2.3.1 Relationship between the iota index and the odds ratio

The $\iota $ index reveals a fundamental connection to the well-established OR, providing a natural bridge between symmetric and asymmetric measures of association. The classical OR for a $2\times 2$ contingency table is defined as

$$\begin{align*}\text{OR}=\frac{\pi_{ab}\cdot\pi_{\bar{a}\bar{b}}}{\pi_{a\bar{b}}\cdot\pi_{\bar{a}b}}. \end{align*}$$

The iota index can be expressed as a directional modification of this OR:

$$\begin{align*}\iota_{A\rightarrow B}=\frac{\pi_{ab}\cdot\pi_{\bar{a}\bar{b}}}{\pi_{a\bar{b}}^{2}}=\text{OR}\cdot\frac{\pi_{\bar{a}b}}{\pi_{a\bar{b}}}. \end{align*}$$

Similarly, the reverse direction yields

$$\begin{align*}\iota_{B\rightarrow A}=\frac{\pi_{ab}\cdot\pi_{\bar{a}\bar{b}}}{\pi_{\bar{a}b}^{2}}=\text{OR}\cdot\frac{\pi_{a\bar{b}}}{\pi_{\bar{a}b}}. \end{align*}$$

This decomposition reveals that the $\iota $ index extends the OR by incorporating the discordance ratio $\pi _{\bar {a}b}/\pi _{a\bar {b}}$ that specifically weights the relative prevalence of the two types of discordant observations. While the OR treats both discordant cells symmetrically, the iota index assigns differential importance based on the hypothesized direction of implication. This relationship between $\iota _{A\rightarrow B}$ and OR provides immediate interpretative value:

• • When $\iota _{A\rightarrow B}>\text {OR}$ : The directional correction factor exceeds unity, indicating that counterexamples to $B\rightarrow A$ (i.e., $\pi _{\bar {a}b}$ ) are more prevalent than counterexamples to $A\rightarrow B$ (i.e., $\pi _{a\bar {b}}$ ), supporting the $A\rightarrow B$ direction.

• • When $\iota _{A\rightarrow B}<\text {OR}$ : The correction factor is less than unity, suggesting evidence against the $A\rightarrow B$ direction.

• • When $\iota _{A\rightarrow B}\approx \text {OR}$ : The correction factor is close to unity, indicating that there is no clear directional preference in the association.

2.3.2 Reconstitution of the odds ratio

A remarkable property emerges when considering both directional indices simultaneously:

$$\begin{align*}\iota_{A\rightarrow B}\cdot\iota_{B\rightarrow A}=\text{OR}\cdot\frac{\pi_{\bar{a}b}}{\pi_{a\bar{b}}}\cdot\text{OR}\cdot\frac{\pi_{a\bar{b}}}{\pi_{\bar{a}b}}=\text{OR}^{2}. \end{align*}$$

Therefore,

$$\begin{align*}\sqrt{\iota_{A\rightarrow B}\cdot\iota_{B\rightarrow A}}=\text{OR}. \end{align*}$$

This elegant result demonstrates that the OR represents the geometric mean of the two directional iota indices. The symmetric association captured by the OR can thus be perfectly decomposed into two asymmetric components whose product preserves the original association strength. This shows that the well-known OR is a purely symmetric measure, while the $\iota $ index provides a decomposition into symmetric and asymmetric components. In psychometrics and cognitive diagnosis, where directionality is central, this decomposition offers an interpretable framework for distinguishing cumulative from reciprocal dependencies.

2.3.3 Properties on the logarithmic scale

The logarithmic transformation also yields interpretable relationships. Define the asymmetry parameter

$$\begin{align*}\Delta=\log\left(\frac{\pi_{\bar{a}b}}{\pi_{a\bar{b}}}\right). \end{align*}$$

On the log scale,

$$ \begin{align*} \iota_{A\rightarrow B}^{\ast} & = \log(\mathrm{OR})+\Delta,\\ \iota_{B\rightarrow A}^{\ast} & = \log(\mathrm{OR})-\Delta. \end{align*} $$

A noteworthy feature follows from the defining expression $\iota ^{\ast}_{A\rightarrow B}=\log \pi _{ab}+\log \pi _{\bar a\bar b}-2\log \pi _{a\bar b}$ : $\iota ^{\ast}$ involves only three of the four cells of the $2\times 2$ table—the discordant cell $\pi _{\bar a b}$ enters $\Delta $ and $\log (\mathrm {OR})$ with opposite signs and cancels. This asymmetry in the roles of A and B is the formal counterpart of the directional-rule semantics: the rule “ $A \Rightarrow B$ ” is evaluated within the $A=1$ row and along its contrapositive $\bar B \Rightarrow \bar A$ , not via the marginal frequency of B.

