### Run Package Examples

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Executes example code for plotting DirichletRegData and fitting DirichReg models.

```R
example(plot.DirichletRegData)
example(DirichReg)
```

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

### Summary of Dirichlet Regression Model Example

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Generates and prints a summary of the Dirichlet regression model 'mod1'.

```R
summary(mod1)
```

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

### Example: Create and Summarize DirichletRegData

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Demonstrates creating a DirichletRegData object from a subset of the ArcticLake data and then summarizing and viewing the first few entries.

```R
# create a DirichletRegData object from the Arctic Lake data
head(ArcticLake[, 1:3])
AL <- DR_data(ArcticLake[, 1:3])
summary(AL)
head(AL)
```

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

### Print Dirichlet Regression Model Object Example

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints the Dirichlet regression model object 'mod1'.

```R
mod1
```

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

### Log-Likelihood Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts and displays the log-likelihood value for the Dirichlet regression model 'mod1'.

```R
logLik(mod1)
```

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

### Dirichlet Regression Model Example

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Demonstrates fitting a Dirichlet regression model using the 'alternative' model type with transformed data.

```R
ALake <- ArcticLake
ALake$AL <- DR_data(ArcticLake[, 1:3])

mod1 <- DirichReg(AL ~ depth + I(depth^2) | depth, data = ALake, model="alternative")
```

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

### Fitted Values Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts and displays the first few fitted values from the Dirichlet regression model 'mod1'.

```R
head(fitted(mod1))
```

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

### Residuals Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts and displays the first few residuals from the Dirichlet regression model 'mod1'.

```R
head(residuals(mod1))
```

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

### Confidence Intervals Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Computes and displays the default confidence intervals for the parameters of the Dirichlet regression model 'mod1'.

```R
confint(mod1)
```

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

### Exponentiated Confidence Intervals Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Computes and displays the confidence intervals for the parameters of the Dirichlet regression model 'mod1', with parameters returned in exponentiated form.

```R
confint(mod1, exp = TRUE)
```

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

### Covariance Matrix Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts and displays the covariance matrix of the parameter estimates for the Dirichlet regression model 'mod1', rounded to 5 decimal places.

```R
round(vcov(mod1), 5)
```

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

### Prediction Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Predicts values for a new dataset with specified depth values using the Dirichlet regression model 'mod1'.

```R
predict(mod1, newdata = data.frame("depth" = seq(10, 100, 10)))
```

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

### Drop-One Analysis Example for Dirichlet Model

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Performs a drop-one analysis on the Dirichlet regression model 'mod1'. Note: may issue a caveat on first use in an R session.

```R
drop1(mod1)
```

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

### Update Dirichlet Regression Model Formula Example

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Shows how to update the formula of an existing Dirichlet regression model object. The 'evaluate = FALSE' argument prevents immediate evaluation.

```R
update(mod1, . ~ . | . + I(depth^2), evaluate = FALSE)
```

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

### Get Fitted Values from DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts fitted values (expected values, alpha, or phi) from a DirichletRegModel object.

```R
## S3 method for class 'DirichletRegModel'
fitted(object, mu = TRUE, alpha = FALSE, phi = FALSE, ...)
```

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

### Calculate Dirichlet Distribution Densities

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Computes densities for the Dirichlet distribution. Use 'sum.up = TRUE' to get the (log-)likelihood. Returns NaN for non-positive alpha values.

```R
ddirichlet(X1, c(5, 5, 10))
ddirichlet(X2, a.mat)

ddirichlet(X2[1:3,], c(1, 2, -1))
ddirichlet(X2[1:3,], c(1, 2, -1), sum.up = TRUE)
```

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

### Get BibTeX Citation for DirichletReg

Source: https://cran.r-project.org/web/packages/DirichletReg/citation.html

Use this R command to retrieve the BibTeX citation for the DirichletReg package. This is useful for including package references in academic papers using LaTeX.

```r
toBibtex(citation("DirichletReg"))
```

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

### Load and Inspect Arctic Lake Data

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Demonstrates how to load and view the first few rows of the ArcticLake dataset, and prepare it for Dirichlet regression analysis.

```R
head(ArcticLake)
AL <- DR_data(ArcticLake[,1:3])
plot(AL)
summary(AL)
```

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

### Fit Dirichlet Regression Models

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Demonstrates fitting common and alternative Dirichlet regression models. Use 'model="alternative"' for precision parameter modeling.

```R
ALake <- ArcticLake
ALake$Y <- DR_data(ALake[,1:3])

# fit a quadratic Dirichlet regression models ("common")
res1 <- DirichReg(Y ~ depth + I(depth^2), ALake)

# fit a Dirichlet regression with quadratic predictor for the mean and
# a linear predictor for precision ("alternative")
res2 <- DirichReg(Y ~ depth + I(depth^2) | depth, ALake, model="alternative")

# test both models
anova(res1, res2)

res1
summary(res2)
```

