3.1 The polychoric model

For ease of exposition, we restrict our presentation to the bivariate polychoric model. The model naturally generalizes to higher dimensions (see, e.g., Muthén (Reference Muthén1984)).

Let there be two ordinal random variables, X and Y, that take values in the sets $\mathcal {X} = \{1,2,\dots ,K_X\}$ and $\mathcal {Y}=\{1,2,\dots ,K_Y\}$ , respectively. The assumption that the sets contain adjacent integers is without loss of generality. Suppose there exist two continuous latent random variables, $\xi $ and $\eta $ , that govern the ordinal variables through the discretization model

(3.1) $$ \begin{align} X= \begin{cases} 1 &\text{if } \xi < a_1,\\ 2 &\text{if } a_1 \leq \xi < a_2,\\ 3 &\text{if } a_2 \leq \xi < a_3,\\ \vdots\\ K_X &\text{if } a_{K_X-1} \leq \xi, \\ \end{cases} \qquad \text{ and } \qquad Y= \begin{cases} 1 &\text{if } \eta < b_1,\\ 2 &\text{if } b_1 \leq \eta < b_2,\\ 3 &\text{if } b_2 \leq \eta < b_3,\\ \vdots\\ K_Y &\text{if } b_{K_Y-1} \leq \eta, \\ \end{cases} \end{align} $$

where the fixed but unobserved parameters $a_1 < a_2 < \dots < a_{K_X-1}$ and $b_1 < b_2 < \dots < b_{K_Y-1}$ are called thresholds.

The primary object of interest is the population correlation between the two latent variables. To identify this quantity from the ordinal variables $(X,Y)$ , one assumes that the continuous latent variables follow a standard bivariate normal distribution with unobserved pairwise correlation coefficient $\rho \in (-1,1)$ , that is,

(3.2) $$ \begin{align} \begin{pmatrix} \xi \\ \eta \end{pmatrix} \sim \text{N}_2 \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \right). \end{align} $$

Combining the discretization model (3.1) with the latent normality model (3.2) yields the polychoric model. In this model, one refers to the correlation parameter $\rho = \mathbb {C}\mathrm {or} \left [\xi ,\ \eta \right ]$ as the polychoric correlation coefficient of the ordinal X and Y. The polychoric model is subject to $d=K_X+K_Y-1$ parameters, namely, the polychoric correlation coefficient from the latent normality model (3.2) and the two sets of thresholds from the discretization model (3.1). These parameters are jointly collected in a d-dimensional vector

$$\begin{align*}\boldsymbol{\theta} = \left(\rho, a_1, a_2,\dots, a_{K_X-1}, b_1, b_2, \dots, b_{K_Y-1}\right)^\top. \end{align*}$$

Under the polychoric model, the probability of observing an ordinal response $(x,y)\in \mathcal {X}\times \mathcal {Y}$ at a parameter vector $\boldsymbol {\theta }$ is given by

(3.3) $$ \begin{align} p_{xy}(\boldsymbol{\theta}) = \mathbb{P}_{\boldsymbol{\theta}} \left[ X = x, Y=y \right] = \int_{a_{x-1}}^{a_x}\int_{b_{y-1}}^{b_y} \phi_2\left(t,s; \rho \right)\text{d} s\ \text{d} t, \end{align} $$

where we use the conventions $a_0=b_0=-\infty , a_{K_X}=b_{K_Y}=+\infty $ , and

$$\begin{align*}\phi_2\left(u,v; \rho \right) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp\left( -\frac{u^2-2\rho uv + v^2}{2(1-\rho^2)} \right) \end{align*}$$

denotes the density of the standard bivariate normal distribution function with correlation parameter $\rho \in (-1,1)$ at some $u,v\in \mathbb {R}$ , with corresponding distribution function

$$\begin{align*}\Phi_2\left(u,v; \rho \right) = \int_{-\infty}^u\int_{-\infty}^v \phi_2\left(t,s; \rho \right)\text{d} s\ \text{d} t. \end{align*}$$

Regarding identification, it is worth mentioning that in the case where both X and Y are dichotomous, the polychoric model is exactly identified by the standard bivariate normal distribution. If at least one of the ordinal variables has more than two response categories, the polychoric model is over-identified, so it could identify more parameters than those in $\boldsymbol {\theta }$ .Footnote ³ We refer to Olsson (Reference Olsson1979, Section 2) for a related discussion.

To distinguish arbitrary parameter values $\boldsymbol {\theta }$ from a specific value under which the polychoric model generates ordinal data, denote the latter by $\boldsymbol {\theta }_{\star} = \left (\rho _{\star}, a_{\star,1},\dots , a_{\star,K_X-1}, b_{\star,1}, \dots , b_{\star,K_Y-1}\right )^\top $ . Given a random sample of ordinal data generated by a polychoric model under parameter value $\boldsymbol {\theta }_{\star}$ , the statistical problem is to estimate the true $\boldsymbol {\theta }_{\star}$ , which is traditionally achieved by the ML estimator (MLE) of Olsson (Reference Olsson1979).Footnote ⁴

3.2 Maximum likelihood estimation

Suppose we observe a sample $\{(X_i, Y_i)\}_{i=1}^N$ of N independent copies of $(X,Y)$ generated by the polychoric model under the true parameter $\boldsymbol {\theta }_{\star}$ . The sample may be observed directly or as a $K_X\times K_Y$ contingency table that cross-tabulates the observed frequencies. Denote by

the observed empirical frequency of a response $(x,y)\in \mathcal {X}\times \mathcal {Y}$ , where the indicator function takes value 1 if an event E is true, and 0 otherwise. The MLE of $\boldsymbol {\theta }_{\star}$ can be expressed as

(3.4) $$ \begin{align} \widehat{\boldsymbol{\theta}}_N^{\mathrm{\ MLE}} = \arg\max_{\boldsymbol{\theta}\in\boldsymbol{\Theta}} \left\{ \sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}} N_{xy} \log\left(p_{xy}(\boldsymbol{\theta})\right) \right\}, \end{align} $$

where the $p_{xy}(\boldsymbol {\theta })$ are the response probabilities in (3.3), and

(3.5) $$ \begin{align} \boldsymbol{\Theta} = \bigg( \Big(\rho, \big(a_i\big)_{i=1}^{K_X-1}, \big(b_j\big)_{j=1}^{K_Y-1}\Big)^\top \ \Big|\ \rho\in(-1,1),\ a_1 < \dots < a_{K_X-1},\ b_1 < \dots < b_{K_Y-1} \bigg) \end{align} $$

is a set of parameters $\boldsymbol {\theta }$ that rules out degenerate cases, such as $\rho = \pm 1$ or thresholds that are not strictly monotonically increasing. This estimator, its computational details, as well as its statistical properties are derived in Olsson (Reference Olsson1979). In essence, if the polychoric model (3.1) is correctly specified—that is, the underlying latent variables $(\xi , \eta )$ are indeed standard bivariate normal—then the estimator $\widehat {\boldsymbol {\theta }}_N^{\mathrm {\ MLE}}$ is consistent for the true $\boldsymbol {\theta }_{\star}$ . In addition, $\widehat {\boldsymbol {\theta }}_N^{\mathrm {\ MLE}}$ is asymptotically normally distributed with mean zero and covariance matrix being equal to the model’s inverse Fisher information matrix, which makes it fully efficient.

