Unanticipated consequences of logarithmic transformation in bivariate allometry

Abstract

Parameters in the two-parameter allometric equation are commonly estimated by fitting a straight line to logarithmic transformations of the original data and by back-transforming the resulting model to the arithmetic scale. However, log transformation distorts the relationship between the predictor and response variables, and this distortion may be sufficient to lead unsuspecting investigators to analyze data that actually are unsuited for allometric research. Two data sets from the current literature are re-examined here to illustrate instances in which log transformation caused ugly data to look deceptively good. One of the investigations focused on the scaling of metabolism to body mass in evolutionary transitions from prokaryotic to protistan to metazoan levels of organization whereas the other addressed the scaling of intestines to body size in rodents. In both instances investigators were led to conclusions that are not supported by the original data. Problems of the sort described here can readily be avoided simply by performing preliminary graphical analysis of observations expressed in the original units and by validating the final model in the arithmetic domain.

Appendix

See Figs. 3, 4 and 5.

Fig. 3

Values for metabolism and mass of inactive (a) and active (b) protistans were taken from Table S1 in DeLong et al. (2010) and transformed to common logarithms. The linear model fitted by ordinary least squares (OLS) to 52 observations for inactive species is $\widehat{{{ \log }\,Y }} = - 3. 1 2 2 + 0. 9 30\,{ \log }\,X$ whereas that fitted to 51 observations for active species is $\widehat{{{ \log }\,Y }} = - 2. 5 3 3 + 0. 90 3 {\text{ log}}\,X$. Equations fitted to the same observations by reduced major axis regression (RMA) are $\widehat{{{ \log }\,Y }} = - 2. 8 3 4 + 0. 9 6 6 {\text{ log}}\,X$ and $\widehat{{{ \log }\,Y }} = - 1. 30 1 + 1.0 8 3 {\text{ log}}\,X$ for inactive and active protistans, respectively. Untransformed values for metabolism and mass of inactive (c) and active (d) protistans are displayed in bivariate plots with linear scales. The lines are based on equations obtained by back-transforming models fitted to logarithms. The OLS line for inactive protistans is an acceptable fit, but that for active organisms is not. Neither of the RMA regressions is a good fit. Observations for metabolism of inactive (e) and active (f) protistans are shown with corresponding masses expressed on logarithmic axes to provide better visual separation of data for small species. Although the models estimated by back-transformation are good fits to observations for the smallest species, all these observations effectively comprise single points when displayed on graphs with linear scales (c, d)

Fig. 4

Values for metabolism and mass of inactive (a) and active (b) metazoans were taken from Table S1 in DeLong et al. (2010) and transformed to common logarithms. The linear model fitted by ordinary least squares (OLS) to 15 observations for inactive species is $\widehat{{{ \log }Y }} = - 3. 4 5 2 + 0. 7 3 7\, {\text{ log}}X$ whereas that fitted to 71 observations for active species is $\widehat{{{ \log }Y }} = - 3.0 7 9 + 0. 7 1 4 \,{\text{ log}}X$. Equations fitted to the same observations by reduced major axis regression (RMA) are $\widehat{{{ \log }Y }} = - 3. 4 4 8 + 0. 7 6 2\, {\text{ log}}X$ and $\widehat{{{ \log }Y }} = - 2. 9 2 2 + 0. 7 8 8\, {\text{ log}}X$ for inactive and active metazoans, respectively. Untransformed values for metabolism and mass of inactive (c) and active (d) metazoans are displayed in bivariate plots with linear scales. The lines are based on equations obtained by back-transforming models fitted to logarithms. The data set for inactive organisms was dominated by a single outlier. The OLS line arguably is a reasonable fit to data for active animals but the RMA line is not. Observations for metabolism of inactive (e) and active (f) metazoans are shown with corresponding masses expressed on logarithmic axes to provide better visual separation of data for small species. The plot for active animals raises the possibility that the three largest species are outliers

Fig. 5

a Common logarithms for body mass and cecum length for 18 species of herbivorous rodent were taken from Table 6 in Lovegrove (2010). A straight line fitted to the transformations by ordinary least squares confirmed the equation reported by the author, namely, $\widehat{{{ \log }Y }} = 1. 7 6 1 + 0. 1 5 6 \,{\text{ log}}X$. A line fitted by generalized linear modeling has the equation $\widehat{{{ \log }Y }} = 1. 80 7 + 0. 1 4 5\, {\text{ log}}X$. Note how slopes for the fitted lines were influenced by observations for the mole rat Bathyergus and the marmot Marmota. b Equations estimated by back-transformation are displayed against the back-drop of observations expressed in the original units. Neither of the tracings is a good fit to observations spanning the full range in body size