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Related to my earlier question, I need to perform regression on a skewed dependent variable (n = 500). Since the residuals weren't normally distributed, I was able to transform the DV non-linearly in a way that it now approaches normality. Residuals are normal when using this transformed variable as a dependent variable.

For the two models, The p-values for the various predictors are very much alike, and the relative sizes of the coefficients are very similar as well.

  • To what extent are those two facts are indicators (or not) of the reliability of the coefficients and p-values obtained in the first model (using raw data)?
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    $\begingroup$ By to produce $beta$'s you mean to move to standardized regression? Probably not, since your covariates are all dichotomous. And I just curious what transformation you actually used, since I posted my tips before this question. $\endgroup$ May 26, 2011 at 5:32
  • $\begingroup$ Nicely phrased question. I'd be very interested to see the coefficients and p-values compared across the 2 models. If they really are quite similar, then I suspect you were very exacting when judging the initial residuals to be too far from normality. I don't see how the 2 models could function so differently, judging by residuals, yet so similarly judging by b's and p's. $\endgroup$
    – rolando2
    May 26, 2011 at 21:20
  • $\begingroup$ @Dmitrij I realized I was using the term "Beta" too loosely. I was referring to the unstandardized coefficients (edited). And for the transformation -- it is not the same DV as for the other post -- I used a square root on a linearly transformed (translated+inverted) score. $\endgroup$ May 27, 2011 at 0:22
  • $\begingroup$ @rolando2 I put a table here that contains the parameters estimates for both the variables (standardized in both cases), their SE and p. $\endgroup$ May 27, 2011 at 0:24
  • $\begingroup$ @dominic Thank you. Something is very fishy. These two models have virtually indistinguishable results, as you said. So I'm thinking that the two response variables and the two sets of residuals must be very similar as well. $\endgroup$
    – rolando2
    May 27, 2011 at 17:39

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This looks like an example of the Li-Duan theorem:

Li, Duan. Regression analysis under link violation. Annals of Statistics, 17:1009-1052, 1989

Which basically says that if your predictor variables are well behaved and you use the wrong link function (transformation on the response) then your coefficient estimates will be off by a multiplicative constant.

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