As a computationally attractive alternative to estimating all parameters in $\boldsymbol {\theta }_{\star}$ simultaneously in problem (3.4), one may consider a “2-step approach” where only the correlation coefficient $\rho _{\star}$ is estimated via ML, but not the thresholds. In this approach, one estimates in a first step the thresholds as quantiles of the univariate standard normal distribution, evaluated at the observed cumulative marginal proportion of each contingency table cell. Formally, in the 2-step approach, thresholds $a_{\star,x}$ and $b_{\star,y}$ are, respectively, estimated via

(3.6) $$ \begin{align} \widehat{a}_{x} = \Phi_1^{-1}\left( \frac{1}{N}\sum_{k=1}^x\sum_{y\in\mathcal{Y}} N_{ky} \right)\qquad\text{and}\qquad \widehat{b}_{y} = \Phi_1^{-1}\left( \frac{1}{N}\sum_{x\in\mathcal{X}}\sum_{l=1}^y N_{xl} \right), \end{align} $$

for $x=1,\dots ,K_X-1$ and $y=1,\dots ,K_Y-1$ , where $\Phi _1^{-1}(\cdot )$ denotes the quantile function of the univariate standard normal distribution. Then, taking these threshold estimates as fixed in the polychoric model, one estimates in a second step the only remaining parameter, correlation coefficient $\rho _{\star}$ , via ML. The main advantage of the 2-step approach is reduced computational time, while it comes at the cost of being theoretically non-optimal because ML standard errors do not apply to the threshold estimators in (3.6) (Olsson, Reference Olsson1979). Using simulation studies, Olsson (Reference Olsson1979) finds that the two approaches tend to yield similar results in practice—both in terms of correlation and variance estimation—for small to moderate true correlations, while there can be small differences for larger true correlations.

Software implementations of polychoric correlation vary with respect to their estimation strategy. For instance, the popular R packages lavaan (Rosseel, Reference Rosseel2012) and psych (Revelle, Reference Revelle2024) only support the 2-step approach, while the package polycor (Fox, Reference Fox2022) supports both the 2-step approach and simultaneous estimation of all model parameters, with the former being the default. Our implementation of ML estimation in package robcat also supports both strategies.

4.1 Partial misspecification of the polychoric model

The polychoric model is partially misspecified when not all unobserved realizations of the latent variables $(\xi ,\eta )$ come from a standard bivariate normal distribution. Specifically, we consider a situation where only a fraction $(1-\varepsilon )$ of those realizations are generated by a standard bivariate normal distribution with true correlation parameter $\rho _{\star}$ , whereas a fixed but unknown fraction $\varepsilon $ are generated by some different but unspecified distribution H. Note that H being unspecified allows its correlation coefficient to differ from $\rho _{\star}$ so that realizations generated by H may be uninformative for the true polychoric correlation coefficient $\rho _{\star}$ , such as, after discretization, careless responses, misresponses, or responses stemming from item misunderstanding.

Formally, we say that the polychoric model is partially misspecified if the latent variables $(\xi ,\eta )$ are jointly distributed according to

(4.1) $$ \begin{align} (u,v) \mapsto G_\varepsilon(u,v) = (1-\varepsilon) \Phi_2\left(u,v; \rho_{\star} \right) +\varepsilon H(u,v), \end{align} $$

for $u,v\in \mathbb {R}$ . Conceptualizing model misspecification in such a manner is standard in the robust statistics literature, going back to the seminal work of Huber (Reference Huber1964).Footnote ⁵ We therefore adopt terminology from robust statistics and call $\varepsilon $ the contamination fraction, the uninformative H the contamination distribution (or simply contamination), and $G_\varepsilon $ the contaminated distribution. Observe that when the contamination fraction is zero, that is, $\varepsilon = 0$ , there is no misspecification so that the polychoric model is correctly specified for all observations. However, neither the contamination fraction $\varepsilon $ nor the contamination distribution H is assumed to be known. Thus, both quantities are left completely unspecified in practice and $\Phi _2\left (u,v; \rho _{\star} \right )$ remains the distribution of interest. That is, we only aim to estimate the model parameters $\boldsymbol {\theta }$ of the polychoric model, while reducing the adverse effects of potential contamination in the observed ordinal data. The contaminated distribution $G_\varepsilon $ , on the other hand, is never estimated. It serves as a purely theoretical construct that we use to study the theoretical properties of estimators of the polychoric model when that model is partially misspecified due to contamination.

Leaving the contamination distribution H and contamination fraction $\varepsilon $ unspecified in the partial misspecification model (4.1) means that we are not making any assumptions on the degree, magnitude, or type of contamination (which is possibly absent altogether). Hence, in our context of responses to rating items, the polychoric model can be misspecified due to an unlimited variety of reasons, for instance, but not limited to careless/inattentive responding (e.g., straightlining, pattern responding, and random-like responding), misresponses, or item misunderstanding.

Although we make no assumption on the specific value of the contamination fraction $\varepsilon $ in the partial misspecification model (4.1), we require the identification restriction $\varepsilon \in [0, 0.5)$ . That is, we require that the polychoric model is correctly specified for the majority of observations, which is standard in the robust statistics literature (e.g., Hampel et al., Reference Hampel, Ronchetti, Rousseeuw and Stahel1986, p. 67). While it is in principle possible to also consider a contamination fraction between $0.5$ and $1$ , one would need to impose certain additional assumptions on the correct model to distinguish it from incorrect ones when the majority of observations are not generated by the correct model. Since we prefer refraining from imposing additional assumptions, we only consider $\varepsilon \in [0, 0.5)$ . We discuss the link between identification and contamination fractions beyond 0.5 in more detail in Section E.1 of the Supplementary Material.

Furthermore, as another, more practical reason for considering $\varepsilon \in [0,0.5)$ , having more than half of all observations in a sample being not informative for the quantity of interest would be indicative of serious data quality issues. When data quality is unreasonably low, it is doubtful whether the data are suitable for modeling analyses in the first place.

4.2 Response probabilities under partial misspecification

Under contaminated distribution $G_\varepsilon $ with contamination fraction $\varepsilon \in [0, 0.5)$ , the probability of observing an ordinal response $(x,y)\in \mathcal {X}\times \mathcal {Y}$ is given by

(4.2) $$ \begin{align} f_{\varepsilon} \left(x,y\right) = \mathbb{P}_{G_\varepsilon} \left[ X = x, Y=y \right] = (1-\varepsilon) p_{xy}(\boldsymbol{\theta}_{\star}) + \varepsilon \int_{a_{\varepsilon,x-1}}^{a_{\varepsilon,x}} \int_{b_{\varepsilon,y-1}}^{b_{\varepsilon,y}} \text{d} H, \end{align} $$

where the unobserved thresholds $a_{\varepsilon ,x},b_{\varepsilon ,y}$ discretize the fraction $\varepsilon $ of latent variables for which the polychoric model is misspecified. The thresholds $a_{\varepsilon ,x},b_{\varepsilon ,y}$ may be different from the true $a_{\star,x},b_{\star,y}$ and/or depend on contamination fraction $\varepsilon $ . However, it turns out that from a theoretical perspective, studying the case where the $a_{\varepsilon ,x},b_{\varepsilon ,y}$ are different from the $a_{\star,x},b_{\star,y}$ is equivalent to a case where they are equal.Footnote ⁶

The population response probabilities $f_{\varepsilon } \left (x,y\right )$ in (4.2) are unknown in practice because they depend on unspecified and unmodeled quantities, namely, the contamination fraction $\varepsilon $ , the contamination distribution H, and the discretization thresholds of the latter. Consequently, we do not attempt to estimate the population response probabilities $f_{\varepsilon } \left (x,y\right )$ . We instead focus on estimating the true polychoric model probabilities $p_{xy}(\boldsymbol {\theta }_{\star})$ while reducing bias stemming from potential contamination in the observed data.

Figure 1 visualizes a simulated example of bivariate data drawn from contaminated distribution $G_\varepsilon $ , where a fraction of $\varepsilon =0.15$ of the data follow a bivariate normal contamination distribution H (orange dots) with mean $(2.5, -2.5)^\top $ , variance $(0.25, 0.25)^\top $ , and zero correlation, whereas the remaining data are generated by a standard bivariate normal distribution with correlation $\rho _{\star}=0.5$ (gray dots). In this example, the data from the contamination distribution H primarily inflate the cell $(x,y) = (5,1)$ after discretization. That is, this cell will have a larger empirical frequency than the polychoric model allows for, since the probability of this cell is nearly zero at the polychoric model, yet many realized responses will populate it. Consequently, due to (partial) misspecification of the polychoric model, an ML estimate of $\rho _{\star}$ on these data might be substantially biased for $\rho _{\star}$ . Indeed, calculating the MLE using the data plotted in Figure 1 yields an estimate of $\widehat {\rho }_N^{\mathrm {\ MLE}}=-0.10$ , which is far off from the true $\rho _{\star}=0.5$ . In contrast, our proposed robust estimator, which is calculated from the exact same information as the MLE and is defined in Section 5, yields a fairly accurate estimate of $0.47$ .