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

### Load and Inspect Blood Samples Data

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Shows how to load and view the first few rows of the BloodSamples dataset, and prepare it for Dirichlet regression analysis.

```R
head(BloodSamples)
Bl <- DR_data(BloodSamples[,1:4])
summary(Bl)
```

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

### Load and Prepare Arctic Lake Data

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Loads the ArcticLake dataset and prepares it for Dirichlet regression by converting the first three columns into a Dirichlet distribution data format.

```R
library("DirichletReg")
head(ArcticLake)

AL <- DR_data(ArcticLake[,1:3])

AL[1:6,]
```

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

### Prepare and Predict with Dirichlet Regression for Arctic Lake

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Prepares the ArcticLake data for a quadratic Dirichlet regression model and generates predictions for a sequence of depths.

```R
AL <- ArcticLake
AL$AL <- DR_data(ArcticLake[,1:3])

dd <- range(ArcticLake$depth)
X <- data.frame(depth=seq(dd[1], dd[2], length.out=200))

pp <- predict(DirichReg(AL~depth+I(depth^2),AL), X)
```

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

### Quick Analyses with DR_data in DirichletReg

Source: https://cran.r-project.org/web/packages/DirichletReg/news.html

This demonstrates a quick analysis using DR_data for preliminary checks. Note that this functionality is intended for temporary use and may be removed in future versions.

```R
DirichReg(DR_data(A) ~ 1)
```

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

### Visualize Arctic Lake Quadratic Model Fit

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Plots the observed sediment composition against depth and overlays the fitted values from the quadratic Dirichlet regression model, including the precision parameter.

```R
par(mar=c(4, 4, 4, 4)+0.1)
plot(rep(ArcticLake$depth,3),as.numeric(AL),
     pch=21, bg=rep(c("#E495A5", "#86B875", "#7DB0DD"),each=39),
     xlab="Depth (m)", ylab="Proportion", ylim=0:1, main="Sediment Composition in an Arctic Lake")

Xnew <- data.frame(depth=seq(min(ArcticLake$depth), max(ArcticLake$depth), length.out=100))
for(i in 1:3)lines(cbind(Xnew, predict(lake2, Xnew)[,i]),col=c("#E495A5", "#86B875", "#7DB0DD")[i],lwd=2)
legend("topleft", legend=c("Sand","Silt","Clay"), lwd=2, col=c("#E495A5", "#86B875", "#7DB0DD"), pt.bg=c("#E495A5", "#86B875", "#7DB0DD"), pch=21,bty="n")

par(new=TRUE)
plot(cbind(Xnew, predict(lake2, Xnew, F,F,T)), lty="24", type="l",ylim=c(0,max(predict(lake2, Xnew, F,F,T)))),axes=F,ann=F,lwd=2)
axis(4)
mtext(expression(paste("Precision (",phi,")",sep="")), 4, line=3)
legend("top",legend=c(expression(hat(mu[c]==hat(alpha)[c]/hat(alpha)[0])),expression(hat(phi)==hat(alpha)[0]))),lty=c(1,2),lwd=c(3,2),bty="n")
```

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

### Visualize Blood Sample Box Plots and Fitted Values

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Generates box plots for each component of the blood sample composition, stratified by disease type, and overlays fitted values from the Dirichlet regression model.

```R
par(mfrow=c(1,4), mar=c(4,4,4,2)+.25)
for(i in 1:4){
  boxplot(Bld$Smp[,i]~Bld$Disease, ylim=range(Bld$Smp[,1:4]), main=paste(names(Bld)[i]), xlab="Disease Type", ylab="Proportion")
  segments(c(-5,1.5),unique(fitted(blood2)[,i]),
           c(1.5,5),unique(fitted(blood2)[,i]),lwd=2,lty=2)
}
```

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

### Fit and Compare Dirichlet Regression Models for Arctic Lake

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Fits a simple Dirichlet regression model and a quadratic model to the Arctic Lake data, then compares them using ANOVA.

```R
lake1 <- DirichReg(AL~depth, ArcticLake)
lake1

coef(lake1)

lake2 <- update(lake1, .~.+I(depth^2)|.+I(depth^2)|.+I(depth^2))
anova(lake1,lake2)

summary(lake2)
```

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

### Arctic Lake Data (Aitchison)

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Dataset containing sediment composition and depth information from Aitchison (2003). Used for analyzing the relation of sediment composition with depth in an Arctic lake.

```APIDOC
## ArcticLake

### Description
These data are taken from Aitchison (2003) and contain information on the relation of sediment composition with depth in an Arctic lake.

### Format
A data frame with 39 observations on the following 4 variables:

`sand`, `silt`, `clay`
    
relative frequencies of sand, silt, and clay 

`depth`
    
water depth in meters

### Source
Aitchison, J. (2003). _The Statistical Analysis of Compositional Data._ The Blackburn Press, Caldwell, NJ.

### Examples
```R
head(ArcticLake)
AL <- DR_data(ArcticLake[,1:3])
plot(AL)
summary(AL)
```
```

