A Note on Applying the BCH Method Under Linear Equality and Inequality Constraints
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Abstract
Researchers often wish to relate estimated scores on latent variables to exogenous covariates not previously used in analyses. The BCH method corrects for asymptotic bias in estimates due to these scores’ uncertainty and has been shown to be relatively robust. When applying the BCH approach however, two problems arise. First, negative cell proportions can be obtained. Second, the approach cannot deal with situations where marginals need to be fixed to specific values, such as edit restrictions. The BCH approach can handle these problems when placed in a framework of quadratic loss functions and linear equality and inequality constraints. This research note gives the explicit form for equality constraints and demonstrates how solutions for inequality constraints may be obtained using numerical methods.
Keywords
Classification Latent class analysis Threestep procedure BCH method1 Introduction
Researchers in many different disciplines apply latent structure models in which observed variables are treated as indicators of an underlying latent variable that cannot be measured directly. An often used strategy in this context consists of three steps (Vermunt 2010). First, the parameters of the measurement model are estimated, describing the relationship between the latent variable and its indicators. Second, each respondent is assigned a latent score based on his/her scores on the indicators. Finally, the relationships between the latent scores and scores on exogenous variables are assessed.
Croon (2002) showed that for general latent structure models, such a strategy leads to inconsistent estimates of the parameters of the joint distribution of the latent variable and the exogenous variables. Bolck et al. (2004) discussed this problem in the context of latent class analysis where observed variables are categorical. They also derived a correction procedure that produces consistent estimates, known as the BCH correction method. Subsequent simulation studies by Vermunt (2010), Bakk et al. (2013), Bakk and Vermunt (2016), and NylundGibson and Masyn (2016) have demonstrated that this procedure produces unbiased parameter estimates and correct inference for a large range of simulation conditions. When applying the BCH correction method in cases of categorical exogenous variables, two problems can arise. First, negative cell proportion estimates can be obtained (Asparouhov and Muthén 2015). Second, the approach cannot deal with situations where marginals need to be constrained. An example is edit restrictions in official statistics, leading to certain marginals being fixed to zero (De Waal et al. 2012), which is also used in combination with latent class modelling (Boeschoten et al. 2017).
In this research, note the BCH method is extended to solve these two problems. We allow for linear equality and inequality constraints by noting the correction method minimizes a quadratic loss function and give a closed form solution for linear equality restrictions. Next, we demonstrate how solutions for inequality constraints may be obtained using numerical methods. We first discuss the threestep approach to the latent class model and the BCH correction method. We then show how to impose linear restrictions and how to extend this to including nonnegativity constraints. At last, the extended BCH method is applied on a dataset from the Political Action Survey. In the Appendix, R code is given to apply the procedure.
2 The ThreeStep Approach to the Latent Class Model and the BCH Correction Method
Let us denote a set of observed exogenous variables Q and an unobserved latent variable X. All variables involved are assumed to be categorical. Let Q = (Q_{1},Q_{2},...,Q_{J}) be the Cartesian product of J different discrete random variables Q_{j}. If the variable Q_{j} is defined for n_{j} categories, the distribution of Q can be specified as a multinomial distribution with \(n={\prod }_{j = 1}^{J}n_{j}\) categories.
3 The Correction Procedure Under Linear Equality Constraints
In some applications, simple linear restrictions may be imposed on the elements of matrix A. For instance, some of the probabilities in the joint distribution of Q and X may be set equal to zero, for example for combinations of Q and X that cannot occur in practice. After imposing such zero constraints, all the nonzero cell probabilities should still add to one. The quadratic loss function φ can be minimized under equality constraints on the unknown elements of matrix A by applying the method of Lagrangian multipliers.
4 The Correction Procedure Under Linear Equality and Inequality Constraints
A second issue with the BCH procedure is that in finite samples the consistent estimate \({\hat {\mathbf {A}}}\) hat may contain negative values. This issue is similar to the occurrence of Heywood cases in factor analysis (Heywood 1931). Such negative values in the probability table estimate \({\hat {\mathbf {A}}}\) may prevent subsequent analyses. We suggest preventing such inadmissible solutions by imposing inequality constraints. The resulting minimization problem is a quadratic program that can be solved by an iterative method.

Subset J_{1} contains the indices of the elements of vector a which are set exactly equal to zero: for those indices j we require a_{j} = 0;

Subset J_{2} contains the indices of the elements of vector a which are required to be nonnegative: for those indices j we require a_{j} ≥ 0.
With this procedure, we are able to find a solution for A (the joint distribution of latent variable X and exogenous covariates Q) where the sum of the elements is equal to 1, where no negative elements are created, and where impossible combinations of scores can be set to have a probability of zero. Having defined b, D_{mat} and H, the solution can be obtained using standard software for quadratic programming, such as the R package quadprog (Turlach and Weingessel 2013).
5 Application
As an illustration, the extended BCH method is applied on a dataset from the Political Action Survey (Barnes et al. 1979; Jennings and Van Deth 1990). The dataset consists of five dichotomous indicators on political involvement and tolerance (“System Responsiveness”; “Ideological Level”; “Repression Potential”; “Protest Approval”; “Conventional Participation”) and three nominal covariates (“Sex”; “Level Of Education”; “Age”). This dataset has previously been used in Hagenaars (1993) and Vermunt and Magidson (2000) and in the Latent GOLD user’s manual (Vermunt and Magidson 2005). The dataset as well as the syntax used in this illustration can be found in Latent GOLD version 5.1 under “syntax examples” → LCA → restrictions → equalities → Model C.
In the first step, a four class restricted model is applied to distinguish between four latent classes on involvement and tolerance. In this model, response probabilities are restricted to be equal for the items “System Responsiveness” and “Conventional Participation,” and the response probability for the variable “Ideological Level” is fixed to 0 by specifying a logit of 100.
Since there are no combinations of scores between “Involvement And Tolerance” and “Age” that are not possible in practice, it is not needed to fix any marginals to zero.
6 Conclusion
We have modified the BCH method to include linear equality and inequality constraints solving the problem of negative solutions and allowing for restrictions on arbitrary cell margins. With these adjustments, analysts interested in relating covariates to assignments on latent class variables will now be able to, for example, impose edit restrictions, further analyse solutions that were previously inadmissible, and analyse datasets involving more complex marginal restrictions. The application demonstrates that when a negative value is obtained using the regular BCH method, this can be solved by using the extended BCH method. In the ??, R code is given to apply the extended BCH method, and an addition to the example is given that demonstates how margins can be fixed to zero using the extended BCH method.
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