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112-579: It is based on the application of related mathematical models to testing data . Because it is often regarded as superior to classical test theory , it is the preferred method for developing scales in the United States, especially when optimal decisions are demanded, as in so-called high-stakes tests , e.g., the Graduate Record Examination (GRE) and Graduate Management Admission Test (GMAT). The name item response theory

224-418: A {\displaystyle a} parameter stretches the horizontal scale, the b {\displaystyle b} parameter shifts the horizontal scale, and the c {\displaystyle c} parameter compresses the vertical scale from [ 0 , 1 ] {\displaystyle [0,1]} to [ c , 1 ] . {\displaystyle [c,1].} This

336-546: A / 4 , {\displaystyle p'(b)=a/4,} meaning that b equals the 50% success level (difficulty), and a (divided by four) is the maximum slope (discrimination), which occurs at the 50% success level. Further, the logit (log odds ) of a correct response is a ( θ − b ) {\displaystyle a(\theta -b)} (assuming c = 0 {\displaystyle c=0} ): in particular if ability θ equals difficulty b, there are even odds (1:1, so logit 0) of

448-401: A i {\displaystyle a_{i}} , the 3PL adds c i {\displaystyle c_{i}} , and the 4PL adds d i {\displaystyle d_{i}} . The 2PL is equivalent to the 3PL model with c i = 0 {\displaystyle c_{i}=0} , and is appropriate for testing items where guessing the correct answer

560-403: A i ( θ − b i ) {\displaystyle p_{i}({\theta })=c_{i}+{\frac {1-c_{i}}{1+e^{-a_{i}({\theta }-b_{i})}}}} where θ {\displaystyle {\theta }} indicates that the person's abilities are modeled as a sample from a normal distribution for the purpose of estimating the item parameters. After

672-607: A i 2 ( p i ( θ ) − c i ) 2 ( 1 − c i ) 2 q i ( θ ) p i ( θ ) . {\displaystyle I(\theta )=a_{i}^{2}{\frac {(p_{i}(\theta )-c_{i})^{2}}{(1-c_{i})^{2}}}{\frac {q_{i}(\theta )}{p_{i}(\theta )}}.} In general, item information functions tend to look bell-shaped. Highly discriminating items have tall, narrow information functions; they contribute greatly but over

784-413: A Chi-square statistic , or a standardized version of it. Two and three-parameter IRT models adjust item discrimination, ensuring improved data-model fit, so fit statistics lack the confirmatory diagnostic value found in one-parameter models, where the idealized model is specified in advance. Data should not be removed on the basis of misfitting the model, but rather because a construct relevant reason for

896-503: A paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into

1008-400: A prior probability distribution (which can be subjective), and then update this distribution based on empirical data. An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending

1120-425: A certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables , state variables , exogenous variables, and random variables . Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants . The variables are not independent of each other as

1232-436: A common approach is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics. Defining a metric to measure distances between observed and predicted data

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1344-546: A computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering , physics models are often made by mathematical methods such as finite element analysis . Different mathematical models use different geometries that are not necessarily accurate descriptions of

1456-521: A confusing distractor. Such valuable analysis is provided by specially-designed psychometric software . Classical test theory is an influential theory of test scores in the social sciences. In psychometrics , the theory has been superseded by the more sophisticated models in item response theory (IRT) and generalizability theory (G-theory). However, IRT is not included in standard statistical packages like SPSS , but SAS can estimate IRT models via PROC IRT and PROC MCMC and there are IRT packages for

1568-667: A correct answer, the greater the ability is above (or below) the difficulty the more (or less) likely a correct response, with discrimination a determining how rapidly the odds increase or decrease with ability. In other words, the standard logistic function has an asymptotic minimum of 0 ( c = 0 {\displaystyle c=0} ), is centered around 0 ( b = 0 {\displaystyle b=0} , P ( 0 ) = 1 / 2 {\displaystyle P(0)=1/2} ), and has maximum slope P ′ ( 0 ) = 1 / 4. {\displaystyle P'(0)=1/4.} The

