Ising: Ising model

Description

This is the family of Ising models fit to dichotomous datasets. Note that the input matters (see also https://arxiv.org/abs/1811.02916) in this model! Models based on a dataset that is encoded with -1 and 1 are not entirely equivalent to models based on datasets encoded with 0 and 1 (non-equivalences occur in multi-group settings with equality constrains).

Usage

Ising(data, omega = "full", tau, beta, beta_model =
                 c("beta", "log_beta"), vars, groups, covs, means,
                 nobs, covtype = c("choose", "ML", "UB"), responses,
                 missing = "listwise", equal = "none",
                 baseline_saturated = TRUE, estimator = "default",
                 optimizer, storedata = FALSE, WLS.W, sampleStats,
                 identify = TRUE, verbose = FALSE, maxNodes = 20,
                 min_sum = -Inf, bootstrap = FALSE, boot_sub,
                 boot_resample, penalty_lambda = NA,
                penalty_alpha = 1, penalize_matrices)

Value

An object of the class psychonetrics

Arguments

data: A data frame encoding the data used in the analysis. Can be missing if covs and nobs are supplied.
omega: The network structure. Either "full" to estimate every element freely, "zero" to set all elements to zero, or a matrix of the dimensions nNode x nNode with 0 encoding a fixed to zero element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
tau: Optional vector encoding the threshold/intercept structure. Set elements to 0 to indicate fixed to zero constrains, 1 to indicate free intercepts, and higher integers to indicate equality constrains. For multiple groups, this argument can be a list or array with each element/column encoding such a vector.
beta: Optional scalar encoding the inverse temperature. 1 indicate free beta parameters, and higher integers to indicate equality constrains. For multiple groups, this argument can be a list or array with each element/column encoding such scalers.
beta_model: How should beta be modeled? Set beta_model = "log_beta" to model the log of beta rather than beta directly.
vars: An optional character vector encoding the variables used in the analyis. Must equal names of the dataset in data.
groups: An optional character vector encoding the variables used in the analyis. Must equal names of the dataset in data.
covs: A sample variance--covariance matrix, or a list/array of such matrices for multiple groups. Make sure covtype argument is set correctly to the type of covariances used.
means: A vector of sample means, or a list/matrix containing such vectors for multiple groups.
nobs: The number of observations used in covs and means, or a vector of such numbers of observations for multiple groups.
covtype: If 'covs' is used, this is the type of covariance (maximum likelihood or unbiased) the input covariance matrix represents. Set to "ML" for maximum likelihood estimates (denominator n) and "UB" to unbiased estimates (denominator n-1). The default will try to find the type used, by investigating which is most likely to result from integer valued datasets.
responses: A vector of dichotemous responses used (e.g., c(-1,1) or c(0,1). Only needed when 'covs' is used.)
missing: How should missingness be handled in computing the sample covariances and number of observations when data is used. Can be "listwise" for listwise deletion, or "pairwise" for pairwise deletion. NOT RECOMMENDED TO BE USED YET IN ISING MODEL.
equal: A character vector indicating which matrices should be constrained equal across groups.
baseline_saturated: A logical indicating if the baseline and saturated model should be included. Mostly used internally and NOT Recommended to be used manually.
estimator: The estimator to be used. Currently implemented are "ML" for maximum likelihood estimation, "FIML" for full-information maximum likelihood estimation, "ULS" for unweighted least squares estimation, "WLS" for weighted least squares estimation, and "DWLS" for diagonally weighted least squares estimation. Only ML estimation is currently supported for the Ising model.
optimizer: The optimizer to be used. Can be one of "nlminb" (the default R nlminb function), "ucminf" (from the optimr package), "nloptr_TNEWTON" (preconditioned truncated Newton via nloptr), and "LBFGS++" (pure C++ L-BFGS-B). Defaults to "nlminb".
storedata: Logical, should the raw data be stored? Needed for bootstrapping (see bootstrap).
WLS.W: Optional WLS weights matrix. CURRENTLY NOT USED.
sampleStats: An optional sample statistics object. Mostly used internally.
identify: Logical, should the model be identified?
verbose: Logical, should messages be printed?
maxNodes: The maximum number of nodes allowed in the analysis. This function will stop with an error if more nodes are used (it is not recommended to set this higher).
min_sum: The minimum sum score that is artifically possible in the dataset. Defaults to -Inf. Set this only if you know a lower sum score is not possible in the data, for example due to selection bias.
bootstrap: Should the data be bootstrapped? If TRUE the data are resampled and a bootstrap sample is created. These must be aggregated using aggregate_bootstraps! Can be TRUE or FALSE. Can also be "nonparametric" (which sets boot_sub = 1 and boot_resample = TRUE) or "case" (which sets boot_sub = 0.75 and boot_resample = FALSE).
boot_sub: Proportion of cases to be subsampled (round(boot_sub * N)).
boot_resample: Logical, should the bootstrap be with replacement (TRUE) or without replacement (FALSE)
penalty_lambda: Numeric penalty strength for penalized ML estimation (PML/PFIML). NA (default) triggers automatic selection via EBIC-based grid search when a penalized estimator is used; set to a specific numeric value to use a fixed penalty strength (0 = no penalty). See find_penalized_lambda and penalize.
penalty_alpha: Elastic net mixing parameter: 1 = LASSO (default), 0 = ridge.
penalize_matrices: Character vector of matrix names to penalize. If missing, defaults are selected based on the model type.

