# Module Kullback-Leibler (MKL) and posterior module Kullback-Leibler (MKLP) This command returns the value of the Kullback-Leibler (MKL) or the posterior Kullback-Leibler (MKLP) weighted likelihood for a given target module and an item bank (both under dichotomous and polytomous IRT models). ```r MKL(itemBank, modules, target.mod, theta = NULL, it.given, x, model = NULL, lower = -4, upper = 4, nqp = 33, type = "MKL", priorDist = "norm", priorPar = c(0, 1), D = 1) ``` ## Arguments - `itemBank`: numeric: a suitable matrix of item parameters. See Details . - `modules`: a binary matrix that specifies the item membership to th emodules. See Details . - `target.mod`: numeric: the module (referred to as its column number in the `modules` matrix) for which the information must be computed. - `theta`: either the provisional ability level or `NULL` (default). See Details . - `it.given`: numeric: a vector of item indicators for all previously administered items. - `x`: numeric: a vector of item responses, coded as 0 or 1 only (for dichotomous items) or from 0 to the number of response categories minus one (for polytomous items). The length of `x` must be equal to the length of `it.given`. - `model`: either `NULL` (default) for dichotomous models, or any suitable acronym for polytomous models. Possible values are `"GRM"`, `"MGRM"`, `"PCM"`, `"GPCM"`, `"RSM"` and `"NRM"`. See Details . - `lower`: numeric: the lower bound for numerical integration (default is -4). - `upper`: numeric: the upper bound for numerical integration (default is 4). - `nqp`: numeric: the number of quadrature points (default is 33). - `type`: character: the type of Kullback-Leibler information to be computed. Possible values are `"MKL"` (default) and `"MKLP"`. See Details . - `priorDist`: character: the prior ability distribution. Possible values are `"norm"` (default) for the normal distribution, and `"unif"` for the uniform distribution. Ignored if `type` is not `"MPWI"`. - `priorPar`: numeric: a vector of two components with the prior parameters. If `priorDist` is `"norm"`, then `priorPar` contains the mean and the standard deviation of the normal distribution. If `priorDist` is `"unif"`, then `priorPar` contains the bounds of the uniform distribution. The default values are 0 and 1 respectively. Ignored if `type` is not `"MPWI"`. - `D`: numeric: the metric constant. Default is `D=1` (for logistic metric); `D=1.702` yields approximately the normal metric (Haley, 1952). Ignored if `model` is not `NULL`. ## Returns The required KL (or KLP) weighted module likelihood for the target module. ## Details This function extends the KL and the KLP methods to select the next item in CAT, to the MST framework. This command serves as a subroutine for the `nextModule` function. Dichotomous IRT models are considered whenever `model` is set to `NULL` (default value). In this case, `itemBank` must be a matrix with one row per item and four columns, with the values of the discrimination, the difficulty, the pseudo-guessing and the inattention parameters (in this order). These are the parameters of the four-parameter logistic (4PL) model (Barton and Lord, 1981). Polytomous IRT models are specified by their respective acronym: `"GRM"` for Graded Response Model, `"MGRM"` for Modified Graded Response Model, `"PCM"` for Partical Credit Model, `"GPCM"` for Generalized Partial Credit Model, `"RSM"` for Rating Scale Model and `"NRM"` for Nominal Response Model. The `itemBank` still holds one row per item, end the number of columns and their content depends on the model. See `genPolyMatrix` for further information and illustrative examples of suitable polytomous item banks. FROM HERE Under polytomous IRT models, let **k** be the number of administered items, and set $x_1, ..., x_k$ as the provisional response pattern (where each response $x_l$ takes values in $\{0, 1, ..., g_l\}$). Set $\hat{\theta}_k$ as the provisional ability estimate (with the first **k** responses). Set $M$ as the number of items in the target module of interest (not yet administered). Set also $L(\theta | x_1, ..., x_k) $ as the likelihood function of the first $k$ items and evaluated at $\theta$. Set finally $P_{jt}(\theta)$ as the probability of answering response category **t** to item **j** of the target module ($j = 1, ..., M$) for a given ability level $\theta$. Then, module Kullack-Leibler (MKL) information is defined as $$ MKL(\theta || \hat{\theta}_k) = \sum_{j=1}^{M} \sum_{t=0}^{g_j} \,P_{jt}(\hat{\theta}_k) \,\log \left( \frac{P_{jt}(\hat{\theta}_k)}{P_{jt}(\theta)}\right). $$ In case of dichotomous IRT models, all $g_l$ values reduce to 1, so that item responses $x_l$ equal either 0 or 1. Set simply $P_j(\theta)$ as the probability of answering item **j** correctly ($j = 1, ..., M$) for a given ability level $\theta$. Then, MKL information reduces to $$ MKL(\theta || \hat{\theta}) = \sum_{j=1}^{M} \left\{P_j(\hat{\theta}) \,\log \left( \frac{P_j(\hat{\theta}_k)}{P_j(\theta)}\right) + [1-P_j(\hat{\theta}_k)] \,\log \left( \frac{1-P_j(\hat{\theta}_k)}{1-P_j(\theta)}\right) \right\}. $$ The quantity that is returned by this `MKL` function is either: the likelihood function weighted by module Kullback-Leibler information (the MKL value): $$ MKL(\hat{\theta}_k) = \int MKL(\theta || \hat{\theta}_k) \, L(\theta | x_1, ..., x_k) \,d\theta $$ or the posterior function weighted by module Kullback-Leibler information (the MKLP value): $$ MKLP(\hat{\theta}) = \int