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

### Fit and Compare Dirichlet Regression Models for Blood Samples

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Fits two Dirichlet regression models to the BloodSamples data using the 'alternative' parametrization, one with a simple disease effect and another with a disease-specific effect, and compares them using ANOVA.

```R
Bld <- BloodSamples
Bld$Smp <- DR_data(Bld[,1:4])

blood1 <- DirichReg(Smp~Disease|1, Bld, model="alternative", base=3)
blood2 <- DirichReg(Smp~Disease|Disease, Bld, model="alternative", base=3)
anova(blood1, blood2)

summary(blood1)
```

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

### Compare OLS and Dirichlet Regression Predictions for Arctic Lake

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Visualizes the predictions from both OLS and Dirichlet regression models for the Arctic Lake data, comparing their performance on a ternary plot.

```R
plot(AL$AL, cex=.1, reset_par=FALSE)
points(toSimplex(AL$AL), pch=16, cex=.5, col=gray(.5))

lines(toSimplex(pp), lwd=3, col=c("#6E1D34", "#004E42")[2])

Dols <- log(cbind(ArcticLake[,2]/ArcticLake[,1],
                  ArcticLake[,3]/ArcticLake[,1]))

ols <- lm(Dols~depth+I(depth^2), ArcticLake)
p2 <- predict(ols, X)
p2m <- exp(cbind(0,p2[,1],p2[,2]))/rowSums(exp(cbind(0,p2[,1],p2[,2])))

lines(toSimplex(p2m), lwd=3, col=c("#6E1D34", "#004E42")[1], lty="21")
```

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

### Model Summaries and Confidence Intervals

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

This snippet shows how to obtain summary statistics for a linear regression model (rlr) and a Dirichlet regression model (rs2), as well as calculate confidence intervals for the Dirichlet regression parameters, both on the original scale and exponentiated.

```R
a <- RS$accuracy
logRa_a <- log(a/(1-a))
rlr <- lm(logRa_a~dyslexia*iq, RS)
summary(rlr)
```

```R
summary(rs2)
```

```R
confint(rs2)
```

```R
confint(rs2, exp=TRUE)
```

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

### Fitting Dirichlet Regression Models

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

This snippet demonstrates fitting two Dirichlet regression models with different parameterizations ('alternative' model) and comparing them using an ANOVA test. Ensure the 'DR_data' function is available for data transformation.

```R
RS <- ReadingSkills
RS$acc <- DR_data(RS$accuracy)

RS$dyslexia <- C(RS$dyslexia, treatment)
rs1 <- DirichReg(acc ~ dyslexia*iq | dyslexia*iq, RS, model="alternative")
rs2 <- DirichReg(acc ~ dyslexia*iq | dyslexia+iq, RS, model="alternative")
anova(rs1,rs2)
```

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

### Serum Protein Composition in Blood Samples

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Dataset containing blood samples' compositions of Albumin, Pre-Albumin, Globulin A, and Globulin B in relation to diseases, from Aitchison (2003).

```APIDOC
## BloodSamples

### Description
These data (Aitchison, 2003) list blood samples' compositions of _Albumin_ , _Pre-Albumin_ , _Globulin A_ , and _Globulin B_ in relation to two types of diseases. 14 patients suffer from disease A, 16 from disease B and 6 are unclassified. 

### Format
A data frame with 36 observations on the following 6 variables.  

`Albumin`, `Pre.Albumin`, `Globulin.A`, `Globulin.B`
    
the amounts of Albumin, Pre-Albumin, Globulin A, and Globulin B. 

`Disease`
    
diagnosis of disease `A`, `B`, or `NA` for unclassified observations. 

`New`
    
a factor indicating whether the observations are old and classified (`No`) or new and unclassified (`Yes`).

### Source
Aitchison, J. (2003). _The Statistical Analysis of Compositional Data._ The Blackburn Press, Caldwell, NJ. 

### Examples
```R
head(BloodSamples)
Bl <- DR_data(BloodSamples[,1:4])
summary(Bl)
```
```

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

### Pure R Versions of Dirichlet Density Functions

Source: https://cran.r-project.org/web/packages/DirichletReg/news.html

Pure R versions of the Dirichlet density functions are available for reference or when C-based routines are not preferred. Refer to '?ddirichlet' for more details.

```R
?ddirichlet
```

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

### print.DirichletRegConfint

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints confidence intervals for Dirichlet regression model parameters.

```APIDOC
## print.DirichletRegConfint

### Description
Prints confidence intervals for Dirichlet regression model parameters.

### Usage
```R
print(x, digits = 3, ...)
```

### Arguments
* `x`: An object of class `DirichletRegConfint`.
* `digits`: The number of digits to display.
* `...`: Further arguments.
```

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

### summary.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Provides a summary of the Dirichlet regression model.

```APIDOC
## summary.DirichletRegModel

### Description
Provides a summary of the Dirichlet regression model.

### Usage
```R
summary(object, ...)
```

### Arguments
* `object`: An object of class `DirichletRegModel`.
* `...`: Further arguments.
```

