ggm_compare_estimate function

GGM Compare: Estimate

Compare partial correlations that are estimated from any number of groups. This method works for continuous, binary, ordinal, and mixed data (a combination of categorical and continuous variables). The approach (i.e., a difference between posterior distributions) was described in \insertCite Williams2019;textualBGGM.

ggm_compare_estimate(
  ...,
  formula = NULL,
  type = "continuous",
  mixed_type = NULL,
  analytic = FALSE,
  prior_sd = sqrt(1/3),
  iter = 5000,
  impute = TRUE,
  progress = TRUE,
  seed = 1
)

Arguments

• ...: Matrices (or data frames) of dimensions n (observations) by p (variables). Requires at least two.
• formula: An object of class formula. This allows for including control variables in the model (i.e., ~ gender). See the note for further details.
• type: Character string. Which type of data for Y ? The options include continuous, binary, ordinal, or continuous. See the note for further details.
• mixed_type: Numeric vector. An indicator of length p for which varibles should be treated as ranks. (1 for rank and 0 to use the 'empirical' or observed distribution). The default is currently to treat all integer variables as ranks when type = "mixed" and NULL otherwise. See note for further details.
• analytic: Logical. Should the analytic solution be computed (default is FALSE)? This is only available for continous data. Note that if type = "mixed" and analytic = TRUE, the data will automatically be treated as continuous.
• prior_sd: The scale of the prior distribution (centered at zero), in reference to a beta distribtuion (defaults to sqrt(1/3)). See note for further details.
• iter: Number of iterations (posterior samples; defaults to 5000).
• impute: Logicial. Should the missing values (NA) be imputed during model fitting (defaults to TRUE) ?
• progress: Logical. Should a progress bar be included (defaults to TRUE) ?
• seed: An integer for the random seed.

Returns

A list of class ggm_compare_estimate containing:

• pcor_diffs partial correlation differences (posterior distribution)
• p number of variable
• info list containing information about each group (e.g., sample size, etc.)
• iter number of posterior samples
• call match.call

Details

This function can be used to compare the partial correlations for any number of groups. This is accomplished with pairwise comparisons for each relation. In the case of three groups, for example, group 1 and group 2 are compared, then group 1 and group 3 are compared, and then group 2 and group 3 are compared. There is a full distibution for each difference that can be summarized (i.e., summary.ggm_compare_estimate) and then visualized (i.e., plot.summary.ggm_compare_estimate). The graph of difference is selected with select.ggm_compare_estimate).

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. 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 handled 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 \insertCite hoff2009first;textualBGGM. 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

Mixed Data :

The mixed data approach was introduced \insertCite @in @hoff2007extending;textualBGGM

(our paper describing an extension to Bayesian hypothesis testing if forthcoming). This is a semi-paramateric copula model based on the ranked likelihood. This is computationally expensive when treating continuous data as ranks. The current default is to treat only integer data as ranks. This should of course be adjusted for continous data that is skewed. This can be accomplished with the argument mixed_type. A 1 in the numeric vector of length pindicates to treat that respective node as a rank (corresponding to the column number) and a zero indicates to use the observed (or "emprical") data.

It is also important to note that type = "mixed" is not restricted to mixed data (containing a combination of categorical and continuous): all the nodes can be ordinal or continuous (but again this will take some time).

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")

Additional GGM Compare Methods

Bayesian hypothesis testing is implemented in ggm_compare_explore and ggm_compare_confirm \insertCite Williams2019_bfBGGM. The latter allows for confirmatory hypothesis testing. An approach based on a posterior predictive check is implemented in ggm_compare_ppc

\insertCite williams2020comparingBGGM. This provides a 'global' test for comparing the entire GGM and a 'nodewise' test for comparing each variable in the network \insertCite Williams2019;textualBGGM.

Examples

# note: iter = 250 for demonstrative purposes

# data: Remove missings for "ordinal"
Y <- bfi[complete.cases(bfi),]

# males and females
Ymale <- subset(Y, gender == 1,
                   select = -c(gender,
                               education))[,1:10]

Yfemale <- subset(Y, gender == 2,
                     select = -c(gender,
                                 education))[,1:10]

# fit model
fit <- ggm_compare_estimate(Ymale,  Yfemale,
                           type = "ordinal",
                           iter = 250,
                           progress = FALSE)

###########################
### example 2: analytic ###
###########################
# only continuous

# fit model
fit <- ggm_compare_estimate(Ymale, Yfemale,
                            analytic = TRUE)

# summary
summ <- summary(fit)

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

# select
E <- select(fit)

# plot select
plt_E <- plot(select(fit))

References

\insertAllCited