fit_acd: Function to estimate approximate cumulative distribution...

In mfp2: Multivariable Fractional Polynomial Models with Extensions

Function to estimate approximate cumulative distribution (ACD)

Description

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).

Usage

fit_acd(x, powers = NULL, shift = 0, scale = 1)

Arguments

x	a numeric vector.
powers	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.
shift	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()).
scale	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()).

Details

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).

Value

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.

References

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.

Examples

set.seed(42)
x = apply_shift_scale(rnorm(100))
y = rnorm(100)
fit_acd(x, y)

mfp2 documentation built on June 8, 2025, 11:04 a.m.