Figure 1

Simulated data with $K_X=K_Y=5$ response options where the polychoric model is misspecified with contamination fraction $\varepsilon =0.15$ .

Note: The gray dots represent random draws of $(\xi ,\eta )$ from the polychoric model with $\rho _{\star}=0.5$ , whereas the orange dots represent draws from a contamination distribution that primarily inflates the cell $(x,y)=(5,1)$ . The contamination distribution is bivariate normal with a mean $(2.5,-2.5)^\top $ , variances $(0.25, 0.25)^\top $ , and zero correlation. The blue lines indicate the locations of the thresholds. In each cell, the numbers in parentheses denote the population probability of that cell under the true polychoric model.

It is worth addressing that there exist nonnormal distributions of the latent variables $(\xi ,\eta )$ that, after discretization with the same thresholds, result in the same response probabilities as under latent normality (Foldnes & Grønneberg, Reference Foldnes and Grønneberg2019). This implies that there may exist contamination distributions H and contamination fractions $\varepsilon> 0$ under which the population probabilities $f_{\varepsilon } \left (x,y\right )$ in (4.2) are equal to the true population probabilities of the polychoric model, $p_{xy}(\boldsymbol {\theta }_{\star})$ , that is, $f_{\varepsilon } \left (x,y\right ) = p_{xy}(\boldsymbol {\theta }_{\star})$ for all $(x,y)\in \mathcal {X}\times \mathcal {Y}$ . In this situation, the polychoric model is misspecified, but the misspecification does not have consequences because the response probabilities remain unaffected. To avoid cumbersome notation in the theoretical analysis of our robust estimator, we assume consequential misspecification throughout this article, that is, $f_{\varepsilon } \left (x,y\right ) \neq p_{xy}(\boldsymbol {\theta }_{\star})$ for some $(x,y)\in \mathcal {X}\times \mathcal {Y}$ whenever $\varepsilon> 0$ . However, it is silently understood that misspecification need not be consequential, in which case there is no issue and both the MLE and our robust estimator are consistent for the true $\boldsymbol {\theta }_{\star}$ .

4.3 Distributional misspecification

A model is distributionally misspecified when all observations in a given sample are generated by a distribution that is different from the model distribution. In the context of the polychoric model, this means that all ordinal observations are generated by a latent distribution that is nonnormal. Let G denote the unknown nonnormal distribution that the latent variables $(\xi , \eta )$ jointly follow under distributional misspecification. The object of interest is the population correlation between latent $\xi $ and $\eta $ under distribution G, for which the normality-based MLE of Olsson (Reference Olsson1979) turns out to be substantially biased in many cases (e.g., Foldnes & Grønneberg, Reference Foldnes and Grønneberg2020, Reference Foldnes and Grønneberg2022; Jin & Yang-Wallentin, Reference Jin and Yang-Wallentin2017; Lyhagen & Ornstein, Reference Lyhagen and Ornstein2023). As such, distributional misspecification is fundamentally different from partial misspecification: In the former, one attempts to estimate the population correlation of the nonnormal and unknown distribution that generated a sample, instead of estimating the polychoric correlation coefficient (which is the correlation under standard bivariate normality). In the latter, one attempts to estimate the polychoric correlation coefficient with a contaminated sample that has only been partly generated by latent normality (that is, the polychoric model). The assumption that the polychoric model is only partially misspecified for some uninformative observations enables one to still estimate the polychoric correlation coefficient of that model, which would not be feasible under distributional misspecification (at least not without additional assumptions).

Despite not being designed for distributional misspecification, the robust estimator introduced in the next section can offer a robustness gain in some situations where the polychoric model is distributionally misspecified. We discuss this in more detail in Section 8.

5.1 The estimator

The proposed estimator is a special case of a class of robust estimators for general models of categorical data called C-estimators (Welz, Reference Welz2024), and is based on the following idea. The empirical relative frequency of a response $(x,y)\in \mathcal {X}\times \mathcal {Y}$ , denoted

is a consistent nonparametric estimator of the population response probability in (4.2),

$$\begin{align*}f_{\varepsilon} \left(x,y\right) = \mathbb{P}_{G_\varepsilon} \left[ X = x, Y=y \right] = (1-\varepsilon) p_{xy}(\boldsymbol{\theta}_{\star}) + \varepsilon \int_{a_{\varepsilon,x-1}}^{a_{\varepsilon,x}} \int_{b_{\varepsilon,y-1}}^{b_{\varepsilon,y}} \text{d} H, \end{align*}$$

as $N\to \infty $ (see, e.g., Chapter 19.2 in Van der Vaart (Reference Van der Vaart1998)). If the polychoric model is correctly specified $(\varepsilon = 0)$ , then $\widehat {f}_{N}(x,y)$ will converge (in probability) to the true model probability $p_{xy}(\boldsymbol {\theta }_{\star})$ because

$$\begin{align*}f_0(x,y) = p_{xy}(\boldsymbol{\theta}_{\star}), \end{align*}$$

for all $(x,y)\in \mathcal {X}\times \mathcal {Y}$ . Conversely, if the polychoric model is misspecified $(\varepsilon> 0)$ , then $\widehat {f}_{N}(x,y)$ may not converge to the true $p_{xy}(\boldsymbol {\theta }_{\star})$ because

$$\begin{align*}f_{\varepsilon} \left(x,y\right) \neq p_{xy}(\boldsymbol{\theta}_{\star}) \end{align*}$$

for some $(x,y)\in \mathcal {X}\times \mathcal {Y}$ , since we assume consequential misspecification.

It follows that if the polychoric model is misspecified, there exists no parameter value $\boldsymbol {\theta }\in \boldsymbol {\Theta }$ for which the nonparametric estimates $\widehat {f}_{N}(x,y)$ converge pointwise to the associated model probabilities $p_{xy}(\boldsymbol {\theta })$ for all $(x,y)\in \mathcal {X}\times \mathcal {Y}$ . Hence, it is indicative of model misspecification if there exists at least one response $(x,y)\in \mathcal {X}\times \mathcal {Y}$ for which $\widehat {f}_{N}(x,y)$ does not converge to any polychoric model probability $p_{xy}(\boldsymbol {\theta })$ , resulting in a discrepancy between $\widehat {f}_{N}(x,y)$ and $p_{xy}(\boldsymbol {\theta })$ .Footnote ⁷ This observation can be exploited in model fitting by minimizing the discrepancy between the empirical relative frequencies, $\widehat {f}_{N}(x,y)$ , and theoretical model probabilities, $p_{xy}(\boldsymbol {\theta })$ , to find the most accurate fit that can be achieved with the polychoric model for the observed data. Specifically, our estimator minimizes with respect to $\boldsymbol {\theta }$ the loss function

(5.1) $$ \begin{align} L\left( \boldsymbol{\theta},\ \widehat{f}_{N} \right) = \sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}} \varphi\left(\frac{\widehat{f}_{N}(x,y)}{p_{xy}(\boldsymbol{\theta})}-1\right)p_{xy}(\boldsymbol{\theta}), \end{align} $$

where $\varphi : [-1,\infty ) \to \mathbb {R}$ is a prespecified discrepancy function that will be defined momentarily. The proposed estimator $\widehat {\boldsymbol {\theta }}_N$ is given by the value minimizing the objective loss over parameter space $\boldsymbol {\Theta }$ ,

(5.2) $$ \begin{align} \widehat{\boldsymbol{\theta}}_N = \arg\min_{\boldsymbol{\theta}\in\boldsymbol{\Theta}} L\Big(\boldsymbol{\theta}, \widehat{f}_{N}\Big). \end{align} $$

For the choice of discrepancy function $\varphi (z) = \varphi ^{\mathrm {MLE}}(z) = (z+1) \log (z+1)$ , it can be easily verified that $\widehat {\boldsymbol {\theta }}_N$ coincides with the MLE $\widehat {\boldsymbol {\theta }}_N^{\mathrm {\ MLE}}$ in (3.4). In the following, we motivate a specific choice of discrepancy function $\varphi (\cdot )$ that makes the estimator $\widehat {\boldsymbol {\theta }}_N$ less susceptible to misspecification of the polychoric model while preserving equivalence with ML estimation in the absence of misspecification.

