# Rasch model estimation

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Rasch_model_estimation
> Markdown URL: https://mediated.wiki/source/Rasch_model_estimation.md
> Source: https://en.wikipedia.org/wiki/Rasch_model_estimation
> Source revision: 1318233942
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

**Estimation of a Rasch model** is used to estimate the parameters of the [Rasch model](/source/Rasch_model). Various techniques are employed to estimate the parameters from matrices of response data. The most common approaches are types of [maximum likelihood](/source/Maximum_likelihood) estimation, such as joint and conditional maximum likelihood estimation. Joint maximum likelihood (JML) equations are efficient, but [inconsistent](/source/Consistency_(statistics)) for a finite number of items, whereas conditional maximum likelihood (CML) equations give consistent and [unbiased](/source/Unbiased_estimate) item estimates. Person estimates are generally thought to have [bias](/source/Bias_of_an_estimator) associated with them, although weighted likelihood estimation methods for the estimation of person parameters reduce the bias.

## Rasch model

The Rasch model for dichotomous data takes the form:

- \Pr \{X_{ni}=1\}=\frac{\exp({\beta_n} - {\delta_i})}{1 + \exp({\beta_n} - {\delta_i})},

where \beta_n is the ability of person n and \delta_i is the difficulty of item i.

## Joint maximum likelihood

Let x_{ni} denote the observed response for person *n* on item *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

- \Lambda = \frac{\prod_{n} \prod_{i} \exp(x_{ni}(\beta_n-\delta_i))}{\prod_{n} \prod_{i}(1+\exp(\beta_n-\delta_i))}.

The log-likelihood function is then

- \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))

where r_n=\sum_i^I x_{ni} is the total raw score for person *n*, s_i=\sum_n^N x_{ni} is the total raw score for item *i*, *N* is the total number of persons and *I* is the total number of items.

Solution equations are obtained by taking partial derivatives with respect to \delta_i and \beta_n and setting the result equal to 0. The JML solution equations are:

- s_i = \sum_{n=1}^N p_{ni},\quad i=1,\dots,I

- r_n = \sum_{i=1}^I p_{ni},\quad n=1,\dots,N

where p_{ni}=\exp(\beta_n-\delta_i)/(1+\exp(\beta_n-\delta_i)).

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, and a more accurate (less biased) estimate of each \delta_i is obtained by multiplying the estimates by (I-1)/I.

## Conditional maximum likelihood

The conditional [likelihood function](/source/Likelihood_function) is defined as

- \Lambda = \prod_{n} \Pr\{(x_{ni})\mid r_n\} =\frac{\exp(\sum_i -s_i\delta_i)}{\prod_{n} \gamma_r}

in which

- \gamma_r = \sum_{(x) \mid r}\exp(-\sum_i x_{ni}\delta_i)

is the [elementary symmetric function](/source/Elementary_symmetric_function) of order *r*, which represents the sum over all combinations of *r* items. For example, in the case of three items,

- \gamma_2 = \exp(-\delta_1-\delta_2)+\exp(-\delta_1-\delta_3)+\exp(-\delta_2-\delta_3).

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](/source/Polytomous_Rasch_model).

## Estimation algorithms

Some kind of [expectation-maximization algorithm](/source/Expectation-maximization_algorithm) is used in the estimation of the parameters of Rasch models. Algorithms for implementing Maximum Likelihood estimation commonly employ [Newton–Raphson](/source/Newton%E2%80%93Raphson) 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.

## See also

- [Expectation-maximization algorithm](/source/Expectation-maximization_algorithm)
- [Rasch model](/source/Rasch_model)

## References

- Linacre, J.M. (1999). ["Understanding Rasch measurement: Estimation methods for Rasch measures"](https://www.researchgate.net/publication/12729624). *Journal of Outcome Measurement*. **3** (4): 382–405. [PMID 10572388](https://pubmed.ncbi.nlm.nih.gov/10572388)
- Linacre, J.M. (2004). "Estimation methods for Rasch measures. Chapter 2". *Introduction to Rasch Measurement*. Maple Grove MN: JAM Press. ISBN 978-0975535110.
- Linacre, J.M. (2004). "Rasch Model Estimation: Further Topics". *Veterinary Record*. **5** (1): 95–110. [doi:10.1136/vr.139.12.290](https://doi.org/10.1136/vr.139.12.290). [PMID 8890464](https://pubmed.ncbi.nlm.nih.gov/8890464)
- Linacre, J.M. (2004). "Rasch model estimation: further topics. Chapter 24". *Introduction to Rasch Measurement*. Maple Grove MN: JAM Press. ISBN 978-0975535110.
- Kiefer, J. & Wolfowitz, J. (1956). ["Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters"](https://archive.org/details/sim_annals-of-mathematical-statistics_1956-12_27_4/page/886). *Annals of Mathematical Statistics*. **27** (4): 887–906. [doi:10.1214/aoms/1177728066](https://doi.org/10.1214/aoms/1177728066)
- von Davier, M. & Rost, J. (1995). ["Polytomous Mixed Rasch Models"](https://archive.org/details/raschmodelsfound0000unse). *Rasch Models*. New York, NY: Springer. [doi:10.1007/978-1-4612-4230-7_20](https://doi.org/10.1007/978-1-4612-4230-7_20). ISBN 0-387-94499-0.
- von Davier, M. (2016). ["The Rasch Model. Chapter 3"](https://www.taylorfrancis.com/chapters/edit/10.1201/9781315374512-12/rasch-model-matthias-von-davier). *Handbook of Item Response Theory, Vol. 1.*. Second ed. van der Linden, W. (ed.). CRC Press. pp. 31–48. [doi:10.1201/9781315374512-12](https://doi.org/10.1201/9781315374512-12). ISBN 9781315374512.

---
Adapted from the Wikipedia article [Rasch model estimation](https://en.wikipedia.org/wiki/Rasch_model_estimation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rasch_model_estimation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.