1.1. Monotone Homogeneity and Monotone Factor Models

We discuss the definitions of three nonparametric IRT models that will be used throughout the article. The first model is monotone homogeneity (MH), which contends that the expected value of each observed binary item score variable increases with a single underlying variable, called the common factor, the latent variable, or the latent dimension. The second model, the monotone factor model (MFM), contends basically the same as the MH model with one or more independent factors to which the items are related in a simple structure. The third model is the higher-order monotone one-factor (HOMOF) model, which is like the MFM, but allows the factors to be correlated with the restriction that they depend on a single higher-order factor. The three models share the assumption of conditional (or local) independence or independent errors, which are similar assumptions. Thus, MH describes a general form of unidimensionality, MFM describes a general form of multidimensionality, and HOMOF is somewhere in between. Applied to intelligence, MH formally resembles Spearman’s theory of a single general intelligence factor, MFM is like Thurstone’s initial theory of multiple independent primary mental abilities, and HOMOF parallels the hierarchical factors of CHC, integrating the other two theories.

We assume the item scores are binary manifest variables, denoted $X=\left(X_1,,,\dots ,,,X_J\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}=\left( X_{1}, \ldots ,X_{J} \right) $$\end{document} . Variable $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} represents the scores (1 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} positive, 0 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} negative) subjects obtained on the i-th item. Factors, latent variables, or dimensions are denoted $\Theta =\left({\Theta }_1,,,\dots ,,,{\Theta }_D\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }=}\left( {\Theta }_{1}, \ldots ,{\Theta }_{D} \right) $$\end{document} . The rest of this section discusses the formal definitions of the three models, which we need to prove the theorems.

We adopt the following assumptions (Holland & Rosenbaum, Reference Holland and Rosenbaum1986; Mokken, Reference Mokken1971; Rosenbaum, Reference Rosenbaum1984):

1. MH1 (conditional independence): $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} is conditionally independent given $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document} .

2. MH2 (monotonicity): $P(X_i=1|\Theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_{i}=1\vert \, \varvec{\Theta } )$$\end{document} is an increasing function of $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document} for all $i=1,\dots ,J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,J$$\end{document} .

3. MH3 (unidimensionality): $D=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D=1$$\end{document} .

We use the term ‘increasing’ synonymous with ‘monotone nondecreasing’. For readers who do not have the information ready, Appendix A provides formulations of the assumptions having greater precision. Following Holland and Rosenbaum (Reference Holland and Rosenbaum1986), we will say that $\left(X,\Theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\textbf {X}}}{,\, }{\varvec{\Theta }})$$\end{document} is a monotone latent variable (MLV) model if MH1 and MH2 hold. Following Mokken (Reference Mokken1971) Mokken and Lewis (Reference Mokken and Lewis1982) and Ellis and Junker (Reference Ellis and Junker1997), we will say that $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} satisfies a unidimensional MLV model or MH model if there exists a variable $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }$$\end{document} such MH1, MH2, and MH3 hold.

Ellis (Reference Ellis2015) studied a narrower class of monotone models. Slightly rephrasing Ellis, we will say that $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} satisfies an MFM if

$\begin{array}{c}X=\varphi \left(\psi ,(\lambda \Theta ,),+,\epsilon \right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\textbf {X}}}=\phi \left( \psi {\varvec{(\lambda \Theta }}\mathbf {)}+{\varvec{\upvarepsilon }} \right) , \end{aligned}$$\end{document}

