# Upon transforming coefficients, how do I transform SE, z value and Pr(>|z|)?

From a Negative Binomial regression, I obtain the following coefficients:

> coef(summary(estimation_negbin$baseline)) Estimate Std. Error z value Pr(>|z|) (Intercept) 1.6249348339 0.0859813660 18.898686 1.169285e-79 num_auth 0.1417207711 0.0265481894 5.338246 9.384996e-08 avg_aff_rank -0.0006997745 0.0001433713 -4.880854 1.056276e-06 num_ack 0.0157015162 0.0032758547 4.793105 1.642194e-06 num_sem 0.0164607240 0.0052609194 3.128868 1.754809e-03 sem_rank -0.0002727935 0.0002860988 -0.953494 3.403398e-01 num_con 0.0177858710 0.0106218552 1.674460 9.404024e-02 num_pages 0.0106210180 0.0021885269 4.853044 1.215805e-06  In order to interpret the marginal effect (at the mean), I apply the following formula to the coefficients: exp(coefficient) -1: > exp(coef(summary(estimation_negbin$baseline)))-1
Estimate   Std. Error       z value     Pr(>|z|)
(Intercept)   4.0780881072 0.0897860210  1.612854e+08 0.000000e+00
num_auth      0.1522548607 0.0269037319  2.071473e+02 9.384997e-08
avg_aff_rank -0.0006995297 0.0001433816 -9.924095e-01 1.056277e-06
num_ack       0.0158254328 0.0032812262  1.196755e+02 1.642196e-06
num_sem       0.0165969481 0.0052747824  2.184811e+01 1.756350e-03
sem_rank     -0.0002727563 0.0002861398 -6.146079e-01 4.054251e-01
num_con       0.0179449816 0.0106784674  4.335913e+00 9.860395e-02
num_pages     0.0106776213 0.0021909234  1.271299e+02 1.215805e-06


The calculation of course changes all other columns, too. But is this correct? Can I apply the transformation I use for the coefficients to these statistics as well? Or rather: Should I apply the same transformation for the standard errors?

• Look closely: all the standard errors are changed, too. Since to first order $\exp(x)-1=x$, values close to $0$ will exhibit little change. In the present case that includes the smaller-sized estimates as well as all the standard errors. Note that applying this transformation to anything but the estimate is statistically meaningless in the first place. You might instead want to ask whether it is possible to estimate standard errors of the transformed estimates and with what accuracy that could be done.
– whuber
Mar 9, 2015 at 22:03
• Thanks for the eye-opener. I'd have been a real puzzle if only the SE had not changed. In fact, I should also have added that I am mainly interested in the SE (because I want to report them). So how do I get SE that belong to the transformed coefficients? Mar 9, 2015 at 22:13
• There is the question you want to ask! Before you edit your post, though, see whether you can find an answer by searching our site for delta method. Adding some terms like "regression" and "error" might help narrow the search.
– whuber
Mar 9, 2015 at 22:16