Latent class analysis (LCA) [1,2], as a mixture modeling, has been widely used to identify homogeneous subpopulations from observed categorical variables under the assumption that the population is heterogeneous. One of the reasons for its popularity is its ability to reveal the characteristics of each homogeneous population identified via statistical modeling. Consideration of the heterogeneity also informs the characteristics of subpopulations unveiled in research areas including social, behavioral, and health sciences. Conceptually, identification of unobserved group characteristics via LCA corresponds to unsupervised learning or cluster analysis in data mining such as K-means or K-median algorithm. However, the popularity of LCA as a statistical model is somewhat different from unsupervised learning due to its strict property implemented in the most common estimation method, maximum likelihood estimation based on the expectation-maximization (EM) algorithm [3]. More precisely, the EM algorithm requires multivariate normality for variance-covariance matrix used in the LCA to estimate parameters. As another way of saying, the multivariate normality property prevents researchers from utilizing the concept of big data in LCA, which often produces estimation issues such as non-positive definiteness in computing a Hessian matrix [4].

As described in the comparison between the mixture-modeling approach and cluster analysis procedure such as K-means [5], the mixture-modeling approach does not always perform better than K-means cluster analysis. Rather, Steinley et al. [5] showed an equivalence in terms of statistical modeling under a certain condition. That is, for two conceptually equivalent clustering methods, it cannot be said that the one is superior to the other, which is consistent with the results from Brusco et al. [6] comparing latent class, K-means, and K-median methods. Although Lubke et al. [7] noted that “Model-based methods have the advantage that more rigorous methods can be applied for the comparison of alternative models” (p. 23), it would not be applicable when we consider big data and/or data mining due to the model complexity and strict assumption of multivariate normality. On the other hand, cluster analysis often takes advantage in estimation due to the simple estimation algorithm using an alternative least square estimation in the distance function from centroids. Steinley et al. [5] also noted that “it is important to realize that increased complexity and flexibility do not necessarily imply that a better solution will be found if the goal of the analysis is to uncover the unknown cluster membership.” (p. 76).

Although Steinley et al. [5] highlighted the advantages of K-means, K-means cluster analysis is not a perfect alternative to a mixture-modeling approach. It often suffers from a poor local optimum and has a limitation that its algorithm works with only convex clustering. Such limitation of K-means cluster analysis can be got rid of by applying fuzzy clustering analysis [8]. When fuzzy clustering methods was applied to identification of homogeneous subgroups, the estimation did show less analytic issues such as non-positive definitie matrix. Furthermore, by utilizing the method of how fuzzy clustering analysis can be accompanied by generalized structured component analysis (GSCA; [9]), Ryoo et al. [8] encompassed the LCA in the GSCA framework, which is called fuzzy clusterwise GSCA. In other words, the merger of fuzzy clustering and GSCA allows researchers not only to effectively classify the homogeneous cluster via fuzzy clustering but also to move forward in the model-based classification via GSCA. One of the advantages of LCA over cluster analysis procedures is the capacity to examine the effects of other variables on the LCA parameters, which is no longer the exclusive property of the fuzzy clusterwise GSCA. Such a role of examining the effects of other variables on the LCA was replaced with GSCA in the fuzzy clusterwise GSCA that is a statistical tool of fitting various component-based structural equation models into data [9,10]. In addition, Ryoo et al. [11] provided more indexes that can be utilized in identifying homogeneous subgroups and the procedure of enumerating the number of clusters, which is out of the scope of this manuscript. The method of fuzzy clusterwise GSCA will be described in Section 2.

Although there are many statistical packages including R for latent class analysis and cluster analysis, the method utilizing both the fuzzy clustering analysis and GSCA for LCA is a new approach and thus, there is no comparable R package for fuzzy clusterwise GSCA. Instead, in this section, we introduce three well-known packages for LCA as a mixture-modeling approach: Mplus, poLCA in R, and SAS procedure LCA as competitors. There are many other software packages available but exhaustive search of packages are out of our scope. We focus on their key features of the three packages, which validates the necessity and coverage of our new package, gscaLCA, in R. Note that most statistical programs for LCA include the capabilities of fitting multi-groups LCA, imposing measurement invariance across groups, and implementing latent class regression (LCR). In addition, binary and multinomial logistic regression options for predicting latent class membership and the ability to take into account sampling weights and clusters are also possible.