where $\varphi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} and $\psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} are component-wise increasing functions, $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document} is a multivariate random vector with independent components, $\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upvarepsilon }}$$\end{document} is a multivariate random vector with components that are independent of each other and of $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document} , the ${\epsilon }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upvarepsilon _{i}$$\end{document} s have log-concave densities (Appendix A), and $\lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\lambda }}$$\end{document} is a nonnegative real matrix with simple structure (i.e., every manifest variable loads positive on one factor and zero on the other factors). As an example of an MFM for binary manifest variables, consider a case where the ${\epsilon }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upvarepsilon _{i}$$\end{document} s have standard normal distributions, $\psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} is the identity function, and the ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{i}$$\end{document} s are step functions with ${\varphi }_i\left(x\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{i}\left( x \right) =0$$\end{document} if $x\le {\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\le \beta _{i}$$\end{document} and ${\varphi }_i\left(x\right)=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{i}\left( x \right) =1$$\end{document} if $x>{\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>\beta _{i}$$\end{document} for some real number ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{i}$$\end{document} . Then, $P\left(X_i=1,|\Theta \right)=\Phi \left({\sum }_{d=1}^D,{\lambda }_{\mathrm{id}}{\Theta }_d,-,{\beta }_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\left( {X_{i}=1} \vert {\varvec{\Theta }}\right) ={\Phi }\left( \mathop {\sum }\nolimits _{d=1}^D {\lambda _{id}{\Theta }_{d}} -\beta _{i} \right) $$\end{document} , where $\Phi (.)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi (.)}$$\end{document} is the standard normal distribution function. Hence, every multidimensional normal ogive IRT model with independent factors and nonnegative loadings with a simple structure is an MFM (also, see Takane & De Leeuw, Reference Takane and de Leeuw1987).

Ellis (Reference Ellis2015) also studied a more general class of models, where the components of $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document} need not be independent but may be the result of a higher-order MFM factor with log-concave disturbances at each level. We call this class of models, with possibly many levels and one factor at the highest level, a HOMOF model. In this class of models, the factor loadings at the lowest level (i.e., $\lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\lambda }}$$\end{document} in $X=\varphi \left(\psi ,(\lambda \Theta ,),+,\epsilon \right))$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}=\phi \left( \psi {\varvec{(\lambda \Theta }}\mathbf {)}+{\varvec{\upvarepsilon }} \right) )$$\end{document} do not necessarily have simple structure.

1.2. Testable Conditions of the Models

In this section, we review statistical inequalities that have been used to test whether MH holds for a given set of manifest variables. These inequalities can be expressed as covariances that are nonnegative. The general result implied by MH is conditional association (CA; Rosenbaum, Reference Rosenbaum1984). Below, we discuss that CA is the hallmark of MH. Coincidentally, CA fares well with Spearman’s (Spearman, Reference Spearman1904) idea that intelligence tests have positive correlations and together measure a single general intelligence factor, and Guttman’s ‘first law of intelligence’, stating that any two intelligence items have a nonnegative correlation in any population that is “not artificially selected” (Guttman & Levy, Reference Guttman and Levy1991), thus suggesting the items should have nonnegative correlations in any subgroup defined by the other items. CA is hard to test fully because it involves many restrictions even for small item sets (De Gooijer & Yuan, Reference De Gooijer and Yuan2011; Ligtvoet, Reference Ligtvoet2022; Yuan & Clarke, 2001). Therefore, we will also discuss conditions that are easier to test, such as the condition that the expected item score increases with the rest score, called marginal monotonicity (MM) (Junker, Reference Junker1993). These conditions can be viewed as incomplete tests of CA (Ligtvoet, Reference Ligtvoet2022). We will now continue this section with the formal definitions.

Following Rosenbaum (Reference Rosenbaum1984), we say that $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} is CA if for every partition $X=(Y,Z)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}=({{\textbf {Y}}},{{\textbf {Z}}})$$\end{document} and every function h, for all increasing functions $\varphi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} and $\psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} ,

$\begin{array}{c}\text{Cov}\left(\varphi ,\left(Y\right),,\psi \left(Y\right)\left|h\right(Z\right))\ge 0.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{Cov}\left( \phi \left( {{\textbf {Y}}} \right) ,\psi ({{\textbf {Y}}})\vert h\mathrm {(}\textbf{Z} \right) )\ge 0. \end{aligned}$$\end{document}

Rosenbaum (Reference Rosenbaum1984) provides examples. Rosembaum’s result that

$\begin{array}{c}\text{MH}\Rightarrow \text{CA}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{MH}\Rightarrow \textrm{CA} \end{aligned}$$\end{document}

is key to this article, in which we develop a practically feasible test for CA. Holland and Rosenbaum (Reference Holland and Rosenbaum1986) generalized Rosenbaum’s (Rosenbaum, Reference Rosenbaum1984) work to non-binary variables. Ellis and Junker (Reference Ellis and Junker1997; also Junker & Ellis, Reference Junker and Ellis1997) furthermore suggested that CA is sufficient to detect multidimensionality in a finite number of items. They studied infinite item sequences and used the condition of vanishing conditional dependence (VCD), which means that certain conditional covariances vanish as $J\to \infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\rightarrow \infty $$\end{document} . They showed that CA and VCD are necessary and sufficient for a unidimensional monotone latent variable model in which the latent variable can be estimated consistently. Since VCD is defined only in the limit as $J\to \infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\rightarrow \infty $$\end{document} , one would expect that if any condition can detect multidimensionality in a finite item set, it will be CA.