These expressions lead to two fundamental properties:

1. 1. Sum property: The sum of the directional log-indices equals twice the log-OR:

   (3) $$ \begin{align} \iota_{A\rightarrow B}^{\ast}+\iota_{B\rightarrow A}^{\ast}=2\log(\mathrm{OR}). \end{align} $$

2. 2. Difference property: The difference isolates pure directionality:

   (4) $$ \begin{align} \iota_{A\rightarrow B}^{\ast}-\iota_{B\rightarrow A}^{\ast}=2\Delta. \end{align} $$

2.4.1 Decomposition into association and asymmetry

The decompositions

$$\begin{align*}\iota_{A\rightarrow B} = \mathrm{OR}_{AB}\times \frac{\pi_{\bar a b}}{\pi_{a\bar b}}, \end{align*}$$

or, on the log scale,

$$\begin{align*}\iota^{\ast}_{A\rightarrow B} = \log(\mathrm{OR}_{AB}) + \log\left(\frac{\pi_{\bar{a}b}}{\pi_{a\bar{b}}}\right), \end{align*}$$

make explicit that $\iota $ combines two conceptually distinct sources of information: (i) the presence of statistical association between A and B, captured by the OR and (ii) directional imbalance among discordant observations, captured by the ratio $\pi _{\bar a b}/\pi _{a\bar b}$ .

Note that, in the case of independence, the OR is one, and the $\iota $ index for $A \rightarrow B$ is

$$\begin{align*}\iota_{A\rightarrow B}=\text{OR}\cdot\frac{\pi_{\bar{a}b}}{\pi_{a\bar{b}}}=\frac{\pi_b / \pi_{\bar{b}}}{\pi_a / \pi_{\bar{a}}}, \end{align*}$$

which depends only on the marginal distributions of A and B. In this situation, $\iota $ no longer reflects a relational property between the two variables, but merely encodes marginal heterogeneity. Consequently, directional imbalance in the absence of association is not substantively interpretable as evidence of directed dependence.

This observation motivates an inferential strategy in which the existence of an association link is treated as a prerequisite for the interpretation of directional asymmetry. Such a strategy mirrors standard practice in other areas of statistics, such as contrast testing in ANOVA, where contrasts are typically interpreted only conditional on a statistically significant omnibus effect. Analogously, in the present framework, directional asymmetry is interpreted only when there is evidence of association, as indicated by a non-unit OR.

These two steps, and their integration into a unified testing procedure, are described in the three following sections.

2.4.2 Test of association (odds ratio)

For an unordered pair $(A,B)$ of events, we first test the null hypothesis of no positive association:

$$\begin{align*}H^{\text{assoc}}_{0}:\ \log(\mathrm{OR}_{AB}) \leq 0 \quad \text{versus} \quad H^{\text{assoc}}_{1}:\ \log(\mathrm{OR}_{AB})> 0, \end{align*}$$

using a one-sided Wald test for the log-OR computed from the $2\times 2$ contingency table. A one-sided test is appropriate here because implication relationships require positive association: whatever the direction of implication ( $A \to B$ or $B \to A$ ), success at one item should make success at the other more likely, implying $\mathrm {OR}> 1$ . Negative associations would indicate a qualitatively different relationship (e.g., mutual exclusivity) that falls outside the scope of implication analysis.

