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Notes on the Estimation of Item Response Theory Models

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New Developments in Quantitative Psychology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 66))

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References

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Author information

Authors and Affiliations

  1. SAS Institute, Cary, NC, USA

    Xinming An & Yiu-Fai Yung

Authors
  1. Xinming An
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  2. Yiu-Fai Yung
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Corresponding author

Correspondence to Xinming An .

Editor information

Editors and Affiliations

  1. Department of Psychology, Arizona State University, Tempe, AZ, USA

    Roger E. Millsap

  2. Department of Methodology and Statistics, Tilburg University, Tilburg, The Netherlands

    L. Andries van der Ark

  3. Department of Educational Psychology, University of Wisconsin, Madison, WI, USA

    Daniel M. Bolt

  4. Department of Psychology, University of Kansas, Lawrence, KS, USA

    Carol M. Woods

Appendices

Appendix 1: Technical Details for EM

The conditional distribution f(η | u i , θ(t)) is

$$\displaystyle{ f(\eta \vert u_{i}{,\theta }^{(t)}) = \frac{f(u_{i}\vert \eta {,\theta }^{(t)})\phi (\eta )} {\int f(u_{i}\vert \eta {,\theta }^{(t)})\phi (\eta )d\eta } = \frac{f(u_{i}\vert \eta {,\theta }^{(t)})\phi (\eta )} {f(u_{i})}. }$$
(14)

Then these conditional expectations involved in the Q function can be expressed as follows:

$$\displaystyle{ E[\log P_{ij}\vert u_{i}{,\theta }^{(t)}] =\int \log P_{ ij}f(\eta \vert u_{i}{,\theta }^{(t)})d\eta, }$$
(15)
$$\displaystyle{ E[\log (1 - P_{ij})\vert u_{i}{,\theta }^{(t)}] =\int \log (1 - P_{ ij})f(\eta \vert u_{i}{,\theta }^{(t)})d\eta, }$$
(16)

and

$$\displaystyle{ E[\log \phi (\eta )\vert u_{i}{,\theta }^{(t)}] =\int \log \phi (\eta )f(\eta \vert u_{ i}{,\theta }^{(t)})d\eta. }$$
(17)

Then we have

$$\displaystyle{ \begin{array}{lll} Q_{1j}& =&\int \sum \limits _{i=1}^{N}\left [u_{ij}\log P_{ij}f(\eta \vert u_{i}{,\theta }^{(t)})+(1-u_{ij})\log (1-P_{ij})f(\eta \vert u_{i}{,\theta }^{(t)})\right ]d\eta \\ & =&\int \left [\log P_{ij}\left [\sum \limits _{i=1}^{N}u_{ij}f(\eta \vert u_{i}{,\theta }^{(t)})\right ]+\log (1-P_{ij})\left [\sum _{i=1}^{N}(1-u_{ij})f(\eta \vert u_{i}{,\theta }^{(t)})\right ]\right ]d\eta \\ & =&\int \left [\log P_{ij}r_{j}{(\theta }^{(t)})+\log (1-P_{ij})[n{(\theta }^{(t)})-r_{j}{(\theta }^{(t)})]\right ]\phi (\eta {\vert \theta }^{(t)}))d\eta,\\ \end{array} }$$
(18)

where \(r_{j}{(\theta }^{(t)}) =\sum _{ i=1}^{N}u_{ij}\frac{f(u_{i}\vert \eta {,\theta }^{(t)})} {f(u_{i})}\), and \(n{(\theta }^{(t)}) =\sum _{ i=1}^{N}\frac{f(u_{i}\vert \eta {,\theta }^{(t)})} {f(u_{i})}\).