In addition to the practical infeasibility of CA due to large numbers of restrictions one must test even for small item sets, a complete test of CA is also impossible because of the sparseness of the data, because most item-score patterns never occur in the most commonly available sample sizes of 100 to 10,000 subjects. Maybe for this reason, many authors have studied weaker restrictions than CA, possibly believing they still capture the gist of CA. Straat et al. (Reference Straat, Van der Ark and Sijtsma2016) acknowledged this limitation and proposed an incomplete strategy. Ligtvoet (Reference Ligtvoet2022) gives an excellent review of weaker conditions, which he describes as “incomplete tests of conditional association”.

An important condition discussed by Ligtvoet (Reference Ligtvoet2022) is multivariate total positivity of order 2 (MTP ${}_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2})$$\end{document} . The ordinary formulation of this condition is given in Appendix A, but for the present purpose it suffices to note that $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} being MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} implies that

$\begin{array}{c}\text{Cov}\left(\varphi ,\left(Y\right),,\psi \left(Y\right)|Z\right)\ge 0.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{Cov}\left( \phi \left( {{\textbf {Y}}} \right) ,\psi ({{\textbf {Y}}})\vert {{\textbf {Z}}} \right) \ge 0. \end{aligned}$$\end{document}

(Note the omission of h() here). Therefore, $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} being MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} means that $\text{Cov}\left(\varphi ,\left(Y\right),,\psi \left(Y\right)|h(Z\right))\ge 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Cov}\left( \phi \left( {{\textbf {Y}}} \right) ,\psi ({{\textbf {Y}}})\vert h\mathrm {(\textbf{Z}} \right) )\ge 0$$\end{document} holds for some, but not all, functions h (Ellis, Reference Ellis2015, p. 264–265). For binary variables, the difference between CA and MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} thus lies in the kind of events on which one may condition: MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} involves conditioning on the finest partition of subgroups that can be made with $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {Z}}}$$\end{document} , whereas CA also involves conditioning on combinations of such groups. Holland and Rosenbaum (Reference Holland and Rosenbaum1986) show that

$\begin{array}{c}\text{CA}\Rightarrow {\text{MTP}}_2.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {CA}\Rightarrow \text {MTP}_{2}. \end{aligned}$$\end{document}

Therefore, a test of ${\text{MTP}}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MTP}_{2}$$\end{document} can be viewed as an incomplete test of CA (Ligtvoet, Reference Ligtvoet2022).

For tests consisting of realistic numbers of items, however, MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} still involves a large number of restrictions (Bartolucci & Forcina, Reference Bartolucci and Forcina2005, p. 35; Ligtvoet, Reference Ligtvoet2022). Therefore, one may want to reduce the number of restrictions to be tested even further by considering properties derived from MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , which include nonnegative partial correlations (NPC; Ellis, Reference Ellis2015; Brusco et al., Reference Brusco, Köhn and Steinley2015) and nonnegative covariances (NNC) (Mokken, Reference Mokken1971).

The important testable property of Manifest Monotonicity (MM) (Junker, Reference Junker1993; Junker & Sijtsma, Reference Junker and Sijtsma2000) means that each item regression on the sum of the other items (known as the rest score) is increasing (Appendix A). Junker (Reference Junker1993, p. 1372), Junker and Sijtsma (Reference Junker and Sijtsma2000) showed that

$\begin{array}{c}\text{MH}\Rightarrow \text{MM}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{MH}\Rightarrow \textrm{MM}. \end{aligned}$$\end{document}

Ligtvoet (Reference Ligtvoet2022) showed that $\text{CA}\Rightarrow \text{MM}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{CA}\Rightarrow \textrm{MM}$$\end{document} , so MM may be viewed as another incomplete test of CA. MM is an important property because it is conceptually like the idea of a monotone IRF, and for the reader who is unfamiliar with these concepts it may be hard to see how one can have MM without MH. Ellis (Reference Ellis2014) gave an example where MM holds while NPC fails, and therefore, MH must fail too.

