make.tran: Response-transformation extensions

In emmeans: Estimated Marginal Means, aka Least-Squares Means

Response-transformation extensions

Description

The make.tran function creates the needed information to perform transformations of the response variable, including inverting the transformation and estimating variances of back-transformed predictions via the delta method. make.tran is similar to make.link, but it covers additional transformations. The result can be used as an environment in which the model is fitted, or as the tran argument in update.emmGrid (when the given transformation was already applied in an existing model).

Usage

make.tran(type = c("genlog", "power", "boxcox", "sympower", "asin.sqrt",
  "atanh", "bcnPower", "scale"), alpha = 1, beta = 0, param, y, ...)

inverse(y)

Arguments

type	The name of the transformation. See Details.
alpha, beta	Numeric parameters needed for the transformation. See Details.
param	If non-missing, this specifies either alpha or c(alpha, beta) (provided for backward compatibility). Also, for the same reason, if alpha is of length more than 1, it is taken as param.
y	A response variable. With type = "scale", the results determine alpha and beta.
...	Additional arguments passed to other functions/methods

Value

A list having at least the same elements as those returned by make.link. The linkfun component is the transformation itself.

inverse returns the reciprocal of its argument. It allows the "inverse" link to be auto-detected as a response transformation.

Details

The make.tran function returns a suitable list of functions for several popular transformations that are not already provided elsewhere. 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 parameters, specified as alpha and beta; in the following discussion, we use \alpha and \beta to denote alpha and beta, and y to denote the response variable. The type argument specifies the following transformations:

"genlog"

Generalized logarithmic transformation: \log_\beta(y + \alpha), where y > -\alpha. When \beta = 0 (the default), we use \log_e(y + \alpha)

"power"

Power transformation: (y-\beta)^\alpha, where y > \beta. When \alpha = 0, \log(y-\beta) is used instead.

"boxcox"

The Box-Cox transformation (unscaled by the geometric mean): ((y - \beta)^\alpha - 1) / \alpha, where y > \beta. When \alpha = 0, \log(y - \beta) is used.

"sympower"

A symmetrized power transformation on the whole real line: |y - \beta|^\alpha\cdot sign(y - \beta). There are no restrictions on y, but we require \alpha > 0 in order for the transformation to be monotone and continuous.

"asin.sqrt"

Arcsin-square-root transformation: \sin^{-1}(y/\alpha)^{1/2}. Typically, alpha will be either 1 (default) or 100.

"atanh"

Arctanh transformation: \tanh^{-1}(y/\alpha). Typically, alpha will be either 1 (default) or 100.

"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^\alpha - 1) / \alpha of the variable z = y + (y^2+\beta^2)^{1/2}. Note that this requires both parameters and that beta > 0.

"scale"

This one is a little different than the others, in that alpha and beta are ignored; instead, they are determined by calling scale(y, ...). The user should give as y the response variable in the model to be fitted to its scaled version.

Note that with the "power", "boxcox", or "sympower" transformations, the argument beta specifies a location shift. In the "genpower" transformation, beta specifies the base of the logarithm – however, quirkily, the default of beta = 0 is taken to be the natural logarithm. For example, make.tran(0.5, 10) sets up the \log_{10}(y + \frac12) transformation. In the "bcnPower" transformation, beta must be specified as a positive value.

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.

Cases where make.tran may not be needed

Often, just the name of the transformation is all that is needed. 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, log1p, sqrt, logit, probit, cauchit, cloglog; as well as (for a response variable y) asin(sqrt(y)), asinh(sqrt(y)), atanh(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".

Note

The genlog transformation is technically unneeded, because a response transformation of the form log(y + c) is now auto-detected by ref_grid.

We modify certain make.link results in transformations where there is a restriction on valid prediction values, so that reasonable inverse predictions are obtained, no matter what. For example, if a sqrt transformation was used but a predicted value is negative, the inverse transformation is zero rather than the square of the prediction. A side effect of this is that it is possible for one or both confidence limits, or even a standard error, to be zero.

Examples

# Fit a model using an oddball transformation:
bctran <- make.tran("boxcox", 0.368)
warp.bc <- with(bctran, 
    lm(linkfun(breaks) ~ wool * tension, data = warpbreaks))
# Obtain back-transformed LS means:    
emmeans(warp.bc, ~ tension | wool, type = "response")

### Using a scaled response...
# Case where it is auto-detected:
mod <- lm(scale(yield[, 1]) ~ Variety, data = MOats)
emmeans(mod, "Variety", type = "response")

# Case where scaling is not auto-detected -- and what to do about it:
copt <- options(contrasts = c("contr.sum", "contr.poly"))
mod.aov <- aov(scale(yield[, 1]) ~ Variety + Error(Block), data = MOats)
emm.aov <- suppressWarnings(emmeans(mod.aov, "Variety", type = "response"))

# Scaling was not retrieved, so we can do:
emm.aov <- update(emm.aov, tran = make.tran("scale", y = MOats$yield[, 1]))
emmeans(emm.aov, "Variety", type = "response")

options(copt)


## Not run: 
### An existing model 'mod' was fitted with a y^(2/3) transformation...
  ptran = make.tran("power", 2/3)
  emmeans(mod, "treatment", tran = ptran)

## End(Not run)

pigs.lm <- lm(inverse(conc) ~ source + factor(percent), data = pigs)
emmeans(pigs.lm, "source", type = "response")

emmeans documentation built on Oct. 18, 2023, 1:13 a.m.