% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/contr.orthonorm.R
\name{contr.orthonorm}
\alias{contr.orthonorm}
\alias{contr.bayes}
\title{Orthonormal Contrast Matrices for Bayesian Estimation}
\usage{
contr.orthonorm(n, contrasts = TRUE, sparse = FALSE)
}
\arguments{
\item{n}{a vector of levels for a factor, or the number of levels.}

\item{contrasts}{a logical indicating whether contrasts should be
    computed.}

\item{sparse}{logical indicating if the result should be sparse
    (of class \code{\link[Matrix:dgCMatrix-class]{dgCMatrix}}), using
    package \href{https://CRAN.R-project.org/package=Matrix}{\pkg{Matrix}}.}
}
\value{
A \code{matrix} with n rows and k columns, with k=n-1 if contrasts is
  \code{TRUE} and k=n if contrasts is \code{FALSE}.
}
\description{
Returns a design or model matrix of orthonormal contrasts such that the
marginal prior on all effects is identical. Implementation from Singmann &
Gronau's \href{https://github.com/bayesstuff/bfrms/}{\code{bfrms}}, following
the description in Rouder, Morey, Speckman, & Province (2012, p. 363).
\cr\cr
Though using this factor coding scheme might obscure the interpretation of
parameters, it is essential for correct estimation of Bayes factors for
contrasts and order restrictions of multi-level factors (where `k>2`). See
info on specifying correct priors for factors with more than 2 levels in
\href{https://easystats.github.io/bayestestR/articles/bayes_factors.html}{the
Bayes factors vignette}.
}
\details{
When `contrasts = FALSE`, the returned contrasts are equivalent to
`contr.treatment(, contrasts = FALSE)`, as suggested by McElreath (also known
as one-hot encoding).
}
\examples{
contr.orthonorm(2) # Q_2 in Rouder et al. (2012, p. 363)

contr.orthonorm(5) # equivalent to Q_5 in Rouder et al. (2012, p. 363)

## check decomposition
Q3 <- contr.orthonorm(3)
Q3 \%*\% t(Q3) ## 2/3 on diagonal and -1/3 on off-diagonal elements
}
\references{
- McElreath, R. (2020). Statistical rethinking: A Bayesian course with
  examples in R and Stan. CRC press.

- Rouder, J. N., Morey, R. D., Speckman, P. L., & Province, J. M. (2012).
  Default Bayes factors for ANOVA designs. *Journal of Mathematical
  Psychology*, 56(5), 356-374. https://doi.org/10.1016/j.jmp.2012.08.001
}