1 Overview

The approximate cumulative distribution (ACD) transformation provides a smooth way to model sigmoid-shaped relationships, including curves that level off at one or both ends (Royston, 2014). It approximates the empirical cumulative distribution function of a continuous predictor.

The MFPA extension combines this transformation with multivariable fractional polynomial (MFP) modelling (Royston and Sauerbrei, 2016). For a predictor selected for ACD assessment, mfp2() considers functions of both the original predictor and its ACD transformation. The usual function-selection procedure is replaced by the function-selection procedure with ACD transformation (FSPA). Other predictors continue through standard MFP selection during backfitting.

Request ACD assessment using:

• acd_vars = "x" in the matrix interface;
• fp(x, acd = TRUE) in the formula interface.

Requesting ACD assessment allows the transformation to be considered; selection may still retain an ordinary FP1 or linear effect, or omit the predictor.

2 Definition of the ACD transformation

For observed predictor values \(x_1,\ldots,x_n\), the transformation has three steps. The description below assumes ordinary continuous data, without structural-zero handling.

3 Interpretation

ACD values are approximately uniformly distributed between zero and one. They need not be exactly uniform, and the smallest and largest observed values need not map exactly to zero and one.

For illustration, consider an uncentred model with an identity link:

\[ \mu(x)=E(Y\mid X=x)=\beta_0+\beta_1 A(x). \]

At ACD values near zero, the fitted mean is near \(\beta_0\); at values near one, it is near \(\beta_0+\beta_1\). Thus, \(\beta_1\) is the signed change in the fitted mean for a one-unit increase in \(A(x)\). Across the actual observed predictor range, the fitted difference is

\[ \mu(x_{\max})-\mu(x_{\min}) =\beta_1\{A(x_{\max})-A(x_{\min})\}. \]

This difference is approximately \(\beta_1\) only when the observed ACD values cover nearly the whole interval. With other links, the same interpretation applies on the linear-predictor scale.

A linear effect of \(A(x)\) typically produces a sigmoid-shaped relationship with the original predictor. The transformation compresses the tails of its distribution, which can reduce the influence of extreme predictor values. It does not guarantee a particular response shape for every distribution or selected function.

4 ACD in the fractional polynomial family

The largest ACD-extended function combines one FP1 term in the shifted original predictor and one FP1 term in its ACD transformation:

\[ \beta_1 h_{p_1}(x+s)+\beta_2 h_{p_2}\{A(x)\}. \]

Following Royston and Sauerbrei (2016), this is denoted \(FP1(p_1,p_2)\). Here the notation represents two components, each containing one FP term. It differs from ordinary FP1 notation, which represents a single power term in one predictor.

The powers \(p_1\) and \(p_2\) are selected for the outcome model. They are separate from the power \(q\) used to construct \(A(x)\). The default power set gives \(8\times8=64\) candidate pairs. A power of zero denotes a logarithm through the definition of \(h_p\) above.

This family also permits simpler functions that omit either component. When both components are retained, the original-predictor term can add a trend, so the complete function need not level off at both ends.

5 Function-selection procedure for ACD (FSPA)

For each predictor requested for ACD assessment, the FSPA considers the following six candidate families. The notation omits the intercept, other predictors, and centring constants.

Here \(A_x=A(x)\), and a dot indicates that the corresponding component is omitted. The families do not form a single nested sequence; for example, M2 and M3 use different components.

6 Simulation example

The following simulation creates a sigmoid relationship between x and a continuous outcome y, with random noise added to the mean.

set.seed(54)

n <- 500
x <- matrix(rnorm(n), ncol = 1, dimnames = list(NULL, "x"))

sigmoid <- function(x) {
  1 / (1 + exp(-1.7 * x))
}

y <- as.numeric(20 * sigmoid(x) + rnorm(n, mean = 0, sd = 0.5))

7 Fitting an ACD model

fit_acd_matrix <- mfp2(
  x, y,
  acd_vars = "x",
  verbose = FALSE
)

fit_acd_matrix$fp_terms
#>   df_setting df_initial select alpha  acd  zero catzero spike prop_zero spike_dec selected
#> x          4          4   0.05  0.05 TRUE FALSE   FALSE FALSE        NA         2     TRUE
#>   searched_fp df_final power1 power2
#> x        TRUE        4      1      1

dat <- data.frame(y = y, x = as.numeric(x))

fit_acd_formula <- mfp2(
  y ~ fp(x, acd = TRUE),
  data = dat,
  verbose = FALSE
)

fit_acd_formula$fp_terms
#>   df_setting df_initial select alpha  acd  zero catzero spike prop_zero spike_dec selected
#> x          4          4   0.05  0.05 TRUE FALSE   FALSE FALSE        NA         2     TRUE
#>   searched_fp df_final power1 power2
#> x        TRUE        4      1      1

8 Comparison with standard MFP

For comparison, fit an ordinary MFP model to the same data without ACD assessment. Its default maximum complexity is FP2.

fit_standard <- mfp2(
  y ~ fp(x, acd = FALSE),
  data = dat,
  verbose = FALSE
)

fit_standard$fp_terms
#>   df_setting df_initial select alpha   acd  zero catzero spike prop_zero spike_dec selected
#> x          4          4   0.05  0.05 FALSE FALSE   FALSE FALSE        NA         2     TRUE
#>   searched_fp df_final power1 power2
#> x        TRUE        4      3      3

9 Plotting the fitted functions

Use plot() to compare the selected functions. Each panel shows the fitted contribution, pointwise 95% confidence intervals, and component-plus-residual points. Because each Gaussian model has only one predictor and an identity link, the fitted curve includes the intercept and the points correspond to the observed outcomes.

p1 <- plot(fit_acd_matrix, terms = "x")[[1]] +
  ggplot2::ylab("Partial predictor + residuals") +
  ggplot2::ggtitle("ACD model")

p2 <- plot(fit_standard, terms = "x")[[1]] +
  ggplot2::ylab("Partial predictor + residuals") +
  ggplot2::ggtitle("Standard MFP model")

patchwork::wrap_plots(p1, p2, ncol = 2)

ACD model compared with a standard MFP model.

Both models use the same simulated observations. Inspect the curves to see how each represents the central transition and the tails, rather than assuming that requesting ACD guarantees a better fit. The intervals are conditional on the selected model and do not account for uncertainty from variable or function selection.

10 Practical remarks

• Use ACD assessment when sigmoid-shaped effects are plausible and the fitted relationship supports them. The ACD option extends the choices available during MFP selection.
• For an eligible ACD predictor, mfp2() sets the maximum df to 4 to allow the two-component reference model. Selection can still choose a simpler form.
• ACD requests are reset when the fitting predictor has fewer than five distinct values. The ordinary MFP rules then apply.
• Supply original predictor values for prediction. The fitted shift, ACD approximation, selected powers, and centring are reused; do not recalculate ranks or refit the ACD transformation in the new data.
• In $fp_terms, power1 describes the original-predictor component and power2 the ACD component. An NA power indicates that the corresponding component is absent. A nonzero ACD flag alone does not establish that the final model retains an ACD component.

11 References

Royston, P. (2014). A smooth covariate rank transformation for use in regression models with a sigmoid dose-response function. Stata Journal, 14(2), 329–341.

Royston, P. and Sauerbrei, W. (2016). mfpa: Extension of mfp using the ACD covariate transformation for enhanced parametric multivariable modeling. Stata Journal, 16(1), 72–87.

Royston, P. and Sauerbrei, W. (2008). Multivariable Model-Building: A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modelling Continuous Variables. John Wiley & Sons.