Latent GOLD®'s cluster module provides the state-of-the-art in cluster analysis based on latent class models. Latent classes are unobservable (latent) subgroups or segments. Cases within the same latent class are homogeneous on certain criteria (variables), while cases in different latent classes are dissimilar from each other in certain important ways.

The traditional latent class model can be used to handle measurement and classification errors in categorical variables, and can accomodate avriables that are nominal, ordinal, continuous, counts, or any combination of these. Covariates can be included directly in the model as well for improved cluster description.

Latent GOLD® improves over traditional ad-hoc types of cluster analysis methods by including model selection criteria and probability-based classification. Posterior membership probabilities are estimated directly from the model parameters and used to assign cases to the classes.

See also Latent GOLD® Tutorial #1

A DFactor model is often used for variable reduction or to define an ordinal attitudinal scale. It contains one or more DFactors which group together variables sharing a common source of variation. Each DFactor is either dichotomous (the default option) or consists of 3 or more ordered levels (ordered latent classes).

In this way, Latent GOLD®’s factor module has several advantages over traditional factor analysis:

• Solutions are immediately interpretable and don’t require rotation
• The factors are assumed to be ordinal and not continuous
• No additional assumptions are required to estimate factor scores
• The observed variables can be nominal, ordinal, continuous, or counts, or any combination of these

See also Latent GOLD® Tutorial 2, Latent GOLD® Tutorial 6A, and Discrete Factor Models References on Discrete Factor models.

A Regression model is used to predict a dependent variable as a function of predictor variables in a homogeneous population.

Latent GOLD® makes it possible to estimate a regression model in a heterogeneous population as well by including a categorical latent variable. Each category of this latent variabe represents a homogeneous subpopulation (segment) having identical regression coefficients.

You can use informative diagnostic statistics to see whether multiple models are needed.

Each case may contain multiple records (regression with repeated measurements) to estimate a LC Growth or Event History model.

The appropriate model is estimated according to the dependent variable scale type:

• Continuous - Linear regression (with normally distributed residuals)
• Dichotomous (specified as nominal, ordinal, or a binomial count) - Binary logistic regression
• Nominal (with more than 2 levels) - Multinomial logistic regression
• Ordinal (with more than 2 ordered levels) -
  Adjacent-category ordinal logistic regression
• Count: Log-linear Poisson regression
• Binomial Count: Binomial logistic regression model

In addition to using predictors to estimate a regression model for each class, covariates can be specified to refine class descriptions and improve classification of cases into the appropriate latent classes.

See also Latent GOLD® Tutorial #3 , Latent GOLD® Tutorial #7A, Latent GOLD® Tutorial #7B, and Latent GOLD® Tutorial #8.

After performing a latent class analysis, you might wish to investigate the relationship between class membership and external variables. A popular three-step approach is to first estimate the latent class model of interest (step 1), then assign individuals to latent classes using their posterior class membership probabilities (step 2), and subsequently investigate the association between the assigned class memberships and external variables (step 3).

In step 2, classification errors are introduced when assigning individuals to latent classes. The estimates of the association with the external variables need to be corrected for classification errors to prevent a downward bias (Bolck, Croon, and Hagenaars, 2004). The Step3 module implements two bias adjustments procedures (Vermunt, 2010).

The Step3 module can be used with external variables predicting the class membership (Covariate option) or with external variables which are predicted by the class membership (Dependent option). These two types of external variables are also referred to as concomitant variables and distal outcomes, respectively.

You will also have the option to use modal or proportional assignment rules for assigning cases to latent classes and obtain an exact equation for scoring new cases.

See also Step-Three Tutorial #1 , Step-Three Tutorial #2 , andStep-Three Tutorial #3

Responses from conjoint/discrete choice data consists of a single choice from each choice task (Choice sets).

Latent class (LC) choice models analyze these data in a way that accounts for heterogeneity by allowing different population segments (latent classes) to express different preferences in making their choices.

For a first choice model, an extended multinomial logit model (MNL) is used to estimate the probability of making a specific choice as a function of choice attributes and individual characteristics (predictors).

Covariates may also be included in the model for improved description/ prediction of the
segments.

