# Bootstrapping improvement testing in unrestricted latent class models

The comparison of latent class models with differing numbers of classes faces the so-called boundary problem. This problem is common in finite mixture models of which latent class models are a special case. The difficulty arises because one of the latent probabilities is set to zero to fit the model with the reduced number of classes under the null hypothesis. In turn, this violates one of the regularity conditions necessary for the difference test statistic, generally the likelihood ratio, to converge to a chi-square distribution. Thus, its distribution is unknown. The parametric bootstrap has been proposed to cope with this situation A simulation study was carried out to assess the Type I Error of test statistics commonly used in improvement tests of latent class models. Conditions varied were: sample size, and 10 one-class models. The statistics investigated were: the Pearson's chi-square, the likelihood ratio, the Cressie-Read (lambda = 2/3), and the Freeman-Tukey. These statistics, when regularity conditions are met, converge to a chi-square distribution with the same degrees of freedom Results indicate that the actual Type I Error rate may be about double the nominal level when testing a one-class model versus a two-class model at the 0.1 level. Likewise, there is strong evidence in favor of the parametric bootstrap as all four statistics maintain the nominal desired level under this procedure. For a few conditions associated with small sample sizes the bootstrapped statistics were conservative