st: Re: reporting log linked, linear, and fractional polynomial results

[Roger Newson <roger.newson@kcl.ac.uk>]

At 08:36 23/10/03 -0400, Buzz Burhans wrote:
> I would appreciate advice on effective ways of reporting on data for 
> similar outcome types from the same trial  which have been modeled using 
> different link functions. The outcomes (plasma metabolites in animals 
> under two treatment regimes) are repeated measurements made over time 
> proximate to parturition, and have variously different profiles of 
> curvilinear increase or decrease.  Simply fitting linear polynomial models 
> failed to adequately satisfy assumptions for residuals for some (but not 
> all) outcomes, so while some were modeled as normally distributed with the 
> identity link, others were modeled using gllamm with family(gamma) 
> link(log), all using adaptive quadrature. I have used logistic regression 
> previously with categorical outcomes, but am unclear about the log link to 
> continuous variables. My questions are as follows:
>
> 1. How to report the results from disparate model types.  My initial 
> thought is to 1) tabulate fitted values and confidence intervals at a set 
> of representative times, with stars for significance of treatment 
> difference , and accompany such a table with plots of fixed effect values 
> (over the entire experimental period).
>
> Does this make sense? I am not sure how to otherwise tabulate coefficients 
> and se, since some refer to outcomes in the original metric, while the log 
> linked ones refer to logged outcomes .
>
> I also considered exponentiating the coefficients and ci, but confess to 
> being a bit unsure about how to express their exponentiated 
> interpretation, given that they are relative to continuous rather than 
> categorical dependant variables.  Is it appropriate to suggest that the 
> exponentiated coefficient describes the proportionate change in the 
> (backtransformed) outcome?
If you fit a GLM (or extension of a GLM) with a log link, then the 
exponentiated parameters are arithmetic means and their ratios. This is 
true whether the Y-variable is a count variable or a continuous variable. 
The ratios may be simple between-group ratios (in the case of categorical 
X-variables) or ratios associated with a unit increase in an X-variable (in 
the case of continuous X-variables). If the continuous X-variable is itself 
a log to the base 2 of another variable (eg X=log_2(W)), then the 
exponentiated parameter is a per-doubling ratio associated with a doubling 
of W.

An alternative to fitting arithmetic means and their ratios might be to 
transform the outcome data to logs and then use a GLM (or extended GLM) 
with an identity link. The exponentiated parameters are then geometric 
means and their ratios. For instance, the exponentiated intercept is a 
geometric mean Y for zero X, and the exponentiated slopes are amounts by 
which the geometric means are multiplied per unit increase in X. The 
geometric mean is often a better proxy for the median than the arithmetic 
mean if the data are positively skewed, eg viral loads, plasma 
triglycerides and house prices. Another possibility is to power-transform 
the data and use a GLM with the corresponding inverse-power link function 
to estimate algebraic means and their differences, but I don't know many 
people who do this.

Either way, it usually makes sense to centre the X-variates at a sensible 
X-value, so that the exponentiated intercept will represent an arithmetic 
or geometric mean expected at a sensible X-value instead of an arithmetic 
or geometric mean expected at a zero X-value (except if a xzero X-value is 
sensible). And, in general, the parameters presented with confidence limits 
should be explicable in words to non-statisticians.

I hope this helps.

Roger


--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
5th Floor, Capital House
42 Weston Street
London SE1 3QD
United Kingdom

Tel: 020 7848 6648 International +44 20 7848 6648
Fax: 020 7848 6620 International +44 20 7848 6620
  or 020 7848 6605 International +44 20 7848 6605
Email: roger.newson@kcl.ac.uk
Website: http://www.kcl-phs.org.uk/rogernewson

Opinions expressed are those of the author, not the institution.

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