### Install DoseFinding from GitHub

Source: https://github.com/openpharma/dosefinding/blob/master/README.md

Installs the development version of the DoseFinding package from GitHub. Requires the devtools package.

```r
# install.packages("devtools")
devtools::install_github("openpharma/DoseFinding")
```

--------------------------------

### Get default bounds for non-linear model parameters with defBnds()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Call `defBnds()` to obtain default lower and upper bounds for nonlinear model parameters, scaled to the study's maximum dose. These bounds are essential for internal use by fitting functions like `fitMod()` and `MCPMod()`.

```r
library(DoseFinding)

# Bounds for a study with maximum dose 200
bnds <- defBnds(mD = 200)
bnds$emax        # c(0.2, 300) — bounds for ED50
bnds$logistic    # 2x2 matrix: rows = ED50, delta
bnds$sigEmax
bnds$exponential
bnds$betaMod
```

--------------------------------

### Define and use built-in dose-response model functions

Source: https://context7.com/openpharma/dosefinding/llms.txt

The DoseFinding package provides direct access to its built-in dose-response model functions, such as `emax()`, `sigEmax()`, `logistic()`, `betaMod()`, `exponential()`, `quadratic()`, and `linlog()`. These can be used for custom calculations, simulations, or visualizations.

```r
library(DoseFinding)

dose <- seq(0, 150, by = 5)

# Emax model: f(d) = e0 + eMax * d / (ed50 + d)
emax(dose, e0 = 0, eMax = 0.4, ed50 = 25)
```

```r
# Sigmoid Emax: f(d) = e0 + eMax / (1 + (ed50/d)^h)
sigEmax(dose, e0 = 0, eMax = 0.4, ed50 = 50, h = 2)
```

```r
# Logistic: S-shaped with left asymptote e0, right asymptote e0+eMax
logistic(dose, e0 = 0, eMax = 0.4, ed50 = 50, delta = 10)
```

```r
# Beta model (unimodal with fixed scal parameter)
betaMod(dose, e0 = 0, eMax = 0.4, delta1 = 0.33, delta2 = 2.31, scal = 200)
```

```r
# Exponential: f(d) = e0 + e1*(exp(d/delta) - 1)
exponential(dose, e0 = 0, e1 = 0.03, delta = 85)
```

```r
# Quadratic: f(d) = e0 + b1*d + b2*d^2
quadratic(dose, e0 = 0, b1 = 0.005, b2 = -0.00002)
```

```r
# Linear in log: f(d) = e0 + delta * log(d + off)
linlog(dose, e0 = 0, delta = 0.2, off = 1)
```

--------------------------------

### Calculate Sample Size with sampSizeMCT() and sampSize()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Calculates the required sample size for multiple contrast tests or general target functions using bisection search. `sampSizeMCT` is a convenience wrapper for MCT.

```r
library(DoseFinding)

fmodels <- Mods(linear = NULL, emax = 25,
                logistic    = c(50, 10.88111),
                exponential = 85,
                betaMod     = rbind(c(0.33, 2.31), c(1.39, 1.39)),
                doses       = c(0, 10, 25, 50, 100, 150),
                placEff     = 0, maxEff = 0.4,
                addArgs     = list(scal = 200))
contMat <- optContr(fmodels, w = 1)

# Convenience wrapper: target 80% mean power, balanced allocation
sSize <- sampSizeMCT(upperN   = 100,
                     contMat  = contMat,
                     sigma    = 1,
                     altModels = fmodels,
                     power    = 0.8,
                     alRatio  = rep(1, 6),
                     alpha    = 0.05)
sSize

```

```r
# General sampSize: custom target function (minimum power over all models)
tFunc <- function(n) min(powMCT(contMat, altModels = fmodels,
                                n = n, sigma = 1, alpha = 0.05))
sampSize(upperN = 150, targFunc = tFunc, target = 0.8,
         alRatio = rep(1, 6), verbose = TRUE)
```

--------------------------------

### Evaluate operating characteristics of a design using planMod()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Use `planMod()` to assess design performance metrics like RMSE, power, and probability of correct EDp identification. It supports asymptotic approximations for speed or simulations for accuracy, and requires a set of candidate models and study parameters.

```r
library(DoseFinding)

doses   <- c(0, 10, 25, 50, 100, 150)
fmodels <- Mods(linear = NULL, emax = 25,
                logistic    = c(50, 10.88111),
                exponential = 85,
                doses       = doses,
                addArgs     = list(scal = 200),
                placEff     = 0, maxEff = 0.4)

# Asymptotic approximation (single model, fast)
plan <- planMod(model      = "emax",
                altModels  = fmodels,
                n          = 50,
                sigma      = 1,
                doses      = doses,
                asyApprox  = TRUE,
                simulation = FALSE)
print(plan)   # dRMSE, Pow(maxDose), P(EDp)
summary(plan) # additional cRMSE, Eff-vs-ANOVA metrics
plot(plan)
```

