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, ... )

Y | Matrix (or data frame) of dimensions |
---|---|

formula | An object of class |

type | Character string. Which type of data for |

mixed_type | Numeric vector. An indicator of length |

analytic | Logical. Should the analytic solution be computed (default is |

prior_sd | Scale of the prior distribution, approximately the standard deviation of a beta distribution (defaults to 0.50). |

iter | Number of iterations (posterior samples; defaults to 5000). |

impute | Logicial. Should the missing values ( |

progress | Logical. Should a progress bar be included (defaults to |

seed | An integer for the random seed. |

... | Currently ignored. |

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.

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`

.

**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"`

)

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
.

# \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)#>#># 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.#>#># 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)) # }