Recurrent Multinomial Models for Categorical Sequences

This paper presents a model of recurrent multinomial sequences. Though there exists a quite considerable literature on modeling autocorrelation in numerical data and sequences of categorical outcomes, there is currently no systematic method of modeling patterns of recurrence in categorical sequences. This paper develops a means of discovering recurrent patterns by employing a more restrictive Markov assumption. The resulting model, which I call the recurrent multinomial model, provides a parsimonious representation of recurrent sequences, enabling the investigation of recurrences on longer time scales than existing models. The utility of recurrent multinomial models is demonstrated by applying them to the case of conversational turn-taking in meetings of the Federal Open Market Committee (FOMC). Analyses are effectively able to discover norms around turn-reclaiming, participation, and suppression and to evaluate how these norms vary throughout the course of the meeting.

Divisible Statistics and Their Partial Sum Processes: Asymptotic Properties and Applications

<p><b>Divisible statistics have been widely used in many areas of statistical analysis. For example, Pearson's Chi-square statistic and the log-likelihood ratio statistic are frequently used in goodness of fit (GOF) and categorical analysis; the maximum likelihood (ML) estimators of the Shannon's and Simpson's diversity indices are often used as measure of diversity; and the spectral statistic plays a key role in the theory of large number of rare events. In the classical multinomial model, where the number of disjoint events N and their probabilities are all fixed, limit distributions of many divisible statistics have gradually been established. However, most of the results are based on the asymptotic equivalence of these statistics to Pearson's Chi-square statistic and the known limit distribution of the latter. In fact, with deeper analysis, one can conclude that the key point is not the asymptotic behavior of the Chi-square statistic, but that of the normalized frequencies. Based on the asymptotic normality of the normalized frequencies in the classical model, a unified approach to the limit theorems of more general divisible statistics can be established, of which the case of the Chi-square statistic is simply a natural corollary.</b></p> <p>In many applications, however, the classical multinomial model is not appropriate, and an extension to new models becomes necessary. This new type of model, called "non-classical" multinomial models, considers the case when N increases and the {Pni} change as sample size n increases. As we will see, in these non-classical models, both the asymptotic normality of the normalized frequencies and the asymptotic equivalence of many divisible statistics to the Chi-square statistic are lost, and the limit theorems established in classical model are no longer valid in non-classical models.</p> <p>The extension to non-classical models not only met the demands of many real world applications, but also opened a new research area in statistical analysis, which has not been thoroughly investigated so far. Although some results on the limit distributions of the divisible statistics in non-classical models have been acquired, e.g., Holst (1972); Morris (1975); Ivchenko and Levin (1976); Ivchenko and Medvedev (1979), they are far from complete. Though not yet attracting much attention by many applied statisticians, another advanced approach, introduced by Khmaladze (1984), makes use of modern martingale theory to establish functional limit theorems of the partial sum processes of divisible statistics successfully. In the main part of this thesis, we show that this martingale approach can be extended to more general situations where both Gaussian and Poissonian frequencies exist, and further discuss the properties and applications of the limiting processes, especially in constructing distribution-free statistics.</p> <p>The last part of the thesis is about the statistical analysis of large number of rare events (LNRE), which is an important class of non-classical multinomial models and presented in numerous applications. In LNRE models, most of the frequencies are very small and it is not immediately clear how consistent and reliable inference can be achieved. Based on the definitions and key concepts firstly introduced by Khmaladze (1988), we discuss a particular model with the context of diversity of questionnaires. The advanced statistical techniques such as large deviation, contiguity and Edgeworth expansion used in establishing limit theorems underpin the potential of LNRE theory to become a fruitful research area in future.</p>

Divisible Statistics and Their Partial Sum Processes: Asymptotic Properties and Applications

<p><b>Divisible statistics have been widely used in many areas of statistical analysis. For example, Pearson's Chi-square statistic and the log-likelihood ratio statistic are frequently used in goodness of fit (GOF) and categorical analysis; the maximum likelihood (ML) estimators of the Shannon's and Simpson's diversity indices are often used as measure of diversity; and the spectral statistic plays a key role in the theory of large number of rare events. In the classical multinomial model, where the number of disjoint events N and their probabilities are all fixed, limit distributions of many divisible statistics have gradually been established. However, most of the results are based on the asymptotic equivalence of these statistics to Pearson's Chi-square statistic and the known limit distribution of the latter. In fact, with deeper analysis, one can conclude that the key point is not the asymptotic behavior of the Chi-square statistic, but that of the normalized frequencies. Based on the asymptotic normality of the normalized frequencies in the classical model, a unified approach to the limit theorems of more general divisible statistics can be established, of which the case of the Chi-square statistic is simply a natural corollary.</b></p> <p>In many applications, however, the classical multinomial model is not appropriate, and an extension to new models becomes necessary. This new type of model, called "non-classical" multinomial models, considers the case when N increases and the {Pni} change as sample size n increases. As we will see, in these non-classical models, both the asymptotic normality of the normalized frequencies and the asymptotic equivalence of many divisible statistics to the Chi-square statistic are lost, and the limit theorems established in classical model are no longer valid in non-classical models.</p> <p>The extension to non-classical models not only met the demands of many real world applications, but also opened a new research area in statistical analysis, which has not been thoroughly investigated so far. Although some results on the limit distributions of the divisible statistics in non-classical models have been acquired, e.g., Holst (1972); Morris (1975); Ivchenko and Levin (1976); Ivchenko and Medvedev (1979), they are far from complete. Though not yet attracting much attention by many applied statisticians, another advanced approach, introduced by Khmaladze (1984), makes use of modern martingale theory to establish functional limit theorems of the partial sum processes of divisible statistics successfully. In the main part of this thesis, we show that this martingale approach can be extended to more general situations where both Gaussian and Poissonian frequencies exist, and further discuss the properties and applications of the limiting processes, especially in constructing distribution-free statistics.</p> <p>The last part of the thesis is about the statistical analysis of large number of rare events (LNRE), which is an important class of non-classical multinomial models and presented in numerous applications. In LNRE models, most of the frequencies are very small and it is not immediately clear how consistent and reliable inference can be achieved. Based on the definitions and key concepts firstly introduced by Khmaladze (1988), we discuss a particular model with the context of diversity of questionnaires. The advanced statistical techniques such as large deviation, contiguity and Edgeworth expansion used in establishing limit theorems underpin the potential of LNRE theory to become a fruitful research area in future.</p>

