Course Text: Pages 11-14
A typical exploratory LC analysis consists of three steps. During Step 1, the number of classes K* is determined, and the model parameters for that K*-class model are estimated. This is similar to Step 1 in a traditional exploratory factor analysis where the number of factors are determined, and estimates for the model parameters (factor loadings and factor correlations) are obtained. Up to now, we have been learning about and discussing topics related to Step 1 of LC modeling.
During Step 2 in LC modeling, known as the Classification step, the class-specific parameter estimates obtained in Step 1, together with the specific response pattern observed for a given case are utilized to compute for that case (i.e., for each respondent), K* posterior membership probabilities. The posterior probabilities (often referred to more simply as ‘the posteriors’) represent the probability of this case belonging to each of the K* latent classes. These posteriors are then used to classify each respondent into the appropriate latent class. This is similar to Step 2 in factor analysis where scores for each factor are estimated for each respondent.
The word posterior is used to differentiate these probabilities from the prior probabilities, (often referred to more simply as ‘priors’), corresponding to the class sizes, which are computed without taking into account individual responses. Transformation of the prior probabilities to the posteriors is an application of Bayes theorem, resulting in ‘scoring equations’ that show how to compute the posteriors as a function of the observed response patterns.
By checking the box for ‘Classification – Posterior’ in the Output tab of LatentGOLD, an output section by that name is produced following model estimation containing tabular classification output. Each observed response pattern forms a row of the table, and the posterior membership probabilities and modal class assignment for cases with these response patterns form columns of the table. In addition, the number of cases having this response pattern is provided in a separate column.
For an example, see pages 16-18 of LG tutorial 1.
Classification information for each individual respondent can also be output to a file using the ClassPred tab of LatentGOLD, as shown in Figure 21 on page 17 of LG Tutorial 1. The format of the output file will be the same as that of the input file. For example, if the input file represents case level data in a SPSS .sav file, the output file will also be a SPSS file, which can contain the case IDs and other variables from the original file if requested on the ClassPred tab,
Scoring equations can also be requested to be output by checking the box marked ‘Scoring Equations’ in the Output tab (see the last paragraph page 21 of LG Tutorial 1. For more detail on the scoring equations, see the section below titled ‘Scoring New Cases’.
During Step 3, the resulting classifications may be related to external variables, not used in the Step 1 modeling process. Specifically, the Step3 module implemented in LatentGOLD 5.1 allows the latent classes to be related to exogenous predictor variables (covariates) in a multinomial logit regression model (Topic H), or used to predict one or more exogenous dependent variables (distal outcomes). In addition, the LG syntax module can be used to relate the latent classes to both covariates and distal outcomes (see Step 3 Tutorial 1).
Scoring New Cases
One-Step Scoring
In addition to tabular classification output being provided for each response pattern and classification output to a file for each individual respondent, the formulae for translating response patterns to posterior membership probabilities can be output in the form of K* multinomial logit scoring equations. This is most useful when you want to classify new cases into the most appropriate class based on their responses.
Version 5.1 and later releases of LatentGOLD contains options to output the scoring equations during Step 1 (i.e., during the Estimation step) by checking the box preceding ‘Scoring Equations’ in the Output tab of LatentGOLD prior to estimating the model. This process is called ‘One-Step scoring’ because the scoring equations are obtained during the Step 1 run where the parameters are estimated. These equations are derived directly from the estimated LC model parameters, and thus have the correct functional form. For the 3-class LC cluster model based on the GSS data, we will see that the scoring equations take the form of the linear multinomial logit model (Exercise E3) while in the case of continuous indicators with class-dependent variances, the equations contain quadratic terms (Exercise L2).
In addition to the tabular output, these parameter-based scoring equations can also be obtained in the form of an SPSS syntax, as a generic set of if-then statements, or as R-code, by checking the box ‘Scoring Syntax’ in the Output tab. Thus, after obtaining a LC model to your liking, if you want to obtain the scoring equations to classify new cases you can do so by re-estimating the model after selecting the desired output option(s).
Step3 Scoring
An alternative way to generate scoring equations is to estimate them as a special case of LatentGOLD’s Step3 module, by entering the indicators from Step 1 as covariates in the Step3 module. The posterior probabilities will then be predicted using a multinomial logit model. When this prediction is perfect, the resulting model fit L2 statistic will equal 0, in which case the scoring equations obtained in Step 3 will be identical to those obtained using One-Step scoring, and will reproduce the posterior membership probabilities perfectly.
Unlike the ‘One-Step scoring’ approach where the proper functional form (e.g., linear, quadratic) is determined automatically by LatentGOLD, the Step 3 approach requires you to select the terms to include in the scoring model. For example, if quadratic terms belong in the model, excluding these terms from the scoring equations will result in a non-zero L2 value, signaling a fit that is less than perfect.
One application of the Step3 module might be to obtain scoring equations (for scoring new cases) that is based on a reduced set of indicators as predictors. For example, Step3 scoring equations might be developed with different subsets of predictors and evaluated using the model fit L2 to see which subset provides the best classifications (e.g., which subset yields the lowest value for L2 and a sufficiently low misclassification error).
In Exercise E3 we will see how to use LatentGOLD to obtain scoring equations using both the One-Step scoring and Step3 scoring approaches. In particular, Step 3 Tutorial 2 begins by obtaining scoring equations from the Step3 scoring module, showing that L2 = 0, and then confirming that the equations are identical to those obtained using One-Step scoring.
Assigned Reading:
E: Course Text (pages 11-14), LatentGOLD Technical Guide (section 6.2)
E: LG tutorial 1: (pages 14-17, 20)