In the existing multidimensional extensions of the log-normal response time (LNRT) model, the log response times are decomposed into a linear combination of several latent traits. These models are fully compensatory as low levels on traits can be counterbalanced by high levels on other traits. We propose an alternative multidimensional extension of the LNRT model by assuming that the response times can be decomposed into two response time components. Each response time component is generated by a one-dimensional LNRT model with a different latent trait. As the response time components—but not the traits—are related additively, the model is partially compensatory. In a simulation study, we investigate the recovery of the model’s parameters. We also investigate whether the fully and the partially compensatory LNRT model can be distinguished empirically. Findings suggest that parameter recovery is good and that the two models can be distinctly identified under certain conditions. The utility of the model in practice is demonstrated with an empirical application. In the empirical application, the partially compensatory model fits better than the fully compensatory model.
AkaikeH. (1992). Information theory and an extension of the maximum likelihood principle. In KotzS.JohnsonN. (Eds.), Breakthroughs in statistics, Vol I, Foundations and basic theory (pp. 610–624). Springer. https://doi.org/10.1007/978-1-4612-0919-5_38
2.
AllassonnièreS.ChevallierJ. (2021). A new class of stochastic EM algorithms. Escaping local maxima and handling intractable sampling. Computational Statistics & Data Analysis, 159, 107159. https://doi.org/10.1016/j.csda.2020.107159
3.
AsmussenS.Rojas-NandayapaL. (2008). Asymptotics of sums of lognormal random variables with Gaussian copula. Statistics and Probability Letters, 78, 2809–2814. https://doi.org/10.1016/j.spl.2008.03.035
4.
BartholomewD.KnottM. (1999). Latent variable models and factor analysis. Arnold.
5.
BatchelderW. (2007). Cognitive psychometrics: Combining two psychological traditions. CSCA Lecture.
6.
BeaulieuN.RajwaniF. (2004). Highly accurate simple closed-form approximations to lognormal sum distributions and densities. IEEE Communications Letters, 8, 709–711. https://doi.org/10.1109/LCOMM.2004.837657
7.
BockR. D.LiebermanM. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197. https://doi.org/10.1007/BF02291262
8.
BrownT. (2015). Confirmatory factor analysis for applied research. Guilford Press.
9.
CohenA. (1987). Three-parameter estimation. In CrowE.ShimizuK. (Eds.), Lognormal distributions: Theory and applications (pp. 113–137). CRC Press.
10.
D’AgostinoR. (1986). Tests for the normal distribution. In D’AgostinoR.StephensM. (Eds.), Goodness-of-fit techniques (pp. 367–420). Marcel Dekker. https://doi.org/10.1201/9780203753064
11.
DeMarsC. (2016). Partially compensatory multidimensional item response theory models: Two alternate model forms. Educational and Psychological Measurement, 76, 231–257. https://doi.org/10.1177/0013164415589595
12.
DomingueB.KanopkaK.StenhaugB.SolandJ.KuhfeldM.WiseS.PiechC. (2021). Variation in respondent speed and its implications: Evidence from an adaptive testing scenario. Journal of Educational Measurement, 58, 335–363. https://doi.org/10.1111/jedm.12291
13.
DouglasJ.KosorokM.ChewingB. (1999). A latent variable model for discrete multivariate psychometric waiting times. Psychometrika, 64, 69–82. https://doi.org/10.1007/BF02294320
ErdfelderE.AuerT.HilbigB.AssfalgA.MoshagenM.NadarevicL. (2009). Multinomial processing tree models: A review of the literature. Journal of Psychology, 217, 108–124. https://doi.org/10.1027/0044-3409.217.3.108
17.
FurneauxW. (1952). Some speed, error and difficulty relationships within a problem-solving situation. Nature, 170, 37–38. https://doi.org/10.1038/170037a0
18.
GaoX.XuH.YeD. (2009). Asymptotic behavior of tail density for sum of correlated lognormal variables. International Journal of Mathematics and Mathematical Sciences, 630857, 1–29. https://doi.org/10.1155/2009/630857
19.
HarikP.ClauserB.GrabovskyI.BaldwinP.MargolisM.BucakD.JodoinM.WalshW.HaistS. (2018). A comparison of experimental and observational approaches to assessing the effects of time constraints in a medical licensing examination. Journal of Educational Measurement, 55, 308–327. https://doi.org/10.1111/jedm.12177
20.
