Simulation results

In each case, the results of simulation study are represented in six tables which include the average parameters estimates, average conditional probabilities, average latent prevalences, average correlation coefficients, and average match proportions for 100 replications separately. We shall explain these results later. The simulation results for 3-class model with 100 sample sizes are presented from Table 7 to Table 12. The simulation results for 3-class model with 500 sample sizes are presented from Table 13 to Table 18. The simulation results for 6-class model with 300 sample sizes are presented from Table 19 to Table 24. The simulation results for 6-class model with 1000 sample sizes are presented from Table 25 to Table 30. The simulation results for 2-class model with 150 sample sizes are presented from Table 31 to Table 36. The simulation results for 2-class model with 700 sample sizes are presented from Table 36 to Table 42. According to Table 7 ~ Table 42, we can see that these results of the k-means method and divisive hierarchical using correlation coefficients measurement are similar to those of k-means method and divisive hierarchical using covariance measurement. So, we shall discard the results of k-means method and divisive hierarchical using covariance measurement in the following discussion.

First we discuss the simulation results which are presented from Table 13 to Table 18 of 3-class model with 500 sample sizes.

Average parameters estimations

Table 12 and Table 13 under the column “TRUE” include all

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in simulated data. All average of

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estimates got from the k-means method using correlation coefficient measurement (K_Corr) and covariance measurement (K_Cova) separately and the divisive hierarchical method using correlation coefficient measurement (D_Corr) and covariance measurement (D_Cova) appeared in Table 12 and Table 13. Table 12 and Table 13 can demonstrate that the parameters estimates got from the k-means method are well compared to the true parameters. But the parameters estimates got from the divisive hierarchical procedure are poor. Furthermore, the divisive hierarchical procedure is sensitive to cluster structure. This means that hierarchical procedure have the chance to perform more well only when there is clear cluster structure than when there is no clear cluster structure.

Table 12 and Table 13 also include the standard errors of parameters estimates in doing multinomial regressions, (4.1) and (4.2), and the average sample standard errors of the parameters estimates for 100 replications. The sample standard errors of the estimates for 100 replications include the variation of doing multinomial regression and creating cluster membership. Because we use the multinomial regression to estimate parameters under the assumption of known cluster membership, the standard errors of parameters estimates in doing multinomial regression did not include the variations of creating cluster membership. Therefore, the standard errors of parameters estimates in doing multinomial regressions should be smaller than the sample standard errors of the estimates for 100 replications. This is demonstrated in

Table 12 and Table 13. However this is not demonstrated in Table 7 and Table 8 for the 3-class model with 100 sample sizes which gave few individuals per parameter.

For the sparse data, the estimated standard errors of parameters estimates in doing multinomial regressions are not accurate. Therefore, the standard errors of parameters estimates in doing multinomial regressions are not always smaller than the sample standard errors of the estimates over 100 replications for the 3-class model with 100 sample sizes.

Average Conditional Probabilities

Table 14 under the column “TRUE” displays the RLCA conditional probabilities evaluated at the sample means of the incorporated covariates:

The average of estimated conditional probabilities over 100 replications with k-means and divisive hierarchical methods appear in Table 14. The estimated conditional probabilities for k-means and divisive hierarchical methods are

j

Overall, the average conditional probabilities for the k-means method are more closed to the true conditional probabilities than the average conditional probabilities for the divisive hierarchical method.

Average Latent Prevalence:

Table 15 under the column “TRUE” displays the sample average of the RLCA

prevalences:

The average of estimated prevalences over 100 replications with k-means and divisive hierarchical methods are also shown in Table 16. The estimated prevalences are

Overall, the average latent prevalences for the k-means method are more closed to the true prevalences than the average latent prevalence for the divisive hierarchical clustering method.

Average Correlation Coefficients

We evaluated theMCov_kof the objects in the same cluster k. Table 16 displays the average of MCov_k over 100 replications in each cluster k. The k-means approach resulted smaller average correlation coefficients than the divisive hierarchical method.

Next, for the 6-class model with 1000 sample sizes, we shall discuss the simulation results which are presented from Table 22 to Table 26. These tables show that the results of whether the k-means procedure or the divisive hierarchical procedure are poor obviously comparing to the 3-class model with 500 sample sizes.

The 2-class model with 150 and 700 sample sizes, we shall discuss the simulation results which are presented from Table 27 to Table 36. It is reasonable that the results of k-means and divisive hierarchical clustering methods for 2-class RLCA models are the same.

When we use maximum likelihood to estimate the parameters in (3.2), the maximum likelihood estimation (MLE) is relative to the number of individuals given in per parameter. For the spare data which gave less individuals per parameter, the MLE can not be obtained or the MLE is not a good estimation .For the three models,

3-class RLCA with 100 sample sizes, 6-class RLCA with 300 sample sizes and 2-class RLCA with 150 sample sizes, which gave less individuals per parameter, the simulation results are not wore than those that gave more individuals per parameter. It demonstrates that our clustering procedure is irrelative to the number of individuals given per parameter.