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1 INTRODUCTION The increasing usability of computers and Webbased assessments requires innovative approaches to the development, delivery, and scoring of tests. Statistical methods play a central role in such frameworks. The item response model (IRM) (LORD; NOVICK, 1968) has been the most common statistical method used. In computerbased adaptive testing (CAT), IRM allows adaptive item selection from an item bank, according to examinee proficiency during test administration. The efficiency of CAT is realized through the targeting of item difficulty to the examinee proficiency (WISE; KINGSBURY, 2000). It implies an item bank or multiple item banks properly developed. A good item bank should cover all aspects of the construct to be measured (content validity) and contain a sufficient number of items to ensure measurement accuracy in the domain, i.e., for all scale values. Items should fulfill requirements set in the American Educational Research Association (AERA), the American Psychological Association (APA), and the National Council on Measurement in Education (NCME) (1999). Stocking (1994) found that doubling the number of item banks reduced test overlap to a much greater extent than doubling the number of items in each bank (apud NYDICK; WEISS, 2009). The development of an item bank for CAT is a complex and multidisciplinary process that follows seven major steps (e.g., BJORNER et al., 2007) represented in Diagram 1, thus, requiring experts from the subject–scientific areas of Construct Framework (steps 1, 2, 5, and 7), Statistics (steps 3, 4, 5, 6, and 7), and Computer Science and Informatics (steps 3, 6, and 7). Once an item bank is available for CAT use, its management requires decisions on several issues such as item bank size and control, security protocols (including item exposure control), statistical modeling, item removal and revision, item addition, maintenance of scale consistency, and use of multiple banks (WISE; KINGSBURY, 2000). Thereafter, CAT administration is basically the repetition of a twophase process. As Wise and Kingsbury (2000) explain, first, an item with difficulty matched to the examinee’s current proficiency estimate is administered. Second, the examinee’s response to the item is scored, and the proficiency estimate is updated. This sequence is repeated until some stopping criterion is met, usually a predetermined maximum number of items or measurement precision. Thus, despite obvious advantages of adaptive testing, there are still some limitations, such as the high cost related to item bank development. However, the cost could be reduced by decreasing expenses on item writing, pretesting, and calibrating new items (VELDKAMP; MATTEUCCI, 2013), involving steps 2, 3, and 4 of Figure 1. Figure 1 Since Classical Test Theory (CTT) methods are less demanding of sample size, the complementarity between CTT and IRMs jointly with the existence of multiple item banks, offer exceptional research opportunities for reducing such costs. As a previous step, this study examines the empirical relationship between indexes and parameters resulting from both approaches in order to justify and support the use of CTT in item pretesting and precalibration, thus reducing the cost of item bank development. Further work remains for demonstration of how any arbitrary scale derived from the precalibration step can be transformed into the scale adopted by the assessment system. Throughout the paper, we will address two research questions: (1) What is the level of association between CTT indexes and IRM parameter estimates? (2) Can CTT provide initial item difficulty estimates for posterior IRM use in CAT? The CTT model and the generalized partial credit model (GPCM) are applied to data collected from the Portuguese student population enrolled in the 4th and 6th grades and to those who were administered with mathematics and motherlanguage tests. The number of students involved is approximately 108,000 in each grade. Estimates of item discrimination and difficulty are obtained and compared. Percentile confidence intervals based on 1000 bootstrap samples are presented for correlation between item difficulty estimates. The study is organized as follows: the next section describes the data and statistical methods used. The results are presented in section three, and conclusions are considered in the last section. 2 METHODOLOGY This section comprises three parts. The first part presents details and characteristics of the data. The second addresses the statistical specification of models in use and explains how to quantify the level of association between estimates obtained from the CTT and the IRM. The third presents a brief description of various steps in the CAT framework. 