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More on interpreting log-transformed data
What can we say about the relationship of Brainweight to Bodyweight?
We can use the regression calculated on the log scale to predict actual Brainweight from
Bodyweight. Calculate the log of Bodyweight, substitute in the prediction formula
derived from the regression formula, calculate the value of predicted LBrainW, then
calculate 10 to the power of this value.
Given a typical human Bodyweight of 62 Kg, what is your prediction of human
Brainweight?
There are problems with this, however, as it leads to biased predictions. This arises
because the mean of the assumed Normal distribution of the transformed data equals
the median, because of symmetry, while the mean of the original skew distribution of the
data exceeds the median. The log transform, being a strictly increasing function,
converts the median to the median, its inverse does the same, so the back transform of
the mean (= median) of the transformed data equals the median of the original data,
which is less than the mean of the original data. Thus, the back-transformed mean gives
a downwards biased estimate of the original mean.
There are ways of adjusting for this but they are relatively complicated and do not
appear in standard textbooks.
It is best to stick to interpretations in the log scale or in terms of a multiplicative model in
the original scale.
Proportionality relations
Further to this, we can demonstrate that, if two species' body weights tend to be in a
certain proportion, then their brain weights are in a corresponding (but different)
proportion. Suppose that the typical body weight of species A, say XA, is k times the
typical body weight of species B, say XB, that is, XA = kXB. If Y denotes brainweight, we
have
log(YA) =  + log(XA),
log(YB) =  + log(XB),
so that
log(YA)  log(YB) = log(XA)  log(XB)],
that is,
Y 
X 
log A    log A    log(k )  log( k  ) ,
 YB 
 XB 
that is,
YA
= k
YB
that is,
YA = k YB.
Special case: k small
If XA is just slightly bigger than XB, then this relationship can be simplified; if we let k = 1 + r,
where r is a small rate of increase in body weight, then the corresponding increase in brain
weight is r, to a good degree of approximation. In this application, this means that a 1%
increase in body weight corresponds to a 0.75% increase in brainweight.
To see this, suppose
logYX =  + logX,
then
logYX(1+r) =  + logX(1+r),
so that
logYX(1+r) – logYX = logX(1+r) – logX,
that is,
log
YX(1 r )
YX
  log
X(1  r )
  log(1  r ) .
X
Taking antilogs,
YX(1r )
YX
= (1+r).
If r is small, the term on the right is, approximately,
1 + r,
so that
YX(1+r) = YX (1 + r).
The interpretation in this special case is not very interesting in this application, as we will want to
able to relate changes in brainweight to changes in body weight going from any species to any
other species, not just species of marginally different body size.
Encephalisation
Closely related to this is the notion of encephalisation, a measure of the extra brain size of a
species relative to its body size. Returning to the log linear regression,
logY =  +  logX + ,
 measure the "extra" on the log scale. If  = 0, this simplifies to
logY =  logX + ,
that is,
log
Y
 ,
X
or
Y
= 10.
X
This is called the encephalisation quotient. A popular value for  is 0.75.

The study of this kind of relationship of body parts is called allometry.
Other applications
In economics, the relation of variables like Demand to corresponding variables like Price are of
interest. If the relationship is linear when both variables are expressed in the log scale, then the
slope () is referred to as the price elasticity of demand.
In finance, if the relationship between the return on a single asset is linearly related to the return
on the overall market when both returns are expressed in the log scale, the slope, , is referred
to as financial elasticity.
More generally, power law relationships of the form
Y = X
are studied in a number of different disciplines.