Integrations in the equations above can be approximated as follows using G–H quadrature. Note that these quadrature points, x g , and weights, w g , correspond to ϕ(η | θ(t)) which is the density function of N(0, Φ(t)).

$$\displaystyle{ \tilde{Q}_{1j} =\sum _{ g=1}^{G}\left [\log P_{ ij}(x_{g})r_{j}(x_{g}{,\theta }^{(t)}) +\log (1 - P_{ ij}(x_{g}))(n(x_{g}{,\theta }^{(t)}) - r_{ j}(x_{g}{,\theta }^{(t)}))\right ]w_{ g}. }$$
(19)

We take the derivatives of Q1j with respect to model parameters

$$\displaystyle{ \frac{\partial \tilde{Q}_{1j}} {\partial \alpha _{j}} =\sum _{ g=1}^{G}\left [\frac{r_{j}(x_{g}{,\theta }^{(t)})} {P_{ij}(x_{g})} -\frac{n(x_{g}{,\theta }^{(t)}) - r_{j}(x_{g}{,\theta }^{(t)})} {1 - P_{ij}(x_{g})} \right ]\frac{\partial P_{ij}(x_{g})} {\partial \alpha _{j}} w_{g}, }$$
(20)
$$\displaystyle{ \frac{\partial \tilde{Q}_{1j}} {\partial \lambda _{j}} =\sum _{ g=1}^{G}\left [\frac{r_{j}(x_{g}{,\theta }^{(t)})} {P_{ij}(x_{g})} -\frac{n(x_{g}{,\theta }^{(t)}) - r_{j}(x_{g}{,\theta }^{(t)})} {1 - P_{ij}(x_{g})} \right ]\frac{\partial P_{ij}(x_{g})} {\partial \lambda _{j}} w_{g}, }$$
(21)
$$\displaystyle{ \frac{{\partial }^{2}\tilde{Q}_{1j}} {\partial \alpha _{j}^{2}} =\sum _{ g=1}^{G}\left [\left [ \frac{-r_{j}} {P_{ij}^{2}} - \frac{n - r_{j}} {{(1 - P_{ij})}^{2}}\right ]{\left [\frac{\partial P_{ij}(x_{g})} {\partial \alpha _{j}} \right ]}^{2} + \left [ \frac{r_{j}} {P_{ij}} - \frac{n - r_{j}} {1 - P_{ij}}\right ]\frac{{\partial }^{2}P_{ij}(x_{g})} {\partial \alpha _{j}^{2}} \right ]w_{g}, }$$
(22)
$$\displaystyle{ \frac{{\partial }^{2}\tilde{Q}_{1j}} {\partial \lambda _{j}^{2}} =\sum _{ g=1}^{G}\left [\left [ \frac{-r_{j}} {P_{ij}^{2}} - \frac{n - r_{j}} {{(1 - P_{ij})}^{2}}\right ]{\left [\frac{\partial P_{ij}(x_{g})} {\partial \lambda _{j}} \right ]}^{2} + \left [ \frac{r_{j}} {P_{ij}} - \frac{n - r_{j}} {1 - P_{ij}}\right ]\frac{{\partial }^{2}P_{ij}(x_{g})} {\partial \lambda _{j}^{2}} \right ]w_{g}, }$$
(23)

and

$$\displaystyle{ \frac{{\partial }^{2}\tilde{Q}_{1j}} {\partial \alpha _{j}\partial \lambda _{j}} =\sum _{ g=1}^{{G}^{d} }\left [\left [ \frac{-r_{j}} {P_{ij}^{2}} - \frac{n - r_{j}} {{(1 - P_{ij})}^{2}}\right ]\left [\frac{\partial P_{ij}(x_{g})} {\partial \alpha _{j}} \frac{\partial P_{ij}(x_{g})} {\partial \lambda _{j}} \right ] + \left [ \frac{r_{j}} {P_{ij}} - \frac{n - r_{j}} {1 - P_{ij}}\right ]\frac{{\partial }^{2}P_{ij}(x_{g})} {\partial \alpha _{j}\partial \lambda _{j}} \right ]w_{g}. }$$
(24)