1680-426: A different model is specified for each administration in order to achieve data-model fit, then a different latent trait is being measured and test scores cannot be argued to be comparable between administrations. One of the major contributions of item response theory is the extension of the concept of reliability . Traditionally, reliability refers to the precision of measurement (i.e., the degree to which measurement

1792-475: A different score value. A common example of this is Likert -type items, e.g., "Rate on a scale of 1 to 5." Another example is partial-credit scoring, to which models like the Polytomous Rasch model may be applied. Dichotomous IRT models are described by the number of parameters they make use of. The 3PL is named so because it employs three item parameters. The two-parameter model (2PL) assumes that

1904-401: A human system, we know that usually the amount of medicine in the blood is an exponentially decaying function, but we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use

2016-428: A large number of misfitting items occur with no apparent reason for the misfit, the construct validity of the test will need to be reconsidered and the test specifications may need to be rewritten. Thus, misfit provides invaluable diagnostic tools for test developers, allowing the hypotheses upon which test specifications are based to be empirically tested against data. There are several methods for assessing fit, such as

2128-429: A lower bound for reliability under rather mild assumptions. Thus, the reliability of test scores in a population is always higher than the value of Cronbach's α {\displaystyle {\alpha }} in that population. Thus, this method is empirically feasible and, as a result, it is very popular among researchers. Calculation of Cronbach's α {\displaystyle {\alpha }}

2240-512: A math item correct. The exact value of the probability depends, in addition to ability, on a set of item parameters for the IRF. For example, in the three parameter logistic model ( 3PL ), the probability of a correct response to a dichotomous item i , usually a multiple-choice question, is: p i ( θ ) = c i + 1 − c i 1 + e −

2352-605: A narrow range. Less discriminating items provide less information but over a wider range. Mathematical model A mathematical model is an abstract description of a concrete system using mathematical concepts and language . The process of developing a mathematical model is termed mathematical modeling . Mathematical models are used in applied mathematics and in the natural sciences (such as physics , biology , earth science , chemistry ) and engineering disciplines (such as computer science , electrical engineering ), as well as in non-physical systems such as

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2464-482: A one-to-one mapping of raw number-correct scores to Rasch θ {\displaystyle {\theta }} estimates. As with any use of mathematical models, it is important to assess the fit of the data to the model. If item misfit with any model is diagnosed as due to poor item quality, for example confusing distractors in a multiple-choice test, then the items may be removed from that test form and rewritten or replaced in future test forms. If, however,

2576-431: A priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually, it

2688-481: A property that does not hold for two-parameter and three-parameter models. Additionally, there is theoretically a four-parameter model (4PL), with an upper asymptote , denoted by d i , {\displaystyle d_{i},} where 1 − c i {\displaystyle 1-c_{i}} in the 3PL is replaced by d i − c i {\displaystyle d_{i}-c_{i}} . However, this

2800-441: A pure chance on a multiple choice item with four possible responses). In the same manner, IRT can be used to measure human behavior in online social networks. The views expressed by different people can be aggregated to be studied using IRT. Its use in classifying information as misinformation or true information has also been evaluated. The concept of the item response function was around before 1950. The pioneering work of IRT as

2912-425: A quality that should be defined or empirically demonstrated in relation to a given purpose or use, but not a quantity that can be measured. 'Local independence' means (a) that the chance of one item being used is not related to any other item(s) being used and (b) that response to an item is each and every test-taker's independent decision, that is, there is no cheating or pair or group work. The topic of dimensionality

3024-407: A single parameter ( b i {\displaystyle b_{i}} ). This results in one-parameter models having the property of specific objectivity, meaning that the rank of the item difficulty is the same for all respondents independent of ability, and that the rank of the person ability is the same for items independently of difficulty. Thus, 1 parameter models are sample independent,

