Graphical VAR
Donny Williams
6/04/2020
Introduction
This vignette describes the implemention for a “graphical VAR” in BGGM. In BGGM, this is fitted as a multivariate regression. The key innovation is a novel prior distribution for the residual covariance matrix. There are a variety of much cooler names than a mere “multivariate regression,” including “VAR” (vector autoregressive models) and “TSCGM” (time series chain graphical model).
R package
# need the developmental version
if (!requireNamespace("remotes")) {
install.packages("remotes")
}
# install from github
remotes::install_github("donaldRwilliams/BGGM")
library(BGGM)
# for comparsion
library(vars)
# for plotting
library(qgraph)
# combine plots
library(cowplot)Data
I use data from the ifit (fit bit) study. The data were gathered over 100 consecutive days on a variety of variables, including the PANAS scale (positive and negative affect) and the number of steps each day. BGGM includes a subset of variables for two individuals.
# data
Y <- subset(ifit, id == 1)[,-1]
# first 3 rows
head(Y, n = 3)
#> interested disinterested excited upset strong stressed steps
#> 72 10 50 2 50 16 7805
#> 75 6 75 0 76 0 18248
># 36 58 38 5 45 1 12139Estimation
The methods in BGGM are organized around Bayesian “estimation” and “hypothesis testing.” This is to reach a broader audience, as former is more similar to classical methods (those more familiar to researchers).
Fit Model
With the data in hand, the model is fitted as follows
# fit model
fit <- var_estimate(Y, beta_sd = 1)Note that beta_sd is the prior distribution for the regression coefficients. A smaller value, say, beta_sd = 0.25, results in a Bayesian ridge regression. Note also this model, including 5000 draws from the posterior, was estimated in less than 1 second.
The results can then be printed
# print
fit
#> BGGM: Bayesian Gaussian Graphical Models
#> ---
#> Vector Autoregressive Model (VAR)
#> ---
#> Posterior Samples: 5000
#> Observations (n): 94
#> Nodes (p): 7
#> ---
#> Call:
#> var_estimate(Y = Y, beta_sd = 10)
#> ---
#> Partial Correlations:
#>
#> interested disinterested excited upset strong stressed steps
#> interested 0.000 -0.170 0.388 -0.217 0.313 0.268 0.089
#> disinterested -0.170 0.000 -0.172 -0.029 0.094 0.160 -0.078
#> excited 0.388 -0.172 0.000 -0.126 0.500 -0.161 -0.016
#> upset -0.217 -0.029 -0.126 0.000 0.118 0.350 -0.039
#> strong 0.313 0.094 0.500 0.118 0.000 -0.010 0.176
#> stressed 0.268 0.160 -0.161 0.350 -0.010 0.000 -0.038
#> steps 0.089 -0.078 -0.016 -0.039 0.176 -0.038 0.000
#> ---
#> Coefficients:
#>
#> interested disinterested excited upset strong stressed steps
#> interested.l1 0.230 -0.009 0.182 -0.102 0.178 0.018 0.113
#> disinterested.l1 -0.051 -0.007 0.056 -0.019 0.049 0.091 -0.023
#> excited.l1 -0.088 -0.196 0.003 0.057 -0.093 0.092 0.106
#> upset.l1 -0.155 0.262 -0.097 0.435 0.057 0.324 -0.091
#> strong.l1 0.026 0.182 0.026 0.048 0.189 -0.073 -0.196
#> stressed.l1 -0.021 -0.014 -0.033 -0.048 -0.079 0.152 0.133
#> steps.l1 -0.157 0.180 -0.211 0.155 -0.092 0.209 0.042
#> ---
#> Date: Thu Jun 04 08:54:04 2020 Note that the coefficients are comparable, given each variable has been standardized (e.g., the predictors and the outcome are standardized). BGGM does not compute the partial directed correlation (PDC) by default (as in graphicalVAR). This is because the standardized effects can readily be tested with the Bayes factor, both across and within each model, whereas this does not seem straightforward for the PDC (which requires a transformation).
Compare to Classical
Here are the estimates from the vars package
t(round(
vars::Bcoef(
vars:::VAR(scale(na.omit(Y)), type = "none")),
digits = 3)
)
#> interested disinterested excited upset strong stressed steps
#> interested.l1 0.229 -0.012 0.184 -0.100 0.180 0.015 0.112
#> disinterested.l1 -0.050 -0.006 0.057 -0.019 0.050 0.092 -0.022
#> excited.l1 -0.088 -0.193 0.002 0.056 -0.091 0.093 0.106
#> upset.l1 -0.155 0.260 -0.096 0.436 0.058 0.321 -0.092
#> strong.l1 0.027 0.182 0.025 0.047 0.188 -0.073 -0.192
#> stressed.l1 -0.021 -0.012 -0.033 -0.046 -0.077 0.152 0.133
#> steps.l1 -0.157 0.183 -0.210 0.153 -0.093 0.207 0.041Recall that the “estimation” methods are similar to, in this case, ordinary least squares. The graphical structure in BGGM is determined with credible intervals, which will be quite similar to using confidence intervals. Hence for those researchers unfamiliar with Bayesian methods the “estimation” methods are perhaps a nice place to start.
