Fits ACD transformation as outlined in Royston (2014). The ACD transformation smoothly maps the observed distribution of a continuous covariate x onto one scale, namely, that of an approximate uniform distribution on the interval (0, 1).
fit_acd(x, powers = NULL, shift = 0, scale = 1)A list is returned with components
acd: the acd transformed input data.
beta0: intercept of estimated model.
beta1: coefficient of estimated model.
power: estimated power.
shift: shift value used for computations.
scale: scaling factor used for computations.
a numeric vector.
a vector of allowed FP powers. The default value is NULL,
meaning that the set \(S = (-2, -1, -0.5, 0, 0.5, 1, 2, 3)\) is used.
a numeric that is used to shift the values of x to positive
values. The default value is 0, meaning no shifting is conducted.
If NULL, then the program will estimate an appropriate shift automatically
(see find_shift_factor()).
a numeric used to scale x. The default value is 1, meaning
no scaling is conducted. If NULL, then the program will estimate
an appropriate scaling factor automatically (see find_scale_factor()).
Briefly, the estimation works as follows. First, the input data are shifted
to positive values and scaled as requested. Then
$$z = \Phi^{-1}(\frac{rank(x) - 0.5}{n}) $$
is computed, where \(n\) is the number of elements in x,
with ties in the ranks handled as averages. To approximate \(z\),
an FP1 model (least squares) is used, i.e.
\(E(z) = \beta_0 + \beta_1 (x)^p\), where \(p\) is chosen such that it
provides the best fitting model among all possible FP1 models.
The ACD transformation is then given as
$$acd(x) = \Phi(\hat{z}),$$
where the fitted values of the estimated model are used.
If the relationship between a response Y and acd(x) is linear,
say, \(E(Y) = \beta_0 + \beta_1 acd(x)\), the relationship between Y
and x is nonlinear and is typically sigmoid in shape.
The parameters \(\beta_0\) and \(\beta_0 + \beta_1\) in such a model are
interpreted as the expected values of Y at the minimum and maximum of x,
that is, at acd(x) = 0 and 1, respectively.
The parameter \(\beta_1\) represents the range of predictions of \(E(Y)\)
across the whole observed distribution of x (Royston 2014).
Royston, P. and Sauerbrei, W. (2016). mfpa: Extension of mfp using the ACD covariate transformation for enhanced parametric multivariable modeling. The Stata Journal, 16(1), pp.72-87.
Royston, P. (2014). A smooth covariate rank transformation for use in regression models with a sigmoid dose–response function. The Stata Journal, 14(2), 329-341.
set.seed(42)
x = apply_shift_scale(rnorm(100))
y = rnorm(100)
fit_acd(x, y)
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