3136-429: A standard weighted linear (Ordinary Least Squares, OLS ) regression and hence can be used to create a weighted index of indicators for unsupervised measurement of an underlying latent concept. For items such as multiple choice items, the parameter c i {\displaystyle c_{i}} is used in attempt to account for the effects of guessing on the probability of a correct response. It indicates

3248-433: A system and to study the effects of different components, and to make predictions about behavior. Mathematical models can take many forms, including dynamical systems , statistical models , differential equations , or game theoretic models . These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models . In many cases,

3360-513: A test consisting of k {\displaystyle k} items u j {\displaystyle u_{j}} , j = 1 , … , k {\displaystyle j=1,\ldots ,k} . The total test score is defined as the sum of the individual item scores, so that for individual i {\displaystyle i} Then Cronbach's alpha equals Cronbach's α {\displaystyle {\alpha }} can be shown to provide

3472-507: A test or research instrument can be claimed to measure a trait. Operationally, this means that the IRT approaches include additional model parameters to reflect the patterns observed in the data (e.g., allowing items to vary in their correlation with the latent trait), whereas in the Rasch approach, claims regarding the presence of a latent trait can only be considered valid when both (a) the data fit

Item response theory - Misplaced Pages Continue

3584-603: A theory occurred during the 1950s and 1960s. Three of the pioneers were the Educational Testing Service psychometrician Frederic M. Lord , the Danish mathematician Georg Rasch , and Austrian sociologist Paul Lazarsfeld , who pursued parallel research independently. Key figures who furthered the progress of IRT include Benjamin Drake Wright and David Andrich . IRT did not become widely used until

3696-414: Is "the correlation between test scores on parallel forms of a test". The problem with this is that there are differing opinions of what parallel tests are. Various reliability coefficients provide either lower bound estimates of reliability or reliability estimates with unknown biases. A third shortcoming involves the standard error of measurement. The problem here is that, according to classical test theory,

3808-538: Is a useful tool for assessing model fit. In statistics, decision theory, and some economic models , a loss function plays a similar role. While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations . Tools from nonparametric statistics can sometimes be used to evaluate how well

3920-403: Is already known from direct investigation of the phenomenon being studied. An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology. It should also be noted that while mathematical modeling uses mathematical concepts and language, it

4032-405: Is assumed that observed score = true score plus some error : Classical test theory is concerned with the relations between the three variables X {\displaystyle X} , T {\displaystyle T} , and E {\displaystyle E} in the population. These relations are used to say something about the quality of test scores. In this regard,

4144-419: Is assumed that, provided sufficient items are tested, the rank-ordering of persons along the latent trait by raw score will not change, but will simply undergo a linear rescaling. By contrast, three-parameter IRT achieves data-model fit by selecting a model that fits the data, at the expense of sacrificing specific objectivity . In practice, the Rasch model has at least two principal advantages in comparison to

4256-472: Is based on the idea that the probability of a correct/keyed response to an item is a mathematical function of person and item parameters . (The expression "a mathematical function of person and item parameters" is analogous to Lewin's equation , B = f(P, E) , which asserts that behavior is a function of the person in their environment.) The person parameter is construed as (usually) a single latent trait or dimension. Examples include general intelligence or

4368-533: Is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation . These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on

4480-587: Is due to the focus of the theory on the item, as opposed to the test-level focus of classical test theory. Thus IRT models the response of each examinee of a given ability to each item in the test. The term item is generic, covering all kinds of informative items. They might be multiple choice questions that have incorrect and correct responses, but are also commonly statements on questionnaires that allow respondents to indicate level of agreement (a rating or Likert scale ), or patient symptoms scored as present/absent, or diagnostic information in complex systems. IRT

4592-459: Is elaborated below. The parameter b i {\displaystyle b_{i}} represents the item location which, in the case of attainment testing, is referred to as the item difficulty. It is the point on θ {\displaystyle {\theta }} where the IRF has its maximum slope, and where the value is half-way between the minimum value of c i {\displaystyle c_{i}} and

