Latent Rank Theory
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Data Format	LRT-SOM		LRT-GTM
	Exametrika	Document	Exametrika	Document
Dichotomous Data		RN08-01		RN08-06
Nominal Polytomous		RN07-03
Ordered Polytomous		RN07-21
Continuous
LRT-GTM Model
The LRT-GTM model is a latent rank model that uses a generative topographic mapping (GTM) mechanism. Latent classes are ordered by adding a smoothing technique to the latent class analysis, which is done using the EM (expectation-maximization) algorithm. Since this model is a batch learning neural network model, its execution time is shorter than that of the LRT-SOM model. Therefore, the LRT-GTM model is more suitable for analyzing large sample size data sets particularly when the sample size is larger than 10,000.
LRT-SOM Model
The LRT-SOM model is a latent rank model that uses a self-organizing map (SOM) mechanism. Since this model is a sequential learning neural network model, its execution time is longer than that of the LRT-GTM model, but the obtained item reference profile (IRP) by the LRT-SOM model is a little smoother than that of the LRT-GTM model. In the original SOM model, the solution is not unique. That is, the result of each calculation differs slightly even when the same data are analyzed. This is because data is input into the model in random order during the sequential learning. However, “predictable randomness” is introduced into the LRT-SOM model. That is, using the same random seed when analyzing the same data, we can make the results consistent between calculations.
Analysis Setting
Estimation Model l The default model is the “dichotomous model.” l Specify the “dichotomous model” when analyzing binary true/false data. l Specify the“nominal model” when analyzing correct and incorrect choices. l Specify the “graded model” when analyzing psychological rating scale (Likert scale) data.
Number of Latent Ranks l The default setting for the number of latent ranks is 10. l A small number of ranks is preferable when the sample size or the number of items is small. l The weakly ordinal alignment condition is easier to satisfy the smaller the number of latent ranks.
Prior Distribution l By checking [Prior Distribution], the latent rank estimate of an examinee with higher score (number-right score) is estimated to be higher.
Monotonicity Constraint l The IRP increases monotonically. l The correct choice for the item category reference profile (ICRP) is monotonic for the nominal model. l The boundary category reference profile (BCRP) is monotonic when analyzing data using the graded model. l The strongly ordinal alignment condition is always satisfied.
Target Latent Rank Distribution l A quasi-equiprobability rank scale is obtained when “uniform distribution” is specified. l The equiprobability rank scale is the scale in which the frequency of each latent rank is equal. l The latent rank distribution is like a normal distribution when “normal distribution” is specified. l The default setting of the target distribution for the LRT-SOM model is “uniform distribution.”
Equating (IRP fixing sheet) l Equating is executed using the concurrent calibration (CC) method. l The CC method is used to estimate the IRPs of only the equated items (the IRPs of the remaining items are fixed). l The ICRPs of the equated items are fixed for both the nominal and graded models.
Output Options
Fit Indices l Fit indices for the complete test and each item are output. l The indices are calculated by comparing the present model results to those of benchmark and null models. l The benchmark model is a LRT-GTM model which number of latent ranks is 50 with monotonicity constraint. l The indices are helpful in determining the number of latent ranks. l Output indices ・Chi-square (df, p-value) ・NFI (normed fit index; Bentler & Bonnet, 1980) : [0, 1] the larger, the better ・RFI (relative fit index; Bollen, 1986) : [0, 1] the larger, the better ・IFI (incremental fit index; Bollen, 1989) : [0, 1] the larger, the better ・TLI (Tucker-Lewis index; Bollen, 1989) : [0, 1] the larger, the better ・CFI (comparative fit index; Bentler, 1990) : [0, 1] the larger, the better ・RMSEA (root mean square error of approximation; Browne & Cudeck, 1993) : [0, ∞] the smaller, the better l Information criteria are relative indices to compare two or more model candidates. The smaller the value, the more efficient the fit of the model to the data. ・AIC (Akaike information criterion; Akaike, 1987) ・CAIC (consistent AIC; Bozdogan, 1987) ・BIC (Bayesian information criterion; Schwarz, 1978)
Observation Ratio Profiles l This option is useful when the test items differ among examinees. l These profiles are used for examining which items are selected depending on the examinees in each latent rank. l This option cannot be used unless “missing indicator” is specified on the item selection form. l Note that a “nonresponse” is considered to be incorrect data and should not be treated as missing data.
IRP Graphs l The IRP of each item is graphed. l The ICRP of each item is also graphed when the nominal model is specified. l The ICRP and BCRP of each item are also graphed when the graded model is specified. l The test reference profile is graphed. l The latent rank distribution and rank membership distribution are graphed.
IRP Coloring l The element of each IRP estimate is colored in accordance with the size of the element. l The element of each BCRP estimate is also colored when the graded model is specified.
Sorting by Correct Response Rate l In worksheet “Items”, the output items are sorted in descending order of correct response rate. l In worksheet “Items”, the output items are sorted in descending order of mean value when the graded model is specified. l This option is useful for creating “can-do chart.”
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