`spBFA`

This is a brief description of how to use the `spBFA`

package within the context of glaucoma progression. To demonstrate the method we use the `VFSeries`

data set from the `womblR`

package. We begin by loading the packages.

# ###Start with a clean space # rm(list = ls()) # # ###Take care of some stuff that I don't want the user to see... # path.package <- "/Users/sam/Documents/Postdoc/Software/spBFA/" # suppressMessages(devtools::load_all(path.package)) #loads scripts # suppressMessages(devtools::document(path.package)) #creates documentation ###Make sure to remove devtools from Suggests line in DESCRIPTION before submission

library(womblR) library(spBFA)

The data object `VFSeries`

has four variables, `Visit`

, `DLS`

, `Time`

and `Location`

. The data object loads automatically; here's what the data looks like,

```
head(VFSeries)
```

The variable `Visit`

represents the visual field test visit number, `DLS`

the observed outcome variable, differential light sensitivity, `Time`

the time of the visual field test (in days from baseline visit) and `Location`

the spatial location on the visual field that the observation occurred. To help illuminate visual field data we can use the `PlotVFTimeSeries`

function from the `womblR`

package. `PlotVFTimeSeries`

is a function that plots the observed visual field data over time at each location on the visual field.

PlotVfTimeSeries(Y = VFSeries$DLS, Location = VFSeries$Location, Time = VFSeries$Time, main = "Visual field sensitivity time series \n at each location", xlab = "Days from baseline visit", ylab = "Differential light sensitivity (dB)", line.col = 1, line.type = 1, line.reg = FALSE)

The figure above demonstrates the visual field from a Humphrey Field Analyzer-II testing machine, which generates 54 spatial locations (only 52 informative locations, note the 2 blanks spots corresponding to the blind spot). At each visual field test a patient is assessed for vision loss.

`spBFA`

We can now begin to think about preparing objects for use in the Bayesian spatial factor analysis function (`bfa_sp`

). According to the manual, we use a `formula`

specification for the observed data, where the data must be first ordered spatially and then temporally (We only use on spatial observation type in this example, thus `O`

is one.). Furthermore, we will remove all locations that correspond to the natural blind spot (which in the Humphrey Field Analyzer-II correspond to locations 26 and 35).

blind_spot <- c(26, 35) # define blind spot VFSeries <- VFSeries[order(VFSeries$Location), ] # sort by location VFSeries <- VFSeries[order(VFSeries$Visit), ] # sort by visit VFSeries <- VFSeries[!VFSeries$Location %in% blind_spot, ] # remove blind spot locations dat <- data.frame(Y = VFSeries$DLS / 10) # create data frame with scaled data

Now that we have assigned the observed outcome we move onto the temporal variable `time`

. For visual field data we define this to be the time from the baseline visit. We obtain the unique days from the baseline visit and scale them to be on the year scale.

Time <- unique(VFSeries$Time) / 365 # years since baseline visit print(Time)

Our example patient has nine visual field visits and the last visit occurred 2.57 years after the baseline visit.

We now specify the adjacency matrix, `W`

, for our areal data analysis. We use the `HFAII_Queen`

adjacency matrix for visual fields, provided in `womblR`

. This is an adjacency matrix that defines an adjacency as edges and corners (i.e., the movements of a queen in chess). The adjacency objects are preloaded and contain the blind spot, so we define our adjacency matrix as follows.

W <- HFAII_Queen[-blind_spot, -blind_spot] # visual field adjacency matrix M <- dim(W)[1] # number of locations

Now that we have specified the data objects, we will customize the objects that characterize Bayesian Markov chain Monte Carlo (MCMC) methods, in particular hyperparameters, starting values, metropolis tuning values and MCMC inputs.

We begin be specifying the hyperparameters for the model. The parameter $\psi$ is uniformly distributed with lower bound, $a_{\psi}$, and upper bound, $b_{\psi}$. The upper bound for $\psi$ cannot be specified arbitrarily since it is important to account for the magnitude of time elapsed. We specify the following upper bound for $\psi$ to dictate a weakly informative prior distribution.