See also Choice Tutorial #1, Choice Tutorial #1A, Choice Tutorial #2, Choice Tutorial #3, and Choice Tutorial #4.

The sequential logit model is used for situations where two or more choices are selected from a choice set. This includes a 1st and 2nd choice, 1st and last choice (best-worst), or other partial rankings as well as a complete ranking of all alternatives.

See also Choice Tutorial #5 and Choice Tutorial #6

The adjacent-category ordinal logit model is used for situations where the response data consists of ratings as opposed to choices.

See also Choice Tutorial #7 and Choice Tutorial #7A

Replication weights may be used to handle designs where respondents
allocate a number of votes (purchases, points) among the various choice alternatives.

The latent Markov model is a popular longitudinal data variant of the standard latent class model; it is in fact a latent class cluster model in which individuals are allowed to switch between clusters across measurement occasions.

The clusters are now called latent states. The Latent Markov model is also referred to as the Latent Transition model.

Latent GOLD® implements the more general mixture Latent Markov model where different latent classes are allowed to have different transition probabilities.

See also Markov Tutorial #1 , Markov Tutorial #2, and Markov Tutorial #3

CFactors can be used to specify continuous latent variable models, such as factor analysis, item response theory models, latent trait models, and regression models with continuous random effects. The CFactors can be included in any LC Cluster, DFactor or LC regression model.

If included, additional information pertaining to the CFactor effects appear in the Parameters output and to CFactor scores in the Standard Classification, the ProbMeans, and the Classification Statistics output.

See also Advanced Tutorial: Latent GOLD 4.5 and IRT Modeling

This advanced option is used to specify a multilevel extension to an LC Cluster
, DFactor or LC Regression model which allows for explanation of the heterogeneity not only at the case level, but also at the group level.

Group-level variation may also be accounted for by specifying group-level latent classes (GClasses) and/or group-level CFactors (GCFactors). In addition, when 2 or more GClasses are specified, group-level covariates (GCovariates) can be included in the model for improved description/ prediction.

The multilevel option can also be used for specifying three-level parametric or nonparametric random-effects regression models or to develop group-level and individual level segments simultaneously.

Two important survey sampling designs are stratified sampling -- sampling cases within strata, and two-stage cluster sampling -- sampling within primary sampling units (PSUs) and subsequent sampling of cases within the selected PSUs. Moreover, sampling weights may exist.

The Survey option takes the sampling design and the sampling weights into account when computing standard errors and related statistics associated with the parameter estimates, and estimates the ‘design effect'.

The Syntax system is an intuitive command language that offers you additional flexibility on top of the graphical user interface (GUI).

Option include:

• More flexible modeling and parameter restrictions by specifying intuitive LG-Equations™
• Additional models compared to the GUI Cluster, DFactor, Regression, Step3, Markov, and Choice modules
• Monte Carlo simulation options
• Multiple imputation options
• N-fold validation and holdout options
• Additional output and saving options
• Options to use saved parameters (e.g., for scoring)

See also Syntax Tutorial #1, Syntax Tutorial #2 and LG Syntax User's Guide.

The ability to include a scale factor in choice models, which may vary across predictor values and/or scale latent classes.

See also Choice Tutorial #8A

Two important applications of Scale Adjusted Latent Class (SALC) models are:

• including scale classes (sClasses) in addition to latent segments (Classes) in choice models, and
• including separate scale factors for best and worst choices with BestWorst data (using the predictor option).See also Choice Tutorial #10A, Choice Tutorial #10B, and Choice Tutorial #11A

Chorus (2010, 2012) proposed a class of choice models based on Random Re-
gret Minimization (RRM) as an alternative to Random Utility Maximization
(RUM).

While the assumed behavioral mechanism underlying RUM-based
models is that individuals select the alternative having the largest utility,
RRM-based models assume that individuals select the alternative having the
smallest potential regret.

A recent study evaluating RRM applications in
various domains showed that latent class approaches, where the decision rule
(RUM or RRM) differs per class, lead to substantial improvements in model
fit compared to models assuming the same decision rule (usually RUM) for
every class (Chorus, van Cranenburgh, and Dekker, 2014).