Item response theory - Misplaced Pages Continue

4704-482: Is free of error). Traditionally, it is measured using a single index defined in various ways, such as the ratio of true and observed score variance. This index is helpful in characterizing a test's average reliability, for example in order to compare two tests. But IRT makes it clear that precision is not uniform across the entire range of test scores. Scores at the edges of the test's range, for example, generally have more error associated with them than scores closer to

4816-648: Is highly unlikely, such as fill-in-the-blank items ("What is the square root of 121?"), or where the concept of guessing does not apply, such as personality, attitude, or interest items (e.g., "I like Broadway musicals. Agree/Disagree"). The 1PL assumes not only that guessing is not present (or irrelevant), but that all items are equivalent in terms of discrimination, analogous to a common factor analysis with identical loadings for all items. Individual items or individuals might have secondary factors but these are assumed to be mutually independent and collectively orthogonal . An alternative formulation constructs IRFs based on

4928-427: Is included in many standard statistical packages such as SPSS and SAS . As has been noted above, the entire exercise of classical test theory is done to arrive at a suitable definition of reliability. Reliability is supposed to say something about the general quality of the test scores in question. The general idea is that, the higher reliability is, the better. Classical test theory does not say how high reliability

5040-640: Is not itself a branch of mathematics and does not necessarily conform to any mathematical logic , but is typically a branch of some science or other technical subject, with corresponding concepts and standards of argumentation. Mathematical models are of great importance in the natural sciences, particularly in physics . Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. It

5152-437: Is often investigated with factor analysis , while the IRF is the basic building block of IRT and is the center of much of the research and literature. The IRF gives the probability that a person with a given ability level will answer correctly. Persons with lower ability have less of a chance, while persons with high ability are very likely to answer correctly; for example, students with higher math ability are more likely to get

5264-467: Is only one of many important statistics), and in many cases, specialized software for classical analysis is also necessary. One of the most important or well-known shortcomings of classical test theory is that examinee characteristics and test characteristics cannot be separated: each can only be interpreted in the context of the other. Another shortcoming lies in the definition of reliability that exists in classical test theory, which states that reliability

5376-415: Is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in

5488-463: Is rarely used. Note that the alphabetical order of the item parameters does not match their practical or psychometric importance; the location/difficulty ( b i {\displaystyle b_{i}} ) parameter is clearly most important because it is included in all three models. The 1PL uses only b i {\displaystyle b_{i}} , the 2PL uses b i {\displaystyle b_{i}} and

5600-466: Is supposed to be. Too high a value for α {\displaystyle {\alpha }} , say over .9, indicates redundancy of items. Around .8 is recommended for personality research, while .9+ is desirable for individual high-stakes testing. These 'criteria' are not based on formal arguments, but rather are the result of convention and professional practice. The extent to which they can be mapped to formal principles of statistical inference

5712-408: Is that the more sophisticated information IRT provides allows a researcher to improve the reliability of an assessment . IRT entails three assumptions: The trait is further assumed to be measurable on a scale (the mere existence of a test assumes this), typically set to a standard scale with a mean of 0.0 and a standard deviation of 1.0. Unidimensionality should be interpreted as homogeneity,

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5824-447: Is the cumulative distribution function (CDF) of the standard normal distribution. The normal-ogive model derives from the assumption of normally distributed measurement error and is theoretically appealing on that basis. Here b i {\displaystyle b_{i}} is, again, the difficulty parameter. The discrimination parameter is σ i {\displaystyle {\sigma }_{i}} ,

5936-471: Is the reciprocal of the test information of at a given trait level, is the SE ( θ ) = 1 I ( θ ) . {\displaystyle {\text{SE}}(\theta )={\frac {1}{\sqrt {I(\theta )}}}.} Thus more information implies less error of measurement. For other models, such as the two and three parameters models, the discrimination parameter plays an important role in

