Estimate the conditional (in)dependence with either an analytic solution or efficiently sampling from the posterior distribution. These methods were introduced in Williams (2018) . The graph is selected with select.estimate and then plotted with plot.select.

estimate(
Y,
formula = NULL,
type = "continuous",
mixed_type = NULL,
analytic = FALSE,
prior_sd = 0.25,
iter = 5000,
impute = TRUE,
progress = TRUE,
seed = 1,
...
)

## Arguments

Y Matrix (or data frame) of dimensions n (observations) by p (variables). An object of class formula. This allows for including control variables in the model (i.e., ~ gender). See the note for further details. Character string. Which type of data for Y ? The options include continuous, binary, ordinal, or mixed. Note that mixed can be used for data with only ordinal variables. See the note for further details. Numeric vector. An indicator of length p for which varibles should be treated as ranks. (1 for rank and 0 to assume normality). The default is currently to treat all integer variables as ranks when type = "mixed" and NULL otherwise. See note for further details. Logical. Should the analytic solution be computed (default is FALSE)? Scale of the prior distribution, approximately the standard deviation of a beta distribution (defaults to 0.50). Number of iterations (posterior samples; defaults to 5000). Logicial. Should the missing values (NA) be imputed during model fitting (defaults to TRUE) ? Logical. Should a progress bar be included (defaults to TRUE) ? An integer for the random seed. Currently ignored.

## Value

The returned object of class estimate contains a lot of information that is used for printing and plotting the results. For users of BGGM, the following are the useful objects:

• pcor_mat Partial correltion matrix (posterior mean).

• post_samp An object containing the posterior samples.

## Details

The default is to draw samples from the posterior distribution (analytic = FALSE). The samples are required for computing edge differences (see ggm_compare_estimate), Bayesian R2 introduced in Gelman et al. (2019) (see predictability), etc. If the goal is to *only* determine the non-zero effects, this can be accomplished by setting analytic = TRUE. This is particularly useful when a fast solution is needed (see the examples in ggm_compare_ppc)

Controlling for Variables:

When controlling for variables, it is assumed that Y includes only the nodes in the GGM and the control variables. Internally, only the predictors that are included in formula are removed from Y. This is not behavior of, say, lm, but was adopted to ensure users do not have to write out each variable that should be included in the GGM. An example is provided below.

Mixed Type:

The term "mixed" is somewhat of a misnomer, because the method can be used for data including only continuous or only discrete variables. This is based on the ranked likelihood which requires sampling the ranks for each variable (i.e., the data is not merely transformed to ranks). This is computationally expensive when there are many levels. For example, with continuous data, there are as many ranks as data points!

The option mixed_type allows the user to determine which variable should be treated as ranks and the "emprical" distribution is used otherwise (Hoff 2007) . This is accomplished by specifying an indicator vector of length p. A one indicates to use the ranks, whereas a zero indicates to "ignore" that variable. By default all integer variables are treated as ranks.

Dealing with Errors:

An error is most likely to arise when type = "ordinal". The are two common errors (although still rare):

• The first is due to sampling the thresholds, especially when the data is heavily skewed. This can result in an ill-defined matrix. If this occurs, we recommend to first try decreasing prior_sd (i.e., a more informative prior). If that does not work, then change the data type to type = mixed which then estimates a copula GGM (this method can be used for data containing only ordinal variable). This should work without a problem.

• The second is due to how the ordinal data are categorized. For example, if the error states that the index is out of bounds, this indicates that the first category is a zero. This is not allowed, as the first category must be one. This is addressed by adding one (e.g., Y + 1) to the data matrix.

Imputing Missing Values:

Missing values are imputed with the approach described in Hoff (2009) . The basic idea is to impute the missing values with the respective posterior pedictive distribution, given the observed data, as the model is being estimated. Note that the default is TRUE, but this ignored when there are no missing values. If set to FALSE, and there are missing values, list-wise deletion is performed with na.omit.

## Note

Posterior Uncertainty:

A key feature of BGGM is that there is a posterior distribution for each partial correlation. This readily allows for visiualizing uncertainty in the estimates. This feature works with all data types and is accomplished by plotting the summary of the estimate object (i.e., plot(summary(fit))). Several examples are provided below.

Interpretation of Conditional (In)dependence Models for Latent Data:

See BGGM-package for details about interpreting GGMs based on latent data (i.e, all data types besides "continuous")

## References

Gelman A, Goodrich B, Gabry J, Vehtari A (2019). “R-squared for Bayesian Regression Models.” American Statistician, 73(3), 307--309. ISSN 15372731, doi: 10.1080/00031305.2018.1549100 .

Hoff PD (2007). “Extending the rank likelihood for semiparametric copula estimation.” The Annals of Applied Statistics, 1(1), 265--283.

Hoff PD (2009). A first course in Bayesian statistical methods, volume 580. Springer.

Williams DR (2018). “Bayesian Estimation for Gaussian Graphical Models: Structure Learning, Predictability, and Network Comparisons.” arXiv. doi: 10.31234/OSF.IO/X8DPR .

## Examples

# \donttest{
# note: iter = 250 for demonstrative purposes

#########################################
### example 1: continuous and ordinal ###
#########################################
# data
Y <- ptsd

# continuous

# fit model
fit <- estimate(Y, type = "continuous",
iter = 250)#> BGGM: Posterior Sampling #> BGGM: Finished
# summarize the partial correlations
summ <- summary(fit)

# plot the summary
plt_summ <- plot(summary(fit))

# select the graph
E <- select(fit)

# plot the selected graph
plt_E <- plot(select(fit))

# ordinal

# fit model (note + 1, due to zeros)
fit <- estimate(Y + 1, type = "ordinal",
iter = 250)#> Warning: imputation during model fitting is
#> currently only implemented for 'continuous' data.#> BGGM: Posterior Sampling #> BGGM: Finished
# summarize the partial correlations
summ <- summary(fit)

# plot the summary
plt <- plot(summary(fit))

# select the graph
E <- select(fit)

# plot the selected graph
plt_E <- plot(select(fit))

##################################
## example 2: analytic solution ##
##################################
# (only continuous)

# data
Y <- ptsd

# fit model
fit <- estimate(Y, analytic = TRUE)

# summarize the partial correlations
summ <- summary(fit)

# plot summary
plt_summ <- plot(summary(fit))

# select graph
E <- select(fit)

# plot the selected graph
plt_E <- plot(select(fit))

# }