--------------------------------

### Perform Multiple Contrast Tests

Source: https://github.com/openpharma/dosefinding/blob/master/README.md

Performs model-based multiple contrast tests using candidate dose-response shapes. Requires the DoseFinding library and data. Set a random seed for reproducible adjusted p-values.

```r
library(DoseFinding)
data(IBScovars)

## set random seed to ensure reproducible adj. p-values for multiple contrast test
set.seed(12)

## perform (model based) multiple contrast test
## define candidate dose-response shapes
models <- Mods(linear = NULL, emax = 0.2, quadratic = -0.17,
               doses = c(0, 1, 2, 3, 4))
## plot models
plot(models)

## perform multiple contrast test
MCTtest(dose, resp, IBScovars, models=models,
                addCovars = ~ gender)
```

--------------------------------

### Bayesian Dose-Response Model Fitting (bFitMod)

Source: https://context7.com/openpharma/dosefinding/llms.txt

Fits dose-response models using MCMC sampling (Bayes) or parametric bootstrap. Input requires dose-group estimates and their covariance matrix from a first-stage fit.

```r
library(DoseFinding)
data(biom)

# First-stage ANOVA fit
anMod <- lm(resp ~ factor(dose) - 1, data = biom)
drFit <- coef(anMod)
S     <- vcov(anMod)
dose  <- sort(unique(biom$dose))

# Bayesian Emax model fit
prior <- list(
  norm = c(0, 10),          # E0: N(0, 10)
  norm = c(0, 100),         # Emax: N(0, 100)
  beta = c(0, 1.5, 0.45, 1.7)  # ED50: Beta on [0, 1.5]
)
gsample <- bFitMod(dose, drFit, S,
                   model = "emax",
                   start = c(0, 1, 0.1),
                   nSim  = 2000,
                   prior = prior)
gsample
head(gsample$samples)     # raw MCMC samples (nSim x n_params)

# Posterior quantiles at specified doses
predict(gsample, doseSeq = c(0, 0.2, 0.5, 1))

# Posterior distribution of TD (target dose)
TD(gsample, Delta = 0.2)  # returns a vector of nSim TD samples

# Plot posterior bands
plot(gsample)

# Bootstrap approach
gsample_bs <- bFitMod(dose, drFit, S,
                      model = "emax",
                      type  = "bootstrap",
                      nSim  = 500,
                      bnds  = defBnds(1)$emax)
plot(gsample_bs)
```

--------------------------------

### Bayesian Multiple Contrast Test (bMCTtest)

Source: https://context7.com/openpharma/dosefinding/llms.txt

Implements the Bayesian MCP-Mod multiple contrast test. Requires RBesT package. The threshold can be derived from a frequentist alpha level or supplied directly.

```r
library(DoseFinding)
if (require("RBesT")) {
  data(biom)
  doses   <- c(0, 0.05, 0.2, 0.6, 1)
  modlist <- Mods(emax = 0.05, linear = NULL,
                  logistic = c(0.5, 0.1),
                  linInt   = c(0, 1, 1, 1),
                  doses    = doses)

  # Informative prior for placebo, vague for all active doses
  plc_prior   <- mixnorm(inf = c(0.8, 0.4, 0.1), rob = c(0.2, 0.4, 10))
  vague_prior <- mixnorm(c(1, 0, 10))
  prior <- list(plc_prior, vague_prior, vague_prior, vague_prior, vague_prior)

  # Run Bayesian MCT test (threshold derived from frequentist alpha=0.025)
  m1 <- bMCTtest(dose, resp, biom, models = modlist, prior = prior)
  print(m1)

  # Supply threshold directly
  m2 <- bMCTtest(dose, resp, biom, models = modlist,
                 prior = prior, critV = 0.99)
}
```

