# (Maximum) Expected Information (MEI) This command returns the expected information (EI) for a given item (under dichotomous and polytomous IRT models), as used for Maximum Expected Information (MEI) criterion. ```r MEI(itemBank, item, x, theta, it.given, model = NULL, method = "BM", priorDist = "norm", priorPar = c(0, 1), D = 1, range = c(-4, 4), parInt = c(-4, 4, 33), infoType = "observed") ``` ## Arguments - `itemBank`: numeric: a suitable matrix of item parameters. See Details . - `item`: numeric: the item (referred to as its rank in the item bank) for which the expected information must be computed. - `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). - `theta`: numeric: the provisional ability estimate. - `it.given`: numeric: a suitable matrix of item parameters for previously administered items. The number of rows of `it.given` must be equal to the length of `x`. - `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 . - `method`: character: the ability estimator. Possible values are `"BM"` (default), `"ML"` and `"WL"`. See Details . - `priorDist`: character: specifies the prior distribution. Possible values are `"norm"` (default), `"unif"` and `"Jeffreys"`. Ignored if `method` is neither `"BM"` nor `"EAP"`. See Details . - `priorPar`: numeric: vector of two components specifying the prior parameters (default is `c(0,1)`) of the prior ability distribution. Ignored if `method` is neither `"BM"` nor `"EAP"`, or if `priorDist="Jeffreys"`. See Details . - `D`: numeric: the metric constant. Default is `D=1` (for logistic metric); `D=1.702` yields approximately the normal metric (Haley, 1952). - `range`: numeric: vector of two components specifying the range wherein the ability estimate must be looked for (default is `c(-4,4)`). Ignored if `method=="EAP"`. - `parInt`: numeric: vector of three components, holding respectively the values of the arguments `lower`, `upper` and `nqp` of the `eapEst` command. Default vector is (-4, 4, 33). Ignored if `method` is not `"EAP"`. - `infoType`: character: the type of information function to be used. Possible values are `"observed"` (default) for observed information function, and `"Fisher"` for Fisher information function. ## Returns The required expected information for the selected item. ## Details The MEI (van der Linden, 1998; van der Linden and Pashley, 2000) can be used as a rule for selecting the next item in the CAT process (see also Choi and Swartz, 2009), both with dichotomous and polytomous IRT models. This command serves as a subroutine for the `nextItem` 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. 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) and let **j** be the item of interest (not previously administered). Set also $P_{jt}(\theta)$ as the probability of answering response category **t** to item **j** for a given ability level $\theta$. Finally, set $\hat{\theta}_{k+1}^t$ as the ability estimates computed under the condition that the response to item **j** is **t** (with $t=0, ..., g_j$). Then, the EI for item **j** equals $$ EI_j = \sum_{t=0}^{g_j} P_{jt}(\hat{\theta}_k)\,I_j(\hat{\theta}_{k+1}^t) $$ where $I_j(\theta)$ is the information function for item **j**. 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 for a given ability level $\theta$, and set $Q_j(\theta)=1-P_j(\theta)$. Finally, set $\hat{\theta}_{k+1}^0$ and $\hat{\theta}_{k+1}^1$ as the ability estimates computed under the condition that the response to item **j** is 0 or 1 respectively (that is, if the response pattern is updated by 0 or 1 for item **j**). Then, the EI for item **j** reduces to $$ EI_j = P_j(\hat{\theta}_k)\,I_j(\hat{\theta}_{k+1}^1) + Q_j(\hat{\theta}_k)\,I_j(\hat{\theta}_{k+1}^0) $$ . Two types of information functions are available. The first one is the observed information function, defined as $$ OI_j(\theta) = -\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j) $$ (van der Linden, 1998), where $L(\theta | x_j)$ is the likelihood related to item **j**. The second one is Fisher information function: $$ I_j(\theta) = -E\,\left[\frac{\partial^2}{\partial\,\theta^2} \,\log L(\theta | x_j))\right]. $$ Under the 1PL and the 2PL models, these