6048-479: Is unclear. Reliability provides a convenient index of test quality in a single number, reliability. However, it does not provide any information for evaluating single items. Item analysis within the classical approach often relies on two statistics: the P-value (proportion) and the item-total correlation ( point-biserial correlation coefficient ). The P-value represents the proportion of examinees responding in

6160-727: Is used to emphasize that discrete item responses are taken to be observable manifestations of hypothesized traits, constructs, or attributes, not directly observed, but which must be inferred from the manifest responses. Latent trait models were developed in the field of sociology, but are virtually identical to IRT models. IRT is generally claimed as an improvement over classical test theory (CTT). For tasks that can be accomplished using CTT, IRT generally brings greater flexibility and provides more sophisticated information. Some applications, such as computerized adaptive testing , are enabled by IRT and cannot reasonably be performed using only classical test theory. Another advantage of IRT over CTT

6272-449: The social sciences (such as economics , psychology , sociology , political science ). It can also be taught as a subject in its own right. The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research . Mathematical models are also used in music , linguistics , and philosophy (for example, intensively in analytic philosophy ). A model may help to explain

6384-403: The speed of light , and we study macro-particles only. Note that better accuracy does not necessarily mean a better model. Statistical models are prone to overfitting which means that a model is fitted to data too much and it has lost its ability to generalize to new events that were not observed before. Any model which is not pure white-box contains some parameters that can be used to fit

6496-468: The 1PL IRT model. However, proponents of Rasch modeling prefer to view it as a completely different approach to conceptualizing the relationship between data and theory. Like other statistical modeling approaches, IRT emphasizes the primacy of the fit of a model to observed data, while the Rasch model emphasizes the primacy of the requirements for fundamental measurement, with adequate data-model fit being an important but secondary requirement to be met before

6608-413: The IRT approach. The first advantage is the primacy of Rasch's specific requirements, which (when met) provides fundamental person-free measurement (where persons and items can be mapped onto the same invariant scale). Another advantage of the Rasch approach is that estimation of parameters is more straightforward in Rasch models due to the presence of sufficient statistics, which in this application means

6720-498: The NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to

6832-411: The Rasch model, and (b) test items and examinees conform to the model. Therefore, under Rasch models, misfitting responses require diagnosis of the reason for the misfit, and may be excluded from the data set if one can explain substantively why they do not address the latent trait. Thus, the Rasch approach can be seen to be a confirmatory approach, as opposed to exploratory approaches that attempt to model

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6944-440: The ability parameter, it is possible to make the 2PL logistic model closely approximate the cumulative normal ogive. Typically, the 2PL logistic and normal-ogive IRFs differ in probability by no more than 0.01 across the range of the function. The difference is greatest in the distribution tails, however, which tend to have more influence on results. The latent trait/IRT model was originally developed using normal ogives, but this

7056-516: The chronology of these models but also contrasts with the more recent psychometric theories, generally referred to collectively as item response theory , which sometimes bear the appellation "modern" as in "modern latent trait theory". Classical test theory as we know it today was codified by Novick (1966) and described in classic texts such as Lord & Novick (1968) and Allen & Yen (1979/2002). The description of classical test theory below follows these seminal publications. Classical test theory

7168-408: The coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of

7280-425: The correlation between parallel test scores is equal to reliability (see Lord & Novick, 1968, Ch. 2, for a proof). Using parallel tests to estimate reliability is cumbersome because parallel tests are very hard to come by. In practice the method is rarely used. Instead, researchers use a measure of internal consistency known as Cronbach's α {\displaystyle {\alpha }} . Consider

7392-441: The data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data. The question of whether

7504-406: The data have no guessing, but that items can vary in terms of location ( b i {\displaystyle b_{i}} ) and discrimination ( a i {\displaystyle a_{i}} ). The one-parameter model (1PL) assumes that guessing is a part of the ability and that all items that fit the model have equivalent discriminations, so that items are only described by