2.1 Data In Portugal, Primary School Assessment Tests (Provas de Aferição do Ensino Básico) are the responsibility of GAVE (Gabinete de Avaliação Educacional), the office of educational assessment, which aims to evaluate how objectives established for each education cycle are achieved. These instruments are yearly administered to all students enrolled in the fourth and sixth years of schooling, in mathematics, and in the mothertongue language (Portuguese), according to provisions of law no. 2351/2007, of February 14, II Series. GAVE tests are always administered to the population nationwide and are based on specific competences of the mathematics and Portuguese subjects presented in the document National Curriculum of Primary School: Key competences and the current syllabus. The mathematics test assesses understanding of concepts and procedures, reasoning and communication abilities, and competence for using mathematics in analysis and problem solving. In the academic year 2006–2007, the mathematics test was administered to 108,441 students attending the 4th grade and also to 108,296 students attending the 6th grade. These tests were composed of two identical parts, including 27 items and containing multiple choice, short answer, completion, and openended questions, covering the following content: numbers and calculation; geometry and measurement; statistics and probabilities; and algebra and functions. From now onward, these tests are called Math4 and Math6 for the 4th and 6th grades, respectively. Portuguese tests involved 108,447 students in the 4th grade and 108,548 students in the 6th grade. Three competences were assessed: reading comprehension, explicit knowledge of language, and written expression. These tests were composed of two parts. The first part mainly contained short answer items, completion, right or wrong association, and multiple choice questions. The second included extensive composition items in which a text of 20–25 lines is produced. Portuguese tests were composed of 27 and 33 items for the 4th and 6th grades, respectively. From now onward, Portuguese tests of the 4th and 6th grades are called Port4 and Port6, respectively. Before statistical modeling, partial scoring of openended answers and extensive composition was performed by experts. The tests’ reliability, as demonstrated by the coefficient of internal consistency, i.e., the coefficient of Kuder–Richardson, is p ≥ 0.85. 2.2 Statistical methods Fundamentals of statistical methods for educational measurement are presented in Statistical Theories of Mental Test Scores by Lord and Novick (1968). According to them, the definition of measurement is “a procedure for the assignment of numbers (scores, measurements) to specified properties of experimental units in such a way as to characterize and preserve specified relationships in the behavioral domain” (p. 17). Two main statistical approaches are used in educational measurement: Classical Test Theory (CTT) and Item Response Models (IRM). Some examples of introductory readings and reviews may be found in Hambleton, Swaminathan, and Rogers (1991), Hambleton (2004) and Klein (2013). The rest of this section presents the model and assumptions underlying classical test theory, explanation and functional specification of the generalized partial credit model, and a brief review of the complementarity of these statistical methods. 2.2.1 Classical test theory It is assumed that variable X represents competencies/skills gained by the student during the learning process. The observable variable X0 is generally obtained by test administration. If tests were instruments with absolute precision, the observed value X0 , regardless of the test used, would be equal to true value X. In a hypothetical situation where the student is tested t times, equation (1) represents the relationship between the true and the observed value, where ε represents the measurement error. Measurement error is assumed to be nonsystematic, homoscedastic, and noncorrelated with the true value X. Characteristics of items are quantified through the discrimination index (ci) and the difficulty index (pi). The discrimination index measures capacity of the item to distinguish the high performance group of students from the low performance group of students, and its values vary from −1 to 1. The difficulty index (pi) is provided by the proportion of correct answers to the item i (e.g., Guilford; Fruchter, 1978). Therefore, high values indicate easy questions. 