sim_alphas {hmcdm}

Generate attribute trajectories under the specified hidden Markov models

Description

Based on the learning model parameters, create cube of attribute patterns of all subjects across time. Currently available learning models are Higher-order hidden Markov DCM('HO_sep'), Higher-order hidden Markov DCM with learning ability as a random effect('HO_joint'), the simple independent-attribute learning model('indept'), and the first order hidden Markov model('FOHM').

Usage

sim_alphas(
  model,
  lambdas = NULL,
  thetas = NULL,
  Q_matrix = NULL,
  Design_array = NULL,
  taus = NULL,
  Omega = NULL,
  N = NA_integer_,
  L = NA_integer_,
  R = NULL,
  alpha0 = NULL
)

Arguments

model

The learning model under which the attribute trajectories are generated. Available options are: 'HO_joint', 'HO_sep', 'indept', 'FOHM'.

lambdas

A vector of transition model coefficients. With 'HO_sep' model specification, lambdas should be a length 4 vector. First entry is intercept of the logistic transition model, second entry is the slope of general learning ability, third entry is the slope for number of other mastered skills, fourth entry is the slope for amount of practice. With 'HO_joint' model specification, lambdas should be a length 3 vector. First entry is intercept of the logistic transition model, second entry is the slope for number of other mastered skills, third entry is the slope for amount of practice.

thetas

A length N vector of learning abilities of each subject.

Q_matrix

A J-by-K Q-matrix

Design_array

A N-by-J-by-L array indicating items administered to examinee n at time point l.

taus

A length K vector of transition probabilities from 0 to 1 on each skill

Omega

A 2^K-by-2^K matrix of transition probabilities from row pattern to column pattern

N

An int of number of examinees.

L

An int of number of time points.

R

A K-by-K dichotomous reachability matrix indicating the attribute hierarchies. The k,k'th entry of R is 1 if k' is prereq to k.

alpha0

Optional. An N-by-K matrix of subjects' initial attribute patterns.

Value

An N-by-K-by-L array of attribute patterns of subjects at each time point.

Examples

## HO_joint ##
N = nrow(Design_array)
J = nrow(Q_matrix)
K = ncol(Q_matrix)
L = dim(Design_array)[3]
class_0 <- sample(1:2^K, N, replace = TRUE)
Alphas_0 <- matrix(0,N,K)
for(i in 1:N){
  Alphas_0[i,] <- inv_bijectionvector(K,(class_0[i]-1))
}
thetas_true = rnorm(N, 0, 1.8)
lambdas_true <- c(-2, .4, .055)
Alphas <- sim_alphas(model="HO_joint", 
                    lambdas=lambdas_true, 
                    thetas=thetas_true, 
                    Q_matrix=Q_matrix, 
                    Design_array=Design_array)

## HO_sep ##
N = dim(Design_array)[1]
J = nrow(Q_matrix)
K = ncol(Q_matrix)
L = dim(Design_array)[3]
class_0 <- sample(1:2^K, N, replace = L)
Alphas_0 <- matrix(0,N,K)
for(i in 1:N){
  Alphas_0[i,] <- inv_bijectionvector(K,(class_0[i]-1))
}
thetas_true = rnorm(N)
lambdas_true = c(-1, 1.8, .277, .055)
Alphas <- sim_alphas(model="HO_sep", 
                     lambdas=lambdas_true, 
                     thetas=thetas_true, 
                     Q_matrix=Q_matrix, 
                     Design_array=Design_array)

## indept ##
N = dim(Design_array)[1]
K = dim(Q_matrix)[2]
L = dim(Design_array)[3]
tau <- numeric(K)
for(k in 1:K){
  tau[k] <- runif(1,.2,.6)
}
R = matrix(0,K,K)
p_mastery <- c(.5,.5,.4,.4)
Alphas_0 <- matrix(0,N,K)
for(i in 1:N){
  for(k in 1:K){
    prereqs <- which(R[k,]==1)
    if(length(prereqs)==0){
      Alphas_0[i,k] <- rbinom(1,1,p_mastery[k])
    }
    if(length(prereqs)>0){
      Alphas_0[i,k] <- prod(Alphas_0[i,prereqs])*rbinom(1,1,p_mastery)
    }
  }
}
Alphas <- sim_alphas(model="indept", taus=tau, N=N, L=L, R=R)

## FOHM ##
N = dim(Design_array)[1]
K = ncol(Q_matrix)
L = dim(Design_array)[3]
TP <- TPmat(K)
Omega_true <- rOmega(TP)
class_0 <- sample(1:2^K, N, replace = L)
Alphas_0 <- matrix(0,N,K)
for(i in 1:N){
  Alphas_0[i,] <- inv_bijectionvector(K,(class_0[i]-1))
}
Alphas <- sim_alphas(model="FOHM", Omega = Omega_true, N=N, L=L)