Define the OR parameter

$$\begin{align*}\mathrm{OR}=\frac{\mu_{ab}\,\mu_{\bar{a}\bar{b}}}{\mu_{a\bar{b}}\,\mu_{\bar{a}b}}, \qquad \log(\mathrm{OR})=\log\mu_{ab}+\log\mu_{\bar{a}\bar{b}}-\log\mu_{a\bar{b}}-\log\mu_{\bar{a}b}. \end{align*}$$

A natural plug-in estimator is

$$\begin{align*}\log(\widehat{\mathrm{OR}}) =\log n_{ab}+\log n_{\bar{a}\bar{b}}-\log n_{a\bar{b}}-\log n_{\bar{a}b}. \end{align*}$$

Under an independent Poisson sampling scheme, using $\operatorname {Var}(\log n_{ij})= 1/\mu _{ij}$ and $\hat {\mu }_{ij}= n_{ij}$ , its asymptotic variance is obtained as

$$\begin{align*}\operatorname{Var}\left[\log(\widehat{\mathrm{OR}})\right] = \frac{1}{n_{ab}}+\frac{1}{n_{\bar{a}\bar{b}}}+\frac{1}{n_{a\bar{b}}}+\frac{1}{n_{\bar{a}b}}. \end{align*}$$

Test $H^{\text {assoc}}_{0}$ with

$$\begin{align*}Z_{\text{OR}}= \frac{\log(\widehat{\mathrm{OR}})} {\sqrt{\frac{1}{n_{ab}}+\frac{1}{n_{\bar{a}\bar{b}}}+\frac{1}{n_{a\bar{b}}}+\frac{1}{n_{\bar{a}b}}}}\sim N(0,1), \end{align*}$$

and $p^{\text {assoc}}=1-\Phi (Z_{\text {OR}})$ the corresponding one-sided p-value.

2.4.3 Test of asymmetry among discordant observations

Conditional on association, directionality is assessed through the imbalance of discordant cells. Specifically, we test

$$\begin{align*}H^{\text{asym}}_{AB}:\ \log\left(\frac{\pi_{\bar a b}}{\pi_{a\bar b}}\right)=0, \end{align*}$$

using a two-sided Wald test based on the estimated log ratio of discordant probabilities.

This test is direction-neutral: rejection indicates the presence of asymmetric dependence, while the sign of the statistic determines the direction of implication.

Define the asymmetry parameter

$$\begin{align*}\Delta=\log\left(\frac{\mu_{\bar{a}b}}{\mu_{a\bar{b}}}\right), \qquad \widehat{\Delta}=\log n_{\bar{a}b}-\log n_{a\bar{b}}. \end{align*}$$

Assuming independent Poisson counts, its asymptotic variance is obtained as

$$\begin{align*}\operatorname{Var}(\widehat{\Delta}) = \frac{1}{n_{\bar{a}b}}+\frac{1}{n_{a\bar{b}}}. \end{align*}$$

We test $H^{\text {asym}}:\ \Delta =0$ using

$$\begin{align*}Z_{\text{asym}}= \frac{\log(n_{\bar{a}b}/n_{a\bar{b}})} {\sqrt{\frac{1}{n_{\bar{a}b}}+\frac{1}{n_{a\bar{b}}}}}\sim N(0,1), \end{align*}$$

and $p^{\text {asym}}=2\bigl (1-\Phi (|Z_{\text {asym}}|)\bigr )$ the corresponding two-sided p-value.

Direction, if any, is then given by $\operatorname {sign}(\widehat {\Delta })$ , with $\widehat {\Delta }>0$ (or $n_{\bar {a}b}>n_{a\bar {b}}$ ) suggesting an $A\rightarrow B$ direction, and $\widehat {\Delta }<0$ (or $n_{\bar {a}b}<n_{a\bar {b}}$ ) suggesting a $B\rightarrow A$ direction.

2.4.4 Joint inference for directed dependence

To formalize the requirement that both positive association and asymmetry be present for a directed relation to be interpretable, we define the composite null hypothesis

$$\begin{align*}H^{\text{dir}}_{AB}:\ \log(\mathrm{OR}_{AB}) \leq 0 \ \text{or}\ \log\left(\frac{\pi_{\bar a b}}{\pi_{a\bar b}}\right)=0, \end{align*}$$

that is, either no positive association or no asymmetry.