2.2.2 Item response models Item response models (IRM) rest on two basic postulates (HAMBLETON; SWAMINATHAN; ROGERS, 1991; HAMBLETON, 2004). According to the first postulate, the examinees’ performance on an item can be explained by their ability; according to the second, the relationship between the probability of a correct answer to the item and the examinee’s ability is described by a function called the item characteristic curve. In this class of models, item response may be dichotomous or polytomous. Additionally, the various IRMs classification depends on the number of latent traits the item represents, giving rise to unidimensional and multidimensional models. The Generalized Partial Credit Model (GPCM) (MURAKI, 1993, 1997; MURAKI; BOCK, 2002) is a unidimensional model for analyzing responses scored in two or more ordered categories. The aim is to extract from an item more information about the examinee’s level than simply whether the examinee correctly answers the item. Items are ranked in which examinees receive partial credit for successfully completing the various levels of performance needed to complete an item. This model relaxes the assumption of items’ uniform discriminating power and includes parameters to represent item difficulty and discrimination. The model is applied to several types of items, such as multiple choice, short answer, completion, and open response items (with the previous items that were gradually scored). Thus, the GPCM suitable for such data is specified by equation (2), where i is the item number (i = 1,.,I; I is the total number of items in the test); Pik (θ) is the probability that an examinee with latent factor θ selecting the kth category from mi possible categories for the polytomous item i; ai is the discrimination parameter for item i, using a logistic metric. In addition, βij = bi  dj, where bi is the difficulty/location parameter of item I, and dj is the parameter of the intercept category, with d1 = 0. According to equation (2), the probability of the student to answer (or to be ranked) in the k category is a conditional probability on the answer to the k1 category. That is to say, the answer to category k has underlying response criteria satisfaction that is associated with the previous category. Estimates are obtained by maximum likelihood procedure, using the EM algorithm. This model, estimation procedures, and maths data were utilized by Ferrão, Costa, and Oliveira (2015) for linking scales and by Ferrão and Prata (2014) for a simulation CAT study. 2.2.3 Complementarity In the paper “The taxonomy of item response models,” Thissen and Steinberg (1988) propose three distinct classes of models with which models are distinguished by their assumptions and constraints on their parameters. Additionally, Goldstein and Wood (1989) present arguments in favor of the unity of item response models by sitting them within an explicit linear modelling framework. The logistic models […] can be seen merely to be one class out of many possible classes of models. […] In practice, the simple identity models used over the effective response range, typically give near equivalent results (p. 163). The paper published by Hambleton and Jones (1993) describes and compares (similarities and differences of) the methodological approaches mentioned above. Two of these approaches are relevant for this paper’s purpose. They concern the relationship between the IRM item difficulty parameter, the CTT index of difficulty, and the relationship between the IRM discrimination parameter and the CTT biserial correlation. Lord (1980) describes a monotonic relationship between the CTT index of item difficulty (pi) and the IRM item difficulty parameter (bi) so that as pi increases, bi decreases when all items discriminate equally. If items have unequal discrimination values, then the relationship between them depends on the item biserial correlation. Lord also demonstrates that, under certain conditions, the item biserial correlation ri and the IRM item discrimination parameter approximately monotonically increase functions of one another, i.e., where ai is the item i discrimination parameter estimate, and ri is the item i biserial correlation. 2.3 Computerbased adaptive testing As aforementioned, in CAT, item response models are applied to establish a relationship between observed responses and ability of the examinee, enabling the item selection adaptively, from an item bank, according to examinee ability during test administration. Thus, the test is tailored to each examinee, and it begins by selecting an initial item. If the examinee answers incorrectly, then an easier item is selected for administration; however, if not, a complex one is administered. Each item is scored, and an estimate of the examinees’ ability is obtained. This process of selection and evaluation is iteratively conducted until a termination criterion is met. Thus, despite being a realtime computing platform, the process implies the existence of a calibrated item bank. Several areas of knowledge are involved in the use of CAT. Figure 2 presents the knowledge areas and their relationships that support the platform. Figure 2 The CAT platform concerns operations from modular structures of Statistical Methods (S), Content (C), and