Based on the description of fuzzy clustering [26] and terminologies [27], we briefly describe fuzzy c means (FCM) algorithm as follows: FCM minimizes the following objective (distance) function, Jm, measured by the sum of squares:

(1)Jm=∑i=1N∑k39=1Kukim‖xi−ck‖2,where m∈(1,∞] is a classification index, uki indicates the membership probability of i^th object in class k, and ck indicates the centroid for class k∈{1,⋯,K}. The m is also known as a fuzzifier, and adjusts the probability of belonging to a class, i.e., m=1 indicates that the membership probability will converge either 0 or 1 such as in K-means, which is excluded in this FCM algorithm. On the other hand, m=∞ indicates that all membership probabilities equal probability in belonging to any of classes, i.e., 1K. Both uki and ck are defined by

ukim=1∑l=1K[‖xi−ck‖2‖xi−cl‖2]1m−1and

ck=∑i=1Nukimxi∑i=1Nukim.

With a termination criterion such that maxki{|(ukim)(L+1)−(ukim)(L)|}<ϵ for a small value ϵ at L+1 respects, the FCM algorithm is done as follows: 1.

(Step 1) Initialize U(0) for U=[ukim],

Generalized structured component analysis (GSCA) [9] is a component-based approach of structural equation modeling (SEM). Different from a maximum likelihood-based SEM (ML-SEM), the component-based SEM (CB-SEM) utilizes the underlying construct as a composite of weighted observed variables, and applies an alternating least square method to estimate model parameters. Here, we briefly describe GSCA as a CB-SEM (see Hwang et al. [9] for more detail). The alternating least square method in the component-based approach is relatively simple and straightforward procedure compared to the ML-based approach because it does not assume the multivariate normality of model parameters but minimizes a sum of squares of residuals computed from sample data directly. Along with regularization such as Ridge and Lasso, the least square method produces a more interpretable and predictive model that has possibly lower prediction error [28]. Such a great property of the least square methods is inherited into GSCA [9].

GSCA consists of three sub models: a measurement model describing observed indicators from each latent construct, a structural model defining the associations among latent constructs, and an weighted relation model defining latent constructs. While the typical SEM models include a measurement model and a structural model assuming the normality of the latent constructs [29], GSCA additionally includes the weighted relation model that represents a formative relation between a component and its indicators. That is, the weighted relation model defines each underlying construct as a weighted composite or component of indicators. In this paper, we used both component and latent variable, interchangeably. Such a function in the weighted relation model plays a key role in parameter estimation without a multivariate normality assumption, which eases the estimation. The three sub models of GSCA can be expressed as follows:

(2)Measurement Model:z=ATγ+ϵ

(3)Structural Model:γ=BTγ+ζ

(4)Weighted Relation Model:γ=WTzwhere z is a J by 1 vector of observed indicators scores from one observation, γ is P by 1 vector of components, A is a P by J matrix of factor loadings, B is a P by P matrix of structural path coefficients, W is a J by P matrix of weights for components. In addition, ϵ presents the residuals of indicators, which is expressed by a J by 1 vector, and ζ presents the residuals of components, which is expressed by a P by 1 vector. The three equations can be merged into one equation as follows:

(5)VTz=CTWTz+e,where V=[IWT]T,C=[ATBT]T, and e=[ϵζ].

While minimizing the residual term e in Eq. (5), the factor loadings, path coefficients, and weights are estimated via the alternating least square method. The details of the estimation procedure along with a matlab code can be found in Hwang et al. [9].

As a similar fashion that Hwang et al. [30] applied fuzzy clustering into latent curve model [31], Ryoo et al. [8] applied fuzzy clustering to latent class model. By applying fuzzy clustering to GSCA in the both models, latent curve model and latent class model, the cluster-level heterogeneity can be taken into account. The distinguished feature between Hwang et al. [30] and Ryoo et al. [8] is that the former focused on continuous indicators whereas the latter focused on discrete/categorical indicators. The fuzzy clusterwise GSCA for LCA follows the four steps: 1.

(Step 1) Identify clusters and estimate membership probabilities, ukim, as the initial procedure, where the initial fuzzy clustering works based on the response data.