The alternative hypothesis corresponds to the simultaneous presence of positive association and asymmetry. A conservative p-value for this intersection alternative is given by

(5) $$ \begin{align} p^{\text{dir}}_{AB}=\max\left(p^{\text{assoc}}_{AB},\ p^{\text{asym}}_{AB}\right), \end{align} $$

which is valid by standard union–intersection testing arguments: the null is rejected only if both component tests provide evidence against their respective null hypotheses.

2.4.5 Illustration

As an illustration, we consider the data reported by Lecoutre and Charron (Reference Lecoutre and Charron2000) on the development of number knowledge among $N = 165$ children (Table 2). The study focuses on fraction calculation and interpretation tasks of two types: (i) part-to-part items (e.g., “there is a 3-to-4 ratio of oranges to apples in this basket of 28 fruits”) and (ii) part-to-whole items (e.g., “3 out of 6 slices have been eaten”). In the literature on numerical cognition, part-to-part fractions are generally considered more difficult than part-to-whole fractions.

Table 2

Lecoutre and Charron’s (Reference Lecoutre and Charron2000) data on fraction mastery

Table 2 Long description

The table cross-tabulates two variables: Part-to-part mastery (rows) and Part-to-whole mastery (columns).

Column Headers:

* Column 1: Empty top-left cell.

* Column 2: Part-to-whole success (b).

* Column 3: Part-to-whole failure (b-bar).

* Column 4: Sums.

Row 1 (Part-to-part success (a)):

* Success in both (n sub a b): 36.

* Part-to-part success but part-to-whole failure (n sub a b-bar): 3.

* Total part-to-part success (n sub a): 39.

Row 2 (Part-to-part failure (a-bar)):

* Part-to-part failure but part-to-whole success (n sub a-bar b): 36.

* Failure in both (n sub a-bar b-bar): 90.

* Total part-to-part failure (n sub a-bar): 126.

Row 3 (Sums):

* Total part-to-whole success (n sub b): 72.

* Total part-to-whole failure (n sub b-bar): 93.

* Grand total (N): 165.

Let A denote success on the part-to-part item and B success on the part-to-whole item. The joint response counts are given in Table 2.

The OR is estimated as

$$\begin{align*}\widehat{\mathrm{OR}}_{AB} =\frac{n_{ab}\,n_{\bar a\bar b}}{n_{a\bar b}\,n_{\bar a b}} =\frac{36\times 90}{3\times 36} =30, \end{align*}$$

yielding $\log (\widehat {\mathrm {OR}}_{AB})=\log (30)\approx 3.40$ .

Under the independent-Poisson sampling scheme, the estimated variance of $\log (\widehat {\mathrm {OR}}_{AB})$ is $1/36+1/90+1/3+1/36=0.40$ , so that the Wald statistic is $Z_{\mathrm {OR}}=3.40/\sqrt {0.40}\approx 5.38$ , corresponding to a one-sided p-value $p^{\mathrm {assoc}}_{AB}\approx 3.7\times 10^{-8}$ . This result provides strong evidence of positive association between the two items.

Directional asymmetry is assessed through the imbalance of discordant outcomes. The estimated asymmetry parameter is $\widehat {\Delta }_{AB}=\log (36/3)=\log (12)\approx 2.48$ , with estimated variance $1/36+1/3\approx 0.36$ , leading to the Wald statistic $Z_{\mathrm {asym}}=2.48/\sqrt {0.36}\approx 4.14$ and a two-sided p-value $p^{\mathrm {asym}}_{AB}\approx 3.5\times 10^{-5}$ .

The overall implication index is then obtained as the sum of both components: $\widehat {\iota }^{\ast}_{A\rightarrow B} = \log (\widehat {\mathrm {OR}}_{AB}) + \widehat {\Delta }_{AB} = 3.40 + 2.48 = 5.88$ .

Following the inferential strategy proposed above, directional dependence is assessed through the composite null hypothesis $H^{\mathrm {dir}}_{AB}:\ \log (\mathrm {OR}_{AB}) \leq 0\ \text {or}\ \log (\pi _{\bar a b}/\pi _{a\bar b})=0$ , with a conservative p-value given by $p^{\mathrm {dir}}_{AB}=\max (p^{\mathrm {assoc}}_{AB},\,p^{\mathrm {asym}}_{AB})\approx 3.5\times 10^{-5}$ .

