F. Interpreting LatentGOLD output

Course Text: Pages 15-20

Parameters and Profile Output

As mentioned earlier (Topic B), the LC cluster model can be expressed in terms of probability parameters or log-linear parameters. Internally, LatentGOLD estimates parameters of the LC cluster model using the log-linear form of the model, and presents these log-linear parameters along with standard errors and p-values in the Parameters Output. LatentGOLD then translates these parameters into the equivalent, more easily interpretable probability Parameters, which are provided in the Profile Output.

The log-linear and probability forms of the LC Cluster model provide equivalent information. See Magidson and Vermunt, 2001 (DFactor.pdf) for technical details of the relationship between the log-linear and probability parameters — section 2.1 provides the log-linear form of the model, while Appendix A (page 255) provides the corresponding probability parameter form for an example with four categorical indicators. This paper also derives the DFactor model as a restricted form of the LC Cluster model.

A special sub-category within the Parameters Output uses Wald tests to assess the difference in the effects of each indicator on each pair of latent classes. This allows testing which pair of classes (clusters) are significantly different in terms of each indicator. For further details of this Paired Comparisons Output and associated Wald tests see section 8.2 of the  LatentGOLD Technical Guide

Output from Latent Class Tree (LCT) Models

Up to now, we have focused on the traditional (standard) approach to latent class modeling. LatentGOLD 6.0 includes the ability to estimate latent class tree (LCT) models which utilize a hierarchical paradigm for performing latent class analysis that is somewhat similar to hierarchical clustering. In this online course you will have the opportunity to experiment with LCT models and view interactive output for these models.

Specifically, in the introductory tutorial for LCT (see Exercise F2), you will develop a 3-class model in the usual way, and then for comparison you will develop a hierarchical 3-class LCT model. The latter model will be developed automatically by LatentGOLD by first splitting the sample into two subgroups (two parent classes which are equivalent to the latent classes from a standard 2-class LC model), and then further splitting parent class #2 into two child classes in order to obtain an acceptable overall model fit with 3-classes.

The three resulting segments are represented as terminal nodes in the tree diagram below, where segment numbers ‘1’, ‘2’ and ‘3’ appear below the terminal nodes associated with the three segments.

The content that appears in each node of the tree (see tree above) provide information on the fit of the tree model. In addition, the more usual Parameter, Profile and other output can be obtained in LatentGOLD 6.0 by clicking on any parent node. For example, clicking on Parent Node 2 (Node 2 is highlighted in the tree above) will display the output associated with the 2-class model where the 2 classes are Child Nodes 21 and 22.

Here is some background on LCT, so you can get the most out of the LatentGOLD interactive implementation.

First, an LCT model is based on a sequential algorithm that proceeds as follows:

• Step 1: Using the full sample at the root of the tree, a standard LC model is estimated with K “basic” or “starting” classes, where K typically is small (say 2, 3, or 4), and thus results from the LC tree model will be more easily interpretable than a traditional LC model with more classes.
• Step 2: Subsequently, K child nodes are formed by splitting the sample based on the posterior membership probabilities from this K-class model.
• Step 3: Specifically, for each of the newly formed child nodes, 1- and 2-class models are estimated.
• Step 4: If the 2-class model fits better than the 1-class model (say based on the BIC*), the node concerned becomes a parent node with its own two child nodes obtained by splitting the sample at this node based on the posterior membership probabilities, and the algorithm returns to step 3. Otherwise, it goes to step 5.
• Step 5: The node concerned serves as an end node.

*Note: By default, LatentGOLD displays the BIC as the content in each tree node of the tree. The content displayed (for all nodes) can be changed by right-clicking on any node to open the ‘Tree Node Display’ and checking ‘LL’ or  any other item(s) to display those items.

In their first proposal, van den Bergh et al., 2017 worked only with binary splits, implying that the LC model at the root node (step 1 above) is also a 2-class model. Later,  van den Bergh et al., 2018 showed that it might be better to use a larger number of classes K at the root node, and proposed a new method to determine the value of K.

It is important to note that the K-class model estimated at the root, defines the structure of the models that will also be used at the next levels. That is, at each of the next levels of the tree, the same model is estimated (with modified weights), but now with the third and next classes restricted to be empty (to have a class proportion equal to 0). This is how the K-class model used at the root is transformed into a model with 2 classes at the next levels of the tree.