1.3. Limitations of Partial Tests of Conditional Association

Ligtvoet (Reference Ligtvoet2022) concluded that existing incomplete tests of CA perform poorly detecting violations of CA. In the data structures he generated, CA was often violated while MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} and MM held. Thus, a researcher who tests MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} and/or MM instead of CA misses the violation of CA. We agree but notice that MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} and MM are not sensitive to violations of dimensionality, necessitating the discussion of the problem from a more theoretical perspective. This discussion is partially inspired by Van den Wollenberg’s (Van den Wollenberg, Reference Van den Wollenberg1982) proof that existing test statistics for the Rasch model were insensitive to violations of unidimensionality.

Ellis (Reference Ellis2015; also proposition A1 of Appendix A) showed that for MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} the problem is that,

$\begin{array}{c}\text{MFM}\Rightarrow {\text{MTP}}_2.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {MFM}\Rightarrow \text {MTP}_{2}. \end{aligned}$$\end{document}

Consequently, any test of MH based on MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} (Bartolucci & Forcina, Reference Bartolucci and Forcina2000, Reference Bartolucci and Forcina2005) logically cannot distinguish MH from MFM. That is, if the test suggests that MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} holds, it is also possible that an MFM that violates MH generated the data. The same conclusion holds for any property derived from MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , which includes NPC (Ellis, Reference Ellis2015; Brusco et al., Reference Brusco, Köhn and Steinley2015), and NNC (Mokken, Reference Mokken1971). Thus, testing MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , NPC, and NNC cannot distinguish unidimensional and multidimensional monotone factor models.

MM is implied by CA but not by MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , and therefore, we need to discuss it separately. For MM, the problem is that

$\begin{array}{c}\text{MFM}\Rightarrow \text{MM}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{MFM}\Rightarrow \textrm{MM}. \end{aligned}$$\end{document}

This follows from Theorem 1 in Appendix B. Consequently, no test of MM (Douglas & Cohen, Reference Douglas and Cohen2001; Junker & Sijtsma, Reference Junker and Sijtsma2000; Molenaar & Sijtsma, Reference Molenaar and Sijtsma2000; Tijmstra et al., Reference Tijmstra, Hessen, van der Heijden and Sijtsma2013; Tijmstra & Bolsinova, Reference Tijmstra and Bolsinova2019) can distinguish MH from MFM. That is, if the test suggests that MM holds, it is also possible that an MFM that violates MH generated the data. Thus, testing MM cannot distinguish unidimensional from multidimensional monotone factor models.

To summarize, based on data it is impossible to distinguish between MH and an MFM if one tests CA only partially with MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , NNC, NPC, and MM. For example, assume that $J=10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J=10$$\end{document} , the first five items satisfying the Rasch model with latent variable ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} , and the last five satisfying the Rasch model with latent variable ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} , and ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} and ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} are independent. This is an MFM, thus satisfying MM and MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , which implies NNC and NPC. Hence, it would be impossible to reject MH if only these conditions are tested. This puts the MH model at a serious disadvantage in comparison with parametric IRT models such as the two-parameter logistic model, where one can easily identify this situation via the M ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} -test (Maydeu-Olivares & Joe, Reference Maydeu-Olivares and Joe2006).

The argument given includes the somewhat artificial case of independent factors, which implies that some items have correlation zero. One might argue that such cases would be excluded with Mokken’s (Reference Mokken1971; Mokken & Lewis, Reference Mokken and Lewis1982) criterion $H>.30$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H>.30$$\end{document} . However, Ellis (Reference Ellis2015) also showed that

$\begin{array}{c}\text{HOMOF}\Rightarrow {\text{MTP}}_2.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{HOMOF}\Rightarrow {\textrm{MTP}_{2}}. \end{aligned}$$\end{document}

Although HOMOF models involve a single higher-order factor, there will generally be multiple first-order factors that are positively correlated with any degree. Consequently, it is possible that such items also satisfy $H>.30$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H>.30$$\end{document} .

