Build contrasts for factors with equal marginal priors on all levels. The 3
functions give the same orthogonal contrasts, but are scaled differently to
allow different prior specifications (see 'Details'). Implementation from
Singmann & Gronau's bfrms,
following the description in Rouder, Morey, Speckman, & Province (2012, p.
363).
contr.equalprior(n, contrasts = TRUE, sparse = FALSE)contr.equalprior_pairs(n, contrasts = TRUE, sparse = FALSE)
contr.equalprior_deviations(n, contrasts = TRUE, sparse = FALSE)
A matrix with n rows and k columns, with k=n-1 if contrasts is
TRUE and k=n if contrasts is FALSE.
When using stats::contr.treatment, each dummy variable is the difference
between each level and the reference level. While this is useful if setting
different priors for each coefficient, it should not be used if one is trying
to set a general prior for differences between means, as it (as well as
stats::contr.sum and others) results in unequal marginal priors on the
means the the difference between them.
library(brms)data <- data.frame(
group = factor(rep(LETTERS[1:4], each = 3)),
y = rnorm(12)
)
contrasts(data$group) # R's default contr.treatment
#> B C D
#> A 0 0 0
#> B 1 0 0
#> C 0 1 0
#> D 0 0 1
model_prior <- brm(
y ~ group, data = data,
sample_prior = "only",
# Set the same priors on the 3 dummy variable
# (Using an arbitrary scale)
prior = set_prior("normal(0, 10)", coef = c("groupB", "groupC", "groupD"))
)
est <- emmeans::emmeans(model_prior, pairwise ~ group)
point_estimate(est, centr = "mean", disp = TRUE)
#> Point Estimate
#>
#> Parameter | Mean | SD
#> -------------------------
#> A | -0.01 | 6.35
#> B | -0.10 | 9.59
#> C | 0.11 | 9.55
#> D | -0.16 | 9.52
#> A - B | 0.10 | 9.94
#> A - C | -0.12 | 9.96
#> A - D | 0.15 | 9.87
#> B - C | -0.22 | 14.38
#> B - D | 0.05 | 14.14
#> C - D | 0.27 | 14.00
We can see that the priors for means aren't all the same (A having a more
narrow prior), and likewise for the pairwise differences (priors for
differences from A are more narrow).
The solution is to use one of the methods provided here, which do result in
marginally equal priors on means differences between them. Though this will
obscure the interpretation of parameters, setting equal priors on means and
differences is important for they are useful for specifying equal priors on
all means in a factor and their differences 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.
NOTE: When setting priors on these dummy variables, always:
Use priors that are centered on 0! Other location/centered priors are meaningless!
Use identically-scaled priors on all the dummy variables of a single factor!
contr.equalprior returns the original orthogonal-normal contrasts as
described in Rouder, Morey, Speckman, & Province (2012, p. 363). Setting
contrasts = FALSE returns the \(I_{n} - \frac{1}{n}\) matrix.
contr.equalprior_pairs
Useful for setting priors in terms of pairwise differences between means -
the scales of the priors defines the prior distribution of the pair-wise
differences between all pairwise differences (e.g., A - B, B - C, etc.).
contrasts(data$group) <- contr.equalprior_pairs
contrasts(data$group)
#> [,1] [,2] [,3]
#> A 0.0000000 0.6123724 0.0000000
#> B -0.1893048 -0.2041241 0.5454329
#> C -0.3777063 -0.2041241 -0.4366592
#> D 0.5670111 -0.2041241 -0.1087736model_prior <- brm(
y ~ group, data = data,
sample_prior = "only",
# Set the same priors on the 3 dummy variable
# (Using an arbitrary scale)
prior = set_prior("normal(0, 10)", coef = c("group1", "group2", "group3"))
)
est <- emmeans(model_prior, pairwise ~ group)
point_estimate(est, centr = "mean", disp = TRUE)
#> Point Estimate
#>
#> Parameter | Mean | SD
#> -------------------------
#> A | -0.31 | 7.46
#> B | -0.24 | 7.47
#> C | -0.34 | 7.50
#> D | -0.30 | 7.25
#> A - B | -0.08 | 10.00
#> A - C | 0.03 | 10.03
#> A - D | -0.01 | 9.85
#> B - C | 0.10 | 10.28
#> B - D | 0.06 | 9.94
#> C - D | -0.04 | 10.18
All means have the same prior distribution, and the distribution of the
differences matches the prior we set of "normal(0, 10)". Success!
contr.equalprior_deviations
Useful for setting priors in terms of the deviations of each mean from the grand mean - the scales of the priors defines the prior distribution of the distance (above, below) the mean of one of the levels might have from the overall mean. (See examples.)
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.equalprior(2) # Q_2 in Rouder et al. (2012, p. 363)
contr.equalprior(5) # equivalent to Q_5 in Rouder et al. (2012, p. 363)
## check decomposition
Q3 <- contr.equalprior(3)
Q3 %*% t(Q3) ## 2/3 on diagonal and -1/3 on off-diagonal elements
Run the code above in your browser using DataLab