ohoegdm {ohoegdm}

Ordinal Higher-Order General Diagnostic Model under the Exploratory Framework (OHOEGDM)

Description

Performs the Gibbs sampling routine for an ordinal higher-order EGDM.

Usage

ohoegdm(
  y,
  k,
  m = 2,
  order = k,
  sd_mh = 0.4,
  burnin = 1000L,
  chain_length = 10000L,
  l0 = c(1, rep(100, sum(choose(k, seq_len(order))))),
  l1 = c(1, rep(1, sum(choose(k, seq_len(order))))),
  m0 = 0,
  bq = 1
)

Arguments

y

Ordinal Item Matrix

k

Dimension to estimate for Q matrix

m

Number of Item Categories. Default is 2 matching the binary case.

order

Highest interaction order to consider. Default model-specified k.

sd_mh

Metropolis-Hastings standard deviation tuning parameter.

burnin

Amount of Draws to Burn

chain_length

Number of Iterations for chain.

l0

Spike parameter. Default 1 for intercept and 100 coefficients

l1

Slab parameter. Default 1 for all values.

m0, bq

Additional tuning parameters.

Details

The estimates list contains the mean information from the sampling procedure. Meanwhile, the chain list contains full MCMC values. Moreover, the details list provides information regarding the estimation call. Lastly, the recovery list stores values that can be used when assessing the method under a simulation study.

Value

A ohoegdm object containing four named lists:

• estimates: Averaged chain iterations

  • thetas: Average theta coefficients

  • betas: Average beta coefficients

  • deltas: Average activeness of coefficients

  • classes: Average class membership

  • m2lls: Average negative two times log-likelihood

  • omegas: Average omega

  • kappas : Average category threshold parameter

  • taus: Average K-vectors of factor intercept

  • lambdas: Average K-vectors of factor loadings

  • guessing: Average guessing item parameter

  • slipping: Average slipping item parameter

  • QS: Average activeness of Q matrix entries

• chain: Chain iterations from the underlying C++ rountine.

  • thetas: Theta coefficients iterations

  • betas: Beta coefficients iterations

  • deltas: Activeness of coefficients iterations

  • classes: Class membership iterations

  • m2lls: Negative two times log-likelihood iterations

  • omegas: Omega iterations

  • kappas : Category threshold parameter iterations

  • taus: K-vector of factor intercept iterations

  • lambdas: K-vector of factor loadings iterations

  • guessing: Guessing item parameter iterations

  • slipping: Slipping item parameter iterations

• details: Properties used to estimate the model

  • n: Number of Subjects

  • j: Number of Items

  • k: Number of Traits

  • m: Number of Item Categories.

  • order: Highest interaction order to consider. Default model-specified k.

  • sd_mh: Metropolis-Hastings standard deviation tuning parameter.

  • l0: Spike parameter

  • l1: Slab parameter

  • m0, bq: Additional tuning parameters

  • burnin: Number of Iterations to discard

  • chain_length: Number of Iterations to keep

  • runtime: Elapsed time algorithm run time in the C++ code.

• recovery: Assess recovery metrics under a simulation study.

  • Q_item_encoded: Per-iteration item encodings from Q matrix.

  • MHsum: Average acceptance from metropolis hastings sampler

Examples

# Simulation Study
if (requireNamespace("edmdata", quietly = TRUE)) {
# Q and Beta Design ----

# Obtain the full K3 Q matrix from edmdata
data("qmatrix_oracle_k3_j20", package = "edmdata")
Q_full = qmatrix_oracle_k3_j20

# Retain only a subset of the original Q matrix
removal_idx = -c(3, 5, 9, 12, 15, 18, 19, 20)
Q = Q_full[removal_idx, ]

# Construct the beta matrix by-hand
beta = matrix(0, 20, ncol = 8)

# Intercept
beta[, 1] = 1

# Main effects
beta[1:3, 2] = 1.5
beta[4:6, 3] = 1.5
beta[7:9, 5] = 1.5

# Setup two-way effects
beta[10, c(2, 3)] = 1
beta[11, c(3, 4)] = 1

beta[12, c(2, 5)] = 1
beta[13, c(2, 5)] = 1
beta[14, c(2, 6)] = 1

beta[15, c(3, 5)] = 1
beta[16, c(3, 5)] = 1
beta[17, c(3, 7)] = 1

# Setup three-way effects
beta[18:20, c(2, 3, 5)] = 0.75

# Decrease the number of Beta rows
beta = beta[removal_idx,]

# Construct additional parameters for data simulation
Kappa = matrix(c(0, 1, 2), nrow = 20, ncol = 3, byrow =TRUE) # mkappa
lambda = c(0.25, 1.5, -1.25) # mlambdas
tau = c(0, -0.5, 0.5) # mtaus


# Simulation conditions ---- 
N = 100        # Number of Observations
J = nrow(beta) # Number of Items
M = 4          # Number of Response Categories
Malpha = 2     # Number of Classes
K = ncol(Q)    # Number of Attributes
order = K      # Highest interaction to consider
sdmtheta = 1   # Standard deviation for theta values

# Simulate data ---- 

# Generate theta values
theta = rnorm(N, sd = sdmtheta)

# Generate alphas 
Zs = matrix(1, N, 1) %*% tau + 
     matrix(theta, N, 1) %*% lambda + 
     matrix(rnorm(N * K), N, K)
Alphas = 1 * (Zs > 0)

vv = gen_bijectionvector(K, Malpha)
CLs = Alphas %*% vv
Atab = GenerateAtable(Malpha ^ K, K, Malpha, order)$Atable

# Simulate item-level data
Ysim = sim_slcm(N, J, M, Malpha ^ K, CLs, Atab, beta, Kappa)

# Establish chain properties 
# Standard Deviation of MH. Set depending on sample size.
# If sample size is:
#  - small, allow for larger standard deviation
#  - large, allow for smaller standard deviation.
sd_mh = .4 
burnin = 50        # Set for demonstration purposes, increase to at least 5,000 in practice.
chain_length = 100 # Set for demonstration purposes, increase to at least 40,000 in practice.

# Setup spike-slab parameters
l0s = c(1, rep(100, Malpha ^ K - 1))
l1s = c(1, rep(1, Malpha ^ K - 1))

my_model = ohoegdm::ohoegdm(
  y = Ysim,
  k = K,
  m = M,
  order = order,
  l0 = l0s,
  l1 = l1s,
  m0 = 0,
  bq = 1,
  sd_mh = sd_mh,
  burnin = burnin,
  chain_length = chain_length
)
}