We conclude that to distinguish unidimensional MFMs from multidimensional MFMs, the testable conditions MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} , NPC, NNC, and MM are logically insufficient. Therefore, we will target specific aspects of CA, which are covariance inequalities that will likely discriminate between MH and multidimensional MFMs. The next section discusses candidate covariance inequalities.

2.1. CARP Inequalities, Definition

Assume that all response probabilities of $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} are known; sample statistics will be discussed later. For a given pair ( $X_i,X_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i},X_{j})$$\end{document} , denote by $X_{-ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}_{-ij}$$\end{document} the variables of $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} except $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} . Our proposal is a generalization of Rosenbaum’s (Reference Rosenbaum1984, p. 427) “Case 2” method to test the covariance of each item pair conditionally on the rest score of the pair. For these covariances, CA implies that

$\begin{array}{c}\text{Cov}(X_i,X_j|{\sum }_{k\ne i,j}^J{X}_k)\ge 0.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{Cov}(X_{i},X_{j}\vert \mathop {\sum }\nolimits _{k\ne i,j}^J X_{k} )\ge 0. \end{aligned}$$\end{document}

We call this the conditioning on rest scores (CRS)inequality, and we call a significance test for the CRS inequality a CRS test. A limitation of the CRS inequality for testing the dimensionality of a set of items is that the rest score used for conditioning is not adapted to the possible multidimensional structure if the item set does not satisfy MH. To obtain greater adaptation, we propose to use weighted rest scores. We use two linear regression analyses, where $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} serve as dependent variables and the other items are independent variables. Write $X_0:=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{0}:=1$$\end{document} , and denote the regression coefficient of $X_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{k}$$\end{document} in the prediction of $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} from $X_{-ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}_{-ij}$$\end{document} as $a_{k.ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k.ij}$$\end{document} , and denote the resulting predicted scores as ${\hat{X}}_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}$$\end{document} ; that is,

$\begin{array}{c}{\hat{X}}_{\mathrm{ij}}={\sum }_{k=0}^J{a}_{k.ij}{X}_k,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{X}_{ij}=\mathop {\sum }\nolimits _{k=0}^J {a_{k.ij}X_{k}}, \end{aligned}$$\end{document}

where the $a_{k.ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k.ij}$$\end{document} are such that they minimize $E\left({\left(X_i,-,{\hat{X}}_{\mathrm{ij}}\right)}^2\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}\left( \left( X_{i}-\hat{X}_{ij} \right) ^{2} \right) $$\end{document} , and with $a_{k.ij}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k.ij}=0$$\end{document} if $k=i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=i$$\end{document} or $k=j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=j$$\end{document} . Similarly, ${\hat{X}}_{\mathrm{ji}}={\sum }_{k=0}^J{a}_{k.ji}{X}_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ji}=\mathop {\sum }\nolimits _{k=0}^J {a_{k.ji}X_{k}} $$\end{document} where the $a_{k.ji}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k.ji}$$\end{document} are such that they minimize $E\left({\left(X_j,-,{\hat{X}}_{\mathrm{ji}}\right)}^2\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}\left( \left( X_{j}-\hat{X}_{ji} \right) ^{2} \right) $$\end{document} , and with $a_{k.ji}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k.ji}=0$$\end{document} if $k=i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=i$$\end{document} or $k=j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=j$$\end{document} . In other words, ${\hat{X}}_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}$$\end{document} is the prediction of $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} if $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} is excluded as predictor, and ${\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ji}$$\end{document} is the prediction of $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} if $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} is excluded as predictor. As the basis for conditioning, we propose the variable

$\begin{array}{c}{\hat{X}}_{\mathrm{ij}}+{\hat{X}}_{\mathrm{ji}}={\sum }_{k=0}^J(a_{k.ij}+a_{k.ji})X_k.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{X}_{ij}+\hat{X}_{ji}=\mathop {\sum }\nolimits _{k=0}^J {(a_{k.ij}+a_{k.ji})X_{k}}. \end{aligned}$$\end{document}

This is the sum of the predicted scores of $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} , where both $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} are excluded from the predictors. It may be noted that we are not assuming linearity or normality here; we are just using the least squares solution as a heuristic tool, without claiming that this produces a good model.