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

### Compare Dirichlet Regression Models

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Perform pairwise likelihood ratio tests (LRT) to compare Dirichlet regression models. The test statistic is computed as LR = -2 * [log(La) - log(Lb)], where La and Lb are the likelihoods of the models being compared.

```R
ALake <- ArcticLake
ALake$AL <- DR_data(ArcticLake[,1:3])
mod0 <- DirichReg(AL ~ 1, ALake)
mod1 <- DirichReg(AL ~ depth, ALake)
mod2 <- DirichReg(AL ~ depth + I(depth^2), ALake)
anova(mod1, mod0, mod2, sorted = TRUE)
```

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

### summary Method for DirichletRegData

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Provides a summary of a DirichletRegData object.

```APIDOC
## summary.DirichletRegData

### Description
Provides a summary of a `DirichletRegData` object.

### Usage
```R
summary(object, ...)
```

### Arguments
* `object` (`DirichletRegData` object): The object to summarize.
* `...`: Further arguments.

### Details
Outputs summary statistics and information about the processed `DirichletRegData` object.
```

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

### Summarize DirichletRegModel Object

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Provides a summary of a DirichletRegModel object.

```R
## S3 method for class 'DirichletRegModel'
summary(object, ...)
```

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

### Prepare Compositional Data with DR_data

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Use DR_data to prepare matrices or data frames with compositional variables for Dirichlet regression. It handles normalization and optional transformations.

```R
DR_data(Y, trafo = sqrt(.Machine$double.eps), base = 1,
    norm_tol = sqrt(.Machine$double.eps))
```

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

### Transform Data to Simplex

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Use these functions to transform matrices with three or four components into a two- or three-dimensional simplex for plotting or analysis. Note that functions other than toSimplex() lack input checks.

```R
toSimplex(x)
toTernary(abc)
toTernaryVectors(c1, c2, c3)

toQuaternary(abcd)
toQuaternaryVectors(c1, c2, c3, c4)
```

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

### print.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints the results of a Dirichlet regression model.

```APIDOC
## print.DirichletRegModel

### Description
Prints the results of a Dirichlet regression model.

### Usage
```R
print(x, digits = max(3, getOption("digits") - 3), ...)
```

### Arguments
* `x`: An object of class `DirichletRegModel`.
* `digits`: The number of digits to display.
* `...`: Further arguments.
```

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

### Plot Arctic Lake Data

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Generates a ternary plot and a depth vs. composition plot for the Arctic Lake dataset.

```R
plot(AL, cex=.5, a2d=list(colored=FALSE, c.grid=FALSE))
```

```R
plot(rep(ArcticLake$depth,3),as.numeric(AL), pch=21, bg=rep(c("#E495A5", "#86B875", "#7DB0DD"),each=39), xlab="Depth (m)", ylab="Proportion", ylim=0:1)
```

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

### BibTeX Entry for DirichletReg Package (Current Version)

Source: https://cran.r-project.org/web/packages/DirichletReg/citation.html

This is the BibTeX entry for the current version of the DirichletReg R package. It includes title, author, year, version, and URL.

```bibtex
@Manual{
    title = {DirichletReg: Dirichlet Regression},
    author = {Marco J. Maier},
    year = {2025},
    note = {R package version 0.7-2},
    url = {https://github.com/maiermarco/DirichletReg},
  }
```

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

### Compute Confidence Intervals for DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Calculates confidence intervals for parameters (beta, gamma, or all) of a DirichletRegModel object. Can return parameters in exponentiated form.

```R
## S3 method for class 'DirichletRegModel'
confint(object, parm, level, ..., type=c("all", "beta", "gamma"), exp = FALSE)
```

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

### Transform Compositional Data for a Simplex

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

These functions transform a matrix with three or four components to fit into a two- or three-dimensional simplex (triangle or tetrahedron). They are useful for plotting compositional data in a reduced dimensional space.

```APIDOC
## Transform Compositional Data for a Simplex

### Description
These functions transform a matrix with three or four components to fit into a two- or three-dimensional simplex (triangle or tetrahedron).

### Usage
```R
toSimplex(x)
toTernary(abc)
toTernaryVectors(c1, c2, c3)
toQuaternary(abcd)
toQuaternaryVectors(c1, c2, c3, c4)
```

### Arguments
`x` | a matrix-like object with 3 or 4 columns.
`abc` | a matrix-like object with 3 columns.
`abcd` | a matrix-like object with 4 columns.
`c1` | a numeric vector with values of the first component.
`c2` | a numeric vector with values of the second component.
`c3` | a numeric vector with values of the third component.
`c4` | a numeric vector with values of the fourth component.

### Details
Most of these functions are only used internally, but sometimes it might be useful to plot “custom” ternary or quaternary graphics. Note that, apart from `toSimplex()`, functions do not have _any_ checks, so it is advisable to use this function if elements are added to plots or own graphics are created.

### Value
The function returns a `matrix` object with coordinates in two or three dimensions
```

