A log-log plot is used when the functional relationship is expected to be
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Log-log plots display data in two dimensions where both axes use logarithmic scales. When one variable changes as a constant power of another, a log-log graph shows the relationship as a straight line. In this post, I’ll show you why these graphs are valuable and how to interpret them.
Example
We’ll use the PanTHERIA database to model the relationship between mammal mass and metabolic with a log-log plot. This dataset includes 572 mammals that range from the masked shrew (4.2 grams) to the common eland (562,000 grams)—which is a much larger sample-size than David Cox’s dataset. Here is the CSV data file so you can try both log-log plot examples for yourself: Mammals.
Most statistical software can create a log-log plot. Here’s what it looks like for the mammal dataset.
The data clearly follow a straight line, which indicates they follow a power law relationship. The p-value for the slope (0.7063) is 0.000 (not shown), indicating that it is statistically significant. The R-squared of 94.3% is impressive, particularly when you consider that different researchers collected these data in various settings and included a wide range of mammals from entirely different habits!
Using the constant and slope, we can rewrite it in the power law form:
Metabolic Rate = 0.5758Mass0.7063
The exponent’s value is consistent with recently published estimates.
When a slope on a log-log plot is between 0 and 1, it signifies that the nonlinear effect of the dependent variable lessens as its value increases. For the mammal data, the exponent (0.7063) is in this range, which indicates that as mammals become more massive, the increase in metabolic rate slows down. In other words, gram for gram, larger mammals use less energy than smaller mammals. Or, a cell in a larger mammal uses less energy than a cell in a smaller mammal. This interpretation fits Cox’s explanation in the show.
The fact that the effect of mass on metabolism decreases has significant ramifications. If the increase in metabolic rate had remained constant (linear), humans would need to consume 16,000 calories a day. However, mammals couldn’t grow more massive than a goat due to overheating problems!
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