7616-473: The difficulty of items or the ability of test-takers. It is a theory of testing based on the idea that a person's observed or obtained score on a test is the sum of a true score (error-free score) and an error score. Generally speaking, the aim of classical test theory is to understand and improve the reliability of psychological tests. Classical test theory may be regarded as roughly synonymous with true score theory . The term "classical" refers not only to

7728-424: The function. The item information function for the two parameter model is I ( θ ) = a i 2 p i ( θ ) q i ( θ ) . {\displaystyle I(\theta )=a_{i}^{2}p_{i}(\theta )q_{i}(\theta ).\,} The item information function for the three parameter model is I ( θ ) =

7840-401: The geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean. Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how

7952-431: The greatly increased complexity, the majority of IRT research and applications utilize a unidimensional model. IRT models can also be categorized based on the number of scored responses. The typical multiple choice item is dichotomous ; even though there may be four or five options, it is still scored only as correct/incorrect (right/wrong). Another class of models apply to polytomous outcomes, where each response has

8064-641: The index of reliability needed in making the correction. Spearman's finding is thought to be the beginning of Classical Test Theory by some (Traub, 1997). Others who had an influence in the Classical Test Theory's framework include: George Udny Yule , Truman Lee Kelley , Fritz Kuder & Marion Richardson involved in making the Kuder–Richardson Formulas , Louis Guttman , and, most recently, Melvin Novick , not to mention others over

8176-473: The item parameters have been estimated, the abilities of individual people are estimated for reporting purposes. a i {\displaystyle a_{i}} , b i {\displaystyle b_{i}} , and c i {\displaystyle c_{i}} are the item parameters. The item parameters determine the shape of the IRF. Figure 1 depicts an ideal 3PL ICC. The item parameters can be interpreted as changing

8288-407: The keyed direction, and is typically referred to as item difficulty . The item-total correlation provides an index of the discrimination or differentiating power of the item, and is typically referred to as item discrimination . In addition, these statistics are calculated for each response of the oft-used multiple choice item, which are used to evaluate items and diagnose possible issues, such as

8400-467: The late 1970s and 1980s, when practitioners were told the "usefulness" and "advantages" of IRT on the one hand, and personal computers gave many researchers access to the computing power necessary for IRT on the other. In the 1990's Margaret Wu developed two item response software programs that analyse PISA and TIMSS data; ACER ConQuest (1998) and the R-package TAM (2010). Among other things,

8512-568: The lowest ability person would be able to discard it, so IRT parameter estimation methods take this into account and estimate a c i {\displaystyle c_{i}} based on the observed data. Broadly speaking, IRT models can be divided into two families: unidimensional and multidimensional. Unidimensional models require a single trait (ability) dimension θ {\displaystyle {\theta }} . Multidimensional IRT models model response data hypothesized to arise from multiple traits. However, because of

8624-409: The maximum value of 1. The example item is of medium difficulty since b i {\displaystyle b_{i}} =0.0, which is near the center of the distribution. Note that this model scales the item's difficulty and the person's trait onto the same continuum. Thus, it is valid to talk about an item being about as hard as Person A's trait level or of a person's trait level being about

8736-674: The middle of the range. Item response theory advances the concept of item and test information to replace reliability. Information is also a function of the model parameters. For example, according to Fisher information theory, the item information supplied in the case of the 1PL for dichotomous response data is simply the probability of a correct response multiplied by the probability of an incorrect response, or, I ( θ ) = p i ( θ ) q i ( θ ) . {\displaystyle I(\theta )=p_{i}(\theta )q_{i}(\theta ).\,} The standard error of estimation (SE)

8848-475: The misfit has been diagnosed, such as a non-native speaker of English taking a science test written in English. Such a candidate can be argued to not belong to the same population of persons depending on the dimensionality of the test, and, although one parameter IRT measures are argued to be sample-independent, they are not population independent, so misfit such as this is construct relevant and does not invalidate

