#' Orthonormal Contrast Matrices for Bayesian Estimation #' #' Returns a design or model matrix of orthonormal contrasts such that the #' marginal prior on all effects is identical. Implementation from Singmann & #' Gronau's [`bfrms`](https://github.com/bayesstuff/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 #' [the #' Bayes factors vignette](https://easystats.github.io/bayestestR/articles/bayes_factors.html). #' #' @inheritParams stats::contr.treatment #' #' @details #' When `contrasts = FALSE`, the returned contrasts are equivalent to #' `contr.treatment(, contrasts = FALSE)`, as suggested by McElreath (also known #' as one-hot encoding). #' #' @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 #' #' @return A `matrix` with n rows and k columns, with k=n-1 if contrasts is #' `TRUE` and k=n if contrasts is `FALSE`. #' #' @aliases contr.bayes #' #' @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 #' @export contr.orthonorm <- function(n, contrasts = TRUE, sparse = FALSE) { contr <- stats::contr.treatment(n, contrasts = FALSE, base = 1, sparse = sparse & !contrasts ) if (contrasts) { n <- ncol(contr) I_a <- diag(n) J_a <- matrix(1, nrow = n, ncol = n) Sigma_a <- I_a - J_a / n contr <- eigen(Sigma_a)\$vectors[, seq_len(n - 1), drop = FALSE] } contr } # ---------- #' @export contr.bayes <- function(n, contrasts = TRUE) { .Deprecated(new = "contr.orthonorm", old = "contr.bayes") contr.orthonorm(n, contrasts = contrasts) }