HartmannR.KlauerC. (2020). Extending rt-MPTS to enable equal process times. Journal of Mathematical Psychology, 96, 102340. https://doi.org/10.1016/j.jmp.2020.102340
21.
HayashiK.BentlerP.YuanK.-H. (2007). On the likelihood ratio test for the number of factors in exploratory factor analysis. Structural Equation Modeling, 14, 505–526. https://doi.org/10.1080/10705510701301891
22.
HenzeN.ZirklerB. (1990). A class of invariant consistent tests for multivariate normality. Communications in Statistics—Theory and Methods, 19, 3595–3617. https://doi.org/10.1080/03610929008830400
23.
JanosW. (1970). Tail of the distribution of sums of log-normal variates. IEEE Transactions on Information Theory, 16, 299–302. https://doi.org/10.1109/TIT.1970.1054458
KimN.BoltD. (2021). A mixture IRTree model for extreme response style: Accounting for response process uncertainty. Educational and Psychological Measurement, 81, 131–154. https://doi.org/10.1177/0013164420913915
KlauerK.KellenD. (2018). RT-MPTs: Process models for response-time distributions based on multinomial processing trees with applications to recognition memory. Journal of Mathematical Psychology, 82, 111–130. https://doi.org/10.1016/j.jmp.2017.12.003
28.
KlineR. (2015). Principles and practice of structural equation modeling. Guilford Press.
LeeY.-H.ChenH. (2011). A review of recent response-time analyses in educational testing. Psychological Test and Assessment Modeling, 53, 359–379.
31.
MardiaK. (1974). Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya, Series B, 36, 115–128. https://doi.org/jstor.org/stable/25051892
32.
MolenaarD.BolsinovaM.VermuntJ. (2018). A semi-parametric within-subject mixture approach to the analyses of responses and response times. British Journal of Mathematical and Statistical Psychology, 71, 205–228. https://doi.org/10.1111/bmsp.12117
33.
MolenaarD.RozsaS.BolsinovaM. (2019). A heteroscedastic hidden Markov mixture model for responses and categorized response times. Behavior Research Methods, 51, 676–696. https://doi.org/10.3758/s13428-019-01229-x
34.
MolenaarD.TuerlinckxF.van der MaasH. (2015). A generalized linear factor model approach to the hierarchical framework for responses and response times. British Journal of Mathematical and Statistical Psychology, 68, 197–219. https://doi.org/10.1111/bmsp.12042
PietersJ.van der VenA. (1982). Precision, speed, and distraction in time-limit tests. Applied Psychological Measurement, 6, 93–109. https://doi.org/10.1177/014662168200600110
37.
R Development Core Team. (2009). R: A language and environment for statistical computing [Computer software manual]. http://www.R-project.org
38.
RaddatzJ.KuhnJ.-T.MollK.HollingH.DobelC. (2017). Comorbidity of arithmetic and reading disorder: Basic number processing and calculation in children with learning impairments. Journal of Learning Disabilities, 50, 298–308. https://doi.org/10.1177/0022219415620899
RangerJ.KuhnJ. (2014). An accumulator model for responses and response times in tests based on the proportional hazards model. British Journal of Mathematical and Statistical Psychology, 67, 388–407. https://doi.org/10.1111/bmsp.12025
41.
RangerJ.KuhnJ. (2018). Estimating diffusion-based item response theory models: Exploring the robustness of three old and two new estimators. Journal of Eductional and Behavioral Statistics, 43, 635–662. https://doi.org/10.3102/1076998618787791
42.
RizopoulosD. (2006). ltm: An R package for latent variable modeling and item response theory analysis. Journal of Statistical Software, 17, 5. https://doi.org/10.18637/jss.v017.i05
43.
RomeoM.DaCostaV.BardouF. (2003). Broad distribution effects in sums of lognormal random variables. The European Physical Journal B – Condensed Matter and Complex Systems, 32, 513–525. https://doi.org/10.1140/epjb/e2003-00131-6
SchweizerK.ZellerF.ReissS. (2020). Higher-order processing and change-to-automaticity as explanations of the item-position effect in reasoning tests. Acta Psychologica, 203, 102991. https://doi.org/10.1016/j.actpsy.2019.102991
46.