The fraction between empirical relative frequencies and model probabilities with value 1 deducted,

$$\begin{align*}\frac{\widehat{f}_{N}(x,y)}{p_{xy}(\boldsymbol{\theta})} -1, \end{align*}$$

is referred to as Pearson residual (PR) (Lindsay, Reference Lindsay1994). It takes values in $[-1,+\infty )$ and can be interpreted as a goodness-of-fit measure. PR values close to 0 indicate a good fit between data and polychoric model at $\boldsymbol {\theta }$ , whereas values toward $-1$ or $+\infty $ indicate a poor fit because empirical response probabilities disagree with their model counterparts. To achieve robustness to misspecification of the polychoric model, responses that cannot be modeled well by the polychoric model, as indicated by their PR being away from 0, should receive less weight in the estimation procedure such that they do not over-proportionally affect the fit. Downweighting when necessary is achieved through a specific choice of discrepancy function $\varphi (\cdot )$ proposed by Welz et al. (Reference Welz, Archimbaud and Alfons2024), which is a special case of a function suggested by Ruckstuhl & Welsh (Reference Ruckstuhl and Welsh2001). The discrepancy function reads

(5.3) $$ \begin{align} \varphi(z) = \begin{cases} (z+1) \log(z+1) &\text{ if } z \in [-1, c],\\ (z+1)(\log(c+1) + 1) - c-1&\text{ if } z> c, \end{cases} \end{align} $$

where $c\in [0,\infty ]$ is a prespecified tuning constant that governs the estimator’s behavior at the PR of each possible response. Figure 2 visualizes this function for the example choice $c=0.6$ as well as the ML discrepancy function $\varphi ^{\mathrm {MLE}}(z) = (z+1) \log (z+1)$ . Note that deducting 1 in (5.1) and adding it again in (5.3) is purely for keeping the interpretation that a PR close to 0 indicates a good fit. We further stress that although the discrepancy function (5.3) can be negative, the loss function (5.1) is always nonnegative (Welz, Reference Welz2024).

Figure 2

Visualization of the robust discrepancy function $\varphi (z)$ in (5.3) for $c = 0.6$ (solid line) and the ML discrepancy function $\varphi ^{\mathrm {MLE}}(z) = (z+1) \log (z+1)$ (dotted line).

For the choice $c=+\infty $ , minimizing the loss (5.1) is equivalent to maximizing the log-likelihood objective in (3.4), meaning that the estimator $\widehat {\boldsymbol {\theta }}_N$ is equal to $\widehat {\boldsymbol {\theta }}_N^{\mathrm {\ MLE}}$ for this choice of c. More specifically, if a PR $z = \frac {\widehat {f}_{N}(x,y)}{p_{xy}(\boldsymbol {\theta })}-1$ of a response $(x,y)\in \mathcal {X}\times \mathcal {Y}$ is such that $z\in [-1, c]$ for fixed $c\geq 0$ , then the estimation procedure behaves at this response like in classic ML estimation. As argued before, in the absence of misspecification, $\widehat {f}_{N}(x,y)$ converges to $p_{xy}(\boldsymbol {\theta }_{\star})$ for all responses $(x,y)\in \mathcal {X}\times \mathcal {Y}$ , therefore, all PRs are asymptotically equal to 0. Hence, if the polychoric model is correctly specified, then estimator $\widehat {\boldsymbol {\theta }}_N$ is asymptotically equivalent to the MLE $\widehat {\boldsymbol {\theta }}_N^{\mathrm {\ MLE}}$ for any tuning constant value $c \geq 0$ . On the other hand, if a response’s PR z is larger than c, that is, $z> c \geq 0$ , then the estimation procedure does not behave like in ML, but the response’s contribution to loss (5.1) is linear rather than super-linear like in ML (Figure 2). It follows that the influence of responses that cannot be fitted well by the polychoric model is downweighted to prevent them from dominating the fit. The tuning constant $c\geq 0$ is the threshold beyond which a PR will be downweighted, so the choice thereof determines what is considered an insufficient fit. The closer to 0 the tuning constant c is chosen, the more robust the estimator is in theory. In Section C of the Supplementary Material, we explore different values of c in simulations and motivate a specific choice that we use for all numerical results in this article, namely, $c=0.6$ .

Note that the discrepancy function $\varphi (\cdot )$ in (5.3) may only downweight overcounts, that is, the empirical probability $\widehat {f}_{N}(x,y)$ exceeding the theoretical probability $p_{xy}(\boldsymbol {\theta })$ for some cell $(x,y)\in \mathcal {X}\times \mathcal {Y}$ . One might wonder why undercounts— $\widehat {f}_{N}(x,y)$ being smaller than $p_{xy}(\boldsymbol {\theta })$ , resulting in negative PRs—are not downweighted as well. Indeed, the discrepancy function in (5.3) does not change its behavior compared to the MLE for PRs below 0. The empirical frequency $\widehat {f}_{N}(x,y)$ is a relative measure, so if a contingency table cell $(x,y)$ has inflated counts, the other cells will have reduced values of $\widehat {f}_{N}$ . If the discrepancy function would downweight undercounts, there is a risk of downweighting non-contaminated cells simply because these cells have reduced $\widehat {f}_{N}$ values if at least one cell is inflated due to contamination. Such behavior could result in bias since non-contaminated cells are not supposed to be downweighted. We refer to Ruckstuhl and Welsh (Reference Ruckstuhl and Welsh2001, p. 1128) for a related discussion.

With the proposed choice of $\varphi (\cdot )$ , we stress that our estimator $\widehat {\boldsymbol {\theta }}_N$ in (5.2) has the same time complexity as ML, namely, $O\big (K_X\cdot K_Y\big )$ , since one needs to calculate the PR of all $K_X\cdot K_Y$ possible responses for every candidate parameter value. Consequently, our proposed estimator does not incur any additional computational cost compared to ML.

5.2 Statistical properties

We first address what quantity is estimated by the proposed estimator before discussing its asymptotic behavior.

5.2.1 Estimand

The estimand of the estimator $\widehat {\boldsymbol {\theta }}_N$ in (5.2) is given by

$$\begin{align*}\boldsymbol{\theta}_0 = \arg\min_{\boldsymbol{\theta}\in\boldsymbol{\Theta}} L\Big(\boldsymbol{\theta}, f_\varepsilon\Big). \end{align*}$$

This minimization problem is simply the population analog to the minimization problem in (5.2) that the sample-based $\widehat {\boldsymbol {\theta }}_N$ solves because the probabilities $f_{\varepsilon } \left (x,y\right )$ are the population analogs to the empirical probabilities $\widehat {f}_{N}(x,y)$ .

If the polychoric model is correctly specified, the estimand $\boldsymbol {\theta }_0$ equals the true parameter $\boldsymbol {\theta }_{\star}$ . Indeed, if $\varepsilon = 0$ , then $f_0(x,y) = p_{xy}(\boldsymbol {\theta }_{\star})$ for all $(x,y)\in \mathcal {X}\times \mathcal {Y}$ , so it follows that the loss

attains its global minimum of zero if and only if $\boldsymbol {\theta } = \boldsymbol {\theta }_{\star}$ , for any choice of $c\geq 0$ . Thus, in the absence of contamination, our estimator estimates the same quantity as the MLE, namely, the true $\boldsymbol {\theta }_{\star}$ . In other words, it obtains the true $\boldsymbol {\theta }_{\star}$ in population when the model is correctly specified, a property known as Fisher consistency. We refer to Welz et al. (Reference Welz, Archimbaud and Alfons2024) for details.

