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

From   Roger Newson <[email protected]>
To   [email protected]
Subject   st: Re: reporting log linked, linear, and fractional polynomial results
Date   Mon, 27 Oct 2003 13:28:07 +0000

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 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
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or 020 7848 6605 International +44 20 7848 6605
Email: [email protected]

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

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