MTP2 and Partial Correlations in Monotone Higher-Order Factor Models

Ellis, Jules L.

doi:10.1007/978-3-319-07503-7_16

Cited by 3 publications

(12 citation statements)

References 19 publications

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“…For a multidimensional vector , Holland and Rosenbaum (1986, Theorem 7) showed that the assumptions of LI and M imply that X satisfies the property of MTP 2 , if is MTP 2 . Also, X is MTP 2 , whenever (X, ) satisfying a particular higher-order factor structure (Ellis, 2015). These results imply that the property of MTP 2 is not confined to unidimensional models only.…”

Section: Multivariate Totally Positivementioning

confidence: 76%

“…which each selected variable taking on the role of X k once (Ellis, 2014). NPC holds, whenever all trivariate distributions of triplets of response variables satisfy MTP 2 (Ellis, 2015).…”

Section: Nonnegative Partial Correlationsmentioning

confidence: 99%

“…In case J = 2, all the properties coincide with Cov(X 1 , X 2 ) ≥ 0. For J = 3, binary random variables, Ellis (2015) showed that the properties CA and MTP 2 coincide. Also, the properties A and SPOD coincide ('Appendix'), as shown in Fig.…”

Section: Relationships Between the Observable Propertiesmentioning

confidence: 99%

“…Holland and Rosenbaum (1986) provide an overview of properties of multivariate positive dependence that are implied by the MH model, with conditional association (CA; Holland and Rosenbaum, 1986;Rosenbaum, 1984) being the most restrictive of these properties for binary response data. Below, we show that the observable property CA also implies MM and NPC (Ellis, 2015). Because the MH model cannot be directly evaluated, we rely on these observable properties to make inferences about the validity of the MH model assumptions (Sijtsma & Van der Ark, 2017).…”

mentioning

confidence: 94%

“…Smits et al (2012) also warn to be cautious about making inferences about the dimensionality of a test based on an automated evaluation of scalability coefficients. Tighter lower bounds for the scalability coefficients can be obtained from the requirement of nonnegative partial correlations (NPC; Ellis, 2014Ellis, , 2015Brusco et al, 2015). Like the scalability coefficients, the partial correlation is implied to be nonnegative under the MH model, but the property NPC takes into consideration the higher-order moments contained in the trivariate distributions of item triplets.…”

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confidence: 99%

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Incomplete Tests of Conditional Association for the Assessment of Model Assumptions

Ligtvoet

2022

Psychometrika

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Many of the models that have been proposed for response data share the assumptions that define the monotone homogeneity (MH) model. Observable properties that are implied by the MH model allow for these assumptions to be tested. For binary response data, the most restrictive of these properties is called conditional association (CA). All the other properties considered can be considered incomplete tests of CA that alleviate the practical limitations encountered when assessing the MH model assumptions using CA. It is found that the assessment of the MH model assumptions with an incomplete test of CA, rather than CA, is generally associated with a substantial loss of information. We also look at the sensitivity of the observable properties to model violation and discuss the implications of the results. It is argued that more research is required about the extent to which the assumptions and the model specifications influence the inferences made from response data.

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Section: Multivariate Totally Positivementioning

confidence: 76%

“…which each selected variable taking on the role of X k once (Ellis, 2014). NPC holds, whenever all trivariate distributions of triplets of response variables satisfy MTP 2 (Ellis, 2015).…”

Section: Nonnegative Partial Correlationsmentioning

confidence: 99%

Section: Relationships Between the Observable Propertiesmentioning

confidence: 99%

mentioning

confidence: 94%

mentioning

confidence: 99%

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Incomplete Tests of Conditional Association for the Assessment of Model Assumptions

Ligtvoet

2022

Psychometrika

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The Sum Scores and Discretization of Variables Under the Linear Normal One-Factor Model

Ligtvoet

2022

Springer Proceedings in Mathematics &Amp; Statistics

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A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models

Ellis

Sijtsma

2023

Psychometrika

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The goodness-of-fit of the unidimensional monotone latent variable model can be assessed using the empirical conditions of nonnegative correlations (Mokken in A theory and procedure of scale-analysis, Mouton, The Hague, 1971), manifest monotonicity (Junker in Ann Stat 21:1359–1378, 1993), multivariate total positivity of order 2 (Bartolucci and Forcina in Ann Stat 28:1206–1218, 2000), and nonnegative partial correlations (Ellis in Psychometrika 79:303–316, 2014). We show that multidimensional monotone factor models with independent factors also imply these empirical conditions; therefore, the conditions are insensitive to multidimensionality. Conditional association (Rosenbaum in Psychometrika 49(3):425–435, 1984) can detect multidimensionality, but tests of it (De Gooijer and Yuan in Comput Stat Data Anal 55:34–44, 2011) are usually not feasible for realistic numbers of items. The only existing feasible test procedures that can reveal multidimensionality are Rosenbaum’s (Psychometrika 49(3):425–435, 1984) Case 2 and Case 5, which test the covariance of two items or two subtests conditionally on the unweighted sum of the other items. We improve this procedure by conditioning on a weighted sum of the other items. The weights are estimated in a training sample from a linear regression analysis. Simulations show that the Type I error rate is under control and that, for large samples, the power is higher if one dimension is more important than the other or if there is a third dimension. In small samples and with two equally important dimensions, using the unweighted sum yields greater power.

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MTP2 and Partial Correlations in Monotone Higher-Order Factor Models

Cited by 3 publications

References 19 publications

Incomplete Tests of Conditional Association for the Assessment of Model Assumptions

Incomplete Tests of Conditional Association for the Assessment of Model Assumptions

The Sum Scores and Discretization of Variables Under the Linear Normal One-Factor Model

A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models

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