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

### Print DirichletRegConfint Object

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints confidence intervals for a DirichletRegConfint object, typically obtained from confint.DirichletRegModel.

```R
## S3 method for class 'DirichletRegConfint'
print(x, digits = 3, ...)
```

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

### confint.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Computes confidence intervals for the parameters of a Dirichlet regression model.

```APIDOC
## confint.DirichletRegModel

### Description
Computes confidence intervals for the parameters of a Dirichlet regression model.

### Usage
```R
confint(object, parm, level, ..., type=c("all", "beta", "gamma"), exp = FALSE)
```

### Arguments
* `object`: An object of class `DirichletRegModel`.
* `parm`: A vector containing names of the parameters to compute confidence intervals for.
* `level`: Confidence level(s), defaults to 0.95.
* `type`: Defines the type of parameter for which confidence values are returned: "all", "beta", or "gamma".
* `exp`: Logical; if TRUE, returns parameters in exponentiated form.
* `...`: Further arguments.
```

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

### Predict and Calculate Dirichlet Probabilities for Blood Samples

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Predicts Dirichlet parameters for two disease groups (A and B) and calculates the probability density of observed blood sample compositions under these predicted parameters.

```R
alpha <- predict(blood2, data.frame(Disease=factor(c("A","B"))), F, T, F)
L <- sapply(1:2, function(i) ddirichlet(DR_data(Bld[31:36,1:4]), unlist(alpha[i,])))
LP <- L / rowSums(L)
dimnames(LP) <- list(paste("C",1:6), c("A", "B"))
print(data.frame(round(LP * 100, 1),"pred."=as.factor(ifelse(LP[,1]>LP[,2], "==> A", "==> B"))),print.gap=2)
```

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

### Print DirichletRegModel Object

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints a DirichletRegModel object. Adjusts digits based on global options.

```R
## S3 method for class 'DirichletRegModel'
print(x, digits = max(3, getOption("digits") - 3), ...)
```

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

### Compare Dirichlet Regression Models using an LRT

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

This function allows for pairwise tests of Dirichlet regression models using a likelihood ratio test (LRT). It helps in selecting the best model among a set of fitted Dirichlet regression models.

```APIDOC
## Compare Dirichlet Regression Models using an LRT

### Description
This function allows for pairwise tests of Dirichlet regression models using a likelihood ratio test (LRT).

### Usage
```R
## S3 method for class 'DirichletRegModel'
anova(object, ..., sorted = FALSE)
```

### Arguments
`object` | the model to be compared against those listed in ...
`...` | models to be tested against the one specified as `object`
`sorted` | should the models be sorted according to their numbers or parameters?

### Details
The test statistic is computed `LR=−2[log⁡(La)−log⁡(Lb)]` where `Li` is the likelihood of model `i` with `df` equal to the difference of the number of parameters in the models.

### Examples
```R
ALake <- ArcticLake
ALake$AL <- DR_data(ArcticLake[,1:3])
mod0 <- DirichReg(AL ~ 1, ALake)
mod1 <- DirichReg(AL ~ depth, ALake)
mod2 <- DirichReg(AL ~ depth + I(depth^2), ALake)
anova(mod1, mod0, mod2, sorted = TRUE)
```
```

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

### rdirichlet Function

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Generates random numbers from a Dirichlet distribution.

```APIDOC
## rdirichlet(n, alpha)

### Description
Generates random numbers from a Dirichlet distribution.

### Arguments
- `n` (numeric): The number of random observations to draw.
- `alpha` (numeric vector or matrix): The Dirichlet distribution's parameters. Can be a vector (one set of parameters for all observations) or a matrix (a different set of parameters for each observation).

### Value
Returns a matrix with random numbers according to the supplied alpha vector or matrix.

### Examples
```R
X1 <- rdirichlet(100, c(5, 5, 10))

a.mat <- cbind(1:10, 5, 10:1)
a.mat
X2 <- rdirichlet(10, a.mat)
# note how the probabilities in the first an last column relate to a.mat
round(X2, 2)
```
```

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

### logLik.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts the log-likelihood of a Dirichlet regression model.

```APIDOC
## logLik.DirichletRegModel

### Description
Extracts the log-likelihood of a Dirichlet regression model.

### Usage
```R
logLik(object, ...)
```

### Arguments
* `object`: An object of class `DirichletRegModel`.
* `...`: Further arguments.
```

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

### Print DirichletRegData Object

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints a DirichletRegData object, showing either the processed or original data. Useful for inspecting the prepared data.

```R
print(x, type = c("processed", "original"), ...)
```

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

### Summarize DirichletRegData Object

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Provides a summary of a DirichletRegData object, offering insights into its structure and properties.

```R
summary(object, ...)
```