In addition to the fuzzy clusterwise GSCA by Hwang et al. [9], the fuzzy clusterwise GSCA for LCA employs the optimal scaling to preserve measurement characteristics of categorical indicators proposed by Young [32] in Step 2. To estimate the parameters of GSCA in Step 2, we minimize the residual sums of their squares weighted with fixed ukim at Step 1 (or Step 3) as below:

(6)ϕ=∑k=1K∑i=1NukimSS(VkTsi−CkTWkTsi),subject to the probabilistic condition, ∑k=1Kukim=1. si is the optimally scaled vector such as si=f(zi) where f is a optimal scaling function. In Step 3, we update ukim by applying the objective function, Jm, in Eq. (1) for the fuzzy clustering. The fuzzifier, m, of ukim is often set up at 2 in practice [26], which is also used as a default in the R package, gscaLCA. The item response probabilities characterizing classes are estimated based on their membership within each class at the end of procedure. The standard error of estimation can also be calculated by the bootstrap method [33].

In addition to R2 type of model evaluation tools, FIT and AFIT, from GSCA [9], the package gscaLCA computes the fuzziness performance index (FPI) and the normalized classification entropy (NCE) recommended by Roubens [34]. They are defined as follows:

(7)FPI=1−K⋅(1N∑i=1N∑k=1Kuki2−1)K−1and

(8)NCE=−1N∑i=1N∑k=1Kukilog⁡ukilog⁡K.

Both FPI and NCE applies the criterion of “smaller is better” between 0 and 1, which helps researchers decide the number of clusters in the process of fitting LCA [9].

Data are the main input to the function gscaLCA, and they should be formatted as a data frame containing indicator variables and covariates. The function gscaLCA requires the indicator variables to be discrete or categorical. It, however, does not requires whether the categorical variables are integer or character. When any indicator variable is continuous, the function is still run by recognizing the type of variable as a categorical variable. Thus, a caution is necessitated. There is an option that a continuous indicator is assigned as a continuous. On the other hand, for the covariates, both discrete and continuous variable are available. When a covariate is categorical numeric variable, it is required to define this variable as a factor. Missing data should be coded as NA in gscaLCA. The missing values will be deleted for the analysis in the gscaLCA algorithm by applying a listwise deletion in the current version.

The package gscaLCA provides two sample datasets that are informative for exploring different situations: (1) categorical variables (binary and more than two categories) for indicators and (2) continuous or categorical variables for covariates.

To estimate LCA based on fuzzy clusterwise GSCA with the gscaLCA algorithm, the default function of gscaLCA can be called with the following arguments:

The command gscaLCA requires main seven options to fit the fuzzy clusterwise GSCA that are specified:

dat: Dataset to be used to fit a model of gscaLCA.

When a model of the fuzzy clusterwise GSCA with covariates is fitted, the additional three arguments are required:

covnames: A character vector. The names of columns of the dataset that indicate covariates in the model fitted.

In these arguments, there are kind of generic arguments such as dat, Boot.num, and multiple.Core that utilizes functions in R. On the other hand, ID.var, num.class, num.factor, covnames, cov.model, and multinomial.ref are unique functionalities associated with the gscaLCA package. Thus, a user considers what those values might be and needs to specify them to run gscaLCA.

The function gscaLCA returns an object involving many different elements. We have selected key features and recommend researchers to check all other arguments by using gscaLCA:

N: The number of observations used after applying listwise deletion when missing values exist.

When the covariates are involved in the analysis, the function gscaLCA returns eight additional elements:

cov_results.multi.hard: This is the main result of the multinomial regression with the hard partitioning.

In addition to the main function gscaLCA, the package gscaLCA provides a function which implements the second and third steps in the algorithm of fuzzy clusterwise GSCA with covariates (gscaLCR). The function is called gscaLCR. As aforementioned, fuzzy clusterwise GSCA with covariate includes three steps. Even if users fit the gscaLCA first without the covariates with the function gscaLCA, steps 2 and 3 of gscaLCR can be executed via the function gscaLCR.

R> gscaLCR(results.obj, covnames, multinomial.ref = "MAX")

The function gscaLCR requires three elements:

results.obj: The result object of gscaLCA.

The output of gscaLCR is the same as gscaLCA.

For the TALIS data, we used the three-class model (num.class = 3) with a single factor (num.factor = "ALLin1"). It is the optimal model with these data that was found through the model comparison. Regarding the number of class, it is expected to have three factors, motivation, pedagogy, and satisfaction, as the variable name and explanation suggests (see Section 4.1.1). Regarding the number of factors, all variables are supposed to be explained by a common latent variable, because TALIS data is a survey data is collected from U.S. teachers with a topic of teaching and learning. The following command with the function gscaLCA was implemented to fit the fuzzy clusterwise GSCA. Once the gscaLCA command is executed, it displays the degree of the completion process as percentages while gscaLCA is running. When the estimation completes, the summary function is available to display the results. The summary function print out the sample size for the analysis, model fit indices, estimated latent class prevalence, and item response probabilities.