Informatics (I), which provide elements to be integrated throughout the Adaptive Test Developer (ATD). The modular structure S comprises statistical methods for item calibration, scoring, scale fitting, and linking, examinees’ ability modeling, test measurement error, and reliability; structure I contains a computer or Web application with interfaces to examinees via desktop or mobile devices. The server connects the database that contains the item bank (module C) and the statistical methods (module S) using the ATD to adapt tests to examinees; structure C includes the item bank (in general, each item record is defined by question, by type of question and field specification, correct answer, its statistical propertiesdiscrimination, difficulty, information, level of exposure to date, and whether it is an anchor item), and the item bank manager, which is software for operations with items. 3 RESULTS CTT and GPCM were applied to Math4, Port4, Math6, and Port6 data. Tables 1 to 4 contain discrimination and difficulty indexes, biserial correlations, and estimates of GCPM discrimination and difficulty parameters. Since intersection parameters are not used for any research questions addressed in this study, their estimates are not presented. The chisquare hypotheses test for goodness of fit suggests this IRM as an adequate model at the 5% level of significance. Table 1 Math4 CTT IRM Item Discrimination Index (c) Difficulty Index (p) Biserial Correlation (r) Discrimination Estimate (a) Difficulty Estimate (b) 1 0.650 0.647 0.532 0.785 0.611 2 0.610 0.630 0.507 0.714 0.569 3 0.569 0.303 0.458 0.340 0.215 4 0.126 0.913 0.172 0.278 5.159 5 0.670 0.667 0.594 0.604 0.670 6 0.414 0.837 0.505 0.981 1.412 7 0.330 0.817 0.381 0.291 1.890 8 0.742 0.366 0.551 0.393 0.241 9 0.242 0.874 0.318 0.219 2.835 10 0.610 0.542 0.461 0.302 0.210 11 0.412 0.787 0.43 0.633 1.478 12 0.215 0.917 0.364 0.821 2.220 13 0.618 0.703 0.545 0.860 0.817 14 0.107 0.947 0.179 0.283 4.001 15 0.373 0.800 0.37 0.535 1.767 16 0.558 0.743 0.544 0.492 0.951 17 0.747 0.582 0.588 0.421 0.467 18 0.500 0.757 0.472 0.530 1.984 19 0.448 0.838 0.506 1.033 1.385 20 0.674 0.475 0.547 0.502 0.544 21 0.585 0.670 0.489 0.364 0.786 22 0.556 0.472 0.409 0.482 0.153 23 0.625 0.674 0.512 0.744 0.751 24 0.359 0.833 0.41 0.355 1.847 25 0.340 0.828 0.363 0.588 1.858 26 0.627 0.428 0.497 0.442 0.274 27 0.639 0.546 0.475 0.615 0.219 Font: Authors (2014). Table 2 Port4 CTT IRM Item Discrimination Index (c) Difficulty Index (p) Biserial Correlation (r) Discrimination Estimate (a) Difficulty Estimate (b) 1 0.380 0.744 0.376 0.270 2.415 2 0.228 0.870 0.305 0.329 3.569 3 0.490 0.246 0.461 0.324 0.116 4 0.312 0.771 0.334 0.297 2.507 5 0.335 0.806 0.380 0.343 3.188 6 0.196 0.162 0.229 0.286 0.640 7 0.438 0.668 0.412 0.299 1.435 8 0.272 0.897 0.417 0.450 3.130 9 0.493 0.461 0.424 0.375 2.328 10 0.565 0.549 0.534 0.239 1.530 11 0.388 0.266 0.352 0.363 1.106 12 0.243 0.879 0.393 0.579 2.343 13 0.486 0.723 0.519 0.535 1.209 14 0.398 0.350 0.331 0.318 0.985 15 0.457 0.701 0.461 0.475 2.603 16 0.360 0.829 0.476 0.313 2.141 17 0.321 0.881 0.534 0.764 1.952 18 0.364 0.333 0.357 0.268 0.739 19 0.419 0.675 0.408 0.303 1.726 20 0.102 0.955 0.283 0.474 3.221 21 0.673 0.307 0.650 0.895 0.427 22 0.433 0.157 0.480 1.286 0.219 23 0.447 0.169 0.494 2.032 0.421 24 0.476 0.187 0.500 1.955 0.549 25 0.438 0.163 0.480 1.554 0.768 26 0.524 0.228 0.507 1.091 0.517 27 0.557 0.283 0.480 0.598 1.055 Font: Authors (2014). Table 3 Math6 CTT IRM Item Discrimination Index (c) Difficulty Index (p) Biserial Correlation (r) Discrimination Estimate (a) Difficulty Estimate (b) 1 0.348 0.841 0.381 0.402 2.034 2 0.647 0.527 0.579 0.531 0.476 3 0.280 0.864 0.342 0.544 2.296 4 0.546 0.683 0.534 0.625 0.888 5 0.352 0.805 0.379 0.508 1.872 6 0.738 0.427 0.721 0.347 0.062 7 0.348 0.841 0.412 0.431 1.664 8 0.381 0.703 0.353 0.197 1.541 9 0.728 0.350 0.762 0.431 0.221 10 0.122 0.181 0.169 0.219 4.170 11 0.569 0.437 0.496 0.300 1.102 12 0.126 0.037 0.319 0.536 2.200 13 0.559 0.362 0.519 0.734 0.573 14 0.632 0.406 0.607 0.307 0.206 15 0.699 0.262 0.818 0.612 0.564 16 0.728 0.360 0.788 0.882 0.138 17 0.544 0.247 0.596 0.364 0.447 18 0.527 0.765 0.552 0.485 1.248 19 0.530 0.348 0.531 0.749 0.629 20 0.175 0.049 0.402 0.462 2.019 21 0.476 0.260 0.486 0.696 1.110 22 0.593 0.298 0.591 0.440 0.017 23 0.461 0.208 0.501 0.336 0.599 24 0.491 0.490 0.439 0.431 0.056 25 0.627 0.402 0.599 0.370 0.134 26 0.472 0.585 0.406 0.228 0.654 27 0.530 0.449 0.505 0.290 0.093 Font: Authors (2014). Table 4 Port6 CTT IRM Item Discrimination Index (c) Difficulty Index (p) Biserial Correlation (r) Discrimination Estimate (a) Difficulty Estimate (b) 1 0.484 0.607 0.416 0.404 0.699 2 0.512 0.618 0.471 0.354 2.931 3 0.258 0.320 0.231 0.175 2.594 4 0.028 0.991 0.141 0.470 6.256 5 0.052 0.978 0.166 0.414 5.447 6 0.249 0.442 0.202 0.207 3.025 7 