The sign of $\widehat {\Delta }_{AB}>0$ indicates that discordant responses are dominated by the pattern $(\bar a,b)$ , corresponding to success on the part-to-whole item without success on the part-to-part item. Equivalently, success on the more difficult part-to-part task is almost always accompanied by success on the easier part-to-whole task.

Taken together, these results indicate a strong asymmetric dependence between the two items, consistent with a cumulative structure in fraction knowledge. Importantly, this conclusion is drawn only after establishing the presence of association between the tasks, thereby avoiding the conflation of directional imbalance with marginal difficulty effects alone.

The decomposition into an asymmetry and an association component suggests a natural two-dimensional representation for pairs of binary variables. By plotting the directional asymmetry and $\log (\widehat {\mathrm {OR}})$ estimates on the first and second axis, respectively, one obtains an immediate visual representation where the Euclidean distance from the origin relates to the overall dependency strength, but: (i) points on the x-axis indicate directional preference and (ii) vertical displacement from the x-axis represents purely symmetric associations (OR). This is illustrated in Figure 1 for Lecoutre and Charron’s (Reference Lecoutre and Charron2000) data discussed above. This representation provides an intuitive framework for exploring asymmetric relationships, where the familiar concept of association strength (through the OR) is augmented with directional information.

Figure 1

Association–directionality plot for Lecoutre and Charron’s (Reference Lecoutre and Charron2000) data on part-to-part and part-to-whole fractions. Implication appears to be in the upper right quadrant, that is, in the part-to-part $\rightarrow $ part-to-whole ( $A\rightarrow B$ ) direction. The shaded bands indicate non-significance regions for association (horizontal, $|Z_{\mathrm {OR}}| < z_{\alpha }$ ) and directionality (vertical, $|Z_{\mathrm {asym}}| < z_{\alpha /2}$ ), computed from the asymptotic standard errors of $\log (\widehat {\mathrm {OR}})$ and $\widehat {\Delta }$ , respectively.

2.4.6 Conditional directed dependence (adjustment for observed covariates)

The inferential framework described above can be extended in a straightforward manner to account for observed covariates. Let $C\in \{0,1\}$ denote a binary covariate and consider the conditional joint distributions of A and B within strata $C=c$ .

For each stratum c, we define the conditional implication index

(6) $$ \begin{align} \iota_{A\rightarrow B\mid C=c} = \frac{\pi_{ab\mid c}\,\pi_{\bar a\bar b\mid c}}{\pi_{a\bar b\mid c}^{2}}, \end{align} $$

which admits the same decomposition as in the marginal case,

$$\begin{align*}\iota_{A\rightarrow B\mid c} = \mathrm{OR}_{AB\mid c} \times \frac{\pi_{\bar a b\mid c}}{\pi_{a\bar b\mid c}}, \end{align*}$$

or, on the log scale,

$$\begin{align*}\iota^{\ast}_{A\rightarrow B\mid c} = \log(\mathrm{OR}_{AB\mid c}) + \log\!\left(\frac{\pi_{\bar a b\mid c}}{\pi_{a\bar b\mid c}}\right). \end{align*}$$

Inference is conducted on both components within each stratum using the same two-step strategy as in the marginal case: a one-sided Wald test for positive association, followed by a two-sided Wald test for asymmetry. The corresponding pooled formula and inferential procedure are described in Appendix A.

3.3.1 The expected structure

In the case of items drawn from two Rasch scales with correlated traits, the joint probabilities for a given respondent and the pair $(\theta _{1},\ \theta _{2})$ of corresponding aptitudes are given by

$$\begin{align*}\begin{array}{llll} \pi_{ab} & =\frac{\exp(\theta_{1}-\delta_{1})\exp(\theta_{2}-\delta_{2})}{\gamma_{12}} & & \pi_{a\bar{b}}=\frac{\exp(\theta_{1}-\delta_{1})}{\gamma_{12}}\\ \pi_{\bar{a}b} & =\frac{\exp(\theta_{2}-\delta_{2})}{\gamma_{12}} & & \pi_{\bar{a}\bar{b}}=\frac{1}{\gamma_{12}} \end{array} \end{align*}$$

with $\gamma _{12}=\left [1+\exp (\theta _{1}-\delta _{1})\right ]\left [1+\exp (\theta _{2}-\delta _{2})\right ]$ the normalizing constant.