The results report that 2,365 observations were used for the analysis, excluding 195 incomplete responses. FIT and AFIT were 0.5046 and 0.5033, respectively. With a single factor, FIT and AFIT are typically lower than the larger number of factors. The indices to evaluate the classification were relatively large (FPI = 0.8623 and NCE = 0.8742), but they are better than when the option num.factor is EACH for these data. The estimated latent class prevalences is 34.59%, 26.22%, and 39.20%. The conditional item response probabilities for each category per variable are also presented in a table. When the standard error and 95% credible interval of the model fit, the prevalence and conditional response probabilities are required, we can print out the objects through the following commands. These standard error was estimated by the bootstrap.

These response probabilities are used to define latent classes. In order to grasp the patterns of the probabilities, a visual representation of profiles based on the probabilities would be more helpful than numeric quantities in the output above. Once the gscaLCA function is executed, the graph is automatically created. When the command summary(T3) is executed, the resulting plot is displayed for all answer categories. When the plots for each answer category are required, it can also be printed by using the command T3$plot. Fig. 2 presents the conditional item response probabilities of the result, T3. Based on the patterns of responses in each class, we define “Pedagogy focused teachers”, “Motivated teachers”, and “Balanced teachers”. For example, for the second response, latent class 1 has relatively higher value in both Pdgg_1, and Pdgg_2 variables, but latent class 1 has relatively smaller value in other variables. Therefore, we define the first latent class as the “Pedagogy focused teachers”. For the second latent class, it has relatively higher value in both Mtv_1, and Mtv_2 variables, but has relatively smaller value in other variables. Therefore, we named the second latent class as the “Motivated teachers”. The third latent class has overall similar values across all five variables, thus, we called the third latent class as the “Balanced teschers”. Lastly, the membership probabilities of observations can be obtained from the membership of the saved objects, T3.

In this example, we demonstrate how to fit the gscaLCA with covaritate by using the AddHealth data. We used the three-class model (num.class = 3) with num.factor = “EACH”. which was found in the previous research, Park et al. [10]. For this example, we considered 5 indicators (Smoking, Alcohol, Drug, Marijuana, and Cocaine) and the gender covariate. The gender covariate was involved in fitting the GSCA model (cov.model = 1). Adding the covariate into the GSCA model depends on researchers' decision in their fields, but we recommend to add the covariates into the GSCA model when the covarites affect not only the membership probabilities but also the relationship between indicators and latent variables. When multiple covariates are considered, the indicators of adding the covariates into the GSCA model are presented as a vector. For example, when researcher uses the second covariate out of three covariates in the GSCA model, the option cov.model with c(0, 1, 0) can be used. Fitting gscaLCA with covariates using the function gscaLCA produces the results as follows:

The results report that 5,065 observations were used for the analysis after applying the listwise deletion. They also show that the model fit indices with the AddHealth data are acceptable. FIT and AFIT were 0.9993 and 0.9993, and they are close to 1. The indices to evaluate the classification were relatively low (FPI = 0.4658 and NCE = 0.5094). The estimated latent class prevalences are 54.30%, 20.23%, and 25.46%. These estimated prevalences are changed depending on whether we consider covariates into a GSCA model or not. The conditional item response probabilities are also presented for each category per variable. Like the example of the TALIS data, the results here provide the 95% standard error by using the following commands:

When a category of response is binary, a graph provided by this package shows the probabilities’ patterns of each category. In the AddHealth data, thus the graph is involved when the response is “Yes”, because the responses are binary (Yes or No),

From the plot shown in Fig. 3, three latent class can be defined as “the smoking and drinking class (Latent Class 1)”, “the binge drinking and heavy smoking class (Latent Class 3)”, and “the heavy substance user class (Latent Class 2)” as previously shown in Park et al. [10]. For the case which needs the graphs for all categories of answers, plots for all categories are provided in the results of gscaLCA. Here, the outcome element of A3_1 contains the graphs for each response in plot. The graphs can be extracted by using the following two commands: the one for the category of “Yes” and the other for the category of “No”, respectively.

To examine the effects of covariates on the membership probabilities, the function summary can be used as demonstrated before. Other options in partitioning and fitting regression (multinomial.soft, binomial.hard, and binomial.soft) are available as well in the function summary. In addition, the following command also provides the results of regressions:

As the example of A3_1, other options are available for hard and soft partitioning and multinomial and binomial regression.