SinharayS.HabermanS. (2014). How often is the misfit of item response theory models practically significant. Educational Measurement: Issues and Practice, 33, 23–35. https://doi.org/10.1111/emip.12024
47.
SullivanT. (2015). Numerical integration. In SullivanT. (Ed.), Introduction to uncertainty quantification (pp. 165–195). Springer. https://doi.org/10.1007/978-3-319-23395-6
48.
SympsonJ. (1977). A model for testing with multidimensional items. In WeissD. (Ed.), Proceedings of the 1977 computerized adaptive testing conference (pp. 82–98). University of Minnesota.
49.
SzyszkowiczS.YanikomerogluH. (2007, June24–28). On the tails of the distribution of the sum of lognormals. In IEEE international conference on communications, IEEE. https://doi.org/10.1109/ICC.2007.881
TinkerM. (1931). The significance of speed in test response. Psychological Review, 38, 450–454. https://doi.org/10.1037/h0072082
52.
TourangeauK.NordC.LêT.SorongonA. G.HagedornM. C.DalyP.NajarianM. (2015). Early childhood longitudinal study, kindergarten class of 2010-11 (ECLS-K: 2011). User’s manual for the ECLS-K: 2011 kindergarten data file and electronic codebook, public version. NCES 2015-074. National Center for Education Statistics.
van der LindenW. (2006). A lognormal model for response times on test items. Journal of Educational and Behavioral Statistics, 31, 181–204. https://doi.org/10.3102/10769986031002181
56.
van der LindenW. (2007). A hierarchical framework for modeling speed and accuracy on test items. Psychometrika, 72, 287–308. https://doi.org/10.1007/s11336-006-1478-z
van der MaasH.MolenaarD.MarisG.KievitR.BorsboomD. (2011). Cognitive psychology meets psychometric theory: On the relation between process models for decision making and latent variable models for individual differences. Psychological Review, 118, 339–356. https://doi.org/10.1037/a0022749
59.
VerdonckS.TuerlinckxF. (2016). Factoring out nondecision time in choice reaction time data: Theory and implications. Psychological Review, 123, 208–218. https://doi.org/10.1037/rev0000019
WeintraubS.DikmenS. S.HeatonR. K.TulskyD. S.ZelazoP. D.BauerP. J.CarlozziN. E.SlotkinJ.BlitzD.Wallner-AllenK.FoxN. A.BeaumontJ. L.MungasD.NowinskiC. J.RichlerJ.DeocampoJ. A.AndersonJ. E.ManlyJJ.BoroshB.HavlikR.ConwayK.EdwardsE.FreundL.KingJ. W.MoyC.WittE.GershonR. C. (2013). Cognition assessment using the NIH toolbox. Neurology, 80(11 Supplement 3), S54–S64. https://doi.org/10.1212/WNL.0b013e3182872ded
62.
WengerM.GibsonB. (2004). Using hazard functions to assess changes in processing capacity in an attentional cuing paradigm. Journal of Experimental Psychology, 30, 708–719. https://doi.org/10.1037/0096-1523.30.4.708
63.
WoodsD.WymaJ.YundE.HerronT.ReedB. (2015). Factors influencing the latency of simple reaction time. Frontiers in Human Neuroscience, 9, 131. https://doi.org/10.3389/fnhum.2015.00131
64.
WuQ.DebeerD.BuchholzJ.HartigJ.JanssenR. (2019). Predictors of individual performance changes related to item positions in PISA assessments. Large-Scale Assessments in Education, 7, 1–21. https://doi.org/10.1186/s40536-019-0073-6
65.
ZhanP.JiaoH.LiaoD. (2018). Cognitive diagnosis modelling incorporating item response times. British Journal of Mathematical and Statistical Psychology, 71, 262–286. https://doi.org/10.1111/bmsp.12114
66.
ZhanP.JiaoH.ManK. (2020). The multidimensional log-normal response time model: An exploration of the multidimensionality of latent processing speed. Acta Psychologica Sinica, 52, 1132–1142. https://doi.org/10.3724/SP.J.1041.2020.01132
67.
ZhanP.JiaoH.WangW.ManK. (2018). A multidimensional hierarchical framework for modeling speed and ability in computer-based multidimensional tests (Tech. Rep. No. 1807.04003). arXiv. https://arxiv.org/abs/1807.04003
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