However, in the presence of misspecification ( $\varepsilon> 0$ ), the sampling distribution differs from the model distribution such that the estimand $\boldsymbol {\theta }_0$ —the parameter that minimizes the loss evaluated at the sampling distribution—is generally different from the true $\boldsymbol {\theta }_{\star}$ (cf. White, Reference White1982).Footnote ⁸ The population estimand being different from the true value translates to biased estimates of the latter as a consequence of the misspecification.

How much the estimand $\boldsymbol {\theta }_0$ differs from the true $\boldsymbol {\theta }_{\star}$ depends on the unknown fraction of contamination $\varepsilon $ , the unknown type of contamination H, as well as the choice of tuning constant c in $\varphi (\cdot )$ . Mainly, the larger $\varepsilon $ (more severe misspecification) and c (less downweighting of hard-to-fit responses), the further $\boldsymbol {\theta }_0$ is away from $\boldsymbol {\theta }_{\star}$ . Figure 3 illustrates this behavior for the polychoric correlation coefficient at an example misspecified distribution that is described in Section 4.2 and in which the true polychoric correlation under the correct model amounts to $\rho _{\star} = 0.5$ . For increasing contamination fractions, the MLE ( $c = +\infty )$ estimates a parameter value that is increasingly farther away from the true $\boldsymbol {\theta }_{\star}$ , where already a contamination fraction of less than $\varepsilon = 0.15$ suffices for a sign flip in the correlation coefficient. Conversely, choosing tuning constant c to be near 0 results in a much less severe bias. For instance, even at contamination fraction $\varepsilon = 0.4$ , the difference between the estimand and the true value is approximately 0.1 or less.

Figure 3

The population estimand $\rho _0$ of the polychoric correlation coefficient for various degrees of contamination fractions $\varepsilon $ (x-axis) and tuning constants c (line colors), for the same contamination distribution as in Figure 1.

Note: The ML estimand corresponds to $c=+\infty $ . There are $K_X=K_Y=5$ response options and the true value corresponds to $\rho _{\star} = 0.5$ (dashed line).

Overall, finite choices of c lead to an estimator that is at least as accurate as the MLE, and more accurate under misspecification of the polychoric model, thereby gaining robustness to misspecification.

A relevant question is whether the true parameter $\boldsymbol {\theta }_{\star}$ can be recovered when $\varepsilon> 0$ such that it can be estimated using $\widehat {\boldsymbol {\theta }}_N$ combined with a bias correction term. To derive such a bias correction term, one would need to impose assumptions on the contamination fraction and type of contamination. However, if one has strong prior beliefs about how the polychoric model is misspecified, modeling them explicitly rather than relying on the polychoric model seems more appropriate. Yet, one’s beliefs about misspecification may not be accurate, so attempts to explicitly model the misspecification may themselves result in a misspecified model. Consequently, robust estimation traditionally refrains from making assumptions on how a model may potentially be misspecified by leaving $\varepsilon $ and H unspecified in the contaminated distribution (4.1). Instead, one may use a robust estimator to identify data points that cannot be modeled with the model at hand, like the one presented in this article.

5.2.2 Asymptotic analysis

It can be shown that under certain standard regularity conditions that do not restrict the degree or type of possible partial misspecification beyond $\varepsilon \in [0,0.5)$ , the robust estimator $\widehat {\boldsymbol {\theta }}_N$ is consistent for estimand $\boldsymbol {\theta }_0$ as well as asymptotically normally distributed. Specifically, under said regularity conditions and fixed tuning constant $c>0$ , Theorem A.1 in the Supplementary Material establishes that

$$\begin{align*}\widehat{\boldsymbol{\theta}}_N \stackrel{\mathbb{P}}{\longrightarrow} \boldsymbol{\theta}_0, \end{align*}$$

as well as

$$\begin{align*}\sqrt{N}\left(\widehat{\boldsymbol{\theta}}_N - \boldsymbol{\theta}_0\right) \stackrel{\mathrm{d}}{\longrightarrow} \text{N}_d \Big(\boldsymbol{0}, \boldsymbol{\Sigma}\left({\boldsymbol{\theta}_0}\right) \Big), \end{align*}$$

as $N\to \infty $ , where “ $\stackrel {\mathbb {P}}{\longrightarrow }$ ” and “ $\stackrel {\mathrm {d}}{\longrightarrow }$ ” denote convergence in probability and distribution, respectively. The asymptotic covariance matrix has a sandwich-type construction

$$\begin{align*}\boldsymbol{\Sigma}\left({\boldsymbol{\theta}}\right) = \boldsymbol{M}\left( \boldsymbol{\theta} \right)^{-1} \boldsymbol{U}\left( \boldsymbol{\theta} \right) \boldsymbol{M}\left( \boldsymbol{\theta} \right)^{-1}, \qquad\boldsymbol{\theta}\in\boldsymbol{\Theta}, \end{align*}$$

where the $d\times d$ matrices

respectively, are the Hessian matrix of the population loss and the covariance matrix (evaluated at $f_{\varepsilon }$ ) of the likelihood score function—that is, the gradient of $\log (p_{XY}(\boldsymbol {\theta }))$ —weighted by stochastic binary weights whether the PR is smaller than or equal to the tuning constant c.Footnote ⁹ We derive closed-form expressions of the matrices $\boldsymbol {M}\left ( \boldsymbol {\theta } \right )$ and $\boldsymbol {U}\left ( \boldsymbol {\theta } \right )$ as well as their properties in Section A of the Supplementary Material.

The asymptotic covariance matrix $\boldsymbol {\Sigma }\left ({\boldsymbol {\theta }_0}\right )$ of our estimator is unobserved in practice because it depends on the unknown quantities $\boldsymbol {\theta }_0$ and $f_{\varepsilon }$ . Yet, $\boldsymbol {\Sigma }\left ({\boldsymbol {\theta }_0}\right )$ can be consistently estimated by replacing $\boldsymbol {\theta }_0$ and $f_{\varepsilon }$ by their corresponding consistent estimators $\widehat {\boldsymbol {\theta }}_N$ and $\widehat {f}_{N}$ , respectively. Details are provided in Section A of the Supplementary Material.

With this limit theory, one can construct standard errors and confidence intervals for every element in $\boldsymbol {\theta }_0$ . Importantly, in the absence of contamination (such that $\boldsymbol {\theta }_0 = \boldsymbol {\theta }_{\star}$ ), the asymptotic covariance matrix $\boldsymbol {\Sigma }\left ({\boldsymbol {\theta }_0}\right )$ of the robust estimator reduces to that of the fully efficient MLE (i.e., inverse Fisher information matrix) as long as $c>0$ . It follows that there is no loss of statistical efficiency when there is no contamination.Footnote ¹⁰ Hence, if the polychoric model is correctly specified, the robust estimator and the MLE are asymptotically first- and second-order equivalent. We refer to Section A of the Supplementary Material for a rigorous exposition and discussion of the robust estimator’s asymptotic properties.

5.3 Implementation

We provide a free and open-source implementation of our proposed methodology in a package for the statistical programming environment R (R Core Team, 2024). The package is called robcat (for “ROBust CATegorical data analysis”; Welz et al., Reference Welz, Alfons and Mair2025), and it is available from CRAN (the Comprehensive R Archive Network) at https://CRAN.R-project.org/package=robcat. To maximize speed and performance, the package is predominantly developed in C++ and integrated to R via Rcpp (Eddelbuettel, Reference Eddelbuettel2013). All numerical results in this article were obtained with this package.

The estimator’s minimization problem in (5.2) can be solved with standard algorithms for numerical optimization. In our experience, using an unconstrained version of the quasi-Newton algorithm L-BFGS-B of Byrd et al. (Reference Byrd, Lu, Nocedal and Zhu1995) works fine. However, additional stability might be gained from imposing the boundary constraint on the correlation coefficient and the monotonicity constraints on the thresholds, see (3.5), for which the simplex algorithm of Nelder & Mead (Reference Nelder and Mead1965) for constrained optimization can be used. In our implementation in package robcat, the default behavior is to first try unconstrained optimization via L-BFGS-B. If numerical instability is encountered or a monotonicity constraint is violated, the constrained optimization algorithm of Nelder & Mead (Reference Nelder and Mead1965) is used instead. While this is the default behavior, the package allows users to freely specify any supported optimization routine.