TimeDist <- as.matrix(dist(Time)) BPsi <- log(0.025) / -min(TimeDist[TimeDist > 0]) APsi <- log(0.975) / -max(TimeDist)

Then, we can create a hyperparameters `list`

object, `Hypers`

, that can be used for `spBFA`

. We first define the number of latent factors (`K`

) and spatial observation types (`O`

).

K <- 10 O <- 1 Hypers <- list(Sigma2 = list(A = 0.001, B = 0.001), Kappa = list(SmallUpsilon = O + 1, BigTheta = diag(O)), Delta = list(A1 = 1, A2 = 20), Psi = list(APsi = APsi, BPsi = BPsi), Upsilon = list(Zeta = K + 1, Omega = diag(K)))

Here, $\sigma2$ has an inverse Gamma distribution with shape and scale, $\kappa$ and $\Upsilon$ have inverse-Wishart distributions with degrees of freedom (scalar) and scale matrix, and $\delta$ has a multiplicative gamma process shrinkage prior (See the help manual for `spBFA`

for further details).

Specify a `list`

object, `Starting`

, that contains the starting values for the hyperparameters.

Starting <- list(Sigma2 = 1, Kappa = diag(O), Delta = 2 * (1:K), Psi = (APsi + BPsi) / 2, Upsilon = diag(K))

Provide tuning parameters for the metropolis steps in the MCMC sampler.

Tuning <- list(Psi = 1)

We set `Tuning`

to the default setting of one and let the pilot adaptation in the burn-in phase tune the acceptance rates to the appropriate range. Finally, we set the MCMC inputs using the `MCMC`

list object.

MCMC <- list(NBurn = 1000, NSims = 1000, NThin = 2, NPilot = 5)

We specify that our model will run for a burn-in period of 1,000 scans, followed by 1,000 scans after burn-in. In the burn-in period there will be 5 iterations of pilot adaptation evenly spaced out over the period. Finally, the final number of samples to be used for inference will be thinned down to 500 based on the thinning number of 2. We suggest running the sampler 250,000 iterations after burn-in, but in the vignette we are limited by compilation time.

We have now specified all model objects and are prepared to implement the `bfa_sp`

regression object. To demonstrate the `bfa_sp`

object we will use all of its options, even those that are being used in their default settings.

```
data(reg.bfa_sp)
```

reg.bfa_sp <- bfa_sp(Y ~ 0, data = dat, dist = W, time = Time, K = 10, starting = Starting, hypers = Hypers, tuning = Tuning, mcmc = MCMC, L = Inf, family = "tobit", trials = NULL, temporal.structure = "exponential", spatial.structure = "discrete", seed = 54, gamma.shrinkage = TRUE, include.space = TRUE, clustering = TRUE) ## Burn-in progress: |*************************************************| ## Sampler progress: 0%.. 10%.. 20%.. 30%.. 40%.. 50%.. 60%.. 70%.. 80%.. 90%.. 100%..

The first line of arguments are the data objects, `formula`

, `data`

, `dist`

, `time`

, and `K`

. These objects must be specified for `bfa_sp`

to run. Note that in the formula we use `~0`

, which means that there are no covariates. The second line of objects are the MCMC characteristics objects we defined previously. These objects do not need to be defined for `spBFA`

to function, but are provided for the user to customize the model to their choosing. If they are not provided, defaults are given. Next, we specify that the infinite mixture model is used `L = Inf`

. `L`

can be any positive integer or infinity. The `family`

is set to `tobit`

since we know that visual field data is censored (`family`

can be `normal`

, `probit`

, `tobit`

, or `binomial`

). If `binomial`

is used, `trials`

must be specified. Our temporal dependence structure is set to be `exponential`

, and the spatial structure is `discrete`

(i.e., areal). Finally, we define the following logicals, `gamma.shrinkage`

(which controls the gamma shrinkage process), `include.space`

(which controls if spatial dependency is included), and `clustering`

(which controls if the Bayesian non-parametric process prior is used). We set the `seed`

to 54.