--------------------------------

### Define Candidate Dose-Response Models with Mods()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Creates a 'Mods' object to define a set of candidate dose-response shapes. Standardized or full model parameters can be supplied. Use this to set up models for further analysis.

```R
library(DoseFinding)

doses <- c(0, 10, 25, 50, 100, 150)

# Define a candidate set with multiple model shapes
models <- Mods(
  linear    = NULL,
  emax      = 25,                             # ED50 guesstimate
  logistic  = c(50, 10.88111),               # ED50, delta
  exponential = 85,                           # delta
  betaMod   = rbind(c(0.33, 2.31),
                    c(1.39, 1.39)),           # two betaMod shapes
  linInt    = rbind(c(0, 1, 1, 1, 1),
                    c(0, 0, 1, 1, 0.8)),     # piecewise-linear shapes
  doses     = doses,
  placEff   = 0,
  maxEff    = 0.4,
  addArgs   = list(scal = 200)               # fixed scal for betaMod
)

# Plot all candidate models (lattice)
plot(models)

# Plot using ggplot2
plotMods(models)

# Extract mean responses at selected doses
getResp(models, doses = c(0, 25, 100, 150))
#>
#>          linear      emax  logistic exponential betaMod1 ...
#> dose=0    0.000     0.000     0.000       0.000    0.000
#> dose=25   0.067     0.250     0.082       0.041    0.289
#> dose=100  0.267     0.360     0.379       0.288    0.393
#> dose=150  0.400     0.375     0.400       0.400    0.340

# Target dose: smallest dose achieving an improvement of 0.3 over placebo
TD(models, Delta = 0.3)
#>
#>    linear      emax  logistic exponential betaMod1 betaMod2 linInt1 linInt2
#>    112.50     62.50     88.52      130.27    62.14    62.14    25.00   50.00

# Effective dose: dose giving 50% of the maximum effect
ED(models, p = 0.5)
```

--------------------------------

### Compute guesstimates for model parameters using guesst()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Use `guesst()` to translate clinical knowledge into nonlinear model parameters. It supports Emax, logistic, and beta models, taking dose-effect pairs as input. The output can be used to build and plot candidate models with `Mods()`.

```r
library(DoseFinding)

# Emax: expect 80% of max effect at dose 0.3
emx <- guesst(d = 0.3, p = 0.8, model = "emax")
# Verify: emax(0.3, 0, 1, emx) should be close to 0.8
emax(0.3, 0, 1, emx)
```

```r
# Logistic model: two (dose, effect%) pairs
lgc <- guesst(d = c(0.2, 0.6), p = c(0.2, 0.95), model = "logistic")
```

```r
# Beta model: one pair + location of maximum
bta <- guesst(d = 0.4, p = 0.8, model = "betaMod",
              dMax = 0.8, scal = 1.2, Maxd = 1)
```

```r
# Build and plot candidate set from guesstimates
models <- Mods(emax    = emx,
               logistic = rbind(lgc),
               betaMod  = bta,
               doses    = c(0, 0.2, 0.4, 0.6, 0.8, 1),
               addArgs  = list(scal = 1.2))
plot(models)
```

--------------------------------

### Perform MCP-Mod Procedure with MCPMod

Source: https://context7.com/openpharma/dosefinding/llms.txt

Execute the complete MCP-Mod procedure, including multiple contrast testing, shape selection, dose-response model fitting, model selection, and target dose estimation. Returns an MCPMod object.

```r
library(DoseFinding)
data(biom)

models <- Mods(linear = NULL, emax = c(0.05, 0.2),
               linInt = c(1, 1, 1, 1),
               doses  = c(0, 0.05, 0.2, 0.6, 1))

# Full MCP-Mod (AIC model selection, TD estimation)
MM <- MCPMod(dose, resp, biom, models, Delta = 0.5)
summary(MM)

# Predictions from all significant models
predict(MM, se.fit = TRUE, doseSeq = c(0, 0.2, 0.4, 0.9, 1),
        predType = "ls-means")

# Plot all fitted models
plot(MM, plotData = "meansCI", CI = TRUE)

# Model averaging via aveAIC weights
MM2 <- MCPMod(dose, resp, biom, models, Delta = 0.5, selModel = "aveAIC")
sq   <- seq(0, 1, length = 11)
pred <- predict(MM2, doseSeq = sq, predType = "ls-means")
modWeights <- MM2$selMod
pred_avg <- do.call("cbind", pred) %*% modWeights   # model-averaged predictions
TDEst    <- MM2$doseEst %*% modWeights               # model-averaged dose estimate

# Two-stage approach (placebo-adjusted estimates from ANCOVA)
data(IBScovars)
anovaMod <- lm(resp ~ factor(dose) + gender, data = IBScovars)
drFit    <- coef(anovaMod)[2:5]
vCov     <- vcov(anovaMod)[2:5, 2:5]
dose2    <- sort(unique(IBScovars$dose))[-1]
models2  <- Mods(emax = c(0.5, 1), betaMod = c(1, 1), doses = c(0, 4))
MM3 <- MCPMod(dose2, drFit, S = vCov, models = models2,
              type = "general", placAdj = TRUE, Delta = 0.2)
```