The variable ${\hat{X}}_{\mathrm{ij}}+{\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}+\hat{X}_{ji}$$\end{document} can attain many different values, producing small conditioning groups. Therefore, we will use deciles or other quantiles of this function. This is like Rosenbaum’s (Reference Rosenbaum1984, p. 428) “Case 5”, which tests the covariance of two subtests conditionally on deciles of the rest score. If $Q_m$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{m}$$\end{document} is an operator that divides any variable into m groups of approximately equal size, that is, $P[Q_m\left(X\right)=k]\approx m^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P[Q_{m}\left( X \right) =k]\approx m^{-1}$$\end{document} , then we obtain

$\begin{array}{c}\mathrm{Cov}[X_i,X_j\left|Q_m\left({\hat{X}}_{\mathrm{ij}},+,{\hat{X}}_{\mathrm{ji}}\right)\right]\ge 0.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathrm {Cov[}X_{i},X_{j}\vert Q_{m}\left( \hat{X}_{ij}+\hat{X}_{ji} \right) ]\ge 0. \end{aligned}$$\end{document}

We refer to covariance inequalities of this form (with or without grouping by $Q_m)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{m})$$\end{document} as conditioning on added regression predictions (CARP) inequalities. Similarly, we refer to the involved conditional covariances as CARP covariances, and to the corresponding correlations as CARP correlations. We call the property that $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} satisfies all CARP inequalities, simply CARP.

CARP is the special case of CA with (if we use the notation used in the definition of CA) $Y=(X_i,X_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {Y}}}=(X_{i},X_{j})$$\end{document} , $\varphi \left(Y\right)=X_i,\psi \left(Y\right)=X_j,$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \left( {{\textbf {Y}}} \right) =X_{i},\, \psi \left( {{\textbf {Y}}} \right) =X_{j},$$\end{document} $Z=X_{-ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {Z}}}={{\textbf {X}}}_{-ij}$$\end{document} , and $h\left(Z\right)={\sum }_{k=0}^J(a_{k.ij}+a_{k.ji})X_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\left( {{\textbf {Z}}} \right) =\, \mathop {\sum }\nolimits _{k=0}^J {(a_{k.ij}+a_{k.ji})X_{k}} $$\end{document} . The latter weighted sum is a function of $X_{-ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}_{-ij}$$\end{document} because $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} has weight $a_{i.ij}+a_{i.ji}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{i.ij}+a_{i. ji}=0$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} has weight $a_{j.ij}+a_{j.ji}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{j. ij}+a_{j.ji}=0$$\end{document} . We assume in this section that all response probabilities of $X$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {X}}}$$\end{document} are known, and therefore, the regression coefficients $a_{k.ij}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k.ij}$$\end{document} are parameters, not sample statistics.

Hence, MH implies CARP. Furthermore, MFM does not imply CARP, which is demonstrated in later simulations, and which can also be seen in theoretical computations of some special cases. Therefore, testing CARP inequalities can reveal some violations of MH that testing MTP ${}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} or MM cannot reveal.

Let us now briefly explain why CARP inequalities may be useful in the assessment of multidimensionality. Suppose that $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} load on different independent latent variables, say, ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} and ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} , and that the other items load on either ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} or ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} . After a suitable transformation of ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} and ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} , we may say that ${\hat{X}}_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}$$\end{document} estimates ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} and ${\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ji}$$\end{document} estimates ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} (set ${\Theta }_1:=E\left({\hat{X}}_{\mathrm{ij}},|,{\Theta }_1\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}:=\mathbb {E}\left( \hat{X}_{ij} \vert {\Theta }_{1}\right) $$\end{document} and ${\Theta }_2:=E\left({\hat{X}}_{\mathrm{ji}},|,{\Theta }_2\right))$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}:=\mathbb {E}\left( {{\, }\hat{X}_{ji}} \vert {\Theta }_{2}\right) )$$\end{document} , so conditioning on ${\hat{X}}_{\mathrm{ij}}+{\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}+\hat{X}_{ji}$$\end{document} tends to create groups with ${\Theta }_1+{\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}+{\Theta }_{2}$$\end{document} approximately equal, which induces a negative correlation between ${\Theta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}$$\end{document} and ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} in these groups (in groups where ${\Theta }_1+{\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1}+{\Theta }_{2}$$\end{document} is constant, ${\Theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{2}$$\end{document} is a decreasing function of ${\Theta }_1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }_{1})$$\end{document} , which in turn tends to create a negative correlation between $X_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i}$$\end{document} and $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{j}$$\end{document} . Theorem 2 of the Appendix states more formally that in this situation, the mean conditional covariance given the unweighted rest scores will be negative or zero, and Theorem 3 of the Appendix states that this will also be true for the mean conditional covariance given the weighted rest score ${\hat{X}}_{\mathrm{ij}}+{\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}+\hat{X}_{ji}$$\end{document} provided that $E\left(X_i,|,{\hat{X}}_{\mathrm{ij}}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}\left( {{\, }X_{i}} \vert \hat{X}_{ij}\right) $$\end{document} and $E\left(X_j,|,{\hat{X}}_{\mathrm{ji}}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}\left( {{\, }X_{j}} \vert \hat{X}_{ji}\right) $$\end{document} are both increasing (i.e., the items have MM with respect to the partial weighted sum score of their respective subtest). In the standard two-dimensional case (defined in section 5.1) with ten items, we computed these correlations using numerical integration, and the outcomes supported our expectation that such correlations tend to be negative or zero. The simulations to assess the power of the CARP tests, reported later in this study, also support this result.

