AlphaMLE function

Maximum likelihood estimation of attribute profile

This function returns the model-based Maximum likelihood estimator(s) of the cognitive diagnostic attribute profile(s). Currently supported cognitive diagnostic models include the DINA, DINO, NIDA, GNIDA, and R-RUM models.

AlphaMLE(Y, Q, par, model = c("DINA", "DINO", "NIDA", "GNIDA", "RRUM"),
undefined.flag = NULL)

Arguments

• Y: A matrix of binary responses. Rows represent persons and columns represent items. 1=correct, 0=incorrect.
• Q: The Q-matrix of the test. Rows represent items and columns represent attributes. 1=attribute required by the item, 0=attribute not required by the item.
• par: A list of parameters. DINA & DINO --- par$slip: a vector of slipping parameters for each item; par$guess: a vector of guessing parameters for each item. NIDA --- par$slip: a vector of slipping parameters for each attribute; par$guess: a vector of guessing parameters for each attribute. GNIDA --- par$slip: a matrix (items by attributes) of slipping parameters; par$guess: a matrix (items by attributes) of guessing parameters. RRUM --- par$pi: a vector of pi parameters for each item; par$r: a matrix (items by attributes) of r parameters.
• model: Currently supports five models: "DINA", "DINO", "NIDA", "GNIDA", and "RRUM". The default is "DINA".
• undefined.flag: A binary vector indicating whether the parameters of each item are undefined. 1=undefined, 0=defined. Generally, this argument is only needed in "JMLE" for the DINA and DINO models, where the data may generate undefined item parameters for some items.

Returns

• alpha.est: A matrix of estimated attribute profiles for all examinees. Rows represent persons and columns represent attributes. 1=examinee masters the attribute, 0=examinee does not master the attribute.

• est.class: The class number (row index in pattern) for each person's attribute profile. It can also be used for locating the log-likelihood value in loglike.matrix for the estimated attribute profile for each person.

• n.tie: Number of ties in the log-likelihood among the candidate attribute profiles for each person. When we encounter ties, one of the tied attribute profiles is randomly chosen.

• class.tie: The class numbers (row index in pattern) of the tied patterns for each person.

• pattern: All possible attribute profiles in the search space.

• loglike.matrix: The matrix of the log-likelihood function values. Rows represent candidate attribute profiles in the same order with the pattern matrix; columns represent different examinees.

See Also

AlphaNP, JMLE, print.AlphaMLE, plot.AlphaMLE

Examples

# Generate item and examinee profiles

natt <- 3
nitem <- 4
nperson <- 5
Q <- rbind(c(1, 0, 0), c(0, 1, 0), c(0, 0, 1), c(1, 1, 1))
alpha <- rbind(c(0, 0, 0), c(1, 0, 0), c(0, 1, 0), c(0, 0, 1), c(1, 1, 1))

# Generate DINA model-based response data

slip <- c(0.1, 0.15, 0.2, 0.25)
guess <- c(0.1, 0.15, 0.2, 0.25)
my.par <- list(slip=slip, guess=guess)

data <- matrix(NA, nperson, nitem)
eta <- matrix(NA, nperson, nitem)

for (i in 1:nperson) {
  for (j in 1:nitem) {
  eta[i, j] <- prod(alpha[i,] ^ Q[j, ])
  P <- (1 - slip[j]) ^ eta[i, j] * guess[j] ^ (1 - eta[i, j])
  u <- runif(1)
  data[i, j] <- as.numeric(u < P)
  }
}

# Using the function to estimate examinee attribute profile

alpha.est.MLE <- AlphaMLE(data, Q, my.par, model="DINA", undefined.flag=NULL)

nperson <- 1   # Choose an examinee to investigate
print(alpha.est.MLE)   # Print the estimated examinee attribute profiles
plot(alpha.est.MLE, nperson)   # Plot the sorted log-likelihood function 
#of different attribute profiles for this examinee
ItemFit(alpha.est.MLE)