--------------------------------

### MCPMod()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Performs the complete MCP-Mod procedure, including multiple contrast testing, selection of significant shapes, fitting of dose-response models, model selection, and target dose estimation. Returns an MCPMod object.

```APIDOC
## MCPMod() — Full MCP-Mod procedure (testing + modeling)

### Description
Performs the complete MCP-Mod procedure: multiple contrast testing, selection of significant shapes, fitting of dose-response models to those shapes, model selection, and target dose estimation. Returns an `MCPMod` object.

### Example Usage
```r
library(DoseFinding)
data(biom)

models <- Mods(linear = NULL, emax = c(0.05, 0.2),
               linInt = c(1, 1, 1, 1),
               doses  = c(0, 0.05, 0.2, 0.6, 1))

# Full MCP-Mod (AIC model selection, TD estimation)
MM <- MCPMod(dose, resp, biom, models, Delta = 0.5)
summary(MM)

# Predictions from all significant models
predict(MM, se.fit = TRUE, doseSeq = c(0, 0.2, 0.4, 0.9, 1),
        predType = "ls-means")

# Plot all fitted models
plot(MM, plotData = "meansCI", CI = TRUE)

# Model averaging via aveAIC weights
MM2 <- MCPMod(dose, resp, biom, models, Delta = 0.5, selModel = "aveAIC")
sq   <- seq(0, 1, length = 11)
pred <- predict(MM2, doseSeq = sq, predType = "ls-means")
modWeights <- MM2$selMod
pred_avg <- do.call("cbind", pred) %*% modWeights   # model-averaged predictions
TDEst    <- MM2$doseEst %*% modWeights               # model-averaged dose estimate

# Two-stage approach (placebo-adjusted estimates from ANCOVA)
data(IBScovars)
anovaMod <- lm(resp ~ factor(dose) + gender, data = IBScovars)
drFit    <- coef(anovaMod)[2:5]
vCov     <- vcov(anovaMod)[2:5, 2:5]
dose2    <- sort(unique(IBScovars$dose))[-1]
models2  <- Mods(emax = c(0.5, 1), betaMod = c(1, 1), doses = c(0, 4))
MM3 <- MCPMod(dose2, drFit, S = vCov, models = models2,
              type = "general", placAdj = TRUE, Delta = 0.2)
```
```

--------------------------------

### Calculate Conditional/Predictive Power with powMCTInterim()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Calculates conditional or predictive power for multiple contrast tests using interim data. Useful for futility analyses or when only partial data is available.

```r
library(DoseFinding)

doses    <- c(0, 0.5, 1, 2, 4, 8)
mods     <- Mods(emax     = c(0.5, 1, 2, 4),
                 sigEmax  = rbind(c(0.5, 3), c(1, 3), c(2, 3), c(4, 3)),
                 quadratic = -0.1,
                 doses    = doses)
w        <- c(1, 0.5, 0.5, 0.5, 1, 1)
contMat  <- optContr(mods, w = w)$contMat

sigma    <- 0.3
n_final  <- round(531 * w / sum(w))
n        <- floor(n_final / 2)
S_0t     <- diag(sigma^2 / n)         # interim covariance
S_01     <- diag(sigma^2 / n_final)   # final covariance
mu_0t    <- 0.05 * doses / (doses + 1) + rnorm(6, 0, sigma / sqrt(n))
mu_assumed <- 0.135 * doses / (doses + 1)

# Predictive power (flat prior over second-stage data)
powMCTInterim(contMat = contMat, S_0t = S_0t, S_01 = S_01,
              mu_0t   = mu_0t, type = "predictive")

# Conditional power (assuming mu_assumed for second stage)
powMCTInterim(contMat = contMat, S_0t = S_0t, S_01 = S_01,
              mu_0t   = mu_0t, type = "conditional",
              mu_assumed = mu_assumed)
```

--------------------------------

### optContr()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Computes optimal contrast coefficients for detecting specified dose-response alternatives. It supports sample size weights or a covariance matrix, and can handle constrained contrasts and placebo-adjusted formulations.

```APIDOC
## optContr() — Calculate optimal contrast vectors

### Description
Computes contrast coefficients that are optimal (maximize the non-centrality parameter) for detecting specified dose-response alternatives. Accepts either sample size weights (`w`) or a covariance matrix (`S`). Also supports constrained contrasts and placebo-adjusted formulations.

### Example Usage
```r
library(DoseFinding)

doses  <- c(0, 10, 25, 50, 100, 150)
models <- Mods(linear = NULL, emax = 25,
               logistic = c(50, 10.88111), exponential = 85,
               betaMod  = rbind(c(0.33, 2.31), c(1.39, 1.39)),
               doses    = doses, addArgs = list(scal = 200))

