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This note sketches two computational shortcuts for estimating unidimensional item response models and multidimensional item response models with between-item dimensionality utilizing an expectation-maximization (EM) algorithm that relies on numerical integration with fixed quadrature points. It is shown that the number of operations required in the E-step can be reduced in situations of many cases and many items by appropriate shortcuts. Consequently, software implementations of a modified E-step in the EM algorithm could benefit from gains in computation time.

This note sketches two computational shortcuts for estimating multidimensional item response theory (IRT) models (

This paper is structured as follows. First, the EM algorithm in IRT models is introduced, and its computational amount is analyzed. Second, we show how computational gains are realized by using so-called pseudo-items instead of original items. Third, we suggest a computational shortcut for multidimensional IRT models whose items possess a simple loading structure (i.e., between item dimensionality). Finally, we conclude with a discussion.

For simplicity, we focus on dichotomous item responses

where

where

In the E-step, individual posterior distributions

For all items, expected counts are computed as

where

We can now compute the total number of operations (multiplications or additions) needed in the E-step. The multiplications of probabilities in the numerator in the fraction in

Notably, additions are typically faster than multiplications. Moreover, calculations on the logarithmic metric prevent numerical underflow (see, e.g.,

Second,

One strategy for improving computational efficiency is to use a parallelized version of an EM algorithm (

A different strand of literature relies on simplifying the model structure in multidimensional IRT models (see

In the following, the concept of pseudo-items is introduced. The main idea is to perform the computationally intensive likelihood computation on pseudo-items instead of original items to reduce computation time. A pseudo-item is composed of

The pseudo-item

In the sequel, we only consider the case of dichotomous items (i.e.,

A computationally more efficient EM algorithm is obtained by evaluating the posterior distributions based on these pseudo-items. Instead of using

The computation based on individual posterior distributions is now given as (insert

The number of operations needed in the posterior computation in

Expected counts can now be computed for pseudo-items

Therefore, for each pseudo-item

We can now define the computational gain

Hence, for a fixed pseudo-item size

In

Due to

We now determine the optimal pseudo-item size

Inserting

By taking the logarithm in

Because the term

We now illustrate the computational gains of the more efficient E-step implementation for sample sizes ranging between

Sample size | _{max} |
_{0} |
---|---|---|

100 | 4 | 3.0 |

1 000 | 6 | 5.0 |

10 000 | 9 | 7.3 |

100 000 | 11 | 9.9 |

1 000 000 | 14 | 12.6 |

10 000 000 | 17 | 15.3 |

We now demonstrate how the proposed shortcut using pseudo-items performs in an experimental implementation using the Rcpp package (

The computational gains were investigated using sample sizes

In

By comparing the computational gains of the computation of the unnormalized likelihood with the prediction of

Sample size: No. of items | Pseudo-item size |
|||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

100: | ||||||||||

50 | 0.5 | 0.3 | 0.3 | 0.4 | 0.3 | 0.2 | 0.2 | 0.1 | 0.1 | |

100 | 0.7 | 0.5 | 0.6 | 0.5 | 0.3 | 0.2 | 0.2 | 0.1 | 0.1 | |

200 | 0.9 | 0.7 | 0.8 | 0.6 | 0.4 | 0.3 | 0.2 | 0.1 | 0.0 | |

1 000: | ||||||||||

50 | 1 | 1.3 | 1.2 | 1.2 | 1.1 | 0.9 | 0.8 | 0.6 | 0.5 | |

100 | 1 | 1.6 | 1.8 | 1.9 | 1.6 | 1.4 | 1.1 | 0.7 | 0.4 | |

200 | 1 | 1.6 | 1.9 | 1.9 | 2.0 | 1.7 | 1.2 | 0.8 | 0.4 | |

10 000: | ||||||||||

50 | 1 | 1.6 | 1.9 | 2.0 | 2.0 | 1.6 | 1.6 | 1.5 | 1.4 | |

100 | 1 | 1.8 | 2.4 | 2.7 | 2.8 | 2.7 | 2.3 | 1.8 | 1.6 | |

200 | 1 | 1.7 | 2.3 | 2.8 | 2.5 | 3.0 | 2.6 | 2.1 | 1.4 | |

100 000: | ||||||||||

50 | 1 | 1.6 | 1.9 | 2.1 | 2.2 | 1.7 | 1.9 | 1.7 | 1.8 | |

100 | 1 | 1.8 | 2.3 | 2.7 | 2.8 | 2.8 | 2.5 | 2.0 | 2.1 | |

200 | 1 | 1.7 | 2.4 | 2.8 | 2.5 | 3.2 | 2.9 | 2.5 | 2.0 | |

100: | ||||||||||

50 | 0.6 | 0.4 | 0.4 | 0.5 | 0.3 | 0.3 | 0.2 | 0.2 | 0.1 | |

100 | 0.9 | 0.6 | 0.8 | 0.7 | 0.4 | 0.3 | 0.2 | 0.2 | 0.1 | |

200 | 1 | 0.9 | 1.1 | 1.0 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | |

1 000: | ||||||||||

50 | 1 | 2.7 | 2.3 | 2.0 | 2.1 | 1.9 | 1.6 | 1.2 | 0.9 | |

100 | 1 | 3.2 | 3.5 | 4.3 | 3.0 | 2.5 | 2.1 | 1.5 | 0.8 | |

200 | 1 | 3.0 | 3.7 | 4.4 | 3.8 | 3.0 | 2.7 | 1.6 | 0.7 | |

10 000: | ||||||||||

50 | 1 | 4.3 | 5.5 | 6.5 | 7.1 | 6.3 | 6.2 | 5.3 | 5.2 | |

100 | 1 | 3.9 | 6.1 | 7.6 | 8.0 | 8.1 | 7.4 | 6.6 | 4.9 | |

200 | 1 | 3.6 | 5.3 | 6.9 | 7.9 | 8.1 | 8.1 | 6.9 | 3.7 | |

100 000: | ||||||||||

50 | 1 | 4.3 | 5.9 | 7.1 | 8.3 | 8.0 | 7.8 | 8.1 | 8.4 | |

100 | 1 | 3.8 | 5.9 | 7.8 | 9.1 | 8.7 | 9.2 | 8.7 | 9.2 | |

200 | 1 | 3.6 | 5.2 | 7.0 | 8.4 | 8.6 | 9.0 | 9.5 | 6.0 | |

100 | 1 | 1.9 | 2.7 | 2.8 | 2.1 | 1.3 | 0.7 | 0.4 | 0.2 | |

1 000 | 1 | 2.0 | 3.0 | 3.9 | 4.6 | 4.8 | 4.0 | 2.7 | 1.6 | |

10 000 | 1 | 2.0 | 3.0 | 4.0 | 5.0 | 5.9 | 6.7 | 7.3 | 6.6 | |

100 000 | 1 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 7.9 | 8.8 |

We also investigated computation gains using 61 quadrature points which is the default for unidimensional IRT model in the popular R package mirt (

It has to be noted that the optimal pseudo-item size from

Overall, it has to be admitted that the proposed shortcut might not result in considerable gains in the E-step of the EM algorithm. However, similar gains would also be obtained in parallel computing if only three or four cores were used.

Assume now that in a multidimensional IRT model, each item loads on exactly one

where

operations are needed for computing the posterior distributions. It turns out that in the computation of the posterior

The expected counts for an item

where

This less demanding computation provides a computational gain of

For a large number of items

Next, we combine the usage of pseudo-items and dimension-wise likelihood computation. For simplicity, we assume an equal number of items per dimension, i.e.,

The computational gain can be computed as

In the minimization of

We now illustrate the computational gains by applying our proposed computational shortcuts. We consider a multidimensional IRT model with between item dimensionality with

In

Sample size | ||||||
---|---|---|---|---|---|---|

DW | DP | DW | DP | DW | DP | |

100 | 10.3 | 19.2 | 22.5 | 24.1 | 19.9 | 20.0 |

1 000 | 10.3 | 23.1 | 22.5 | 24.5 | 19.9 | 20.0 |

10 000 | 10.3 | 25.6 | 22.5 | 24.6 | 19.9 | 20.0 |

100 000 | 10.3 | 27.2 | 22.5 | 24.7 | 19.9 | 20.0 |

1 000 000 | 10.3 | 28.3 | 22.5 | 24.8 | 19.9 | 20.0 |

10 000 000 | 10.3 | 29.1 | 22.5 | 24.8 | 19.9 | 20.0 |

This note shows that by using two computational shortcuts, the number of operations required in an E-step in unidimensional or multidimensional models can be substantially reduced. First, the evaluation of individual posterior distributions can be more efficiently implemented by using pseudo-items that are composed of a set of original items. By pursuing this strategy, additional item response probabilities—evaluated at the

In our evaluation, we only considered the case of dichotomous item responses. In principle, the same strategy can be applied for datasets containing polytomous item responses. However, using pseudo-items for polytomous items with many categories will typically not result in such substantial computational gains as for dichotomous items because a larger number of probabilities have to be computed. The handling of ignorable missing item responses is relatively simple because one could treat them as a special case of polytomous item responses and define a corresponding probability of a missing category as one. In this strategy, missing item responses are essentially omitted from the computation. For planned missingness designs (

Our proposed computational shortcuts can be combined with parallel computing (

The author has no funding to report.

The author has declared that no competing interests exist.

I thank the associate editor and two anonymous reviewers for their helpful comments which substantially improved the manuscript.