MKL(\theta || \hat{\theta}_k) \, \pi(\theta) \,L(\theta | x_1, ..., x_k) \,d\theta $$ where $\pi(\theta)$ is the prior distribution of the ability level. These integrals are approximated by the `integrate.mstR` function. The range of integration is set up by the arguments `lower`, `upper` and `nqp`, giving respectively the lower bound, the upper bound and the number of quadrature points. The default range goes from -4 to 4 with length 33 (that is, by steps of 0.25). The argument `type` defines the type of information to be computed. The default value, `"MKL"`, computes the MKL value, while the MKLP value is obtained with `type="MKLP"`. For the latter, the `priorDist` and `priorPar` arguments fix the prior ability distribution. The normal distribution is set up by `priorDist="norm"` and then, `priorPar` contains the mean and the standard deviation of the normal distribution. If `priorDist` is `"unif"`, then the uniform distribution is considered, and `priorPar` fixes the lower and upper bounds of that uniform distribution. By default, the standard normal prior distribution is assumed. This argument is ignored whenever `method` is not `"MKLP"`. The provisional response pattern and the related administered items are provided by the vectors `x` and `it.given` respectively. The target module (for which the maximum information is computed) is given by its column number in the `modules` matrix, through the `target.mod` argument. The provisioal (ad-interim) ability level can be provided through the `theta` argument. If not provided or set to `NULL` (default value), it is then internally computed as the ML estimate of ability for the given response pattern `x` and the previously administered items `it.given`. ## References Barton, M.A., and Lord, F.M. (1981). **An upper asymptote for the three-parameter logistic item-response model**. Research Bulletin 81-20. Princeton, NJ: Educational Testing Service. Haley, D.C. (1952). **Estimation of the dosage mortality relationship when the dose is subject to error**. Technical report no 15. Palo Alto, CA: Applied Mathematics and Statistics Laboratory, Stanford University. ## Author(s) David Magis Department of Psychology, University of Liege, Belgium david.magis@uliege.be ## See Also `Ii`, `nextModule`, `integrate.mstR`, `genPolyMatrix` ## Examples ```r ## Dichotomous models ## # Generation of an item bank under 2PL, made of 7 successive modules that target # different average ability levels and with different lengths # (the first generated item parameters hold two modules of 8 items each) it <- rbind(genDichoMatrix(16, model = "2PL"), genDichoMatrix(6, model = "2PL", bPrior = c("norm", -1, 1)), genDichoMatrix(6, model = "2PL", bPrior = c("norm", 1, 1)), genDichoMatrix(9, model = "2PL", bPrior = c("norm", -2, 1)), genDichoMatrix(9, model = "2PL", bPrior = c("norm", 0, 1)), genDichoMatrix(9, model = "2PL", bPrior = c("norm", 2, 1))) it <- as.matrix(it) # Creation of the 'module' matrix to list item membership in each module modules <- matrix(0, 55, 7) modules[1:8, 1] <- modules[9:16, 2] <- modules[17:22, 3] <- 1 modules[23:28, 4] <- modules[29:37, 5] <- modules[38:46, 6] <- 1 modules[47:55, 7] <- 1 # Creation of the response pattern for module 1 and true ability level 0 x <- genPattern(th = 0, it = it[1:8,], seed = 1) # MKL for module 3 MKL(it, modules, target.mod = 3, it.given = 1:8, x = x) # Same with pre-estimation of ability by ML (same result) th <- thetaEst(it[1:8,], x, method = "ML") MKL(it, modules, target.mod = 3, it.given = 1:8, x = x, theta = th) # Same with pre-estimation of ability by EAP (different result) th <- thetaEst(it[1:8,], x, method = "EAP") MKL(it, modules, target.mod = 3, it.given = 1:8, x = x, theta = th) # MKLP for module 3 MKL(it, modules, target.mod = 3, it.given = 1:8, x = x, type = "MKLP") # MKL for for module 3, different integration range MKL(it, modules, target.mod = 3, it.given = 1:8, x = x, lower = -2, upper = 2, nqp = 20) # MKLP for module 3, uniform prior distribution on the range [-2,2] MKL(it, modules, target.mod = 3, it.given = 1:8, x = x, type = "MKLP", priorDist = "unif", priorPar = c(-2, 2)) ## Polytomous models ## # Same structure as above but parameters are now generated from PCM with at most # 4 categories it.pol <- genPolyMatrix(55, model = "PCM", nrCat = 4) it.pol <- as.matrix(it) # Creation of the response pattern for module 1 and true ability level 0 x <- genPattern(th = 0, it = it.pol[1:8,], seed = 1) # MKL for module 3 MKL(it.pol, modules, target.mod = 3, it.given = 1:8, x = x, model = "PCM") # Same with pre-estimation of ability by ML (same result) th <- thetaEst(it.pol[1:8,], x, method = "ML", model = "PCM") MKL(it.pol, modules, target.mod = 3, it.given = 1:8, x = x, theta = th, model = "PCM") # Same with pre-estimation of ability by EAP (different result) th <- thetaEst(it.pol[1:8,], x, method = "EAP", model = "PCM") MKL(it.pol, modules, target.mod = 3, it.given = 1:8, x = x, theta = th, model = "PCM") # MKLP for module 3 MKL(it.pol, modules, target.mod = 3, it.given = 1:8, x = x, type = "MKLP", model = "PCM") # MKL for for module 3, different integration range MKL(it.pol, modules, target.mod = 3, it.given = 1:8, x = x, lower = -2, upper = 2, nqp = 20, model = "PCM") # MKLP for module 3, uniform prior distribution on the range [-2,2] MKL(it.pol, modules, target.mod = 3, it.given = 1:8, x = x, type = "MKLP", priorDist = "unif", priorPar = c(-2, 2), model = "PCM") ```

MKL function