Three final things to note regarding to the LCT implementation in LatentGOLD 6.0 are:

• At each split, the latent classes are reordered according to their size (from large to small) to deal with label switching. This prevents obtaining a seemingly different tree when rerunning the same LCT model.
• In the LG GUI modules, a LCT model is estimated behind the scenes using the LG Syntax system. Thus, the output obtained at each node of the tree (e.g., for Parameters), will have the Syntax type format.
• The coding for the tree latent variable is set to dummy first (or coding=first). This is done to assure that one obtains meaningful Parameters when K is 3 or larger. When estimating such models with the Syntax module, this coding can be suppressed using the “coding=…” option for the tree latent variable.

The primary example used by Van den Berg et. al. (2017, 2018) is available as a demo dataset (see exercise F3). In addition, our Latent Class Tree Tutorial introduces the interactive LC Tree modeling capabilities implemented in LatentGOLD in a simple example with 5 dichotomous indicators. In practice, LC tree models are most useful when the number of indicators is large, as in the examples provided by Van den Berg et. al.

GUI vs. Syntax module differences in building LC Tree models.

While our tutorial (below) uses the Model tab menu in the LC Cluster GUI module to estimate LCT models, with the more general LG Syntax, you would use the keyword tree to indicate a LCT model is to be developed, (and for models with more than one latent variable, which latent variable is the one that should be used for splitting).

The simplest example of a LCT syntax model is:

latent Cluster nominal 3 tree;

When running this model, a LC Tree will be formed that splits at the root into 3 classes using the posteriors from a standard 3-class model. At the next levels, the splitting procedure continues using binary splits, and stops when one of the stopping criteria is reached (see tree options below).

It is also possible to construct the tree exactly as one wishes, that is, by specifying which nodes should be split using the syntax. This is achieved with the option split=(list of nodes), for example,

latent Cluster nominal 3 tree split=(1 3 31);

This specification implies that Cluster 1 and 3 (Node 1 and 3) from the initial LC model should be split, as well as the first child node obtained when splitting Cluster 3 (Node 31).

Further description of LCT options

Whether to split a node (yes or no) will typically be decided based on the increase in LL between the 1- and 2-class model. The minimally required LL difference can be defined by the user (option minLLdiff=<number>) or be derived from the BIC, AIC, or AIC3 (options BIC, AIC and AIC3). An alternative criterion that one can select for this purpose is the minimum entropy R-squared value threshold for the 2-class model to achieve in order to split into these 2 classes (minentropyR2=<number>).

Alternative criteria for not splitting can also be specified. These include when the sample at a node is too small (minN=<number>), when the largest BVR in the 1-class model is too small (minBVR=<number>), when a certain maximum number of levels is reached (maxlevels=<number>), or when a certain maximum number of nodes is reached (maxnodes=<number>). Note that these latter stop criteria differ from those discussed above in that these are evaluated before new 1- and 2-class models are estimated at the node concerned, except for the stop criterion based on the BVRs, which requires estimating the 1-class model.

In this course you will have the opportunity to experiment with LCT models and view new output for LCT models that will appear in LatentGOLD 6.0. In particular:

Parameters and Profile output listings are extended to include LCT output:

• Output from the first (parent) level of the Tree is displayed in the usual way in the standard Parameters and Profile Output listings.
• Output from child classes spawned by splitting of each parent class is provided in a separate output section.

Exercise F2 introduces this new output along with a new graphical tree display using a simple example.

Assigned Reading:

Updated Sage Article:

Magidson, J., and Vermunt, J.K. (2004). Latent class models. In: Kaplan., D (Ed.), The Sage Handbook of Quantitative Methodology for the Social Sciences, Chapter 10, 175-198. Thousand Oaks: Sage Publications.

F1: Significance Tests, section 2.2: (Update Sage article pages 8-9) F2: Graphical Displays, section 2.4: (pages 11-12, 53)

van den Bergh et al., 2017

van den Bergh et al., 2018

LG Tutorial #1 :   LG Tutorial1

F1: Parameters Output, (Tutorial 1 pages 10-11)

F2: Pairwise Comparisons (Tutorial 1 page 11-12)

F3: ProbMeans Output, (Tutorial 1page 15)

LG Tutorial on Estimating a LC Tree Model: Latent Class Tree Tutorial