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

### Visualize Blood Sample Predictions on Ternary Plot

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

Plots observed blood sample compositions and overlays predicted values for disease groups A and B on a ternary plot, using a color gradient to represent the predicted proportions.

```R
B2 <- DR_data(BloodSamples[,c(1,2,4)])
plot(B2, cex=.001, reset_par=FALSE)

div.col <- colorRampPalette(c("#023FA5", "#c0c0c0", "#8E063B"))(100)

# expected values
temp <- (alpha/rowSums(alpha))[,c(1,2,4)]
points(toSimplex(temp/rowSums(temp)), pch=22, bg=div.col[c(1,100)], cex=2, lwd=.25)
```

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

### ddirichlet_R Function

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Computes the density function of the Dirichlet distribution using R implementation.

```APIDOC
## ddirichlet_R(x, alpha, log = FALSE, sum.up = FALSE)

### Description
Computes the density function of the Dirichlet distribution, implemented purely in R.

### Arguments
- `x` (matrix): A matrix containing observations.
- `alpha` (numeric vector or matrix): The Dirichlet distribution's parameters.
- `log` (logical): If `TRUE`, logarithmic densities are returned.
- `sum.up` (logical): If `TRUE`, the (log-)likelihood is returned.

### Value
As `ddirichlet`, only implemented purely in R.

### Examples
```R
X1 <- rdirichlet(100, c(5, 5, 10))

ddirichlet_R(X1, c(5, 5, 10))
```
```

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

### DR_data Function

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prepares a matrix with compositional variables for further processing in the DirichletReg package. It handles normalization and optional transformations.

```APIDOC
## DR_data

### Description
This function prepares a matrix with compositional variables for further processing in the DirichletReg package.

### Usage
```R
DR_data(Y, trafo = sqrt(.Machine$double.eps), base = 1,
    norm_tol = sqrt(.Machine$double.eps))
```

### Arguments
* `Y` (matrix or data.frame): Compositional variables with nonnegative values. Can also be a vector in some cases.
* `trafo` (logical or numeric): Controls transformation of variables to shrink extreme values. If numeric, it acts as a threshold.
* `base` (numeric): The "base" component to use in the reparametrized model. Defaults to 1.
* `norm_tol` (numeric): Tolerance for testing near equality to 1 for row sums of Y. Defaults to `sqrt(.Machine$double.eps)`.

### Details
#### `Y`
If row sums of `Y` do not equal 1, normalization is applied. Rows with `NA` values will result in `NA` for the entire row. Beta-distributed variables can be supplied as a single vector, with the second variable generated as `1 - Y`.

#### `trafo`
Applies a transformation to components `y` using the formula `y* = y(n-1) + 1/d / n`, where `n` is the number of observations and `d` is the number of dimensions. This is a generalization of methods used in `betareg`.

#### `base`
Sets the omitted component for the "alternative" (mean/precision) model. By default, it's the first variable in `Y`.

### Value
Returns a `matrix` object of class `DirichletRegData` with attributes like original data, dimensions, observation counts, normalization status, transformation status, and base component.

### Examples
```R
# create a DirichletRegData object from the Arctic Lake data
head(ArcticLake[, 1:3])
AL <- DR_data(ArcticLake[, 1:3])
summary(AL)
head(AL)
```
```

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

### Drop Variables from DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Performs a drop-one analysis for a DirichletRegModel object, testing the significance of variables. Use with caution as it may issue a caveat on first use.

```R
## S3 method for class 'DirichletRegModel'
drop1(object, scope, test = c("LRT", "none"), k = 2, sort = TRUE, ...)
```

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

### drop1.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Performs a sequential analysis of competing models by dropping terms from a Dirichlet regression model.

```APIDOC
## drop1.DirichletRegModel

### Description
Performs a sequential analysis of competing models by dropping terms from a Dirichlet regression model.

### Usage
```R
drop1(object, scope, test = c("LRT", "none"), k = 2, sort = TRUE, ...)
```

### Arguments
* `object`: An object of class `DirichletRegModel`.
* `scope`: Defines the scope of variables to be dropped.
* `test`: Defines the type of test for `drop1`.
* `k`: Number for the weighting of parameters.
* `sort`: If TRUE, p-values will be sorted in decreasing order.
* `...`: Further arguments.
```

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

### ddirichlet Function

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Computes the density function of the Dirichlet distribution.

```APIDOC
## ddirichlet(x, alpha, log = FALSE, sum.up = FALSE)

### Description
Computes the density function of the Dirichlet distribution.

### Arguments
- `x` (matrix): A matrix containing observations.
- `alpha` (numeric vector or matrix): The Dirichlet distribution's parameters.
- `log` (logical): If `TRUE`, logarithmic densities are returned.
- `sum.up` (logical): If `TRUE`, the (log-)likelihood is returned.

### Value
Returns a vector of densities (if `sum.up = FALSE`) or the (log-)likelihood (if `sum.up = TRUE`) for the given data and alphas. Returns `NaN` if any element of `alpha` is `≤0`.

### Examples
```R
X1 <- rdirichlet(100, c(5, 5, 10))

ddirichlet(X1, c(5, 5, 10))

