^2)/n1 +
(s2^2)/n2, where s1^2 and s2^2 are the respective sample variances.
(Those sample variances are also used in calculating the number of
degrees of freedom for the t-test, which will generally not be an

In "Lisa's formula" the denominator is the square root of [se(B1)]^2 +
[se(B2)]^2, which can be written var(b1) + var(b2).  It uses the fact
that, since B1 and B2 come from different groups (and can be assumed
to be independent), var(B1 - B2) = var(B1) + var(B2).  Lisa is
replacing the true variances by their estimates from the two
regressions (the squares of the respective standard errors) and
referring z to the standard normal distribution.  The standard errors
of B1 and B2 already incorporate n1 and n2, along with other features
of the regression data.

David Hoaglin

On Thu, Aug 2, 2012 at 7:42 PM, Fitzgerald, James <> wrote:
> Lisa,
> Do I need to divide the squared standard errors by n of each sample?
> The formula you provided appears to be Welch's t-test formula, but Welch's formula would be:
> z = (B1 - B2) / √(seB1^2/n1 + seB2^2/n2)
> Welch, B. L. (1947). "The generalization of "Student's" problem when several different population variances are involved". Biometrika 34 (1–2): 28–35
> Regards
> James
> ________________________________________
> From: Lisa Marie Yarnell []
> Sent: 01 August 2012 04:29
> To:; Fitzgerald, James
> Subject: Re: st: Comparing coefficients across sub-samples
> Hi James,
> Typically the effect of a predictor in two different groups can be compared with the unstandardized beta. You can do a statistical test of the difference in the betas using the z-score formula below.  I usually just calculate the difference between unstandardized betas from two different models by hand, though Stata might have a command to do this for you.  Is that what you are looking for: the Stata command?
>             (b1 – b2)                       b1 and b2 are the unstandardized regression weights that you want
> z = --------------------                                    to test the difference between
>       √(seb12 + seb22)                   seb1 and seb2are the standard errors of these unstandardized
>       ↑                                                    regression weights, found next to the weights themselves
> This is a square root sign!                      in your SPSS output.  Remember to square them.
> Take the square root of the
> entire value in parentheses.
> In terms of comparing the *magnitude* of the effect in the two different subsamples, it is more correct to do this qualitatively by comparing the *standardized* beta for the variable of interest against effect size rules of thumb for small/medium/large (which sometimes differ by discipline, such as social sciences/education/engineering).  Just report the standardized beta as the effect size in each group; it would be a qualitative statement about the effect in each group.
> Here are rules that I have:
> Standardized regression coefficients:
> * Keith’s (2006) rules for effects on school learning: .05 = too small to be considered meaningful, .above .05 = small but meaningful effect, .10 = moderate effect, .25 = large effect.
> * Cohen’s (1988) rules of thumb: .10 = small, .30 = medium, >  (or equal to) .50 = large
> Lisa

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