The functions emmeans, ref_grid, and related ones
automatically detect response transformations that are recognized by
examining the model formula. These are log, log2, log10,
sqrt, logit, probit, cauchit, cloglog; as
well as (for a response variable y) asin(sqrt(y)),
asinh(sqrt(y)), and sqrt(y) + sqrt(y+1). In addition, any
constant multiple of these (e.g., 2*sqrt(y)) is auto-detected and
appropriately scaled (see also the tran.mult argument in
update.emmGrid).
A few additional character strings may be supplied as the tran
argument in update.emmGrid: "identity",
"1/mu^2", "inverse", "reciprocal", "log10", "log2", "asin.sqrt",
and "asinh.sqrt".
More general transformations may be provided as a list of functions and
supplied as the tran argument as documented in
update.emmGrid. The make.tran function returns a
suitable list of functions for several popular transformations. Besides being
usable with update, the user may use this list as an enclosing
environment in fitting the model itself, in which case the transformation is
auto-detected when the special name linkfun (the transformation
itself) is used as the response transformation in the call. See the examples
below.
Most of the transformations available in "make.tran" require a parameter,
specified in param; in the following discussion, we use \(p\) to
denote this parameter, and \(y\) to denote the response variable.
The type argument specifies the following transformations:
"genlog"Generalized logarithmic transformation: \(log(y +
p)\), where \(y > -p\)
"power"Power transformation: \(y^p\), where \(y > 0\).
When \(p = 0\), "log" is used instead
"boxcox"The Box-Cox transformation (unscaled by the geometric
mean): \((y^p - 1) / p\), where \(y > 0\). When \(p = 0\), \(log(y)\)
is used.
"sympower"A symmetrized power transformation on the whole real
line:
\(abs(y)^p * sign(y)\). There are no restrictions on \(y\), but we
require \(p > 0\) in order for the transformation to be monotone and
continuous.
"asin.sqrt"Arcsin-square-root transformation:
\(sin^(-1)(y/p)^{1/2)}\). Typically, the parameter \(p\) is equal to 1 for
a fraction, or 100 for a percentage.
"bcnPower"Box-Cox with negatives allowed, as described for the
bcnPower function in the car package. It is defined as the Box-Cox
transformation \((z^p - 1) / p\) of the variable \(z = y + (y^2+g^2)^(1/2)\).
This requires param to have two elements:
the power \(p\) and the offset \(g > 0\).
"scale"This one is a little different than the others, in that
param is ignored; instead, param is determined by calling
scale(y, ...). The user should give as y the response variable in the
model to be fitted to its scaled version.
The user may include a second element in param to specify an
alternative origin (other than zero) for the "power", "boxcox",
or "sympower" transformations. For example, type = "power",
param = c(1.5, 4) specifies the transformation \((y - 4)^1.5\).
In the "genpower" transformation, a second param element may be
used to specify a base other than the default natural logarithm. For example,
type = "genlog", param = c(.5, 10) specifies the \(log10(y + .5)\)
transformation. In the "bcnPower" transformation, the second element
is required and must be positive.
For purposes of back-transformation, the sqrt(y) + sqrt(y+1)
transformation is treated exactly the same way as 2*sqrt(y), because
both are regarded as estimates of \(2\sqrt\mu\).