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LETTER TO THE EDITOR |
Department of Zoology
University of Oxford
South Parks Road
Oxford OX1 3PS
United Kingdom
E-mail: fiona.mathews{at}zoology.ox.ac.uk
Center for Veterinary Medicine
Food and Drug Administration
Laurel, MD
Division of Public Health and Primary Health Care
University of Oxford
Institute of Health Sciences
Oxford
United Kingdom
Dear Sir:
Bodnar and Nelson’s assertion that "inappropriate statistical modeling strategy may explain null findings" is incorrect. In fact, fractional polynomial regression and related procedures are valuable modeling tools where it is important to understand the precise shapes of dose-response relations (1). In particular, these tools improve on the practice of splitting continuous variables into categories for the estimation of risk of particular outcomes. It is self-evident that wildly different risk ratios can be generated from the same data set depending on the positioning of the cutoff for the referent category. In the worst-case scenario, data-driven placement of cutoffs can lead to seriously biased estimates of risk. However, such concerns are irrelevant to our article (2). There is also no basis for the assertion that our work suffers from bias. Unlike many intensive studies, our well-characterized sample was selected to be representative of the wider population, and our analyses were appropriate.
We investigated whether there are important relations between maternal nutritional status and birth and placental weights. In common with most epidemiologic research, we had little or no a priori expectations of the shape of such relations. However, we operated under the reasonable assumption that the relations were monotonic (checked by visual inspection of the data) (3) and had no special interest in the shape of the relation per se. As emphasized in many articles about nonlinear regression (eg, 1, 4), the most parsimonious explanations of data are usually linear, and nonlinear terms should be accepted only if there is convincing evidence for them. We systematically examined residuals and conducted visual inspections of the raw data to determine whether important nonlinear relations were likely. We found it strange that Bodnar and Nelson assumed that such procedures had not been conducted, particularly given the specific statement in our article that "the fit of models was ascertained by examination of residuals" (2). These plots, which included standardized residuals against each nutrient and against the fitted values, did not indicate any important nonlinearity.
For the sake of argument, we reanalyzed our birth weight data, fitting first-degree polynomial models (more complex models cannot be fitted and discriminated between without specialist software). The family of 8 power transformations from –2 to 3 were used [1, 3], where –2 is 1/(x2) and 3 is x3. Thus, we ran 136 models (8 x 17) for biomarkers in early pregnancy and 96 models for later pregnancy. Only for one nutrient in late pregnancy (ß-cryptoxanthin) did a nonlinear model offer any improvement over the linear model. In this case, the inverse term was the best predictor of birth weight (P = 0.016 after adjustment formaternal height and smoking), whereas the linear term was not significant. However, the change in the R2 value associated with the inclusion of the factor in the model was tiny (0.008). The 2 regression lines are shown in Figure 1
, and the scatter in the data is obvious. Note that had there been strong nonlinear trends in the data (as opposed to statistically significant relations, which are detectable in a data set such as ours because of the sample size), systematic patterns would have been evident with inspection of residuals from the linear analysis. Bodnar and Nelson particularly raise the case of hemoglobin and birth weight. Our new models confirm that the relation was linear, as reported in our article. This finding agrees with most previous published research, whereas the U-shaped relation was reported only once previously.
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