0.223 0.892 0.342 0.485 2.872 8 0.417 0.743 0.407 0.269 2.272 9 0.161 0.921 0.269 0.487 3.279 10 0.452 0.439 0.379 0.211 0.715 11 0.479 0.686 0.439 0.376 1.328 12 0.451 0.529 0.386 0.159 1.844 13 0.299 0.668 0.263 0.259 1.659 14 0.207 0.192 0.207 0.109 3.398 15 0.289 0.825 0.330 0.415 2.405 16 0.513 0.558 0.470 0.506 0.312 17 0.118 0.059 0.212 0.423 2.028 18 0.142 0.040 0.348 0.407 1.471 19 0.283 0.111 0.402 0.416 0.560 20 0.217 0.092 0.327 0.617 0.977 21 0.391 0.207 0.463 0.343 0.988 22 0.475 0.506 0.422 0.419 2.155 23 0.389 0.667 0.371 0.428 2.911 24 0.405 0.279 0.380 0.460 0.605 25 0.508 0.579 0.464 0.410 1.261 26 0.052 0.978 0.200 0.428 3.202 27 0.603 0.469 0.517 0.559 1.223 28 0.509 0.228 0.524 0.813 0.768 29 0.420 0.175 0.496 1.159 0.427 30 0.415 0.168 0.493 1.401 0.610 31 0.391 0.141 0.497 1.295 0.761 32 0.354 0.128 0.447 1.160 0.339 33 0.585 0.279 0.567 0.514 0.603 Font: Authors (2014). Regarding Math4 test, joint analysis of item properties based on CTT and IRM, presented in Table 1, indicates that most items discriminate; items 4, 9, and 14 slightly discriminate; and item 19 is very discriminatory in both approaches. Concerning the difficulty parameter, we verify that approximately 44% of items are easy, whereas items 4 and 14 are very easy. In general, results demonstrate that the tests are mainly composed of discriminative and very discriminative items and, additionally, items of all difficulty levels. A joint analysis of Port4 reveals that, in general, the test items do discriminate, with the exception of item 6, which slightly discriminates, and item 26, which is very discriminative and has a medium difficulty level. Additionally, items 2, 5, 8, 12, 16, 17, and 20 are very easy. Concerning Math6 items, analysis based on CTT and IRM shows that the most discriminative items are 13, 15, 16, 19, and 21; the least discriminative items are 10 and 20. The difficulty index and parameter indicate that the easiest items are items 1, 3, 5, 7, and 18; the most difficult items are 10, 12, and 20. In particular, item 12 is slightly discriminative according to the CTT approach and discriminative according to the GPCM approach. For Port6 items, analysis based on the two approaches reveals that items 3, 4, 5, 7, 9, 13, 14, 15, 17, 18, 19, and 26 discriminate slightly and that there is one set of six very easy items (4, 5, 7, 9, 15, and 26) and a set of three very difficult items (14, 17, and 18). The relationship between the biserial correlation (r) and the discrimination parameter estimate (a), given by formula (3), indicates a moderate correlation varying from 0.4 to 0.5. Concerning difficulty, the correlation between p and b is very strong since it ranges from −0.8 to −0.9, i.e., the correlation is −0.83 in Mathematics 4th grade, −0.88 in Portuguese 4th grade, −0.88 in Mathematics 6th grade, and −0.80 in Portuguese 6th grade. Percentile confidence intervals of 95% based on 1000 bootstrap samples are presented in Table 5. The intervals confirm that in the population, the correlation is strong since its absolute value is always greater than 0.71. In this sense, results support this study’s purpose of providing empirical evidence on the complementarity between the two statistical approaches regarding the estimate of item difficulty. Table 5 Subject/Grade Correlation 95% Confidence Interval Lower Upper Mathematics / 4th grade 0.826 0.927 0.766 Portuguese / 4th grade 0.883 0.938 0.809 Mathematics / 6th grade 0.879 0.977 0.799 Portuguese / 6th grade 0.805 0.885 0.712 Font: Authors (2014). 4 CONCLUSION The results obtained in this study show a very strong correlation between the CTT index of difficulty and the IRM item difficulty parameter estimate. The correlation is −0.83 in Mathematics 4th grade, −0.88 in Portuguese 4th grade, −0.88 in Mathematics 6th grade, and −0.80 in Portuguese 6th grade. The results also suggest that the level of association does not depend on subject or on grade. A moderate relationship between the IRM estimate of discrimination and the approximation given by the biserial function was verified. In addition, it was shown that even when items do not discriminate equally, a monotonic relationship exists between the CTT index of item difficulty and the IRM item difficulty parameter. Therefore, CTT may be utilized as initial estimates for item pretesting and precalibration in item bank development, particularly supporting implementation of Webbased adaptive tests. Since the sample size required for item pretesting and calibration is a crucial aspect for development of item banks, these are promising results for the future of computer or Webbased testing. Further work is needed to determine whether changes in pretesting and in algorithms related to adaptive test design and administration affect score precision and reliability.
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