Stephenson et al.’s ecological fallacy 
Eastaugh CS*a, Thurnher Cb, Hasenauer Hb and Vanclay JKa 
a   Forest Research Centre, School of Science, Environment and Engineering, Southern Cross University, PO Box 157, 
Lismore, NSW 2480, Australia. 
b  Institute of Silviculture, University of Natural Resources and Life Sciences (BOKU), Peter-Jordan Strasse 82, 1190 Vienna, 
Austria. 
* chris.eastaugh@scu.edu.au 
 
After more than a century of research the typical growth pattern of a tree was thought to be fairly 
well understood. Following germination height growth accelerates for some time, then increment 
peaks and the added height each year becomes less and less. The cross sectional area (basal area) of 
the tree follows a similar pattern, but the maximum basal area increment occurs at some time after 
the maximum height increment. An increase in basal area in a tall tree will add more volume to the 
stem than the same increase in a short tree, so the increment in stem volume (or mass) peaks very 
late. Stephenson et al. challenge this paradigm, and suggest that mass increment increases 
continuously. Their analysis methods however are a textbook example of the ‘ecological fallacy’, and 
their conclusions therefore unsupported. 
The ecological fallacy1 was most famously described by the sociologist William Robinson (1913-
1996)1, who pointed out that correlations present in aggregated data do not imply that the same 
correlations are present in the individuals3. In the context of the Stephenson et al. paper4, this 
means that the fact that large trees of a species tend to have higher mass increment than small trees 
in no way implies that increment in individuals is a monotonic function of tree size. Robinson gave a 
mathematical proof of the fallacy, but we can demonstrate the effect here using tree measurements 
pertinent to Stephenson et al.’s study. 
Adolf Ritter von Guttenberg (1839-1917) compiled one of the classic datasets of forestry science5,6. 
In 95 stands of Norway Spruce (Picea abies [L] Karst) across the Austrian Alps he felled 107 average 
trees and cut the stems into sections of between 1.0 and 4.0 metres length. The volume of each 
section was calculated individually and summed to give a total stem volume. By counting and 
measuring the annual growth rings in each section he was able to record a quite detailed growth 
history for each tree. We use here the data from the Hinterberg forest region in Styria, and select 
the six trees taken from Site Class 1 plots (the most productive). Tree ages at felling were 140 and 
150 years. Aboveground biomass (including branches and foliage) is derived from the volume, 
diameter and height data using relationships from the standard Austrian biomass functions7. 
When we aggregate the data for these six trees and fit a piecewise linear regression (using 
Stephenson et al.’s software scripts8) we see that the results do indeed suggest monotonic mass 
accumulation (fig 1a). However, when we look at the course of mass increment for the individual 
trees (fig 1b), we see that typically growth accelerates while the tree is young or maturing, then 
levels and falls as maturity turns to senescence9. While Stephenson et al.’s analysis can tell us that 
the growth rate of large trees is typically greater than the growth rate of small trees, it does not 
follow that any individual tree will have a continuously increasing growth rate.  

 
Figure 1 Mass growth data from von Guttenberg's Hinterberg Site Class I trees5. Panel 'a' shows a 4-bin piecewise linear 
regression through the data cloud, while panel 'b' shows the actual trajectories of growth of the individual trees 
 
Stephenson et al.’s data are not timeseries of individual trees as we show here, but are isolated 
observations from many different trees. In effect, this is as if they had a random sample from a point 
cloud such as that in figure 1a, with single observations from a wide range of possible timeseries. It is 
unsurprising that a regression through such a cloud has an increasing trend, but it is simply false to 
infer that this trend applies to the individual trees.  Stephenson et al.’s conclusions that rates of tree 
carbon accumulation increase continuously with tree size are therefore invalid. 
 

 [1]  Piantadosi, S., Byar, D.P., Green, S.B. The ecological fallacy. American Journal of Epidemiology 
127(5), 893-904 (1988) 
[2]  Robinson, W.S. Ecological correlations and the behavior of individuals. American Sociological 
Review 15(3), 351–357 (1950).  
[3]  Thorndike, E.L. On the fallacy of imputing the correlations found for groups to the individuals or 
smaller groups composing them. The American Journal of Psychology  52(1) 122-124 (1939) 
[4]  Stephenson et al.  Rate of tree carbon accumulation increases continuously with tree size. 
Nature doi:10.1038/nature12914 (2014). 
[5]  von Guttenberg, A.R. Wachstum und Ertrag der Fichte im Hochgebirge. (Verlag Franz Deuticke, 
Wien, 1915). 
[6]  Zeide, B. Analysis of growth equations. Forest Science 39(3), 594-616 (1993). 
[7]  Thurnher, C., Gerritzen, T., Maroschek, M., Lexer, M.J. & Hasenauer, H. Analysing different 
carbon estimation methods for Austrian forests. Austrian Journal of Forest Science 130(3), 141-166 
(2013). 
[8]   Condit, R. CTFS Tutorials 
http://ctfs.arnarb.harvard.edu/Public/CTFSRPackage/index.php/web/tutorials/growthfitAnalysis/ind
ex.html (2011) 
[9]   Assmann, E. The Principles of Forest Yield Study (Pergamon press, Oxford, 1970). 
 
 
The authors declare no competing financial interests. 
 