An important user choice is that of the tuning constant c in discrepancy function (5.3). The closer c is to 0, the more robust the estimator will be to possible misspecification of the polychoric model (see, e.g., Figure 3). On the other hand, in the presence of model misspecification, the more robust the estimator is made, the larger its estimation variance becomes. Moreover, if the model is correctly specified, then Welz et al. (Reference Welz, Archimbaud and Alfons2024) shows that the most robust choice, $c=0$ , is associated with two drawbacks, namely, asymptotic nonnormality as well as certain finite sample issues. We therefore suggest choosing a value slightly larger than 0. In simulation experiments (see Section C of the Supplementary Material), we find that the estimator is relatively insensitive to the specific choice of $c> 0$ , as long as it is reasonably small (for robustness) yet sufficiently far away from 0 (to avoid the aforementioned issues). The choice $c = 0.6$ thereby yields a good compromise so that we use this value for all applications in this article. However, we acknowledge that a detailed study, preferably founded in statistical theory, is necessary to provide guidelines on the choice of c. We will explore this in future work.

Furthermore, a two-step estimation procedure like in (3.6) is not recommended for robust estimation. The possible presence of responses that have not been generated by the polychoric model can inflate the empirical cumulative marginal proportion of some responses, which may result in a sizable bias of threshold estimates (3.6), possibly translating into biased estimates of polychoric correlation coefficients in the second stage. Our robust estimator therefore estimates all model parameters (thresholds and polychoric correlation) simultaneously.

6.1 Individual polychoric correlation coefficient

Let there be $K_X=K_Y = 5$ response categories for each of the two rating variables and define the true thresholds in the discretization process (3.1) as

$$\begin{align*}a_{\star,1}=b_{\star,1} = -1.5,\quad a_{\star,2}=b_{\star,2} = -0.5,\quad a_{\star,3}=b_{\star,3}=0.5,\quad a_{\star,4}=b_{\star,4}=1.5, \end{align*}$$

and let the true polychoric correlation coefficient in the latent normality model (3.2) be $\rho _{\star}=0.5$ . To simulate partial misspecification of the polychoric model according to (4.1), we let a fraction $\varepsilon $ of the data be generated by a particular contamination distribution H—which is left unspecified and therefore not explicitly modeled by our estimator—namely, a bivariate normal distribution with mean $(2.5,-2.5)^\top $ , variances $(0.25, 0.25)^\top $ , and zero covariance (and therefore zero correlation). We discretize the realizations of the contamination distribution according to the same thresholds $a_{\star,1}, \dots , a_{\star,4}, b_{\star,1}, \dots , b_{\star,4}$ as the uncontaminated realizations. This contamination distribution will inflate the empirical frequency of contingency table cells $(x,y) \in \{(5,1), (4,1), (5,2)\}$ , in the sense that they have a higher realization probability than under the true polychoric model.Footnote ¹¹ In fact, the data plotted in Figure 1 were generated by this process for contamination fraction $\varepsilon = 0.15$ , and one can see in this figure that particularly cell $(x,y) = (5,1)$ is sampled frequently although it only has a near-zero probability at the true polychoric model. The data points causing these three cells to be inflated are instances of negative leverage points. Here, such leverage points drag correlational estimates away from a positive value toward zero or, if there are sufficiently many of them, even a negative value. For intuition, one may think of such points as the responses of careless or inattentive participants whose responses are not based on item content. Although careless responding is only one special case of the unlimited and unrestricted variety of uninformative responses generated by H, we use careless responding as an illustrative running example throughout our simulations.

For contamination fraction $\varepsilon \in \{0, 0.01, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.49\}$ , we sample $N=1{,}000$ ordinal responses from this data-generating process. We estimate the true parameter $\boldsymbol {\theta }_{\star}$ with our proposed estimator with tuning constant set to $c=0.6$ , the MLE (Olsson, Reference Olsson1979), and, for comparison, the Pearson sample correlation calculated on the integer-valued responses.

Let $\widehat {\rho }_N$ be the estimate on a simulated dataset and $\widehat {\text {SE}}\left (\widehat {\rho }_N\right )$ the estimated standard error of $\widehat {\rho }_N$ , which is constructed using the limit theory developed in Theorem A.1 in the Supplementary Material. As performance measures, we calculate the average bias of the correlation estimates, the average bias of the standard error estimates (using the standard deviation of the correlation estimates across repetitions as an approximation of the true standard error), as well as coverage and average length of confidence intervals at significance level $\alpha =0.05$ . The coverage is defined as the proportion (across repetitions) of confidence intervals $\left [\widehat {\rho }_N\mp q_{1-\alpha /2}\cdot \widehat {\text {SE}}\left (\widehat {\rho }_N\right )\right ]$ that contain the true $\rho _{\star}$ , where $q_{1-\alpha /2}$ is the $(1-\alpha /2)$ quantile of the standard normal distribution. The length of a confidence interval is given by $2 \cdot q_{1-\alpha /2}\cdot \widehat {\text {SE}}\left (\widehat {\rho }_N\right )$ .

Figure 4 visualizes the bias of each estimator with respect to the true polychoric correlation $\rho _{\star}$ across the 5,000 simulated datasets. An analogous plot for the whole parameter $\boldsymbol {\theta }_{\star}$ can be found in Section D.1 of the Supplementary Material; the results are similar to those of $\rho _{\star}$ . Additional performance measures are shown in Table 1. For all considered contamination fractions, the estimates of the MLE and sample correlation are somewhat similar, which is expected because these two estimators are known to yield similar results when there are five or more rating options and the discretization thresholds are symmetric (cf. Rhemtulla et al., Reference Rhemtulla, Brosseau-Liard and Savalei2012). In the absence of contamination, the MLE and the robust estimator yield accurate estimates. Both estimators are nearly equivalent to one another in the sense that their point estimates, standard deviation, and coverage at significance level $\alpha =0.05$ are very similar. However, when contamination is introduced, MLE, sample correlation, and the robust estimator yield noticeably different results. Already at the small contamination fraction $\varepsilon =0.01$ (corresponding to only ten observations), MLE and sample correlation are noticeably biased, resulting in poor coverage of only about 0.45 and 0.04, respectively. Increasing the contamination fraction to the still relatively small value of $\varepsilon = 0.05$ , MLE and sample correlation start to be substantially biased with average biases of $-0.25$ and $-0.28$ , respectively, leading to zero coverage. The biases of these two methods further deteriorate as the contamination fraction is gradually increased. At $\varepsilon \geq 0.15$ , MLE and sample correlation produce estimates that are not only severely biased but also sign-flipped: while the true correlation is positive (0.5), both estimates are negative. In stark contrast, the proposed robust estimator remains accurate throughout nearly all considered contamination fractions. At the small $\varepsilon = 0.01$ , the robust estimator is nearly unaffected, while at $\varepsilon = 0.15$ , it only exhibits a minor bias of about $-0.02$ . Even at extreme contamination $\varepsilon = 0.4$ , its bias amounts to less than 0.1. In addition, the coverage of the robust method remains above or close to 0.9 for contamination fractions $\varepsilon \leq 0.2$ .

Figure 4

Boxplot visualization of the bias of three estimators of the polychoric correlation coefficient, $\widehat {\rho }_N - \rho _{\star}$ , for various contamination fractions in the misspecified polychoric model across 5,000 repetitions.

Note: The estimators are the robust estimator with $c=0.6$ (left), the MLE (center), and the Pearson sample correlation (right). Diamonds represent the respective average bias. The dashed line denotes value 0 and the dotted line $-\rho _{\star} = -0.5$ , the latter of which indicating a sign flip in the correlation estimate.

