Ising: Ising model

Description

This is the family of Ising models fit to datasets with two or more ordered integer response options. The classical case is a dichotomous dataset (e.g., encoded with -1 and 1 or with 0 and 1), but the same parameterization---one threshold (tau) per variable and a pairwise network (omega)---is also used for any number of ordered response options (e.g., c(-1,0,1) or seq(-5,5); the values need not be integers), encoded identically across all variables. Only the partition function changes: it sums over all length(responses)^nNode response patterns rather than only 2^nNode. 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 constraints).

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,
                 maxStates = 2^maxNodes, 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 constraints. 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 constraints. 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 constraints. For multiple groups, this argument can be a list or array with each element/column encoding such scalars.
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 analysis. Must equal names of the dataset in data.
groups: An optional string indicating the name of the group variable in the 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 the response options used, encoded identically across all variables (e.g., c(-1,1), c(0,1), c(-1,0,1) or seq(-5,5)). Automatically detected from the data when not supplied; required when covs is used. Any number of distinct response options is allowed, and the values need not be integers.
missing: How should missingness be handled when data is used. Only "listwise" (listwise deletion) is currently supported for the Ising model; other options (e.g. "pairwise") are rejected at construction because they produce undefined sufficient statistics.
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 (exact) maximum likelihood estimation and "PML" for penalized maximum likelihood estimation; no other estimators are 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. Used to set the default of maxStates (2^maxNodes); for binary data this reproduces the historical node limit. It is not recommended to set this higher.
maxStates: The maximum number of response patterns the exact ML estimator may enumerate. Exact ML estimation of the Ising model sums over every possible response pattern when computing the partition function, expected values and (expected) Hessian, so the cost grows as length(responses)^nNode. The function stops with an error when length(responses)^nNode exceeds maxStates. The default 2^maxNodes reproduces the historical binary node limit while accounting for more than two response options. Raise this only if the computation is feasible (note that the number of states---and hence the run time---grows very quickly with both the number of nodes and the number of response options).
min_sum: The minimum sum score that is artificially 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.

The responses \(y_i\) need not be dichotomous: they may take any number of ordered values (the same set for every variable, and not necessarily integers), supplied through responses or detected automatically from the data. The Hamiltonian above is unchanged; only the partition function \(Z\) (and the expected values and Hessian derived from it) sums over all length(responses)^nNode response patterns instead of only 2^nNode. With two response options the model is the usual Ising model and reproduces the same estimates as before. With more than two response options the model remains a maximum-entropy distribution matching the means (via tau) and the pairwise products (via omega); it does not model higher-order interactions or the within-variable second moments, and the enumeration cost grows steeply with the number of nodes and response options (see maxStates).

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)
# }

Run the code above in your browser using DataLab