8960-449: The model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning , the optimization of parameters is called training , while the optimization of model hyperparameters is called tuning and often uses cross-validation . In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting . A crucial part of

9072-467: The model describes well the properties of the system between data points is called interpolation , and the same question for events or data points outside the observed data is called extrapolation . As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics , we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles traveling at speeds close to

9184-700: The model's user. Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models . Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. Mathematical modeling problems are often classified into black box or white box models, according to how much

9296-553: The model. In black-box models, one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively,

9408-493: The model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability . Thomas Kuhn argues that as science progresses, explanations tend to become more complex before

9520-427: The model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below

9632-408: The modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. Usually, the easiest part of model evaluation is checking whether a model predicts experimental measurements or other empirical data not used in the model development. In models with parameters,

9744-411: The most important concept is that of reliability . The reliability of the observed test scores X {\displaystyle X} , which is denoted as ρ X T 2 {\displaystyle {\rho _{XT}^{2}}} , is defined as the ratio of true score variance σ T 2 {\displaystyle {\sigma _{T}^{2}}} to

9856-432: The next quarter century after Spearman's initial findings. Classical test theory assumes that each person has a true score , T , that would be obtained if there were no errors in measurement. A person's true score is defined as the expected number-correct score over an infinite number of independent administrations of the test. Unfortunately, test users never observe a person's true score, only an observed score , X . It

9968-432: The normal probability distribution; these are sometimes called normal ogive models . For example, the formula for a two-parameter normal-ogive IRF is: p i ( θ ) = Φ ( θ − b i σ i ) {\displaystyle p_{i}(\theta )=\Phi \left({\frac {\theta -b_{i}}{\sigma _{i}}}\right)} where Φ

10080-403: The observed data. The presence or absence of a guessing or pseudo-chance parameter is a major and sometimes controversial distinction. The IRT approach includes a left asymptote parameter to account for guessing in multiple choice examinations, while the Rasch model does not because it is assumed that guessing adds randomly distributed noise to the data. As the noise is randomly distributed, it

10192-417: The observed score variance σ X 2 {\displaystyle {\sigma _{X}^{2}}} : Because the variance of the observed scores can be shown to equal the sum of the variance of true scores and the variance of error scores, this is equivalent to This equation, which formulates a signal-to-noise ratio, has intuitive appeal: The reliability of test scores becomes higher as

10304-434: The open source statistical programming language R (e.g., CTT). While commercial packages routinely provide estimates of Cronbach's α {\displaystyle {\alpha }} , specialized psychometric software may be preferred for IRT or G-theory. However, general statistical packages often do not provide a complete classical analysis (Cronbach's α {\displaystyle {\alpha }}

10416-506: The probability that very low ability individuals will get this item correct by chance, mathematically represented as a lower asymptote . A four-option multiple choice item might have an IRF like the example item; there is a 1/4 chance of an extremely low ability candidate guessing the correct answer, so the c i {\displaystyle c_{i}} would be approximately 0.25. This approach assumes that all options are equally plausible, because if one option made no sense, even

10528-399: The proportion of error variance in the test scores becomes lower and vice versa. The reliability is equal to the proportion of the variance in the test scores that we could explain if we knew the true scores. The square root of the reliability is the absolute value of the correlation between true and observed scores. Reliability cannot be estimated directly since that would require one to know

10640-510: The purpose of IRT is to provide a framework for evaluating how well assessments work, and how well individual items on assessments work. The most common application of IRT is in education, where psychometricians use it for developing and designing exams , maintaining banks of items for exams, and equating the difficulties of items for successive versions of exams (for example, to allow comparisons between results over time). IRT models are often referred to as latent trait models . The term latent

10752-451: The purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what

10864-540: The quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the physical sciences , a traditional mathematical model contains most of the following elements: Mathematical models are of different types: In business and engineering , mathematical models may be used to maximize

