### Abstract

This article proposes a new, more efficient method to compute the minus two log likelihood, its gradient, and the Hessian for structural equation models (SEMs) in reticular action model (RAM) notation. The method exploits the beneficial aspect of RAM notation that the matrix derivatives used in RAM are sparse. For an SEM with K variables, P parameters, and P′ entries in the symmetrical or asymmetrical matrix of the RAM notation filled with parameters, the asymptotical run time of the algorithm is O(P ′ K ^{2} + P ^{2} K ^{2} + K ^{3}). The naive implementation and numerical implementations are both O(P ^{2} K ^{3}), so that for typical applications of SEM, the proposed algorithm is asymptotically K times faster than the best previously known algorithm. A simulation comparison with a numerical algorithm shows that the asymptotical efficiency is transferred to an applied computational advantage that is crucial for the application of maximum likelihood estimation, even in small, but especially in moderate or large, SEMs.

Original language | English (US) |
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Pages (from-to) | 385-395 |

Number of pages | 11 |

Journal | Behavior research methods |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2014 |

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### All Science Journal Classification (ASJC) codes

- Experimental and Cognitive Psychology
- Developmental and Educational Psychology
- Arts and Humanities (miscellaneous)
- Psychology (miscellaneous)
- Psychology(all)

### Cite this

*Behavior research methods*,

*46*(2), 385-395. https://doi.org/10.3758/s13428-013-0384-4

}

*Behavior research methods*, vol. 46, no. 2, pp. 385-395. https://doi.org/10.3758/s13428-013-0384-4

**Efficient Hessian computation using sparse matrix derivatives in RAM notation.** / von Oertzen, Timo; Brick, Timothy R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Efficient Hessian computation using sparse matrix derivatives in RAM notation

AU - von Oertzen, Timo

AU - Brick, Timothy R.

PY - 2014/6

Y1 - 2014/6

N2 - This article proposes a new, more efficient method to compute the minus two log likelihood, its gradient, and the Hessian for structural equation models (SEMs) in reticular action model (RAM) notation. The method exploits the beneficial aspect of RAM notation that the matrix derivatives used in RAM are sparse. For an SEM with K variables, P parameters, and P′ entries in the symmetrical or asymmetrical matrix of the RAM notation filled with parameters, the asymptotical run time of the algorithm is O(P ′ K 2 + P 2 K 2 + K 3). The naive implementation and numerical implementations are both O(P 2 K 3), so that for typical applications of SEM, the proposed algorithm is asymptotically K times faster than the best previously known algorithm. A simulation comparison with a numerical algorithm shows that the asymptotical efficiency is transferred to an applied computational advantage that is crucial for the application of maximum likelihood estimation, even in small, but especially in moderate or large, SEMs.

AB - This article proposes a new, more efficient method to compute the minus two log likelihood, its gradient, and the Hessian for structural equation models (SEMs) in reticular action model (RAM) notation. The method exploits the beneficial aspect of RAM notation that the matrix derivatives used in RAM are sparse. For an SEM with K variables, P parameters, and P′ entries in the symmetrical or asymmetrical matrix of the RAM notation filled with parameters, the asymptotical run time of the algorithm is O(P ′ K 2 + P 2 K 2 + K 3). The naive implementation and numerical implementations are both O(P 2 K 3), so that for typical applications of SEM, the proposed algorithm is asymptotically K times faster than the best previously known algorithm. A simulation comparison with a numerical algorithm shows that the asymptotical efficiency is transferred to an applied computational advantage that is crucial for the application of maximum likelihood estimation, even in small, but especially in moderate or large, SEMs.

UR - http://www.scopus.com/inward/record.url?scp=84901380218&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901380218&partnerID=8YFLogxK

U2 - 10.3758/s13428-013-0384-4

DO - 10.3758/s13428-013-0384-4

M3 - Article

C2 - 24197708

AN - SCOPUS:84901380218

VL - 46

SP - 385

EP - 395

JO - Behavior Research Methods

JF - Behavior Research Methods

SN - 1554-351X

IS - 2

ER -