X2 <- rdirichlet(10, cbind(1:10, 5, 10:1))
ddirichlet(X2, cbind(1:10, 5, 10:1))

ddirichlet(X2[1:3,], c(1, 2, -1))
ddirichlet(X2[1:3,], c(1, 2, -1), sum.up = TRUE)
```
```

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

### Residual Plots for OLS and Dirichlet Regression

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

This code generates a series of residual plots for both OLS (rlr) and Dirichlet regression (rs2) models. It includes Normal Q-Q plots and plots of residuals against fitted values and IQ scores, with points colored by group (dyslexia status).

```R
gcol <- c("#E495A5", "#39BEB1")[3-as.numeric(RS$dyslexia)]
tmt <- c(-3,3)

par(mfrow=c(3,2), cex=.8)

qqnorm(residuals(rlr,"pearson"), ylim=tmt, xlim=tmt, pch=21, bg=gcol, main="Normal Q-Q-Plot: OLS Residuals",cex=.75,lwd=.5)
abline(0,1, lwd=2)
qqline(residuals(rlr,"pearson"), lty=2)

qqnorm(residuals(rs2,"standardized")[,2], ylim=tmt, xlim=tmt, pch=21, bg=gcol, main="Normal Q-Q-Plot: DirichReg Residuals",cex=.75,lwd=.5)
abline(0,1, lwd=2)
qqline(residuals(rs2,"standardized")[,2], lty=2)

plot(ReadingSkills$iq, residuals(rlr,"pearson"), pch=21, bg=gcol, ylim=c(-3,3),main="OLS Residuals",xlab="IQ",ylab="Pearson Residuals",cex=.75,lwd=.5)
abline(h=0,lty=2)
plot(ReadingSkills$iq, residuals(rs2,"standardized")[,2], pch=21, bg=gcol ,ylim=c(-3,3),main="DirichReg Residuals",xlab="IQ",ylab="Standardized Residuals",cex=.75,lwd=.5)
abline(h=0,lty=2)

plot(fitted(rlr), residuals(rlr,"pearson"), pch=21, bg=gcol ,ylim=c(-3,3),main="OLS Residuals",xlab="Fitted",ylab="Pearson Residuals",cex=.75,lwd=.5)
abline(h=0,lty=2)
plot(fitted(rs2)[,2], residuals(rs2,"standardized")[,2], pch=21, bg=gcol ,ylim=c(-3,3),main="DirichReg Residuals",xlab="Fitted",ylab="Standardized Residuals",cex=.75,lwd=.5)
abline(h=0,lty=2)
```

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

### Predict New Values using DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Predicts new values (expected values, alpha, or phi) for a DirichletRegModel object using new data.

```R
## S3 method for class 'DirichletRegModel'
predict(object, newdata, mu = TRUE, alpha = FALSE, phi = FALSE, ...)
```

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

### print Method for DirichletRegData

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Prints a DirichletRegData object, showing either the processed or original data.

```APIDOC
## print.DirichletRegData

### Description
Prints a `DirichletRegData` object.

### Usage
```R
print(x, type = c("processed", "original"), ...)
```

### Arguments
* `x` (`DirichletRegData` object): The object to print.
* `type` (character): Specifies whether to display the `"processed"` or `"original"` data. Defaults to `"processed"`.
* `...`: Further arguments.

### Details
Displays either the (possibly normalized or transformed) `"processed"` data or the `"original"` data based on the `type` argument.
```

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

### Visualizing Dirichlet Regression and OLS Predictions

Source: https://cran.r-project.org/web/packages/DirichletReg/vignettes/DirichletReg-vig.html

This code generates a plot comparing predicted reading accuracy from a Dirichlet regression model (rs2) and an OLS regression model, with different colors and line types distinguishing groups and model types. It also plots precision (phi) for the Dirichlet model.

```R
g.ind <- as.numeric(RS$dyslexia)
g1 <- g.ind == 1   # normal
g2 <- g.ind != 1   # dyslexia
par(mar=c(4,4,4,4)+0.25)
plot(accuracy~iq, RS, pch=21, bg=c("#E495A5", "#39BEB1")[3-g.ind],
cex=1.5, main="Dyslexic (Red) vs. Control (Green) Group",
xlab="IQ Score",ylab="Reading Accuracy", xlim=range(ReadingSkills$iq))

x1 <- seq(min(RS$iq[g1]), max(RS$iq[g1]), length.out=200)
x2 <- seq(min(RS$iq[g2]), max(RS$iq[g2]), length.out=200)
n <- length(x1)
X <- data.frame(dyslexia=factor(rep(0:1, each=n), levels=0:1, labels=c("no", "yes")),iq=c(x1,x2))
pv <- predict(rs2, X, TRUE, TRUE, TRUE)

lines(x1, pv$mu[1:n,2], col=c("#E495A5", "#39BEB1")[2],lwd=3)
lines(x2, pv$mu[(n+1):(2*n),2], col=c("#E495A5", "#39BEB1")[1],lwd=3)

a <- RS$accuracy
logRa_a <- log(a/(1-a))
rlr <- lm(logRa_a~dyslexia*iq, RS)
olds <- 1/(1+exp(-predict(rlr, X)))
lines(x1, ols[1:n], col=c("#AD6071", "#00897D")[2],lwd=3,lty=2)
lines(x2, ols[(n+1):(2*n)], col=c("#AD6071", "#00897D")[1],lwd=3,lty=2)

### precision plot
par(new=TRUE)
plot(x1, pv$phi[1:n], col=c("#6E1D34", "#004E42")[2], lty="11", type="l",ylim=c(0,max(pv$phi)),axes=F,ann=F,lwd=2, xlim=range(RS$iq))
lines(x2, pv$phi[(n+1):(2*n)], col=c("#6E1D34", "#004E42")[1], lty="11", type="l",lwd=2)
axis(4)
mtext(expression(paste("Precision (",phi,")",sep="")), 4, line=3)
legend("topleft",legend=c(expression(hat(mu)),expression(hat(phi)),"OLS"),lty=c(1,3,2),lwd=c(3,2,3),bty="n")
```