# Equal group sizes (balanced design)
contMat <- optContr(models, w = rep(50, 6))
print(contMat)

# Placebo-adjusted contrasts (for two-stage general approach)
dosPlac <- doses[-1]
S <- diag(5) + matrix(1, 5, 5)   # proportional to cov-matrix
contMat0 <- optContr(models, doses = dosPlac, S = S, placAdj = TRUE)
colSums(contMat0$contMat)         # columns do NOT sum to 0 (by design)
```
```

--------------------------------

### Configure R Markdown Vignette Settings

Source: https://github.com/openpharma/dosefinding/blob/master/vignettes/children/settings.txt

Sets up knitr chunk options for rendering vignettes, including graphics device, resolution, and figure aspect ratio. Also sets the default ggplot2 theme.

```r
library(ggplot2)
knitr::opts_chunk$set(echo = TRUE, message = FALSE, cache = FALSE,
                      comment = NA,
                      dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 7, out.width = "85%", fig.align = "center")
options(rmarkdown.html_vignette.check_title = FALSE)
theme_set(theme_bw())
```

--------------------------------

### guesst() — Compute guesstimates for model parameters

Source: https://context7.com/openpharma/dosefinding/llms.txt

Computes nonlinear model parameter estimates from prior knowledge expressed as expected percentages of maximum effect at specific doses. This function helps translate clinical knowledge into model parameters for use in Mods().

```APIDOC
## guesst()

### Description
Computes the nonlinear model parameter θ₂ from prior knowledge expressed as expected percentages of the maximum effect at specific doses. This helps practitioners translate clinical knowledge into model parameters for use in `Mods()`.

### Usage
```r
guesst(d, p, model, dMax, scal, Maxd)
```

### Parameters
* **d** (numeric or numeric vector) - Dose(s).
* **p** (numeric or numeric vector) - Expected percentage(s) of maximum effect at the given dose(s).
* **model** (character) - The name of the dose-response model.
* **dMax** (numeric, optional) - Maximum dose for beta models.
* **scal** (numeric, optional) - Scaling parameter for beta models.
* **Maxd** (numeric, optional) - Dose at which maximum effect is achieved for beta models.

### Examples
```r
# Emax model: expect 80% of max effect at dose 0.3
emx <- guesst(d = 0.3, p = 0.8, model = "emax")

# Logistic model: two (dose, effect%) pairs
lgc <- guesst(d = c(0.2, 0.6), p = c(0.2, 0.95), model = "logistic")

# Beta model: one pair + location of maximum
bta <- guesst(d = 0.4, p = 0.8, model = "betaMod", dMax = 0.8, scal = 1.2, Maxd = 1)
```
```

--------------------------------

### Visualize Power Curves with powN()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Evaluates power or any scalar target function over a range of sample sizes to visualize power curves. Useful for study design planning.

```r
library(DoseFinding)

fmodels <- Mods(linear = NULL, emax = 25,
                logistic    = c(50, 10.88111),
                exponential = 85,
                doses       = c(0, 10, 25, 50, 100, 150),
                placEff     = 0, maxEff = 0.4,
                addArgs     = list(scal = 200))
contMat <- optContr(fmodels, w = 1)

# Power for each model + min/mean/max across n=10,...,100
pVsN <- powN(upperN = 100, lowerN = 10, step = 10,
             contMat   = contMat,
             sigma     = 1,
             altModels = fmodels,
             alpha     = 0.05,
             alRatio   = rep(1, 6))

# Trellis/lattice plot of power curves
plot(pVsN, superpose = TRUE, line.at = 0.8)
```

--------------------------------

### Perform Multiple Contrast Test with MCTtest()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Performs the multiple contrast test (MCT) to detect a dose-response signal. Supports standard ANCOVA ('normal') and a general formulation for non-normal endpoints ('general'). Use this to test for dose-response effects.

```R
library(DoseFinding)
data(biom)  # built-in dataset: 100 patients, doses 0, 0.05, 0.2, 0.6, 1

# Define candidate models
modlist <- Mods(emax    = 0.05,
                linear  = NULL,
                logistic = c(0.5, 0.1),
                linInt  = c(0, 1, 1, 1),
                doses   = c(0, 0.05, 0.2, 0.6, 1))

# Standard normal-ANCOVA test
m1 <- MCTtest(dose, resp, biom, models = modlist)
print(m1)
#>
#>  Multiple Contrast Test
#>   Contrasts:
#>              emax   linear logistic linInt
#> dose=0.00  -0.616 -0.447   -0.556  -0.447
#> ...
#>   Multiple Contrast Test:
#>             t-Stat adj-p
#> linInt      3.453 0.002
#> emax        3.310 0.003
#> ...

# With covariates
data(IBScovars)
modlist2 <- Mods(emax = 0.05, linear = NULL, doses = c(0, 1, 2, 3, 4))
MCTtest(dose, resp, IBScovars, models = modlist2, addCovars = ~gender)

# General approach: supply pre-computed estimates and covariance matrix
ancMod <- lm(resp ~ factor(dose) + gender, data = IBScovars)
drEst <- coef(ancMod)[2:5]        # placebo-adjusted estimates
vc    <- vcov(ancMod)[2:5, 2:5]
MCTtest(1:4, drEst, S = vc, models = modlist2, placAdj = TRUE,
        type = "general", df = Inf)

# Retrieve multiplicity-adjusted p-values
attr(m1$tStat, "pVal")
```