The conditional implication intensity, for a given pair $(\theta _{1},\theta _{2})$ , is

$$ \begin{align*} \iota_{X_{1}\rightarrow X_{2}|\theta_{1},\theta_{2}} & =\exp(\theta_{2}-\delta_{2})\exp\left[-(\theta_{1}-\delta_{1})\right] \\ & =\exp(\theta_{2}-\theta_{1})\exp(\delta_{1}-\delta_{2}). \end{align*} $$

Under the common assumption of Gaussian distributions for person parameters, that is, $\theta _{1}\sim N(\mu _{1},\sigma _{1}^{2})$ and $\theta _{2}\sim N(\mu _{2},\sigma _{2}^{2})$ , we have $\theta _{2}-\theta _{1}\sim N(\mu _{2}-\mu _{1},\sigma _{1}^{2}+\sigma _{2}^{2}-2\sigma _{12})$ , where $\sigma _{12}$ denotes the covariance between both latent scores. The variable $Z=\exp (\theta _{2}-\theta _{1})$ is by definition log-normally distributed with parameters $\mu _{2}-\mu _{1}$ and $\frac {\sigma _{1}^{2}+\sigma _{2}^{2}-2\sigma _{12}}{2}$ . From the properties of the lognormal, the corresponding expectation is $E(Z)=\exp \left (\mu _{2}-\mu _{1}+\tfrac {\sigma _{1}^{2}+\sigma _{2}^{2}}{2}-\sigma _{12}\right )$ .

Finally, we have, marginally,

$$ \begin{align*} \iota_{X_{1}\rightarrow X_{2}} & =\exp(\delta_{1}-\delta_{2})\int Zf(Z)dZ \\ & =\exp\left[(\delta_{1}-\delta_{2})+(\mu_{2}-\mu_{1})+\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2}-\sigma_{12}\right], \end{align*} $$

and on the log-scale,

(11) $$ \begin{align} \iota_{X_{1}\rightarrow X_{2}}^{\ast} & =\delta_{1}-\delta_{2}+(\mu_{2}-\mu_{1})+\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2}-\sigma_{12}. \end{align} $$

This last expression helps clarify that, in the presence of two Rasch scales, the $\iota ^*$ index decomposes into two components: a directional component, reflecting item and scale shifts, and a symmetric component that reflects the latent covariance structure.

Both components can be separated. Writing the symmetric counterpart of (11) gives

$$\begin{align*}\iota_{X_{2}\rightarrow X_{1}}^{\ast}=\delta_{2}-\delta_{1}+(\mu_{1}-\mu_{2})+\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2}-\sigma_{12}. \end{align*}$$

Analogously to Equations (3) and (4), (half) summing and differencing both expressions then yield the pure association and directionality quantities:

$$ \begin{align*} A & =\frac{1}{2}\left(\iota_{X_{1}\rightarrow X_{2}}^{\ast}+\iota_{X_{2}\rightarrow X_{1}}^{\ast}\right)=\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2}-\sigma_{12}, \\ D & =\frac{1}{2}\left(\iota_{X_{1}\rightarrow X_{2}}^{\ast}-\iota_{X_{2}\rightarrow X_{1}}^{\ast}\right)=(\delta_{1}-\delta_{2})+(\mu_{2}-\mu_{1}). \end{align*} $$

We note that only the A component depends upon the latent ability.

3.3.2 Extraction from empirical data

From Equations (3) and (4), the half-sum and half-difference within a pair of empirical $\iota ^*$ are interpretable as a difficulty difference ( $\hat {\Delta }$ ) and an association strength ( $\log (\widehat {\mathrm {OR}})$ ). Given the marginal nature of observed counts, some attenuation is expected on the association component, which depends on the latent ability variance. It is shown in Appendix B that, under the Rasch model, the expected marginal log-OR for any item pair within a scale is approximately constant, and given by the special case of Formula (B.1) with $a_i=a_j=1$ :

