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Growth mixture modeling is a popular analytic tool for longitudinal data analysis. It detects latent groups based on the shapes of growth trajectories. Traditional growth mixture modeling assumes that outcome variables are normally distributed within each class. When data violate this normality assumption, however, it is well documented that the traditional growth mixture modeling mislead researchers in determining the number of latent classes as well as in estimating parameters. To address nonnormal data in growth mixture modeling, robust methods based on various nonnormal distributions have been developed. As a new robust approach, growth mixture modeling based on conditional medians has been proposed. In this article, we present the results of two simulation studies that evaluate the performance of the median-based growth mixture modeling in identifying the correct number of latent classes when data follow the normality assumption or have outliers. We also compared the performance of the median-based growth mixture modeling to the performance of traditional growth mixture modeling as well as robust growth mixture modeling based on t distributions. For identifying the number of latent classes in growth mixture modeling, the following three Bayesian model comparison criteria were considered: deviance information criterion, Watanabe-Akaike information criterion, and leave-one-out cross validation. For the median-based growth mixture modeling and t -based growth mixture modeling, our results showed that they maintained quite high model selection accuracy across all conditions in this study (ranged from 87 to 100%). In the traditional growth mixture modeling, however, the model selection accuracy was greatly influenced by the proportion of outliers. When sample size was 500 and the proportion of outliers was 0.05, the correct model was preferred in about 90% of the replications, but the percentage dropped to about 40% as the proportion of outliers increased to 0.15. 

Selected response items and constructed response (CR) items are often found in the same test. Conventional psychometric models for these two types of items typically focus on using the scores for correctness of the responses. Recent research suggests, however, that more information may be available from the CR items than just scores for correctness. In this study, we describe an approach in which a statistical topic model along with a diagnostic classification model (DCM) was applied to a mixed item format formative test of English and Language Arts. The DCM was used to estimate students’ mastery status of reading skills. These mastery statuses were then included in a topic model as covariates to predict students’ use of each of the latent topics in their written answers to a CR item. This approach enabled investigation of the effects of mastery status of reading skills on writing patterns. Results indicated that one of the skills, Integration of Knowledge and Ideas, helped detect and explain students’ writing patterns with respect to students’ use of individual topics. 

Abstract The prebiotic synthesis of ribonucleotides is likely to have been accompanied by the synthesis of noncanonical nucleotides including the threo-nucleotide building blocks of TNA. Here, we examine the ability of activated threo-nucleotides to participate in nonenzymatic template-directed polymerization. We find that primer extension by multiple sequential threo-nucleotide monomers is strongly disfavored relative to ribo-nucleotides. Kinetic, NMR and crystallographic studies suggest that this is due in part to the slow formation of the imidazolium-bridged TNA dinucleotide intermediate in primer extension, and in part because of the greater distance between the attacking RNA primer 3′-hydroxyl and the phosphate of the incoming threo-nucleotide intermediate. Even a single activated threo-nucleotide in the presence of an activated downstream RNA oligonucleotide is added to the primer 10-fold more slowly than an activated ribonucleotide. In contrast, a single activated threo-nucleotide at the end of an RNA primer or in an RNA template results in only a modest decrease in the rate of primer extension, consistent with the minor and local structural distortions revealed by crystal structures. Our results are consistent with a model in which heterogeneous primordial oligonucleotides would, through cycles of replication, have given rise to increasingly homogeneous RNA strands.