John Tukey once said, “The best thing about being a statistician is that you get to play in everyone’s backyard.” I enthusiastically agree!
I frequently enjoy reading and watching science-related material. This invariably raises questions, involving other "backyards," that I can better understand using statistics. For instance, see my post about the statistical analysis of dolphin sounds.
The latest topic that grabbed my attention was an apparent error in the BBC program Wonders of Life. In the episode “Size Matters,” Professor Brian Cox presents a graph with a linear regression line that illustrates the relationship between the size of mammals and their metabolic rate.
How Does the Size of Mammals Affect Their Lives?
Brian Cox, a theoretical physicist, is a really smart guy and one of my favorite science presenters. So, I was surprised when his interpretation of the linear regression model seemed incorrect. Below is a closer look at the graph he presents, and his claim.
Cox points out the straight line and states, “That implies, gram-for-gram, large animals use less energy than small animals . . . because the slope is less than one.”
For linear regression, the slope being less than 1 is irrelevant. Instead, the fact that it is a straight line indicates that the same relationship applies for both small and large mammals. If you increase mass by 1 unit for a small mammal and for a large mammal, metabolism increases by the same average amount for both sizes. In other words, gram-for-gram, size doesn’t seem to matter!
However, it’s unlikely that Cox would make such a fundamental mistake, so I conducted some research. It turns out that biologists use a log-log plot to model the relationship between the mass of mammals and their basal metabolic rate.
Perhaps Cox actually presented a log-log plot but glossed over the details?
If so, this dramatically changes the interpretation of this graph, because log-log plots transform both axes in order to model curvature while using linear regression. If the slope on a log-log plot of metabolic rate by mass is between 0 and 1, it indicates that the nonlinear effect of mass on metabolic rate lessens as mass increases.
This description fits Cox’s statements about the slope and how mass effects metabolic rate.
Let’s test Cox’s hypothesis ourselves! Thanks to the PanTHERIA database*, we can fit the same type of log-log plot using similar data.
Log-Log Plot of Mammal Mass and Basal Metabolic Rate
I’ll use the Fitted Line Plot in Minitab statistical software to fit a regression line to 572 mammals, ranging from the masked shrew (4.2 grams) to the common eland (562,000 grams). You can find the data here.
In Minitab, go to Stat > Regression > Fitted Line Plot. Metabolic rate is our response variable and adult mass is our predictor. Go to Options and check all four boxes under Transformations to produce the log-log plot.
The slope (0.7063) is significant (p = 0.000) and its value is consistent with recently published estimates. Because the slope is between 0 and 1, it confirms Cox’s interpretation. Gram-for-gram, larger animals use less energy than smaller animals. In order to function, a cell in a larger animal requires less energy than a cell in a smaller animal.
I’m quite amazed that the R-squared is 94.3%! Usually you only see R-squared values this high for a low-noise physical process. Instead, these data were collected by a variety of researchers in different settings and cover a huge range of mammals that live in completely different habitats.
So Cox presented the correct interpretation after all: for mammals, size matters. However, he presented an oversimplified version of the underlying analysis by not mentioning the double-log transformations. This is television, after all!
There are important implications based on the fact that this model is curved rather than linear. If the increase in metabolic rate remained constant as mass increased (the linear model), we’d have to eat 16,000 calories a day to sustain ourselves. Further, mammals wouldn’t be able to grow larger than a goat due to overheating!
Basal Metabolic Rate and Longevity
Cox also presented the idea that mammals with a slower metabolism live longer than those with a faster metabolism. However, he didn’t present data or graphs to support this contention. Fortunately, we can test this hypothesis as well.
In Minitab, I used Calc > Calculator to divide metabolic rate by grams. This division allows us to make the gram-for-gram comparison. Time for another fitted line plot with a double-log transformation!
The negative slope is significant (0.000) and tells us that as metabolic rate per gram increases, longevity decreases. The R-squared is 45.8%, which is pretty good considering that we’re looking at just one of many factors than can impact maximum lifespan!
However, it is not a linear relationship because this is a log-log plot. Maximum longevity asymptotically approaches a minimum value around 13 months as metabolism increases. The graph below shows the curvilinear relationship using the natural scale.
A one-unit increase in the slower metabolic rates (left side of x-axis) produces much larger drops in longevity than a on-unit increase in the faster metabolic rates (right side of x-axis).
Once again, we’ve shown that size does matter! Larger mammals tend to have a slower metabolism. And animals that have a slower metabolism tend to live longer. That’s fortunate for us because without our slower metabolism, we’d only live about a year!
*Kate E. Jones, Jon Bielby, Marcel Cardillo, Susanne A. Fritz, Justin O'Dell, C. David L. Orme, Kamran Safi, Wes Sechrest, Elizabeth H. Boakes, Chris Carbone, Christina Connolly, Michael J. Cutts, Janine K. Foster, Richard Grenyer, Michael Habib, Christopher A. Plaster, Samantha A. Price, Elizabeth A. Rigby, Janna Rist, Amber Teacher, Olaf R. P. Bininda-Emonds, John L. Gittleman, Georgina M. Mace, and Andy Purvis. 2009. PanTHERIA: a species-level database of life history, ecology, and geography of extant and recently extinct mammals. Ecology 90:2648.