Overview -------- edit Learn more This section **needs additional citations for verification **. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. _(December 2015)_ _(Learn how and when to remove this message )_ The concept of the item response function was around before 1950. The pioneering work of IRT as a theory occurred during the 1950s and 1960s. Three of the pioneers were the Educational Testing Service psychometrician Frederic M. Lord ,[\[4\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-4) 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 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 "R (programming language)") \-package TAM (2010). Among other things, 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 "Test (student assessment)") , 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).[\[5\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-5) IRT models are often referred to as _latent trait models_. The term _latent_ 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 is that the more sophisticated information IRT provides allows a researcher to improve the reliability "Reliability (psychometric)") of an assessment . IRT entails three assumptions: 1. A unidimensional trait denoted by θ {\\displaystyle {\\theta }} ; 2. Local independence of items; 3. The response of a person to an item can be modeled by a mathematical _item response function_ (IRF). 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, 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 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 item response function -------------------------- edit 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 a math item correct. The exact value of the probability depends, in addition to ability, on a set of _item parameters_ for the IRF. ### Three parameter logistic model edit  Figure 1: Example of 3PL IRF, with dotted lines overlaid to demonstrate parameters. 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 − 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 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. In brief, the parameters are interpreted as follows (dropping subscripts for legibility); _b_ – difficulty, item location; _a_ – discrimination, scale, slope; _c_ – pseudo-guessing, chance, asymptotic minimum. IRT models ---------- edit 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 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 a different score value.[\[7\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-7) [\[8\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-8) 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. ### Logistic and normal IRT models edit An alternative formulation constructs IRFs based on the normal probability distribution; these are sometimes called _normal ogive "Ogive (statistics)") 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** 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}} , 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.[\[10\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-10) This means it is technically possible to estimate a simple IRT model using general-purpose statistical software. With rescaling of the ability parameter, it is possible to make the 2PL logistic model closely approximate the cumulative normal ogive.[\[11\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-11) 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 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_,_[\[12\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-12) the normal-ogive model is no more computationally demanding than logistic models.[\[13\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-13) Scoring ------- edit The person parameter θ {\\displaystyle {\\theta }} represents the magnitude of _latent trait_ of the individual, which is the human capacity or attribute measured by the test.[\[23\]](https://en.wikipedia.org/wiki/Item_response_theory?utm_source=chatgpt.com#cite_note-23) It might be a cognitive ability, physical ability, skill, knowledge, attitude, personality characteristic, etc. The estimate of the person parameter - the "score" on a test with IRT - is computed and interpreted in a very different manner as compared to traditional scores like number or percent correct. The individual's total number-correct score is not the actual score, but is rather based on the IRFs, leading to a weighted score when the model contains item discrimination parameters. It is actually obtained by multiplying the item response function for each item to obtain a _likelihood function_, the highest point of which is the _maximum likelihood estimate_ of θ {\\displaystyle {\\theta }} . While scoring is much more sophisticated with IRT, for most tests, the correlation between the theta estimate and a traditional score is very high; often it is 0.95 or more.\_[citation needed\ _\] A graph of IRT scores against traditional scores shows an ogive shape implying that the IRT estimates separate individuals at the borders of the range more than in the middle.