Table 1

Results for the robust estimator with $c=0.6$ , the MLE, and the Pearson sample correlation, for various contamination fractions across 5,000 simulated datasets

Note: The true polychoric correlation coefficient is $\rho _{\star}=0.5$ . We compute the average of point estimates $\widehat {\rho }_N$ of the polychoric correlation coefficient, the average bias ( $\widehat {\rho }_N - \rho _{\star}$ ), the standard deviation of the $\widehat {\rho }_N$ (SE; an approximation of the true standard error), the average standard error estimate $\widehat {\text {SE}}$ , the corresponding average bias ( $\widehat {\text {SE}} - \text {SE}$ ), confidence interval coverage with respect to the true $\rho _{\star}$ at nominal level 95%, and the average length of the respective confidence intervals.

It is worth noting that the confidence intervals of the robust estimator grow wider with increasing contamination fraction $\varepsilon $ . We also observe that the standard deviation of the robust estimates over the repetitions grows similarly. This indicates that the derived asymptotic distribution used to estimate standard errors matches well with the simulated distribution of the estimator across the repetitions. Indeed, the bias of standard error estimation remains at near-zero for the robust estimator, except for extremely large contamination fractions ( $\varepsilon \geq 0.4$ ). We investigate this in more detail in Section D.1 of the Supplementary Material.

6.2 Polychoric correlation matrix

The goal of this simulation study is to robustly estimate a polychoric correlation matrix, that is, a matrix comprising of pairwise polychoric correlation coefficients. The simulation design is based on Foldnes & Grønneberg (Reference Foldnes and Grønneberg2020).

Let there be r observed ordinal random variables and assume that a latent variable underlies each ordinal variable. In accordance with the multivariate polychoric model (e.g., Muthén, Reference Muthén1984), the latent variables are assumed to jointly follow an r-dimensional normal distribution with mean zero and a covariance matrix with unit diagonal elements so that the covariance matrix is a correlation matrix. Each individual latent variable is discretized to its corresponding observed ordinal variable akin to the discretization process (3.1).

Following the five-dimensional simulation design in Foldnes & Grønneberg (Reference Foldnes and Grønneberg2020), there are $r=5$ ordinal variables with a polychoric correlation matrix as in Table 2 such that the pairwise correlations vary from a low 0.2 to a moderate 0.56.Footnote ¹² For all latent variables, the discretization thresholds are set to, in ascending order, $\Phi _1^{-1}(0.1) = -1.28,\ \Phi _1^{-1}(0.3) = -0.52,\ \Phi _1^{-1}(0.7) = 0.56$ , and $\Phi _1^{-1}(0.9) = 1.28$ , such that each ordinal variable can take five possible values. A visualization of the implied distribution of each ordinal variable can be found in Figure 5 in Foldnes & Grønneberg (Reference Foldnes and Grønneberg2020).

Table 2

Correlation matrix of $r=5$ latent variables as in Foldnes & Grønneberg (Reference Foldnes and Grønneberg2020)

Note: In line with the multivariate polychoric correlation model (e.g., Muthén, Reference Muthén1984), the latent variables are jointly normally distributed with mean zero and this correlation matrix as covariance matrix.

Figure 5

Absolute average bias (top) and confidence interval coverage (bottom) at nominal level 95% (dashed horizontal lines) of the robust estimator with $c=0.6$ (left) and the MLE (right) for each unique pairwise polychoric correlation coefficient in the true correlation matrix (Table 2), expressed as a function of the contamination fraction $\varepsilon $ (x-axis).

Note: Results are aggregated over 5,000 repetitions.

As contamination distribution, we choose an r-dimensional Gumbel distribution comprising of mutually independent Gumbel marginal distributions, each with location and scale parameters equal to 0 and 3, respectively. To obtain ordinal observations, the unobserved realizations from this distribution are discretized via the same threshold values as realizations from the model (normal) distribution. As such, the uninformative ordinal observations generated by this contaminated distribution emulate the erratic behavior of a careless respondent. Unlike in the previous simulation design (Section 6.1), the uninformative responses are not concentrated around a few response options, but may occur in every response option.

For contamination fraction $\varepsilon \in \{0, 0.01, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.49\}$ , we sample $N=1{,}000$ ordinal five-dimensional responses from this data-generating process and use them to estimate the polychoric correlation matrix in Table 2 via our robust estimator (again with tuning constant $c=0.6$ ) as well as the MLE.

Figure 5 visualizes the absolute average bias as well as confidence interval coverage at 95% nominal level (calculated over the 5,000 repetitions) of the robust estimator and the MLE for each pairwise polychoric correlation coefficient. As expected, when the model is correctly specified ( $\varepsilon = 0$ ), both estimators coincide with accurate estimates. However, in the presence of contamination ( $\varepsilon> 0$ ), the two estimators deviate. The MLE exhibits a notable bias for all correlation coefficients, which increases gradually with increasing contamination fraction. The magnitude of the bias tends to be larger for pairs with larger true correlation, such as $0.56$ for pair $(2,1)$ , than for pairs with weaker true correlation, such as $0.20$ for pair $(5,4)$ . In addition, coverage of the MLE drops quickly for many pairs and gradually for the remaining ones. Conversely, the robust estimator remains nearly unaffected for a broad range of contamination fractions for each correlation coefficient ( $\varepsilon \leq 0.2$ or $\varepsilon \leq 0.3$ depending on the variable pair), with bias only somewhat increasing afterward. Furthermore, its coverage remains close to 0.9 or higher even at the high contamination fraction of $\varepsilon = 0.3$ . This reflects the excellent performance of the proposed estimator with respect to robustness to uninformative responses. Section D.2 of the Supplementary Material contains additional evaluations. In essence, the robust estimator yields accurate standard errors, but its confidence intervals tend to be wider than those of the MLE in the presence of contamination.

We stress that polychoric correlation matrices need not be positive definite (e.g., Mair, Reference Mair2018, p. 22), although all estimated polychoric correlation matrices in this simulation turned out positive definite. If an estimated polychoric correlation matrix is not positive definite, one may opt to apply a smoothing procedure like in Yuan et al. (Reference Yuan, Wu and Bentler2011) or Bock et al. (Reference Bock, Gibbons and Muraki1988).

6.3 Discussion and additional simulation experiments

The two simulation studies above demonstrate that already a small degree of partial misspecification due to uninformative responses, such as careless responses, can render the commonly employed MLE unreliable, while the proposed robust estimator retains good accuracy and coverage even in the presence of a considerable number of uninformative responses. On the other hand, when the polychoric model is correctly specified, the MLE and the robust estimator produce equivalent estimates.

To further evaluate our robust estimator and investigate its limitations, we conduct additional simulation experiments in Section E of the Supplementary Material.

The first experiment, described in Section E.2 of the Supplementary Material, is a generalization of the design in Section 6.1 with different mean shifts in the contamination distribution H. For small mean shifts, the proposed estimator does not improve upon the MLE, but the bias of both estimators remains reasonable. The larger the mean shift, the larger the detrimental effect on the MLE and the higher the robustness gain of our proposed estimator.

The second experiment, described in Section E.3 of the Supplementary Material, focuses on correlation shifts in the contamination distribution H. Specifically, the contamination distribution is the same as the true model distribution except for a sign-flipped correlation coefficient $-\rho _{\star}$ . For moderate correlation $\rho _{\star}$ , the proposed estimator does not improve upon the MLE due to substantial overlap between the true model distribution and the contamination. However, the gain in robustness increases substantially for higher correlation coefficients $\rho _{\star}$ . We expect the gain in robustness to increase for a higher number of response options and decrease for fewer response options. In the most extreme case of two dichotomous rating variables, no improvement can be expected.

7.1 Background and study design

Each marker is an item comprising a single English adjective (such as “bold” or “timid”) asking respondents to indicate how accurately the adjective describes their personality using a 5-point Likert-type rating scale (very inaccurate, moderately inaccurate, neither accurate nor inaccurate, moderately accurate, and very accurate). Here, each Big Five personality trait is measured with six pairs of adjectives that are polar opposites to one another (such as “talkative” vs. “silent”), that is, 12 items in total for each trait. It seems implausible that an attentive respondent would choose to agree (or disagree) to both items in a pair of polar opposite adjectives. Consequently, one would expect a strongly negative correlation between polar adjectives if all respondents respond attentively (Arias et al., Reference Arias, Garrido, Jenaro, Martinez-Molina and Arias2020).

