Model
We consider a latin square experiment, with a 9x9 latin square
design, following the procedure of Fuglstad et
al. (2020). We assume we have the following model: \[
y_{i,j} = \alpha + \beta \cdot k[i,j] + a_i + b_j + c_{k[i,j]} +
\varepsilon_{i,j}, \quad i,j = 1, \dots, 9,
\] where
- \(a\) is an intercept with \(\alpha \sim N(0, 1000^2)\),
- \(b \cdot k[i,j]\) is a linear
effect of treatment and \(\beta \sim N(0,
1000^2)\),
- \(\mathbf{a} = (a_1, \dots, a_9)^\top \sim
N_9(\mathbf{0}, \sigma_{a}^2 \mathbf{I}_9)\) is a row
effect,
- \(\mathbf{b} = (b_1, \dots, b_9)^\top \sim
N_9(\mathbf{0}, \sigma_{b}^2 \mathbf{I}_9)\) is a column
effect,
- \(\mathbf{\varepsilon} =
(\varepsilon_{1,1},\varepsilon_{1,2} \dots, \varepsilon_{9,9})^\top \sim
N_{81}(\mathbf{0}, \sigma_{\varepsilon}^2 \mathbf{I}_{81})\) is
residual noise,
- the treatment effect consists of:
- a smooth signal \(\mathbf{c}^{(1)} =
(c_1^{(1)}, \dots, c_9^{(1)}) \sim (\mathbf{0}, \sigma_{c^{(1)}}^2
\mathbf{Q}_{\mathrm{RW2}}^{-1})\) where \(\sigma_{c^{(1)}}^2\) is the variance and
\(\mathbf{Q}_{\mathrm{RW2}}^{-1}\) is
the covariance matrix describing the intrinsic second-order random walk,
and
- random noise \(\mathbf{c}^{(2)} =
(c_1^{(2)}, \dots, c_9^{(2)}) \sim N_9(\mathbf{0}, \sigma_{c^{(2)}}^2
\mathbf{I}_9)\).
We remove implicit intercept and linear effect by requiring \(\sum_{i=1}^9 c_i^{(1)} = 0\) and \(\sum_{i=1}^9 i c_i^{(1)} = 0\).
The model is specified as:
formula <- y ~ lin + mc(row) + mc(col) + mc(iid, constr = TRUE) +
mc(rw2, model = "rw2", constr = TRUE, lin_constr = TRUE)
We use the dataset latin_square in
makemyprior, and present three priors. We do not carry out
inference, as it takes time and will slow down the compilation of the
vignettes by a lot, but include code so the user can carry out the
inference themselves.
Prior 1
We want to avoid overfitting of the model, and use a prior with
shrinkage towards the residuals in the top split with median of \(0.25\). We do not have any preference for
the attribution of the row, column and treatment effects, and use an
ignorant Dirichlet prior for the middle split. In the bottom split we
again we want to avoid overfitting, and use a prior with shrinkage
towards the unstructured treatment effect and a median corresponding to
75% unstructured treatment effect. We do not want to say anything about
the scale of the total variance, and use the default Jeffreys’
prior.
prior1 <- make_prior(
formula, latin_data,
prior = list(tree = "s1 = (iid, rw2); s2 = (row, col, s1); s3 = (s2, eps)",
w = list(s1 = list(prior = "pc1", param = 0.75),
s2 = list(prior = "dirichlet"),
s3 = list(prior = "pc0", param = 0.25))))
prior1
#> Model: y ~ lin + mc(row) + mc(col) + mc(iid, constr = TRUE) + mc(rw2,
#> model = "rw2", constr = TRUE, lin_constr = TRUE)
#> Tree structure: iid_rw2 = (iid,rw2); row_col_iid_rw2 = (row,col,iid_rw2); eps_row_col_iid_rw2 = (eps,row_col_iid_rw2)
#>
#> Weight priors:
#> w[iid/iid_rw2] ~ PC1(0.75)
#> (w[row/row_col_iid_rw2], w[col/row_col_iid_rw2]) ~ Dirichlet(3)
#> w[eps/eps_row_col_iid_rw2] ~ PC1(0.75)
#> Total variance priors:
#> V[eps_row_col_iid_rw2] ~ Jeffreys'
plot_prior(prior1) # or plot(prior)
plot_tree_structure(prior1)
Inference can be carried out by running:
posterior1 <- inference_stan(prior1, iter = 15000, warmup = 5000,
seed = 1, init = "0", chains = 1)
plot_posterior_stan(posterior1, param = "prior", prior = TRUE)
For inference with INLA we need to include the linear constraint in a
different way:
formula_inla <- y ~ lin + mc(row) + mc(col) + mc(iid, constr = TRUE) +
mc(rw2, model = "rw2", constr = TRUE, extraconstr = list(A = matrix(1:9, 1, 9), e = matrix(0, 1, 1)))
prior1_inla <- make_prior(
formula_inla, latin_data,
prior = list(tree = "s1 = (iid, rw2); s2 = (row, col, s1); s3 = (s2, eps)",
w = list(s1 = list(prior = "pc1", param = 0.75),
s2 = list(prior = "dirichlet"),
s3 = list(prior = "pc0", param = 0.25))))
posterior1_inla <- inference_inla(prior1_inla)
plot_posterior_stdev(posterior1_inla)
Prior 2
We can imagine we do not not want to include the residual effect in
the tree with the row, column and treatment effects, and assume we have
prior knowledge that the latent variance \(\sigma_{a}^2 + \sigma_{b}^2 + \sigma_{c^{(1)}}^2 +
\sigma_{c^{(2)}}^2\) is not not larger than \(0.25\), and use a PC prior for variance
that fulfills \(\text{P}(\text{total st.dev.}
> sqrt(0.2)) = 0.05\). We assume we have knowledge saying that
the same is true for the residual variance. We assume the latent
variance is distributed in the same way as in Prior
1.
