Augmenting designs with controlled efficiency loss

In optedr: Calculating Optimal and D-Augmented Designs for Single- and Multi-Factor Models

knitr::opts_chunk$set(
  collapse = TRUE,
  comment  = "#>",
  fig.width  = 6,
  fig.height = 4,
  out.width  = "90%"
)

library(optedr)

Motivation

In practice an experiment is rarely designed from scratch. A researcher may already have data collected at certain conditions and want to add new observations to improve estimation — without discarding what has already been measured. The key question is: where can new points be placed so that the efficiency of the augmented design stays above an acceptable threshold?

optedr answers this question with two functions used in sequence:

1. get_augment_region() — computes the candidate region: the set of design points whose addition keeps the D-efficiency of the augmented design above a user-specified threshold delta_val.
2. augment_design() — adds a chosen point to the initial design and rescales the weights.

Both functions support the same optimality criteria as opt_des() and work for any number of factors.

Key parameters

| Parameter | Role | |-----------|------| | init_design | Current design (data frame with Point/Weight in 1D, or factor columns + Weight in multi-D) | | alpha | Fraction of total weight assigned to the new point after augmentation | | delta_val | Minimum acceptable D-efficiency of the augmented design | | calc_optimal_design | If TRUE, also computes the optimal design and uses it as the reference for efficiency | | new_points | Data frame of points to add (non-interactive mode); omit for interactive mode | | par_int | Indices of parameters of interest (Ds-Optimality only) | | n_lhs | Number of Latin-Hypercube candidates for the region search (multi-D) |

One-factor augmentation

Step 1: compute the candidate region

We start with a uniform three-point design for Antoine's equation and look for points that keep the D-efficiency of the augmented design above 85 %.

init_des <- data.frame(
  Point  = c(30, 60, 90),
  Weight = c(1/3, 1/3, 1/3)
)

region <- get_augment_region(
  criterion           = "D-Optimality",
  init_design         = init_des,
  alpha               = 0.25,
  model               = y ~ 10^(a - b / (c + x)),
  parameters          = c("a", "b", "c"),
  par_values          = c(8.07131, 1730.63, 233.426),
  design_space        = c(1, 100),
  calc_optimal_design = FALSE,
  delta_val           = 0.85
)
print(region)

region$region is a data frame of candidate intervals. Each row gives a lower and upper bound on the design space where the new point can be placed.

Step 2: choose a point and augment

new_pt <- mean(region$region[1:2])

augmented <- augment_design(
  criterion           = "D-Optimality",
  init_design         = init_des,
  alpha               = 0.25,
  model               = y ~ 10^(a - b / (c + x)),
  parameters          = c("a", "b", "c"),
  par_values          = c(8.07131, 1730.63, 233.426),
  design_space        = c(1, 100),
  calc_optimal_design = FALSE,
  delta_val           = 0.85,
  new_points          = data.frame(Point = new_pt, Weight = 1)
)
print(augmented)
cat("Sum of weights:", sum(augmented$Weight), "\n")

Comparing efficiency before and after

result_opt <- opt_des(
  "D-Optimality",
  y ~ 10^(a - b / (c + x)), c("a", "b", "c"),
  c(8.07131, 1730.63, 233.426), c(1, 100)
)

eff_before <- design_efficiency(init_des, result_opt)
eff_after  <- design_efficiency(augmented, result_opt)

cat("Efficiency before augmenting:", round(eff_before * 100, 2), "%\n")
cat("Efficiency after augmenting: ", round(eff_after  * 100, 2), "%\n")
cat("Gain:                        ", round((eff_after - eff_before) * 100, 2),
    "percentage points\n")

Using the optimal design as reference (calc_optimal_design = TRUE)

When calc_optimal_design = TRUE, the function internally computes the optimal design and uses it to define the efficiency threshold. This is the recommended mode when no optimal design has been computed yet:

region_opt <- get_augment_region(
  criterion           = "D-Optimality",
  init_design         = init_des,
  alpha               = 0.25,
  model               = y ~ 10^(a - b / (c + x)),
  parameters          = c("a", "b", "c"),
  par_values          = c(8.07131, 1730.63, 233.426),
  design_space        = c(1, 100),
  calc_optimal_design = TRUE,
  delta_val           = 0.85
)

Two-factor augmentation

In multi-dimensional spaces get_augment_region() samples candidate points with a Latin Hypercube (controlled by n_lhs) and returns a data frame of candidates together with their estimated efficiency gain. A heatmap of the efficiency function is displayed automatically.