functions are identical (Veerkamp, 1996). See also `OIi`. The observed and Fisher information functions are specified by the `infoType` argument, with respective values `"observed"` and `"Fisher"`. By default, the observed information function is considered (Choi and Swartz, 2009; van der Linden, 1998). The estimator of provisional ability is defined by means of the arguments `method`, `priorDist`, `priorPar`, `D`, `range` and `parInt` of the `thetaEst` function. See the corresponding help file for further details. The provisional response pattern and the related item parameters are provided by the arguments `x` and `it.given` respectively. The target item (for which the maximum information computed) is given by its number in the item bank, through the `item` argument. Note that the provisional response pattern `x` can also be set to `NULL` (which is useful when the number `nrItems` of starting items is set to zero). In this case, `it.given` must be a matrix with zero rows, such as e.g., `itemBank[NULL,]`. ## 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. Choi, S. W., and Swartz, R. J. (2009). Comparison of CAT item selection criteria for polytomous items. **Applied Psychological Measurement, 32**, 419-440. tools:::Rd_expr_doi("10.1177/0146621608327801") 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. Magis, D. and Barrada, J. R. (2017). Computerized Adaptive Testing with R: Recent Updates of the Package **catR**. **Journal of Statistical Software**, **Code Snippets**, **76(1)**, 1-18. tools:::Rd_expr_doi("10.18637/jss.v076.c01") Magis, D., and Raiche, G. (2012). Random Generation of Response Patterns under Computerized Adaptive Testing with the R Package **catR**. **Journal of Statistical Software**, **48 (8)**, 1-31. tools:::Rd_expr_doi("10.18637/jss.v048.i08") van der Linden, W. J. (1998). Bayesian item selection criteria for adaptive testing. **Psychometrika, 63**, 201-216. tools:::Rd_expr_doi("10.1007/BF02294775") van der Linden, W. J., and Pashley, P. J. (2000). Item selection and ability estimation in adaptive testing. In W. J. van der Linden and C. A. W. Glas (Eds.), **Computerized adaptive testing. Theory and practice** (pp. 1-25). Boston, MA: Kluwer. Veerkamp, W. J. J. (1996). **Statistical inference for adaptive testing**. Internal report. Enschede, The Netherlands: University of Twente. ## Author(s) David Magis Department of Psychology, University of Liege, Belgium david.magis@uliege.be ## See Also `Ii`, `OIi`, `nextItem`, `thetaEst`, `genPolyMatrix` ## Examples ```r ## Dichotomous models ## # Loading the 'tcals' parameters data(tcals) # Selecting item parameters only bank <- as.matrix(tcals[,1:4]) # Selection of two arbitrary items (15 and 20) of the # 'tcals' data set it.given <- bank[c(15, 20),] # Creation of a response pattern x <- c(0, 1) # MEI for item 1, provisional ability level 0 MEI(bank, 1, x, 0, it.given) # With Fisher information instead MEI(bank, 1, x, 0, it.given, infoType = "Fisher") # With WL estimator instead MEI(bank, 1, x, 0, it.given, method = "WL") ## Polytomous models ## # Generation of an item bank under GRM with 100 items and at most 4 categories m.GRM <- genPolyMatrix(100, 4, "GRM") m.GRM <- as.matrix(m.GRM) # Selection of two arbitrary items (15 and 20) it.given <- m.GRM[c(15, 20),] # Generation of a response pattern (true ability level 0) x <- genPattern(0, it.given, model = "GRM") # EPV for item 1, provisional ability level 0 MEI(m.GRM, 1, x, 0, it.given, model = "GRM") # With WL method MEI(m.GRM, 1, x, 0, it.given, model = "GRM", method = "WL") # With Fisher information MEI(m.GRM, 1, x, 0, it.given, model = "GRM", infoType = "Fisher") # Loading the cat_pav data data(cat_pav) cat_pav <- as.matrix(cat_pav) # Selection of two arbitrary items (15 and 20) it.given <- cat_pav[c(15, 20),] # Generation of a response pattern (true ability level 0) x <- genPattern(0, it.given, model = "GPCM") # EPV for item 1, provisional ability level 0 MEI(cat_pav, 1, x, 0, it.given, model = "GPCM") # With WL method MEI(cat_pav, 1, x, 0, it.given, model = "GPCM", method = "WL") # With Fisher information MEI(cat_pav, 1, x, 0, it.given, model = "GPCM", infoType = "Fisher") ```

MEI function