(12) $$ \begin{align} \mathbb{E}[\log(\mathrm{OR})_{\mathrm{within}}] \approx \frac{(\pi^2/3)\,\sigma^2}{\sigma^2 + \pi^2/3}, \end{align} $$

where $\sigma ^2$ is the latent ability variance. In the multiple Rasch scale case, the expected marginal log-OR also depends upon the latent correlation between the scales and may be approximated by (Formula (B.2))

(13) $$ \begin{align} \mathbb{E}[\log(\mathrm{OR})_{\mathrm{between}}] \approx \frac{(\pi^2/3)\;\rho\,\sigma_1\sigma_2}{\sqrt{(\sigma_1^2 + \pi^2/3)(\sigma_2^2 + \pi^2/3)}}. \end{align} $$

Figure 3 illustrates the quality of these approximations, in a $(\hat {\Delta },\log (\widehat {\mathrm {OR}}))$ representation of a simulated dataset that mixes two 8-item Rasch scales, equally spaced in $[-2;2]$ , with two levels of latent correlation ( $\rho =0$ and $\rho =0.5$ ). Within-scale pairs—which share a common latent trait—exhibit elevated $\log (\widehat {\mathrm {OR}})$ due to the positive marginal association induced by marginalization over $\theta $ . In contrast, cross-scale pairs under independence show near-zero $\log (\widehat {\mathrm {OR}})$ (see gray triangles aligned along the x-axis in panels (a) and (c)). But these cross-scale association values increase with trait correlation, reducing the difference with the within log-OR values (panels (b) and (d)) and thereby degrading separability. Expected within- and between-scale log-OR levels, computed from these approximations, are plotted as horizontal gray lines on the figure.

Figure 3

Directionality ( $\hat {\Delta }$ ) versus Association ( $\log (\widehat {\mathrm {OR}})$ ) for 120 item pairs from two simulated 8-item Rasch scales, varying latent correlation ( $\rho =0,0.5$ ) and variance ( $\sigma _1=1,2$ ). The constant vertical offset between within-scale and between-scale pairs reflects the difference in marginal association (augmented by variance and diminished by correlation). The approximately constant expected marginal within- and between-scale log-OR levels (obtained from the within- and between-scale approximations of Appendix B) are plotted as gray horizontal lines.

Figure 3 Long description

A four-panel scatter plot grid labeled a through d. Each panel has a horizontal x-axis labeled with the symbol phi ranging from negative 4 to 4 and a vertical y-axis labeled log O R ranging from negative 0.5 to 2.5. Three data series are present in each panel: Within scale 1 represented by blue circles, Within scale 2 represented by red squares, and Between scales represented by gray triangles.

* Panel a: sigma sub 1 equals 1, sigma sub 2 equals 1, rho equals 0. Blue circles and red squares are clustered together between log O R 0.5 and 1.0. Gray triangles are clustered lower, around log O R 0.0.

* Panel b: sigma sub 1 equals 1, sigma sub 2 equals 1, rho equals 0.5. The blue and red clusters remain between 0.5 and 1.0, but the gray triangles shift upward toward log O R 0.5, narrowing the gap between series.

* Panel c: sigma sub 1 equals 2, sigma sub 2 equals 1, rho equals 0. Blue circles shift significantly upward to cluster around log O R 1.8. Red squares remain around 0.7, and gray triangles remain around 0.0.

* Panel d: sigma sub 1 equals 2, sigma sub 2 equals 1, rho equals 0.5. Blue circles remain high at 1.8. Red squares and gray triangles both shift slightly upward, with gray triangles now overlapping the lower range of the red squares around log O R 0.6.

In all panels, gray dashed horizontal lines indicate the expected marginal levels for each scale grouping.

Figure 4

Individual item layout reconstructed from separate unidimensional MDS analyses i) on the $|\hat {\Delta }|$ matrix (x-axis) and ii) on the $C-\log (\widehat {\mathrm {OR}})$ matrix (y-axis), for the simulated data of Figure 3. Horizontal dashed lines illustrate a separation obtained through k-means clustering. Scale separability increases with difference in latent variance and decreases with latent correlation. In the absence of latent correlation (panels a and c), no between-scale links appear; in the correlated case (panels b and d), undirected between-scale links (black) tend to connect items of similar difficulty across the two scales, while directed links (dark gray) arise when a difficulty difference is also present.