--------------------------------

### planMod() — Evaluate operating characteristics of a design

Source: https://context7.com/openpharma/dosefinding/llms.txt

Evaluates performance metrics for a model-based dose-response analysis under different true scenarios, including RMSE for dose-response curve estimation, power to detect an effect at the top dose, and probability of correctly identifying the EDp. Uses either asymptotic approximations or simulation.

```APIDOC
## planMod()

### Description
Evaluates performance metrics for a model-based dose-response analysis under different true scenarios: RMSE for dose-response curve estimation, power to detect an effect at the top dose, and probability of correctly identifying the EDp. Uses either asymptotic approximations or simulation.

### Usage
```r
planMod(model, altModels, n, sigma, doses, asyApprox, simulation)
```

### Parameters
* **model** (character) - The name of the candidate model.
* **altModels** (list or object) - A list of alternative models to compare against.
* **n** (numeric) - Sample size.
* **sigma** (numeric) - Standard deviation of the error term.
* **doses** (numeric vector) - The doses to be used in the study.
* **asyApprox** (logical) - Whether to use asymptotic approximation.
* **simulation** (logical) - Whether to perform simulation.

### Examples
```r
doses   <- c(0, 10, 25, 50, 100, 150)
fmodels <- Mods(linear = NULL, emax = 25,
                logistic    = c(50, 10.88111),
                exponential = 85,
                doses       = doses,
                addArgs     = list(scal = 200),
                placEff     = 0, maxEff = 0.4)

# Asymptotic approximation (single model, fast)
plan <- planMod(model      = "emax",
                altModels  = fmodels,
                n          = 50,
                sigma      = 1,
                doses      = doses,
                asyApprox  = TRUE,
                simulation = FALSE)
print(plan)   # dRMSE, Pow(maxDose), P(EDp)
summary(plan) # additional cRMSE, Eff-vs-ANOVA metrics
plot(plan)
```
```

--------------------------------

### defBnds() — Default bounds for non-linear model parameters

Source: https://context7.com/openpharma/dosefinding/llms.txt

Returns a list of reasonable lower and upper bounds for the nonlinear parameters of each built-in model, scaled to the maximum dose in the study. These bounds are used internally by fitMod(), MCPMod(), and bFitMod().

```APIDOC
## defBnds()

### Description
Returns a list of reasonable lower/upper bounds for the nonlinear parameters of each built-in model, scaled to the maximum dose in the study. These bounds are used internally by `fitMod()`, `MCPMod()`, and `bFitMod()`.

### Usage
```r
defBnds(mD, model)
```

### Parameters
* **mD** (numeric) - Maximum dose in the study.
* **model** (character, optional) - The name of the model for which to get bounds. If NULL, bounds for all models are returned.

### Examples
```r
# Bounds for a study with maximum dose 200
bnds <- defBnds(mD = 200)
bnds$emax        # c(0.2, 300) — bounds for ED50
bnds$logistic    # 2x2 matrix: rows = ED50, delta
bnds$sigEmax
bnds$exponential
bnds$betaMod
```
```

--------------------------------

### Calculate Optimal Contrast Vectors with optContr

Source: https://context7.com/openpharma/dosefinding/llms.txt

Use optContr to compute contrast coefficients that are optimal for detecting specified dose-response alternatives. It accepts sample size weights (w) or a covariance matrix (S) and supports constrained contrasts and placebo-adjusted formulations.

```r
library(DoseFinding)

doses  <- c(0, 10, 25, 50, 100, 150)
models <- Mods(linear = NULL, emax = 25,
               logistic = c(50, 10.88111), exponential = 85,
               betaMod  = rbind(c(0.33, 2.31), c(1.39, 1.39)),
               doses    = doses, addArgs = list(scal = 200))

# Equal group sizes (balanced design)
contMat <- optContr(models, w = rep(50, 6))
print(contMat)
#> Optimal contrasts
#>           linear    emax  logistic exponential betaMod1 betaMod2
#> dose=0   -0.408  -0.342    -0.378      -0.289   -0.342   -0.342
#> ...

# Plot contrasts (lattice)
plot(contMat)
# Plot using ggplot2
plotContr(contMat)

# Placebo-adjusted contrasts (for two-stage general approach)
dosPlac <- doses[-1]
S <- diag(5) + matrix(1, 5, 5)   # proportional to cov-matrix
contMat0 <- optContr(models, doses = dosPlac, S = S, placAdj = TRUE)
colSums(contMat0$contMat)         # columns do NOT sum to 0 (by design)
```