The following are the returned objects from `reg.bfa_sp`

.

```
names(reg.bfa_sp)
```

The object `reg.bfa_sp`

contains raw MCMC samples for all parameters, metropolis acceptance rates and final tuning parameters (`metropolis`

) and model runtime (`runtime`

). The objects `datobj`

and `dataug`

can be ignored as they are for later use in secondary functions.

Before analyzing the raw MCMC samples from our model we want to verify that there are no convergence issues. We begin by loading the `coda`

package.

```
library(coda)
```

Then we convert the raw `spBFA`

MCMC objects to `coda`

package `mcmc`

objects. We look at $\sigma^2(\mathbf{s}_1)$ only for learning purposes.

Sigma2_1 <- as.mcmc(reg.bfa_sp$sigma2[, 1])

We begin by checking traceplots of the parameter.

par(mfrow = c(1, 1)) traceplot(Sigma2_1, ylab = expression(paste(sigma^2 ~ "(" ~ s[1]~ ")")), main = expression(paste("Posterior" ~ sigma^2 ~ "(" ~ s[1]~ ")")))

From the figure, it is clear that the traceplots exhibit some poor behavior. However, these traceplots are nicely behaved considering the number of iterations the MCMC sampler ran. The traceplots demonstrate that the parameters have converged to their stationary distribution, but still need more samples to rid themselves of autocorrelation. Finally, we present the corresponding test statistics from the Geweke diagnostic test.

geweke.diag(Sigma2_1)$z

Since the test statistic is not terribly large in the absolute value there is not strong evidence that our model did not converge.

The `diagnostics`

function in the `spBFA`

package can be used to calculate various diagnostic metrics. The function takes in the `spBFA`

regression object.

Diags <- spBFA::diagnostics(reg.bfa_sp, diags = c("dic", "dinf", "waic"), keepDeviance = TRUE)

The `diagnostics`

function calculates diagnostics that depend on both the log-likelihood and posterior predictive distribution. So, if any of these diagnostics are specified, one or both of these must be sampled from. The `keepDeviance`

and `keepPPD`

indicate whether or not these distributions should be saved for the user. We indicate that we would like the output to be saved for the log-likelihood (i.e., deviance). We explore the output by looking at the traceplot of the deviance.

Deviance <- as.mcmc(Diags$deviance) traceplot(Deviance, ylab = "Deviance", main = "Posterior Deviance")

This distribution has possible convergence issues, however this is not concerning given the number of MCMC iterations run.

```
print(Diags)
```

unlist(Diags$dic) unlist(Diags$dinf) unlist(Diags$waic)

The `spBFA`

package provides the `predict.spBFA`

function for sampling from the posterior predictive distribution at future time points of the observed data. This is different from the posterior predictive distribution obtained from the `diagnostics`

function, because that distribution is for the observed time points and is automatically obtained given the posterior samples from `spBFA`

. We begin by specifying the future time point we want to predict as 3 years.

NewTimes <- 3

Then, we use `predict.spBFA`

to calculate the future posterior predictive distribution.

Predictions <- predict(reg.bfa_sp, NewTimes)

We can see that `predict.spBFA`

returns a `list`

containing a matrix of predictions corresponding to each future time point. The name of each matrix is the numeric time point for each future visit.

```
names(Predictions)
```

You can plot a heat map representation of the posterior distribution of the change points using the function `PlotSensitivity`

from `womblR`

.

PlotSensitivity(Y = apply(Predictions$Y$Y10, 2, mean) * 10, main = "Posterior mean prediction\n at 3 years", legend.lab = "Posterior Mean", legend.round = 2, bins = 250, border = FALSE, zlim = c(0, 40))

This figure shows the posterior predicted mean at 3 years over the visual field. The `PlotSensitivity`

function can be used for plotting any observations on the visual field surface. We can also plot the standard deviation.

PlotSensitivity(Y = apply(Predictions$Y$Y10 * 10, 2, sd), main = "Posterior standard deviation\n (SD) at 3 years", legend.lab = "Posterior SD", legend.round = 2, bins = 250, border = FALSE, zlim = c(0, 40))

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