A Generic Nomogram Predicting the Stage of Liver Fibrosis Based on Serum Biochemical Indicators Among Chronic Hepatitis B Patients

Introduction: Liver fibrosis staging is of great importance for reducing unnecessary injuries and prompting treatment in chronic viral hepatitis B patients. Liver biopsy is not suitable to act a screening method although it is a gold standard because of various shortcomings. This study aimed to establish a predictive nomogram as a convenient tool to effectively identify potential patients with different stages of liver fibrosis for patients with chronic hepatitis B.Methods: A nomogram for multinomial model was developed in a training set to calculate the probability for each stage of fibrosis and tested in a validation set. Fibrosis stages were subgrouped as followed: severe fibrosis/cirrhosis (F3–F4), moderate fibrosis (F2), and nil-mild fibrosis (F0–F1). The indicators were demographic characteristics and biochemical indicators of patients. Continuous indicators were divided into several groups according to the optimal candidate value generated by the decision tree.Results: This study recruited 964 HBV patients undergoing percutaneous liver biopsy. The multinomial model with 10 indicators was transformed into the final nomogram. The calibration plot showed a good agreement between nomogram-predicted and observed probability of different fibrosis stages. Areas under the receiver operating characteristics (AUROCs) for severe fibrosis/cirrhosis were 0.809 for training set and 0.879 for validation set. For moderate fibrosis, the AUROCs were 0.75 and 0.781. For nil-mild fibrosis, the AUROCs were 0.792 and 0.843. All the results above showed great predictive performance in predicting the stage of fibrosis by our nomogram.Conclusion: Our model demonstrated good discrimination and extensibility in internal and external validation. The proposed nomogram in this study resulted in great reliability and it can be widely used as a convenient and efficient way.

Does foreign language alter moral judgments? No evidence from two pre-registered studies with the CNI-model

Recent studies suggest that processing moral dilemmas in a foreign language instead of the native language increases the likelihood of moral judgments in line with the utilitarian principle. The goal of our research was to investigate the replicability and robustness of this moral foreign-language effect and to explore its underlying mechanisms by means of the CNI model—a multinomial model that allows to estimate the extent to which moral judgments are driven by people’s sensitivity to consequences (C-parameter), their sensitivity to norms (N-parameter), and their general preference for action or inaction (I-parameter). In two pre-registered studies, German participants provided moral judgments to dilemmas that were either presented in German or English. In Experiment 1, participants judged eight different dilemmas in four versions each (i.e., 32 dilemmas in total). In Experiment 2, participants judged four different dilemmas in one of the four versions (i.e., 4 dilemmas in total). Neither of the two studies replicated the moral foreign-language effect. Moreover, we also did not find reliable language effects on the three parameters of the CNI-model. We conclude that if there is a moral foreign-language effect, it must be very fragile and context specific.

Checking for model failure and for prior-data conflict with the constrained multinomial model

Multinomial models can be difficult to use when constraints are placed on the probabilities. An exact model checking procedure for such models is developed based on a uniform prior on the full multinomial model. For inference, a nonuniform prior can be used and a consistency theorem is proved concerning a check for prior-data conflict with the chosen prior. Applications are presented and a new elicitation methodology is developed for multinomial models with ordered probabilities.

Statistical Inference From Stem Cell Barcoding Data Using Adaptive Approximate Bayesian Computation

Background: Barcodes that can be supplied to cells by transduction of a library of unique DNA sequences allow identification of heterogeneity in cell populations and lineage tracing applications. Estimation of the number of hematopoietic stem cell (HSC) clones is important since it also allows to approximate the number of hematopoietic stem cells from which the circulating blood cells descend. This problem is similar to the species problem, well-known to ecologists. However, an additional ”degree of freedom” exists, since different HSC generally give rise to clones with different growth rates. This adds credibility to sampling models based on different versions of Dirichlet-multinomial distributions. Results: We developed a truncated population approximate Bayesian computation (ABC) algorithm which is derived from sequential Monte Carlo ABC (SMC-ABC) and applied the method to the symmetric Dirichlet-multinomial model proposed by Zhang et al. (2005) and asymmetric Dirichlet-multinomial model we proposed. Methodology was tested using simulated and real-life data. Conclusions: Results suggest that flexibility of the asymmetric Dirichlet-multinomial helps to obtain insight into heterogeneity of proliferating cell systems such as HSC. Estimates based on experimental data approach the correct count of murine HSC.