--------------------------------

### fitMod()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Fits a non-linear dose-response model to observed data using ordinary or generalized least squares. Supports various built-in models and returns a DRMod object with methods for coefficients, variance-covariance, prediction, plotting, and model fit statistics.

```APIDOC
## fitMod() — Fit a non-linear dose-response model

### Description
Fits one of the built-in dose-response models ("linear", "quadratic", "emax", "sigEmax", "logistic", "exponential", "betaMod", "linlog", "linInt") to observed data via ordinary or generalized least squares. Returns a `DRMod` object with `coef`, `vcov`, `predict`, `plot`, `AIC`, and `logLik` methods.

### Example Usage
```r
library(DoseFinding)
data(IBScovars)

# Fit Emax model (normal OLS)
fitemax <- fitMod(dose, resp, data = IBScovars, model = "emax",
                  bnds = c(0.01, 4))
summary(fitemax)

# Predictions with standard errors
predict(fitemax, predType = "ls-means", doseSeq = c(0, 1, 2, 4), se.fit = TRUE)

# Target dose from fitted model
TD(fitemax, Delta = 0.2)
ED(fitemax, p = 0.5)

# Plot with confidence band and dose-group means+CIs
plot(fitemax, plotData = "meansCI", CI = TRUE)

# General approach for binary data (two-stage)
data(migraine)
PFrate     <- migraine$painfree / migraine$ntrt
doseVec    <- migraine$dose
fitBin     <- glm(PFrate ~ as.factor(doseVec) - 1, family = binomial,
                  weights = migraine$ntrt)
gEst <- coef(fitBin)
vCov  <- vcov(fitBin)
gfit  <- fitMod(doseVec, gEst, S = vCov, model = "emax",
                bnds = c(0, 100), type = "general")
gAIC(gfit)

# Include covariate
fitemax2 <- fitMod(dose, resp, data = IBScovars, model = "emax",
                   addCovars = ~gender, bnds = c(0.01, 4))
AIC(fitemax2)
```
```

--------------------------------

### Fit Dose-Response Models with fitMod

Source: https://context7.com/openpharma/dosefinding/llms.txt

Fit various dose-response models to observed data using ordinary or generalized least squares. The function returns a DRMod object with methods for coefficients, variance-covariance, prediction, plotting, and model fit statistics.

```r
library(DoseFinding)
data(IBScovars)

# Fit Emax model (normal OLS)
fitemax <- fitMod(dose, resp, data = IBScovars, model = "emax",
                  bnds = c(0.01, 4))
summary(fitemax)
#> Dose Response Model
#> Model: emax  Fit-type: normal
#> Coefficients with approx. stand. error:
#>       Estimate Std. Error
#> e0      0.372      0.059
#> eMax    0.451      0.118
#> ed50    0.789      0.380

coef(fitemax)
vcov(fitemax)

# Predictions with standard errors
predict(fitemax, predType = "ls-means", doseSeq = c(0, 1, 2, 4), se.fit = TRUE)

# Target dose from fitted model
TD(fitemax, Delta = 0.2)
ED(fitemax, p = 0.5)

# Plot with confidence band and dose-group means+CIs
plot(fitemax, plotData = "meansCI", CI = TRUE)

# General approach for binary data (two-stage)
data(migraine)
PFrate     <- migraine$painfree / migraine$ntrt
doseVec    <- migraine$dose
fitBin     <- glm(PFrate ~ as.factor(doseVec) - 1, family = binomial,
                  weights = migraine$ntrt)
drEst <- coef(fitBin)
vCov  <- vcov(fitBin)
gfit  <- fitMod(doseVec, drEst, S = vCov, model = "emax",
                bnds = c(0, 100), type = "general")
gAIC(gfit)

# Include covariate
fitemax2 <- fitMod(dose, resp, data = IBScovars, model = "emax",
                   addCovars = ~gender, bnds = c(0.01, 4))
AIC(fitemax2)
```

