function,trait value¼aMb, whereaandbare constants. Thus, log (trait value)
plotted against logMgives a straight line with slopeb.
A special case of scaling is when two traits (e.g. body mass at hatch and at
maturity) have the same units and the value of one trait is directly proportional
to the value of another: here, their ratio is a dimensionless constant or an
invariant (Charnov, 1993 ), and a log-log plot of these values produces a slope
of 1. Such invariants could underpin the form and shape of trade-offs that
constrain life-history evolution, and if they extend across species could provide
the basis of a general theory of life-history evolution, rather than a diverse array
of species-specific models.Methodological issues
Determining the scaling exponent
A first issue is to ensure that the appropriate regression model is used, as the
value of the scaling exponent is sensitive to whether the error is assumed to be
confined just to they-variable (as in a Type I regression), or is shared between the
covariates (as in a Type II regression; McArdle, 2003 ). Thus, if there is a similar
degree of error in both covariates, a Type II regression should be used.Identifying invariant quantities in life histories
A major problem is that the method generally used for identifying invariant
quantities is flawed: thus, the very existence of invariant quantities will be in
doubt until an alternative method is devised (Neeet al., 2005). The difficulty is
that while it is true that for an invariant quantity the log-log regression of trait
values will always produce a slope of 1, it is not true that a slope of 1 always
implies invariance (Neeet al., 2005). The problem arises when the variable on the
y-axis is a fraction of thex-variable, for instance mass at birth versus mass at
maturity, so that the variables have been regressed against themselves. De Jong
(2005) provides a graphical illustration of how combining data from very differ-
ent invariant relationships for a variety of species into a single relationship for
all species can produce the illusion of a single invariant quantity for all the
species with a slope of 1, and high r^2. A high r^2 does not imply invariance, as the
high r^2 can arise when there is a wide range of values on thex-axis (Neeet al.,
2005 ). This problem of detecting invariance may be more widespread than it
first appears, as it can occur even when one variable does not appear initially to
be a fraction of the other. For instance, the proposed invariant relationship
between yearly clutch sizeband adult mortality ratezhas been shown by Nee
et al.(2005) to involve correlation with itself. Specifically, if the animal produces
Eeggs in total over its adult lifetime ofYyears, then its yearly clutch sizebisE/Y
and its annual mortality rate is1/Y: therefore,b¼zE, a regression of ln(b) against
ln(z) is expected to have a slope of 1.0, and r^2 ¼var[ln(z)]/(var[ln(z)]þvar[ln(E)]),
which will be high ifzhas a wide range of values.44 D. ATKINSON AND A. G. HIRST