Author

Sacha Epskamp <mail@sachaepskamp.com>

Details

The Ising Model takes the following form:

\(\Pr(\boldsymbol{Y} = \boldsymbol{y}) = \frac{\exp\left( -\beta H\left(\boldsymbol{y}; \boldsymbol{\tau}, \boldsymbol{\Omega}\right)\right)}{Z(\boldsymbol{\tau}, \boldsymbol{\Omega})}\)

With Hamiltonian:

\(H\left(\boldsymbol{y}; \boldsymbol{\tau}, \boldsymbol{\Omega}\right) = -\sum_{i=1}^{m} \tau_i y_{i} - \sum_{i=2}^{m} \sum_{j=1}^{i-1} \omega_{ij} y_i y_j.\)

And Z representing the partition function or normalizing constant.

References

Epskamp, S., Maris, G., Waldorp, L. J., & Borsboom, D. (2018). Network Psychometrics. In: Irwing, P., Hughes, D., & Booth, T. (Eds.), The Wiley Handbook of Psychometric Testing, 2 Volume Set: A Multidisciplinary Reference on Survey, Scale and Test Development. New York: Wiley.

Examples

Run this code

# \donttest{
library("dplyr")
data("Jonas")

# Variables to use:
vars <- names(Jonas)[1:10]

# Arranged groups to put unfamiliar group first (beta constrained to 1):
Jonas <- Jonas[order(Jonas$group),]

# Form saturated model:
model1 <- Ising(Jonas, vars = vars, groups = "group")

# Run model:
model1 <- model1 %>% runmodel
# Note: SEs are approximated automatically because there are zeroes
# in the crosstables of the "Knows Jonas" group, which makes the
# Fisher information matrix singular. A warning is shown when
# this fallback occurs.

# Prune-stepup to find a sparse model:
model1b <- model1 %>% prune(alpha = 0.05) %>%  stepup(alpha = 0.05)

# Equal networks:
model2 <- model1 %>% groupequal("omega") %>% runmodel

# Prune-stepup to find a sparse model:
model2b <- model2 %>% prune(alpha = 0.05) %>% stepup(mi = "mi_equal", alpha = 0.05)

# Equal thresholds:
model3 <- model2 %>% groupequal("tau") %>% runmodel

# Prune-stepup to find a sparse model:
model3b <- model3 %>% prune(alpha = 0.05) %>% stepup(mi = "mi_equal", alpha = 0.05)

# Equal beta:
model4 <- model3 %>% groupequal("beta") %>% runmodel

# Prune-stepup to find a sparse model:
model4b <- model4 %>% prune(alpha = 0.05) %>% stepup(mi = "mi_equal", alpha = 0.05)

# Compare all models:
compare(
  `1. all parameters free (dense)` = model1,
  `2. all parameters free (sparse)` = model1b,
  `3. equal networks (dense)` = model2,
  `4. equal networks (sparse)` = model2b,
  `5. equal networks and thresholds (dense)` = model3,
  `6. equal networks and thresholds (sparse)` = model3b,
  `7. all parameters equal (dense)` = model4,
  `8. all parameters equal (sparse)` = model4b
) %>% arrange(BIC)
# }

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