Back-transform standard errors

Main Page | Report this Page

Science Forum Index » Statistics - Math Forum » Back-transform standard errors

Page 1 of 1

Author

Message

Henrik

Posted: Fri Mar 21, 2008 2:35 am

Guest

Dear all,

I am performing a one-way ANOVA where the response variable is log-
transformed. The results of the model (output from summary-function in
R), i.e. the parameter estimates for the intercept and the difference
between the intercept and the other groups are given with standard
errors. I want to back-transform the parameter estimates and their
standard errors to their original scale. For the parameter estimates I
just take the antilog, but is it as simple to back-transform the
standard errors?

Thanks in advance!

Henrik Pärn

Paul Rubin

Posted: Fri Mar 21, 2008 10:03 am

Guest

Henrik wrote:

Quote:

I'm not sure you are correct about antilogging the parameter estimates.
I'm used to using regression rather than ANOVA, so I'll frame it in
regression terms. The issue is the distributional assumption about the
disturbances. In ordinary linear regression, the disturbances are
typically assumed to be additive with mean zero. In log-linear
regression, the disturbances are typically assumed to be multiplicative
(so that the logged equation has additive disturbances). If the
universe were just, multiplicative disturbances with mean 1 would have
logs with mean zero, but the universe is known to be unjust. A typical
assumption is that the original disturbances are lognormal with mean 1,
in which case the logged equation has normal disturbances with,
unfortunately, a nonzero mean (it's a function of the variance, but
off-hand I don't recall the precise value). The result is that the
coefficient estimates in the logged equation are unbiased estimators of
the logs of the original coefficients, *except* that the constant term
is biased. I seem to recall the usual trick is to add half the square
of the residual standard error to the constant before antilogging, but I
won't swear my memory is accurate.

On the bright side, in my personal experience the adjustment tends to be
pretty minor, so if you just antilog the parameters I think one of them
will be biased, but quite possibly by an undetectable amount.

/Paul

David Winsemius

Posted: Fri Mar 21, 2008 4:20 pm

Guest

Henrik <henrik.parn@bio.ntnu.no> wrote in news:571f7dbe-f0ab-4c47-8017-
d5cb79798739@s12g2000prg.googlegroups.com:

Quote:

If you were doing Poisson regression (glm w/ link = log), then the 95% CI
around the parameter estimates would be:
( exp( beta - 1.96*se(beta) ), exp( beta + 1.96*se(beta) )
This results in CI's that are not symmetric around the parameters. I
think you would do something similar here. The deviance change is a
better method for doing inference about "significance".

--
David Winsemius

David Winsemius

Posted: Thu Apr 17, 2008 10:08 am

Guest

David Winsemius <doe_snot@comcast.n0T> wrote in
news:Xns9A68B07F04E17dwtttttt@216.196.97.136:

Quote:

Henrik <henrik.parn@bio.ntnu.no> wrote in
news:571f7dbe-f0ab-4c47-8017-
d5cb79798739@s12g2000prg.googlegroups.com:

Dear all,

I am performing a one-way ANOVA where the response variable is log-
transformed. The results of the model (output from summary-function
in R), i.e. the parameter estimates for the intercept and the
difference between the intercept and the other groups are given
with standard errors. I want to back-transform the parameter
estimates and their standard errors to their original scale. For
the parameter estimates I just take the antilog, but is it as
simple to back-transform the standard errors?

If you were doing Poisson regression (glm w/ link = log), then the
95% CI around the parameter estimates would be:
( exp( beta - 1.96*se(beta) ), exp( beta + 1.96*se(beta) )
This results in CI's that are not symmetric around the parameters. I
think you would do something similar here. The deviance change is a
better method for doing inference about "significance".

I have seen what appears to me to be credible opinion that suggests
that simple back transformation is not correct for either the parameter
estimates (as noted above by Paul Rubin) since the mean of a log-normal
distribution is not the same as the log of the mean. If you transformed
Y to log(Y) and make estimates, then exp(Y) is an estimate of the
median. You need to add 0.5*var(log(Y)) before the back-tranformation
and then add or subtract a term proportional to the observed sd(log
(Y)), so your CI estimates per Rubén Roa-Ureta should be:

LB=exp(mean(logY)+0.5*var(logY)+sd(logY)*H_alpha/sqrt(n-1))
UB=exp(mean(logY)+0.5*var(logY)+sd(logY)*H_(1-alpha)/sqrt(n-1))

"... where the quantiles H_alpha and H_(1-alpha) are quantiles of the
distribution of linear combinations of the normal mean and variance
(Land, 1971, Ann. Math. Stat. 42:1187-1205, and Land, 1975, Sel. Tables
Math. Stat. 3:385-419)."

See the discussion yesterday on gmane.comp.lang.r.general with subject
"Confidence intervals of log transformed data" with the substantive
opinions offered by Rubén Roa-Ureta and Tobias Verbeke.

--
David Winsemius

Page 1 of 1 All times are GMT - 5 Hours The time now is Thu Jul 24, 2008 6:16 am Science Forum Index