Initial design and candidate region

init_2d <- data.frame(
  x1     = c(0.8, 10, 5),
  x2     = c(10, 0.8, 5),
  Weight = c(1/3, 1/3, 1/3)
)

result_2D <- opt_des(
  criterion    = "D-Optimality",
  model        = y ~ Vmax * x1 * x2 / ((K1 + x1) * (K2 + x2)),
  parameters   = c("Vmax", "K1", "K2"),
  par_values   = c(1, 1, 1),
  design_space = list(x1 = c(0.1, 10), x2 = c(0.1, 10))
)

region_2d <- get_augment_region(
  criterion           = "D-Optimality",
  init_design         = init_2d,
  alpha               = 0.25,
  model               = y ~ Vmax * x1 * x2 / ((K1 + x1) * (K2 + x2)),
  parameters          = c("Vmax", "K1", "K2"),
  par_values          = c(1, 1, 1),
  design_space        = list(x1 = c(0.1, 10), x2 = c(0.1, 10)),
  calc_optimal_design = FALSE,
  delta_val           = 0.85
)

region_2d$region is a data frame of sampled candidates, each with an efficiency column. Pick the candidate that maximises efficiency:

best_2d <- region_2d$region[which.max(region_2d$region$efficiency), ]

eff_antes <- suppressMessages(design_efficiency(init_2d, result_2D))

aug_2d <- augment_design(
  criterion           = "D-Optimality",
  init_design         = init_2d,
  alpha               = 0.25,
  model               = y ~ Vmax * x1 * x2 / ((K1 + x1) * (K2 + x2)),
  parameters          = c("Vmax", "K1", "K2"),
  par_values          = c(1, 1, 1),
  design_space        = list(x1 = c(0.1, 10), x2 = c(0.1, 10)),
  calc_optimal_design = FALSE,
  delta_val           = 0.85,
  new_points          = data.frame(x1 = best_2d$x1, x2 = best_2d$x2, Weight = 1)
)

eff_despues <- suppressMessages(design_efficiency(aug_2d, result_2D))

cat("Efficiency before:", round(eff_antes  * 100, 2), "%\n")
cat("Efficiency after: ", round(eff_despues * 100, 2), "%\n")
print(aug_2d)

Three-factor augmentation

For three or more factors the candidate region is displayed as a scatter-matrix coloured by candidate/non-candidate status, with the current design shown as triangles.

init_3d <- data.frame(
  x1     = c(0.8, 10,  10,  0.8, 10),
  x2     = c(10,  0.8, 10,  10,  0.8),
  x3     = c(10,  10,  0.8, 0.8, 10),
  Weight = rep(0.2, 5)
)

region_3d <- get_augment_region(
  criterion           = "D-Optimality",
  init_design         = init_3d,
  alpha               = 0.45,
  model               = y ~ Vmax * x1 * x2 * x3 / ((K1+x1) * (K2+x2) * (K3+x3)),
  parameters          = c("Vmax", "K1", "K2", "K3"),
  par_values          = c(1, 1, 1, 1),
  design_space        = list(x1 = c(0.1, 10), x2 = c(0.1, 10), x3 = c(0.1, 10)),
  calc_optimal_design = FALSE,
  delta_val           = 0.93
)
cat("Number of candidate points:", nrow(region_3d$region), "\n")
plot(region_3d$plot)

Augmenting with Ds-Optimality

When the goal is to augment while preserving estimation quality for a subset of parameters, use criterion = "Ds-Optimality" and pass par_int:

region_ds <- get_augment_region(
  criterion           = "Ds-Optimality",
  init_design         = init_2d,
  alpha               = 0.25,
  model               = y ~ Vmax * x1 * x2 / ((K1 + x1) * (K2 + x2)),
  parameters          = c("Vmax", "K1", "K2"),
  par_values          = c(1, 1, 1),
  design_space        = list(x1 = c(0.1, 10), x2 = c(0.1, 10)),
  calc_optimal_design = FALSE,
  par_int             = c(1),
  delta_val           = 0.85,
  n_lhs               = 5000
)

best_ds <- region_ds$region[which.max(region_ds$region$efficiency), ]

aug_ds <- augment_design(
  criterion           = "Ds-Optimality",
  init_design         = init_2d,
  alpha               = 0.25,
  model               = y ~ Vmax * x1 * x2 / ((K1 + x1) * (K2 + x2)),
  parameters          = c("Vmax", "K1", "K2"),
  par_values          = c(1, 1, 1),
  design_space        = list(x1 = c(0.1, 10), x2 = c(0.1, 10)),
  calc_optimal_design = FALSE,
  par_int             = c(1),
  delta_val           = 0.85,
  new_points          = data.frame(x1 = best_ds$x1, x2 = best_ds$x2, Weight = 1),
  n_lhs               = 5000
)
print(aug_ds)

Interactive mode

Omitting new_points (and delta_val) from both functions triggers an interactive session where the package plots the candidate region and asks the user to type a point. This mode is documented in ?augment_design.

optedr documentation built on June 23, 2026, 5:07 p.m.