Figure 4 Long description

The four panels are arranged in a two-by-two grid, labeled a through d. Each panel shares the same axes: the x-axis represents Difficulty ranging from negative 2 to 2, and the y-axis represents Associative locations ranging from negative 2 to 2.

* Panel a (top-left): Sigma sub 1 equals 1, sigma sub 2 equals 1, rho equals 0. Items i 1 through i 8 form a horizontal cluster at y equals 0.5, connected by light blue curved links. Items i 9 through i 16 form a horizontal cluster at y equals negative 0.5, connected by dark red curved links. No links exist between the two clusters.

* Panel b (top-right): Sigma sub 1 equals 1, sigma sub 2 equals 1, rho equals 0.5. The item clusters are positioned similarly to panel a. However, black symmetric links and dark gray directed links now connect items between the top and bottom clusters, primarily between items of similar difficulty.

* Panel c (bottom-left): Sigma sub 1 equals 2, sigma sub 2 equals 1, rho equals 0. The top cluster (i 1 to i 8) is shifted upward to y equals 1.2, while the bottom cluster (i 9 to i 16) is shifted downward to y equals negative 1.2. There are no links between the clusters.

* Panel d (bottom-right): Sigma sub 1 equals 2, sigma sub 2 equals 1, rho equals 0.5. The clusters are vertically separated as in panel c. Dense networks of dark gray directed links connect the two clusters, showing a high degree of cross-scale interaction despite the increased vertical distance.

A legend in the top-right of panel b identifies the link types: light blue for Within scale A, dark red for Within scale B, black for Between symmetric, and dark gray for Between directed.

For graph plotting, an optimal item layout may be obtained by extracting individual item locations from these two sets of item pair relationships. This is obtained through two separate unidimensional MDS on the properly transformed $\hat {\Delta }$ and $\log (\widehat {\mathrm {OR}})$ matrices. On the $\hat {\Delta }$ component, since $|\hat {\Delta }_{ij}|$ represents the magnitude of the difficulty difference regardless of direction, it has a natural distance interpretation; we therefore take absolute values. From the results in the previous section, we expect a unidimensional MDS on this matrix to recover the true latent Rasch difficulties with good precision, and this is observed in practice. On the second component, the symmetric $\log (\widehat {\mathrm {OR}})$ association matrix can be transformed into distances using any monotone decreasing function of the form $d_{ij}=C-\log (\widehat {\mathrm {OR}}_{ij})$ , for any $(i,j)$ item pair. We use $C=\max \bigl (\log (\widehat {\mathrm {OR}}_{ij})\bigr )$ in practice. This matrix can also be submitted to a one-dimensional MDS, to obtain “associative coordinates” at the item level, where similar association levels will translate into close values on the y-axis.

From the properties discussed above (within-scale constant association, and between-scale lower association), we expect items to form separate horizontal alignments if they belong to different Rasch scales, with a vertical offset determined by the difference in latent variances and moderated by scale correlation. The offset becomes larger as difference in latent variance increases, but smaller as scale correlation increases. Separability is, therefore, expected to be better in the independence and/or heterogeneity of variance cases. The inflating effect of variance heterogeneity, the moderating effect of scale correlation on the vertical offset, and the alignment property are all clearly visible in Figure 4 for the simulated data of Figure 3. Separability is easily achieved using a clustering analysis on the y-axis coordinates. The result of a k-means algorithm is plotted as horizontal lines on the figure for illustration.

In the scale independence case (panel (a) of Figure 4), directed arrows (in dark gray) appear only within each scale, following the difficulty ordering, and no between-scale links appear at all. In the correlated case (panels (b) and (d), $\rho = 0.5$ ), between-scale links do appear, since the shared latent covariance raises the cross-scale log-OR above zero, passing the first gate of the association test. Two types of between-scale links are visible: (i) undirected links (solid black), which tend to connect items at similar difficulty levels across the two scales—items that are vertically aligned in the plot—reflecting a purely associative relationship with no net asymmetry and (ii) directed arrows (dark gray), which appear when a non-negligible difficulty difference exists between a pair of cross-scale items, so that the asymmetric component is also significant.