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

### Plot Dirichlet-Distributed Data

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

With this function you can plot Dirichlet-distributed data in 2, 3 and 4 dimensions. It supports various customization options for visualizing the data distribution.

```APIDOC
## Plot Dirichlet-Distributed Data

### Description
With this function you can plot Dirichlet-distributed data in 2, 3 and 4 dimensions.

### Usage
```R
## S3 method for class 'DirichletRegData'
plot(x, dims, ticks = TRUE, ref.lines = NULL, dim.labels, a2d = list(colored = TRUE, c.grid = TRUE, col.scheme = c("dims", "entropy"), entropy.contours = FALSE, entropy.colors = FALSE), a3d = list(rgl = TRUE, ...), rug = TRUE, reset_par = TRUE, ...)
```

### Arguments
`x` | data prepared with `DR_data`
`dims` | select two, three, or four Dimensions of your data `x` to plot
`ticks` | display ticks?
`ref.lines` | .
`dim.labels` | a character vector giving labels for the dimensions/variables
`a2d` | a named list of settings for ternary plots (3 variables), see Details
`a3d` | a named list of settings for quaternary plots (4 variables), see Details
`rug` | display a rug for a one-dimensional plot (2 variables)
`reset_par` | reset graphical parameters of `DR_data` after creating a two-dimensional plot (2 variables), see Details
`...` | further graphical arguments as `col`, `pch`, `cex`, ...

### Examples
```R
# plot of "Sand" in the Arctic Lake data set
plot(DR_data(ReadingSkills[, 1]), main="Reading Accuracy")

# ternary plot of Arctic Lake data
plot(DR_data(ArcticLake[, 1:3]), a2d = list(colored = FALSE))
```
```

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

### nobs.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Returns the number of observations in a Dirichlet regression model.

```APIDOC
## nobs.DirichletRegModel

### Description
Returns the number of observations in a Dirichlet regression model.

### Usage
```R
nobs(object, ...)
```

### Arguments
* `object`: An object of class `DirichletRegModel`.
* `...`: Further arguments.
```

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

### update.DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Updates a Dirichlet regression model.

```APIDOC
## update.DirichletRegModel

### Description
Updates a Dirichlet regression model.

### Usage
```R
update(object, formula., ..., evaluate = TRUE)
```

### Arguments
* `object`: An object of class `DirichletRegModel`.
* `formula.`: The new formula to be updated.
* `evaluate`: If FALSE, the updated call will be returned, but not evaluated.
* `...`: Further arguments.
```

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

### Generate Dirichlet Distribution Random Numbers

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Generates random numbers from a Dirichlet distribution. Supports a single alpha vector or a matrix of alpha parameters for each observation.

```R
X1 <- rdirichlet(100, c(5, 5, 10))

a.mat <- cbind(1:10, 5, 10:1)
a.mat
X2 <- rdirichlet(10, a.mat)
# note how the probabilities in the first an last column relate to a.mat
round(X2, 2)
```

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

### Extract Log-Likelihood from DirichletRegModel

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Extracts the log-likelihood of a DirichletRegModel object.

```R
## S3 method for class 'DirichletRegModel'
logLik(object, ...)
```

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

### Plot Dirichlet-Distributed Data

Source: https://cran.r-project.org/web/packages/DirichletReg/refman/DirichletReg.html

Visualize Dirichlet-distributed data in 2, 3, or 4 dimensions. Customize plots with options for dimensions, ticks, reference lines, dimension labels, and specific settings for 2D (ternary) and 3D (quaternary) plots.

```R
# plot of "Sand" in the Arctic Lake data set
plot(DR_data(ReadingSkills[, 1]), main="Reading Accuracy")

# ternary plot of Arctic Lake data
plot(DR_data(ArcticLake[, 1:3]), a2d = list(colored = FALSE))
```