prior2 <- make_prior(
formula, latin_data,
prior = list(tree = "s1 = (iid, rw2); s2 = (row, col, s1); (eps)",
w = list(s1 = list(prior = "pc1", param = 0.75),
s2 = list(prior = "dirichlet")),
V = list(s2 = list(prior = "pc", param = c(sqrt(0.2), 0.05)),
eps = list(prior = "pc", param = c(sqrt(0.2), 0.05)))))
prior2
#> Model: y ~ lin + mc(row) + mc(col) + mc(iid, constr = TRUE) + mc(rw2,
#> model = "rw2", constr = TRUE, lin_constr = TRUE)
#> Tree structure: iid_rw2 = (iid,rw2); row_col_iid_rw2 = (row,col,iid_rw2); (eps)
#>
#> Weight priors:
#> w[iid/iid_rw2] ~ PC1(0.75)
#> (w[row/row_col_iid_rw2], w[col/row_col_iid_rw2]) ~ Dirichlet(3)
#> Total variance priors:
#> sqrt(V)[row_col_iid_rw2] ~ PC0(0.447213595499958, 0.05)
#> Independent variance priors:
#> sigma[eps] ~ PC0(0.447213595499958, 0.05)
plot_prior(prior2) # or plot(prior2)
plot_tree_structure(prior2)
Inference can be carried out by running:
posterior2 <- inference_stan(prior2, iter = 15000, warmup = 5000,
seed = 1, init = "0", chains = 1)
plot_posterior_stan(posterior2)
Prior 3
In the last prior, we use component-wise priors on row, column,
treatment and residual variance, but use a PC prior with shrinkage
towards the unstructured treatment effect to avoid overfitting. We use a
strong prior, and say that that the all variances are not much larger
than \(0.1\).
prior3 <- make_prior(
formula, latin_data,
prior = list(tree = "(row); (col); (iid); (rw2); (eps)",
V = list(
row = list(prior = "pc", param = c(sqrt(0.1), 0.05)),
col = list(prior = "pc", param = c(sqrt(0.1), 0.05)),
iid = list(prior = "pc", param = c(sqrt(0.1), 0.05)),
rw2 = list(prior = "pc", param = c(sqrt(0.1), 0.05)),
eps = list(prior = "pc", param = c(sqrt(0.1), 0.05))
)))
prior3
#> Model: y ~ lin + mc(row) + mc(col) + mc(iid, constr = TRUE) + mc(rw2,
#> model = "rw2", constr = TRUE, lin_constr = TRUE)
#> Tree structure: (row); (col); (iid); (rw2); (eps)
#>
#> Independent variance priors:
#> sigma[row] ~ PC0(0.316227766016838, 0.05)
#> sigma[col] ~ PC0(0.316227766016838, 0.05)
#> sigma[iid] ~ PC0(0.316227766016838, 0.05)
#> sigma[rw2] ~ PC0(0.316227766016838, 0.05)
#> sigma[eps] ~ PC0(0.316227766016838, 0.05)
plot_prior(prior3) # or plot(prior3)
plot_tree_structure(prior3)
Inference can be carried out by running:
posterior3 <- inference_stan(prior3, iter = 15000, warmup = 5000,
seed = 1, init = "0", chains = 1)
plot_posterior_stan(posterior3)
sessionInfo()
#> R version 4.3.2 (2023-10-31)
#> Platform: aarch64-apple-darwin20 (64-bit)
#> Running under: macOS Sonoma 14.1.1
#>
#> Matrix products: default
#> BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
#>
#> locale:
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#>
#> time zone: Europe/Oslo
#> tzcode source: internal
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] makemyprior_1.2.2
#>
#> loaded via a namespace (and not attached):
#> [1] Matrix_1.6-1.1 gtable_0.3.4 jsonlite_1.8.8 highr_0.10
#> [5] dplyr_1.1.4 compiler_4.3.2 promises_1.2.1 tidyselect_1.2.0
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#> [13] scales_1.3.0 yaml_2.3.8 fastmap_1.1.1 mime_0.12
#> [17] lattice_0.21-9 ggplot2_3.4.4 R6_2.5.1 labeling_0.4.3
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#> [53] ellipsis_0.3.2 tools_4.3.2 pkgconfig_2.0.3 htmltools_0.5.7