Arias et al. (Reference Arias, Garrido, Jenaro, Martinez-Molina and Arias2020) collect measurements of three Big Five traits in this way, namely, extroversion, neuroticism, and conscientiousness.Footnote ¹³ The sample that we shall use, Sample 1 in Arias et al. (Reference Arias, Garrido, Jenaro, Martinez-Molina and Arias2020), consists of $N=725$ online respondents who are all U.S. citizens, native English speakers, and tend to have relatively high levels of reported education (about 90% report to hold an undergraduate or higher degree). Concerned about respondent inattention in their data, Arias et al. (Reference Arias, Garrido, Jenaro, Martinez-Molina and Arias2020) construct a factor mixture model for detecting inattentive/careless participants. Their model crucially relies on response inconsistencies to polar opposite adjectives and is designed to primarily detect careless straightlining responding. They find that careless responding is a sizable problem in their data. Their model finds evidence of straightliners, and the authors conclude that if unaccounted for, they can substantially deteriorate the fit of theoretical models, produce spurious variance, and overall jeopardize the validity of research results.

Due to the suspected presence of careless respondents, we apply our proposed method to estimate the polychoric correlation coefficients between all $\binom {12}{2}=66$ unique item pairs in the neuroticism scale to obtain an estimate of the scale’s (polychoric) correlation matrix. The results of the remaining two scales are qualitatively similar and are reported in Section F of the Supplementary Material. We estimate the polychoric correlation matrix via the MLE and via our proposed robust alternative with tuning parameter $c=0.6$ . As a robustness check, we further investigate the effect of the choice of c.

8.1 Distributional vs. partial misspecification

Huber and Ronchetti (Reference Huber and Ronchetti2009, p. 4) note that, although conceptually distinct, robustness to distributional misspecification and partial misspecification are “practically synonymous notions.” Hence, despite distributional misspecification not being covered by the partial misspecification framework under which we study the theoretical properties of our proposed estimator, our estimator could still offer a gain in robustness compared to the MLE of the polychoric model under distributional misspecification. Specifically, the robust estimator may still be useful if the central part of the nonnormal distribution G is not too different from a standard bivariate normal distribution. Intuitively, if the difference between G and a standard normal distribution is mainly in the tails, G can be approximated by a contaminated distribution as in (4.1), with the standard normal distribution covering the central part and some contamination distribution H covering the tails. The polychoric MLE tries to treat influential observations from the tails—which cannot be fitted well by the polychoric model—as if they were normally distributed, resulting in a possibly large estimation bias. In contrast, the robust estimator uses the normal distribution only for observations from the central part—which may fit the polychoric model well enough—and downweights observations from the tails. Thus, as long as such a contaminated normal distribution is a decent approximation of the nonnormal distribution G, the robust estimator should perform reasonably well. However, if G cannot be approximated by such a contaminated normal distribution, neither the polychoric MLE nor our estimator can be expected to perform well. Overall, though, our estimator could offer an improvement in terms of robustness to distributional misspecification.

In the following, we perform a simulation study to investigate the performance of our estimator when the polychoric model is distributionally misspecified.

8.2 Simulation study

To simulate ordinal variables that were generated by a nonnormal latent distribution G, we employ the VITA simulation method of Grønneberg & Foldnes (Reference Grønneberg and Foldnes2017). For a pre-specified value of the population correlation $\rho _G$ , the VITA method models the latent random vector $(\xi ,\eta )$ such that the individual variables $\xi $ and $\eta $ both possess standard normal marginal distributions with population correlation set equal to $\rho _G = \mathbb {C}\mathrm {or}_{G} \left [\xi ,\ \eta \right ]$ , but are not jointly normally distributed. Instead, their joint distribution G is equal to a pre-specified nonnormal copula distribution, such as the Clayton or Gumbel copula. Grønneberg & Foldnes (Reference Grønneberg and Foldnes2017) show that discretizing such VITA latent variables yields ordinal observations that could not have been generated by a standard bivariate normal distribution, thereby ensuring proper violation of the latent normality assumption.

To investigate the robustness of our estimator to distributional misspecification, we use the VITA implementation in package covsim (Grønneberg et al., Reference Grønneberg, Foldnes and Marcoulides2022) to generate draws of the latent variables $(\xi ,\eta )$ such that the latent variables are jointly distributed according to a Clayton copula G with population correlation $\rho _G \in \{0.9, 0.3\}$ (see Figure 8 for visualizations). Following the discretization process (3.1), we discretize both latent variables via discretization thresholds

$$ \begin{align*} \begin{array}{r r r r r} a_1 = -1.5, & a_2 = 0,& a_3 = 0.5, &a_4 = 1, & \quad \text{and}\\ b_1 = -1, & b_2 =1,& b_3 =1.5, &b_4 =2, & \end{array} \end{align*} $$

such that both resulting ordinal variables have five response options each. We generate $N=1{,}000$ ordinal responses according to this data generation process and compute across 5,000 repetitions the same estimators and performance measures as in the simulations in Section 6.

Figure 8

Bivariate density plots of the standard normal distribution (left) and Clayton copula with standard normal marginals (right), for population correlations 0.9 (top) and 0.3 (bottom).

Figure 9 visualizes the bias of the robust estimator and the polychoric MLE under both Clayton copulas across the repetitions.Footnote ¹⁵ For correlation 0.9, the polychoric MLE exhibits a noteworthy bias, whereas the robust estimator remains accurate, albeit with a larger estimation variance as compared to most simulation configurations of partial misspecification (cf. Section 6). Conversely, for the weaker correlation 0.3, both estimators are fairly accurate with average biases of about $-0.015$ . Table 5 contains additional performance measures regarding inference. For $\rho _G = 0.3$ , the average standard error estimate of the robust method is accurate, but for $\rho _G = 0.9$ , it notably overestimates. We therefore also computed the median of the standard error estimates in the latter case: at 0.017, it is fairly close to the true standard error of 0.014. It turns out that there is a small number of simulated datasets with a large majority of empty cells in the contingency table, resulting in numerical instability of the standard error estimates and inflating their average. As discussed in Section 4.3, distributional misspecification is not covered by our partial misspecification framework, so it is not surprising that standard errors derived under partial misspecification are not always valid. Bootstrap inference may therefore be an attractive alternative. Nevertheless, unlike the polychoric MLE with coverage of only about 13% for $\rho _G = 0.9$ , the robust estimator maintains high coverage of over 90%.

Table 5

Results for the robust estimator with $c=0.6$ and the polychoric MLE across 5,000 simulated datasets under distributional misspecification via a Clayton copula with true population correlation $\rho _G\in \{0.9, 0.3\}$

Note: See Table 1 for explanatory notes on the performance measures.

Figure 9

Boxplot visualization of the bias of the robust estimator and the polychoric MLE, $\widehat {\rho }_N - \rho _G$ , under distributional misspecification via a Clayton copula with correlation $\rho _G = 0.9$ (left) and $\rho _G = 0.3$ (right), across 5,000 repetitions.

Note: Diamonds represent the respective average bias. The tuning constant of the robust estimator is set to $c=0.6$ .

These results suggest that at least for point estimation, the Clayton copula with correlation 0.9 might be reasonably approximable by a contaminated normal distribution where the normal distribution covers the center of the probability mass and some contamination distribution covers the tails (cf. Section 8.1). Indeed, Figure 8 indicates that the normal distribution and Clayton copula at correlation 0.9 seem to behave similarly in the center, but deviate from one another toward the tails. On the other hand, at correlation 0.3, the two densities do not appear to be drastically different from one another, which may explain why both the polychoric MLE and the robust estimator work reasonably well for such distributional misspecification. If the latent distribution is not too different from a normal distribution, then the polychoric model may offer a satisfactory fit despite being technically misspecified.

Overall, this simulation study demonstrates that in some cases of distributional misspecification, robustness can be gained with our estimator, compared to the polychoric MLE. However, it also demonstrates that there are cases of distributional misspecification for which the polychoric MLE still works quite well such that the robust estimator offers little gain. Nevertheless, the fact that in some situations robustness can be gained under distributional misspecification represents an overall gain in robustness.