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Rasch model estimation
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<h2 id="rasch-model">Rasch model</h2> <p>The Rasch model for dichotomous data takes the form: </p> Pr { X n i = 1 } = exp ( β n − δ i ) 1 + exp ( β n − δ i ) , {\displaystyle \Pr\{X_{ni}=1\}={\frac {\exp({\beta _{n}}-{\delta _{i}})}{1+\exp({\beta _{n}}-{\delta _{i}})}},} <p>where β n {\displaystyle \beta _{n}} is the ability of person n {\displaystyle n} and δ i {\displaystyle \delta _{i}} is the difficulty of item i {\displaystyle i} . </p> <h2 id="joint-maximum-likelihood">Joint maximum likelihood</h2> <p>Let x n i {\displaystyle x_{ni}} denote the observed response for person <i>n</i> on item <i>i</i>. The probability of the observed data matrix, which is the product of the probabilities of the individual responses, is given by the likelihood function </p> Λ = ∏ n ∏ i exp ( x n i ( β n − δ i ) ) ∏ n ∏ i ( 1 + exp ( β n − δ i ) ) . {\displaystyle \Lambda ={\frac {\prod _{n}\prod _{i}\exp(x_{ni}(\beta _{n}-\delta _{i}))}{\prod _{n}\prod _{i}(1+\exp(\beta _{n}-\delta _{i}))}}.} <p>The log-likelihood function is then </p> log Λ = ∑ n N β n r n − ∑ i I δ i s i − ∑ n N ∑ i I log ( 1 + exp ( β n − δ i ) ) {\displaystyle \log \Lambda =\sum _{n}^{N}\beta _{n}r_{n}-\sum _{i}^{I}\delta _{i}s_{i}-\sum _{n}^{N}\sum _{i}^{I}\log(1+\exp(\beta _{n}-\delta _{i}))} <p>where r n = ∑ i I x n i {\displaystyle r_{n}=\sum _{i}^{I}x_{ni}} is the total raw score for person <i>n</i>, s i = ∑ n N x n i {\displaystyle s_{i}=\sum _{n}^{N}x_{ni}} is the total raw score for item <i>i</i>, <i>N</i> is the total number of persons and <i>I</i> is the total number of items. </p><p>Solution equations are obtained by taking partial derivatives with respect to δ i {\displaystyle \delta _{i}} and β n {\displaystyle \beta _{n}} and setting the result equal to 0. The JML solution equations are: </p> s i = ∑ n = 1 N p n i , i = 1 , … , I {\displaystyle s_{i}=\sum _{n=1}^{N}p_{ni},\quad i=1,\dots ,I} r n = ∑ i = 1 I p n i , n = 1 , … , N {\displaystyle r_{n}=\sum _{i=1}^{I}p_{ni},\quad n=1,\dots ,N} <p>where p n i = exp ( β n − δ i ) / ( 1 + exp ( β n − δ i ) ) {\displaystyle p_{ni}=\exp(\beta _{n}-\delta _{i})/(1+\exp(\beta _{n}-\delta _{i}))} . </p><p>The resulting estimates are biased, and no finite estimates exist for persons with score 0 (no correct responses) or with 100% correct responses (perfect score). The same holds for items with extreme scores, no estimates exists for these as well. This bias is due to a well known effect described by Kiefer & Wolfowitz (1956). It is of the order ( I − 1 ) / I {\displaystyle (I-1)/I} , and a more accurate (less biased) estimate of each δ i {\displaystyle \delta _{i}} is obtained by multiplying the estimates by ( I − 1 ) / I {\displaystyle (I-1)/I} . </p> <h2 id="conditional-maximum-likelihood">Conditional maximum likelihood</h2> <p>The conditional likelihood function is defined as </p> Λ = ∏ n Pr { ( x n i ) ∣ r n } = exp ( ∑ i − s i δ i ) ∏ n γ r {\displaystyle \Lambda =\prod _{n}\Pr\{(x_{ni})\mid r_{n}\}={\frac {\exp(\sum _{i}-s_{i}\delta _{i})}{\prod _{n}\gamma _{r}}}} <p>in which </p> γ r = ∑ ( x ) ∣ r exp ( − ∑ i x n i δ i ) {\displaystyle \gamma _{r}=\sum _{(x)\mid r}\exp(-\sum _{i}x_{ni}\delta _{i})} <p>is the <a href="/facts/Elementary_symmetric_function/9RFXe9b6">elementary symmetric function</a> of order <i>r</i>, which represents the sum over all combinations of <i>r</i> items. For example, in the case of three items, </p> γ 2 = exp ( − δ 1 − δ 2 ) + exp ( − δ 1 − δ 3 ) + exp ( − δ 2 − δ 3 ) . {\displaystyle \gamma _{2}=\exp(-\delta _{1}-\delta _{2})+\exp(-\delta _{1}-\delta _{3})+\exp(-\delta _{2}-\delta _{3}).} <p>Details can be found in the chapters by von Davier (2016) for the dichotomous Rasch model and von Davier & Rost (1995) for the polytomous Rasch model. </p> <h2 id="estimation-algorithms">Estimation algorithms</h2> <p>Some kind of <a href="/facts/Expectation-maximization_algorithm/32P2bdBc">expectation-maximization algorithm</a> is used in the estimation of the parameters of Rasch models. Algorithms for implementing Maximum Likelihood estimation commonly employ <a href="/facts/Newton%E2%80%93Raphson/VlRhI2FL">Newton–Raphson</a> iterations to solve for solution equations obtained from setting the partial derivatives of the log-likelihood functions equal to 0. Convergence criteria are used to determine when the iterations cease. For example, the criterion might be that the mean item estimate changes by less than a certain value, such as 0.001, between one iteration and another for all items. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Expectation-maximization_algorithm/32P2bdBc">Expectation-maximization algorithm</a></li> <li><a href="/facts/Rasch_model/8NzYrqmR">Rasch model</a></li></ul> <ul><li>Linacre, J.M. (2004). <i>Estimation methods for Rasch measures</i>. Chapter 2 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN: JAM Press.</li> <li>Linacre, J.M. (2004). <i>Rasch model estimation: further topics</i>. Chapter 24 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN: JAM Press.</li> <li>von Davier M., Rost J. (1995) Polytomous Mixed Rasch Models. In: Fischer G.H., Molenaar I.W. (eds) Rasch Models. Springer, New York, NY. <a href="https://doi.org/10.1007/978-1-4612-4230-7_20">https://doi.org/10.1007/978-1-4612-4230-7_20</a></li> <li>von Davier, M. (2016). The Rasch Model. Chapter 3 in: van der Linden, W. (ed.) Handbook of Item Response Theory, Vol. 1. Second Edition. CRC Press, p. 31-48. <a href="https://www.taylorfrancis.com/chapters/edit/10.1201/9781315374512-12/rasch-model-matthias-von-davier">https://www.taylorfrancis.com/chapters/edit/10.1201/9781315374512-12/rasch-model-matthias-von-davier</a></li></ul>