--------------------------------

### Bootstrap Model Averaging (maFitMod)

Source: https://context7.com/openpharma/dosefinding/llms.txt

Fits dose-response models using a bagging (bootstrap model averaging) approach. Selects the best-fitting model by gAIC for each bootstrap sample. Supports predict, plot, TD, and ED methods.

```r
library(DoseFinding)
data(biom)

anMod <- lm(resp ~ factor(dose) - 1, data = biom)
drFit <- coef(anMod)
S     <- vcov(anMod)
dose  <- sort(unique(biom$dose))

mFit <- maFitMod(dose, drFit, S,
                 models = c("emax", "sigEmax", "linear"),
                 nSim   = 500)
mFit

# Median prediction across bootstrap models
plot(mFit, plotData = "meansCI")

# Effective dose (p=0.9) using median ensemble prediction
ED(mFit, direction = "increasing", p = 0.9)
```

--------------------------------

### Calculate Optimal Experimental Design with optDesign()

Source: https://context7.com/openpharma/dosefinding/llms.txt

Calculates D-optimal or TD-optimal experimental designs for estimating dose-response model parameters or target doses. It determines optimal allocation weights to candidate doses.

```r
library(DoseFinding)

doses  <- c(0, 10, 25, 50, 100, 150)
models <- Mods(linear = NULL, emax = 25,
               logistic = c(50, 10.88111),
               exponential = 85,
               doses   = doses, addArgs = list(scal = 200),
               placEff = 0, maxEff = 0.4)

# Equal model probabilities
probs <- rep(1/5, 5)

# D-optimal design (maximize determinant of information matrix)
des_D <- optDesign(models, probs = probs, designCrit = "Dopt")
des_D

```

```r
# TD-optimal design
des_TD <- optDesign(models, probs = probs,
                    designCrit = "TD", Delta = 0.3)

# Mixture of D-optimal and TD-optimal
des_mix <- optDesign(models, probs = probs,
                     designCrit = "Dopt&TD", Delta = 0.3)

# Plot design overlaid on candidate models
plot(des_D, models)

# Round to finite sample size (total N=150)
rndDesign(des_D, n = 150)
```

--------------------------------

### Power of Multiple Contrast Test (powMCT)

Source: https://context7.com/openpharma/dosefinding/llms.txt

Calculates the power to detect a dose-response signal for a given contrast matrix, sample size, and standard deviation. Supports normal balanced/unbalanced designs or general covariance matrices.

```r
library(DoseFinding)

doses   <- c(0, 10, 25, 50, 100, 150)
fmodels <- Mods(linear = NULL, emax = 25,
                logistic    = c(50, 10.88111),
                exponential = 85,
                betaMod     = rbind(c(0.33, 2.31), c(1.39, 1.39)),
                doses       = doses,
                addArgs     = list(scal = 200),
                placEff     = 0, maxEff = 0.4)

contMat <- optContr(fmodels, w = 1)

# Power for balanced design, 50 patients/arm, sigma=1
powMCT(contMat, altModels = fmodels, n = 50, alpha = 0.05, sigma = 1)
#>    linear      emax  logistic exponential betaMod1 betaMod2
#>     0.872     0.913     0.878       0.769    0.911    0.902

# Power using a general covariance matrix
S <- diag(6) / 50
powMCT(contMat, altModels = fmodels, S = S, alpha = 0.05, df = 6*50 - 6)
```

--------------------------------

### Calculate Optimal Designs for Dose Estimation

Source: https://github.com/openpharma/dosefinding/blob/master/README.md

Calculates optimal designs for target dose (TD) estimation using specified dose levels and candidate models. The plot function can visualize these designs, and optDesign computes the optimal allocation of subjects.

```r
## Calculate optimal designs for target dose (TD) estimation
doses <- c(0, 10, 25, 50, 100, 150)
fmodels <- Mods(linear = NULL, emax = 25, exponential = 85,
                logistic = c(50, 10.8811),
                doses = doses, placEff=0, maxEff=0.4)
plot(fmodels, plotTD = TRUE, Delta = 0.2)

weights <- rep(1/4, 4)
optDesign(fmodels, weights, Delta=0.2, designCrit="TD")
```

--------------------------------

### Fit Non-linear Emax Dose-Response Model

Source: https://github.com/openpharma/dosefinding/blob/master/README.md

Fits a non-linear Emax dose-response model to the data. The function fitMod is used for this purpose, and the results can be visualized with plot, including confidence intervals and data means.

```r
## fit non-linear emax dose-response model
fitemax <- fitMod(dose, resp, data=IBScovars, model="emax",
                  bnds = c(0.01,5))
## display fitted dose-effect curve
plot(fitemax, CI=TRUE, plotData="meansCI")
```