`R/contr.orthonorm.R`

`contr.orthonorm.Rd`

Returns a design or model matrix of orthonormal contrasts such that the
marginal prior on all effects is identical (see 'Details'). Implementation
from Singmann & Gronau's `bfrms`

,
following the description in Rouder, Morey, Speckman, & Province (2012, p.
363).

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.

contr.orthonorm(n, contrasts = TRUE, sparse = FALSE)

n | a vector of levels for a factor, or the number of levels. |
---|---|

contrasts | a logical indicating whether contrasts should be computed. |

sparse | logical indicating if the result should be sparse
(of class |

A `matrix`

with n rows and k columns, with k=n-1 if contrasts is
`TRUE`

and k=n if contrasts is `FALSE`

.

When `contrasts = FALSE`

, the returned contrasts are equivalent to
`contr.treatment(, contrasts = FALSE)`

, as suggested by McElreath (also known
as one-hot encoding).

It is recommended to set 0-centered identically scaled priors of the dummy coded variables produced by this method. These priors then represent the distance the mean of one of the levels might have from the overall mean.

This method guarantees that any set of contrasts between the *k* groups will
have the same multivariate prior regardless of level order; However,
different contrasts within a set contrasts can have different univariate
prior shapes/scales.

For example the contrasts `A - B`

will have the same prior as `B - C`

, as
will `(A + C) - B`

and `(B + A) - C`

, but `A - B`

and `(A + C) - B`

will
differ.

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

contr.orthonorm(2) # Q_2 in Rouder et al. (2012, p. 363)#> [,1] #> [1,] -0.7071068 #> [2,] 0.7071068contr.orthonorm(5) # equivalent to Q_5 in Rouder et al. (2012, p. 363)#> [,1] [,2] [,3] [,4] #> [1,] 0.0000000 0.8944272 0.0000000 0.0000000 #> [2,] 0.0000000 -0.2236068 -0.5000000 0.7071068 #> [3,] 0.7071068 -0.2236068 -0.1666667 -0.4714045 #> [4,] -0.7071068 -0.2236068 -0.1666667 -0.4714045 #> [5,] 0.0000000 -0.2236068 0.8333333 0.2357023## check decomposition Q3 <- contr.orthonorm(3) Q3 %*% t(Q3) ## 2/3 on diagonal and -1/3 on off-diagonal elements#> [,1] [,2] [,3] #> [1,] 0.6666667 -0.3333333 -0.3333333 #> [2,] -0.3333333 0.6666667 -0.3333333 #> [3,] -0.3333333 -0.3333333 0.6666667