The goal of `sim2Dpredictr`

is to facilitate
straightforward simulation of spatially dependent predictors (continuous
or binary), which may then be used to simulate continuous, binary,
count, or categorical (\(> 2\)
categories) outcomes within a (generalized) linear model framework. A
real-world example is when using medical images to model/predict
(scalar) clinical outcomes; such a scenario motivated the development of
`sim2Dpredictr`

, which was used to simulate data to evaluate
the performance of methods for high-dimensional data analysis and
prediction (Leach, Aban, and Yi 2022; Leach et al. 2022).

In the first step, we simulate the predictors, i.e., the \(\mathbf{X}_i\) part of a GLM, \(g(E[Y_i]) = \mathbf{X}_i\mathbf{\beta}\), where \(g(\cdot)\) is an appropriate link function.

Continuous predictors are simulated using Multivariate Normal (MVN)
distributions with a focus on specific correlation structures;
alternatively, one can specify conditional dependence via a precision
matrix, specifically for a Conditional Autoregressive (CAR) model. Tools
are included for easily constructing and taking the Cholesky
decomposition of a covariance or precision matrix with either base
`R`

or the `R`

package `spam`

, which
makes this process faster when the matrix is sparse. The Boolean Model
and thresholding of MVN’s are used to simulate spatially dependent
binary maps. The package also includes a tool for easily specifying a
parameter vector with spatially clustered non-zero elements. These
simulation tools are designed for, but not limited to, testing the
performance of variable selection methods when predictors are spatially
correlated.

In the second step we use the predictor vectors \(\mathbf{X}_i\) to generate scalar outcomes, i.e., the \(Y_i\) part of the GLM. The default approach is to use the inverse link function to define subject specific means, \(\mu_i = E[Y_i] = g^{-1}(\mathbf{X}_i\mathbf{\beta})\). For Normally distributed outcomes, \(g(\cdot)\) is the identity link, so \(\mu_i = \mathbf{X}_i\mathbf{\beta}\) and a separate dispersion parameter, \(\sigma^2\), specifies the variance. In general, the variance may be a function of the mean, as well as some other dispersion parameter. We can draw directly from the desired distributions using (some function of) \(\mu_i\) (and if necessary, dispersion, \(\sigma^2\)) as the parameter(s) for that distribution to obtain outcomes, \(Y_i\). Alternatively, outcomes can be initially generated as continuous and then a threshold applied to obtain binary/categorical data if using the inverse link function is too computationally expensive.

`sim2Dpredictr`

is available on CRAN:

`install.packages("sim2Dpredictr")`

You can install the latest version of `sim2Dpredictr`

from
GitHub with:

`::install_github("jmleach-bst/sim2Dpredictr") devtools`

A simple demonstration is as follows; suppose each subject has a
\(5 \times 5\) standardized
continuous-valued predictor image, and a binary outcome. We can generate
a spatial cluster of non-zero parameter values with
`beta_builder()`

, simulate and take the Cholesky
decomposition of a correlation (or covariance) matrix with
`chol_s2Dp()`

, and generate both the images and outcomes with
`sim_Y_MVN_X()`

.

```
library(sim2Dpredictr)
# Construct spatially clusterd non-zero parameters.
<- sim2Dpredictr::beta_builder(row.index = c(1, 1, 2),
Bex col.index = c(1, 2, 1),
im.res = c(3, 3),
B0 = 0, B.values = rep(1, 3))
# Construct and take Cholesky decomposition of correlation matrix.
<- sim2Dpredictr::chol_s2Dp(corr.structure = "ar1",
Rex im.res = c(3, 3), rho = 0.5,
use.spam = TRUE)
# Simulate a dataset with spatially dependent design matrix and binary outcomes.
<- sim2Dpredictr::sim_Y_MVN_X(N = 3, B = Bex$B,
sim.dat R = Rex$R, S = Rex$S,
dist = "binomial")
sim.dat#> Y X1 X2 X3 X4 X5 X6 X7
#> 1 0 0.5198678 -1.1035062 -0.805563 -1.3767652 -1.70525968 -0.6388318 -1.5858393
#> 2 0 0.1798315 0.5728877 1.463024 -0.9348662 -0.05594379 0.8592733 0.2619289
#> 3 1 0.1996273 1.7450479 2.500030 1.4882150 1.61903239 1.2503542 0.3439712
#> X8 X9 subjectID
#> 1 -1.72102736 0.06016316 1
#> 2 0.08300277 -0.28904666 2
#> 3 0.63695327 0.85655952 3
```

Once the dependence framework and non-zero parameter vector is set,
`sim_Y_MVN_X()`

can be used to draw as many data sets as
necessary, upon each of which variable selection methods are applied;
summaries from each analyzed data set can be obtained and then used to
evaluate variable selection performance. The documentation provides
details about how to use these functions (and others) to create desired
simulations.

Leach, Justin M, Inmaculada Aban, and Nengjun Yi. 2022. “Incorporating
Spatial Structure into Inclusion Probabilities for Bayesian Variable
Selection in Generalized Linear Models with the Spike-and-Slab Elastic
Net.” *Journal of Statistical Planning and Inference* 217:
141–52. https://doi.org/10.1016/j.jspi.2021.07.010.

Leach, Justin M, Lloyd J Edwards, Rajesh Kana, Kristina Visscher,
Nengjun Yi, and Inmaculada Aban. 2022. “The Spike-and-Slab Elastic Net
as a Classification Tool in Alzheimer’s Disease.” *PLoS ONE* 17:
e0262367. https://doi.org/10.1371/journal.pone.0262367.