When we developed the test, we initially created a slightly different method, which can produce smaller correlations than the CARP correlations (the computation of the next example can be found in the Supplementary Material). For example, take two uncorrelated standard normal dimensions each with five Rasch items, all having ${\beta }_i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{i}=0$$\end{document} . Then, using numerical integration, one can obtain a correlation of -.204 in the union of the two most extreme vigintile groups of ${\hat{X}}_{\mathrm{ij}}-{\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}-\hat{X}_{ji}$$\end{document} ; that is, $\phi \left(X_i,X_j,|,[Q_{20}\left({\hat{X}}_{\mathrm{ij}},-,{\hat{X}}_{\mathrm{ji}}\right)=1]\cup [Q_{20}\left({\hat{X}}_{\mathrm{ij}},-,{\hat{X}}_{\mathrm{ji}}\right)=20]\right)=-.204$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \left( {X_{i},X_{j}} \vert {[Q_{20}\left( \hat{X}_{ij}-\hat{X}_{ji} \right) =1]\, \cup [Q_{20}\left( \hat{X}_{ij}-\hat{X}_{ji} \right) =20]}\right) =-.204$$\end{document} . This is not aCARP correlation, because we condition on ${\hat{X}}_{\mathrm{ij}}-{\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}-\hat{X}_{ji}$$\end{document} instead of ${\hat{X}}_{\mathrm{ij}}+{\hat{X}}_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{X}_{ij}+\hat{X}_{ji}$$\end{document} . Although this correlation is smaller than the CARP correlations we obtained, a statistical test based on this conditional correlation of $-.204$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-.204$$\end{document} turns out to be less powerful because the 90% observations with $2\le Q_{20}\left({\hat{X}}_{\mathrm{ij}},-,{\hat{X}}_{\mathrm{ji}}\right)\le 19$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\, \le \, Q_{20}\left( \hat{X}_{ij}-\hat{X}_{ji} \right) \le 19$$\end{document} are discarded. Simulations showed that a CARP test has greater power in this case. The next section focusses on a test statistic that can be used to test CARP.

Our approach is almost the opposite of the DETECT and DIMTEST procedures for investigating an item set’s dimensionality (Stout et al., Reference Stout, Habing, Douglas, Kim, Roussos and Zhang1996; Zhang & Stout, Reference Zhang and Stout1999a, Reference Zhang and Stoutb). DETECT and DIMTEST look for large conditional covariances, averaged over item pairs, as a sign that unidimensionality is violated. Unlike CARP, these approaches do not use rigorously established inequalities for the conditional covariances, but rather assume that they are approximately equal to certain theoretical conditional covariances given $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta }$$\end{document} .

