# 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. ```r 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 ```r # 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) ```

AlphaMLE function