Summarize Model
The model can also be summarized with
print(
summary(fit, cred = 0.95),
param = "pcor"
)
#> BGGM: Bayesian Gaussian Graphical Models
#> ---
#> Vector Autoregressive Model (VAR)
#> ---
#> Partial Correlations:
#>
#> Relation Post.mean Post.sd Cred.lb Cred.ub
#> interested--disinterested -0.170 0.108 -0.382 0.044
#> interested--excited 0.388 0.085 0.219 0.546
#> disinterested--excited -0.172 0.104 -0.369 0.049
#> interested--upset -0.217 0.106 -0.417 0.000
#> disinterested--upset -0.029 0.101 -0.239 0.161
#> excited--upset -0.126 0.098 -0.315 0.066
#> interested--strong 0.313 0.090 0.135 0.480
#> disinterested--strong 0.094 0.112 -0.120 0.318
#> excited--strong 0.500 0.078 0.337 0.645
#> upset--strong 0.118 0.109 -0.100 0.325
#> interested--stressed 0.268 0.102 0.058 0.460
#> disinterested--stressed 0.160 0.100 -0.049 0.351
#> excited--stressed -0.161 0.099 -0.358 0.031
#> upset--stressed 0.350 0.091 0.166 0.519
#> strong--stressed -0.010 0.107 -0.212 0.201
#> interested--steps 0.089 0.108 -0.123 0.297
#> disinterested--steps -0.078 0.108 -0.284 0.125
#> excited--steps -0.016 0.100 -0.207 0.182
#> upset--steps -0.039 0.107 -0.245 0.178
#> strong--steps 0.176 0.101 -0.024 0.364
#> stressed--steps -0.038 0.108 -0.236 0.193
#> --- The coefficients can also be printed by changing param to either all or beta, The summary can also be plotted. Here are the coefficients
plts <- plot(summary(fit, cred = 0.95))
cowplot::plot_grid(
cowplot::plot_grid(
plts$beta_plt$interested,
plts$beta_plt$disinterested,
plts$beta_plt$excited,
nrow = 1),
cowplot::plot_grid(
plts$beta_plt$upset,
plts$beta_plt$strong,
plts$beta_plt$stressed,
nrow = 1
),
nrow = 2)
There is a plot for the partial correlations in the object plts.
Select Graph
The graphs are selected with
select(fit, cred = 0.95)
#> BGGM: Bayesian Gaussian Graphical Models
#> ---
#> Vector Autoregressive Model (VAR)
#> ---
#> Posterior Samples: 5000
#> Credible Interval: 95 %
#> ---
#> Call:
#> var_estimate(Y = Y, beta_sd = 10)
#> ---
#> Partial Correlations:
#>
#> interested disinterested excited upset strong stressed steps
#> interested 0.000 0 0.388 -0.217 0.313 0.268 0
#> disinterested 0.000 0 0.000 0.000 0.000 0.000 0
#> excited 0.388 0 0.000 0.000 0.500 0.000 0
#> upset -0.217 0 0.000 0.000 0.000 0.350 0
#> strong 0.313 0 0.500 0.000 0.000 0.000 0
#> stressed 0.268 0 0.000 0.350 0.000 0.000 0
#> steps 0.000 0 0.000 0.000 0.000 0.000 0
#> ---
#> Coefficients:
#>
#> interested disinterested excited upset strong stressed steps
#> interested.l1 0 0.000 0 0.000 0 0.000 0
#> disinterested.l1 0 0.000 0 0.000 0 0.000 0
#> excited.l1 0 0.000 0 0.000 0 0.000 0
#> upset.l1 0 0.262 0 0.435 0 0.324 0
#> strong.l1 0 0.000 0 0.000 0 0.000 0
#> stressed.l1 0 0.000 0 0.000 0 0.000 0
#> steps.l1 0 0.000 0 0.000 0 0.209 0
#> ---
Plot Graph
For plotting, I use the qgraph package.
par(mfrow=c(1,2))
qgraph::qgraph(sel$pcor_weighted_adj, title = "Partials")
qgraph::qgraph(sel$beta_weighted_adj, title = "Coefficients")
Predictability
Finally, it is also possible to compute predictability, in this case Bayesian \(R^2\)
r2 <- predictability(fit)
# print
r2
#> BGGM: Bayesian Gaussian Graphical Models
#> ---
#> Metric: Bayes R2
#> Type: continuous
#> ---
#> Estimates:
#>
#> Node Post.mean Post.sd Cred.lb Cred.ub
#> interested 0.144 0.057 0.050 0.271
#> disinterested 0.166 0.061 0.060 0.302
#> excited 0.127 0.054 0.039 0.250
#> upset 0.220 0.070 0.093 0.368
#> strong 0.116 0.051 0.035 0.232
#> stressed 0.227 0.069 0.102 0.373
#> steps 0.105 0.047 0.032 0.210
The object r2 can also be plotted
plot(r2, type = "ridgeline")
Explore
Bayesian (exploratory) testing to come…
Confirm
Bayesian (confirmatory) testing to come…