5.1. Method

5.1.2. Optimum Size of Training Sample

Training samples in cross-validation often contain at least 70% of the observations of the whole sample. We did some simulations to find out whether we must maintain that percentage here. We used the standard two-dimensional case with $J\in \{12,24\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\in \{12,\, 24\}$$\end{document} and $N\in \{500,1000,2000,5000\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \{500,\, 1000,\, 2000,\, 5000\}$$\end{document} , and $\ell \in \{.1,.2,.3,.4,.5,.6,.7,.8,.9\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in \{.1,\,.2,\,.3,\,.4,\,.5,.6,.7,\,.8,.9\}$$\end{document} , using 1000 samples per $(J,N,\ell )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(J,N,\, \ell )$$\end{document} cell. For each combination of J and N, we fitted the power of the CARP method found in the simulation by a quadratic regression on $\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} . From the estimated regression coefficients, we computed the value of $\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} for which the quadratic curve has its maximum. Table 1 shows that for small samples ( $N=500)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=500)$$\end{document} , the estimated optimum was close to $\ell =.5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell =.5$$\end{document} , and for large samples the optimum was rather $\ell =.2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell =.2$$\end{document} or $\ell =.3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell =.3$$\end{document} .

Table 1. Estimated optimum values of training proportion $\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} for varying sample size N and test length J.

7.1. Aggregation of CARP Tests across Item Pairs

We discuss how one can combine tests of multiple item pairs without prior selection. If one applies a CARP test to all item pairs of a scale, this produces a sequence of $J(J-1)/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J(J-1)/2$$\end{document} p-values. To keep the family-wise Type I error rate (FWER) under control, a multiple testing correction may be in order. Alternatively, one could choose to control the false discovery rate (FDR), which generally leads to tests with higher power (Benjamini & Hochberg, Reference Benjamini and Hochberg1995; Benjamini & Yekutieli, Reference Benjamini and Yekutieli2001). However, applying either correction method to all $J(J-1)/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J(J-1)/2$$\end{document} p-values results in unnecessary loss of power if many of the involved covariances are positive. Instead, one may consider only the item pairs that have a negative conditional covariance in the training sample and test their conditional covariances in the test sample. We collect the set of item pairs $\left(i,,,j\right),i>j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( i,j \right) ,i>j$$\end{document} with a negative conditional covariance in the training sample in set $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document} , and let $S:=\left(S\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:=\left| \mathcal {S} \right| $$\end{document} denote its size. Since this statistic is independent of the data in the test sample, one may apply the Bonferroni correction with this number; that is, reject the null hypothesis for pair (i, j) iff $(i,j)\in S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(i,j)\in \mathcal {S}$$\end{document} and $p_{\mathrm{ij}}\le \alpha /S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}\le \alpha /S$$\end{document} . To check that this correction with S controls the FWER, assume that under the null hypothesis the distribution of each $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} dominates the Uniform(0, 1) distribution in the sense that $P\left(p_{\mathrm{ij}},\le ,x\right)\le x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}\left( p_{ij}\le x \right) \le x$$\end{document} for all $x\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \mathbb {R}$$\end{document} where $P$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}$$\end{document} denotes the probability measure of the $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} . This condition is called supra-uniformity by Ellis et al. (2020). The condition is satisfied if $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} dominates the Uniform(0, 1) distribution in likelihood ratio order (Whitt, Reference Whitt1980), and it is equivalent to $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} having a density that is increasing on the interval (0, 1). This is true if the test is conducted with $p_{\mathrm{ij}}={\Phi }^{-1}\left(Z_{\mathrm{ij}}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}={\Phi }^{-1}\left( Z_{ij} \right) $$\end{document} with $Z_{\mathrm{ij}}\sim N({\mu }_{\mathrm{ij}},1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{ij}\sim \mathcal {N}(\mu _{ij},1)$$\end{document} and ${\mu }_{\mathrm{ij}}\ge 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{ij}\ge 0$$\end{document} . Let $R_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{ij}$$\end{document} denote the event that the null hypothesis is rejected for pair (i, j), then, for a fixed pair (i, j) we have $P\left(R_{\mathrm{ij}},|(i,j)\in S\right)\le \alpha /S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}\left( \mathcal {R}_{ij} \vert {(i,j)\in \mathcal {S}}\right) \le \alpha /S$$\end{document} and $P\left(R_{\mathrm{ij}},|(i,j)\notin S\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}\left( \mathcal {R}_{ij} \vert {{\, \, \, }(i,j)\notin \mathcal {S}}\right) =0$$\end{document} ; therefore, the FWER is