In the above equations, we have

$$\displaystyle{ \frac{\partial P_{ij}(x_{g})} {\partial \alpha _{j}} = -\phi (\alpha _{j} -\lambda _{j}x_{g}) = -\frac{\partial Q_{ij}(x_{g})} {\partial \alpha _{j}}, }$$
(25)
$$\displaystyle{ \frac{\partial P_{ij}(x_{g})} {\partial \lambda _{j}} =\phi (\alpha _{j} -\lambda _{j}x_{g})x_{g} = -\frac{\partial Q_{ij}(x_{g})} {\partial \lambda _{j}}, }$$
(26)
$$\displaystyle{ \frac{{\partial }^{2}P_{ij}(x_{g})} {\partial \alpha _{j}^{2}} = -\frac{\partial \phi (\alpha _{j} -\lambda _{j}x_{g})} {\partial \alpha _{j}} =\phi (\alpha _{j} -\lambda _{j}x_{g})(\alpha _{j} -\lambda _{j}x_{g}) = -\frac{{\partial }^{2}Q_{ij}(x_{g})} {\partial \alpha _{j}^{2}}, }$$
(27)
$$\displaystyle{ \frac{{\partial }^{2}P_{ij}(x_{g})} {\partial \alpha _{j}\partial \lambda _{j}} = -\frac{\partial \phi (\alpha _{j} -\lambda _{j}x_{g})} {\partial \lambda _{j}} = -\phi (\alpha _{j} -\lambda _{j}x_{g})(\alpha _{j} -\lambda _{j}x_{g})x_{g} = -\frac{{\partial }^{2}Q_{ij}(x_{g})} {\partial \alpha _{j}\partial \lambda _{j}}, }$$
(28)

and

$$\displaystyle{ \frac{{\partial }^{2}P_{ij}(x_{g})} {\partial \lambda _{j}^{2}} = \frac{\partial \phi (\alpha _{j} -\lambda _{j}x_{g})x_{g}} {\partial \lambda _{j}} =\phi (\alpha _{j} -\lambda _{j}x_{g})(\alpha _{j} -\lambda _{j}x_{g})x_{g}^{2} = -\frac{{\partial }^{2}Q_{ ij}(x_{g})} {\partial \lambda _{j}^{2}}.}$$
(29)

Appendix 2: Technical Details for the Quasi-Newton Algorithm

For our objective function, \(Log\tilde{L}(\theta )\), the first derivative with respect to θ j , the latent trait for the jth item, is

$$\displaystyle{ \frac{\partial \log \tilde{L}(\theta \vert U)} {\partial \theta _{j}} =\sum _{ i=1}^{N}\left [{(\tilde{L}_{ i})}^{-1}\frac{\partial \tilde{L}_{i}} {\partial \theta _{j}} \right ] =\sum _{ i=1}^{N}\left [{(\tilde{L}_{ i})}^{-1}\sum _{ g=1}^{G}\left [\frac{\partial f_{i}(x_{g})} {\partial \theta _{j}} w_{g}\right ]\right ], }$$
(30)

where

$$\displaystyle{ \tilde{L}_{i} =\sum _{ g=1}^{G}\left [\prod _{ j=1}^{J}{(P_{ ij}(x_{g}))}^{u_{ij} }{(Q_{ij}(x_{g}))}^{1-u_{ij} }\right ]w_{g} =\sum _{ g=1}^{G}f_{ i}(x_{g})w_{g}, }$$
(31)
$$\displaystyle{ \frac{\partial f_{i}(x_{g})} {\partial \theta _{j}} = \frac{\partial [P_{ij}{(x_{g})}^{u_{ij}}Q_{ij}{(x_{g})}^{1-u_{ij}}]} {\partial \theta _{j}} \frac{f_{i}(x_{g})} {P_{ij}{(x_{g})}^{u_{ij}}Q_{ij}{(x_{g})}^{1-u_{ij}}}. }$$
(32)

For the probit link

$$\displaystyle{ \frac{\partial P_{ij}(x_{g})} {\partial \alpha _{j}} = -\phi (\alpha _{j} -\lambda _{j}x_{g}) = -\frac{\partial Q_{ij}(x_{g})} {\partial \alpha _{j}}, }$$
(33)
$$\displaystyle{ \frac{\partial P_{ij}(x_{g})} {\partial \lambda _{j}} =\phi (\alpha _{j} -\lambda _{j}x_{g})x_{g} = -\frac{\partial Q_{ij}(x_{g})} {\partial \lambda _{j}}. }$$
(34)

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An, X., Yung, YF. (2013). Notes on the Estimation of Item Response Theory Models. In: Millsap, R.E., van der Ark, L.A., Bolt, D.M., Woods, C.M. (eds) New Developments in Quantitative Psychology. Springer Proceedings in Mathematics & Statistics, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9348-8_19

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