10976-415: The same as Item Y's difficulty, in the sense that successful performance of the task involved with an item reflects a specific level of ability. The item parameter a i {\displaystyle a_{i}} represents the discrimination of the item: that is, the degree to which the item discriminates between persons in different regions on the latent continuum. This parameter characterizes

11088-566: The shape of the standard logistic function : P ( t ) = 1 1 + e − t . {\displaystyle P(t)={\frac {1}{1+e^{-t}}}.} In brief, the parameters are interpreted as follows (dropping subscripts for legibility); b is most basic, hence listed first: If c = 0 , {\displaystyle c=0,} then these simplify to p ( b ) = 1 / 2 {\displaystyle p(b)=1/2} and p ′ ( b ) =

11200-405: The slope of the IRF where the slope is at its maximum. The example item has a i {\displaystyle a_{i}} =1.0, which discriminates fairly well; persons with low ability do indeed have a much smaller chance of correctly responding than persons of higher ability. This discrimination parameter corresponds to the weighting coefficient of the respective item or indicator in

11312-442: The speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics. Many types of modeling implicitly involve claims about causality . This is usually (but not always) true of models involving differential equations. As

11424-406: The standard deviation of the measurement error for item i , and comparable to 1/ a i {\displaystyle a_{i}} . One can estimate a normal-ogive latent trait model by factor-analyzing a matrix of tetrachoric correlations between items. This means it is technically possible to estimate a simple IRT model using general-purpose statistical software. With rescaling of

11536-408: The standard error of measurement is assumed to be the same for all examinees. However, as Hambleton explains in his book, scores on any test are unequally precise measures for examinees of different ability, thus making the assumption of equal errors of measurement for all examinees implausible (Hambleton, Swaminathan, Rogers, 1991, p. 4). A fourth, and final shortcoming of the classical test theory

11648-404: The state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of

11760-447: The strength of an attitude. Parameters on which items are characterized include their difficulty (known as "location" for their location on the difficulty range); discrimination (slope or correlation), representing how steeply the rate of success of individuals varies with their ability; and a pseudoguessing parameter, characterising the (lower) asymptote at which even the least able persons will score due to guessing (for instance, 25% for

11872-494: The system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations . A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, Boolean values or strings , for example. The variables represent some properties of

11984-437: The system, for example, the measured system outputs often in the form of signals , timing data , counters, and event occurrence. The actual model is the set of functions that describe the relations between the different variables. General reference Philosophical Classical test theory Classical test theory (CTT) is a body of related psychometric theory that predicts outcomes of psychological testing such as

12096-399: The test or the model. Such an approach is an essential tool in instrument validation. In two and three-parameter models, where the psychometric model is adjusted to fit the data, future administrations of the test must be checked for fit to the same model used in the initial validation in order to confirm the hypothesis that scores from each administration generalize to other administrations. If

12208-492: The true scores, which according to classical test theory is impossible. However, estimates of reliability can be acquired by diverse means. One way of estimating reliability is by constructing a so-called parallel test . The fundamental property of a parallel test is that it yields the same true score and the same observed score variance as the original test for every individual. If we have parallel tests x and x', then this means that and Under these assumptions, it follows that

12320-427: The underlying process, whereas neural networks produce an approximation that is opaque. Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify

12432-419: Was born only after the following three achievements or ideas were conceptualized: 1. a recognition of the presence of errors in measurements, 2. a conception of that error as a random variable, 3. a conception of correlation and how to index it. In 1904, Charles Spearman was responsible for figuring out how to correct a correlation coefficient for attenuation due to measurement error and how to obtain

12544-415: Was considered too computationally demanding for the computers at the time (1960s). The logistic model was proposed as a simpler alternative, and has enjoyed wide use since. More recently, however, it was demonstrated that, using standard polynomial approximations to the normal CDF , the normal-ogive model is no more computationally demanding than logistic models. The Rasch model is often considered to be

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