Growth curves mixture model with serial covariance structure

Growth curve mixture models with unknown covariance structures

Score limitation at the top of a scale is commonly termed “ceiling effect.” Ceiling effects can lead to serious artifactual parameter estimates in most data analysis. This study examines the consequences of ceiling effects in longitudinal data analysis and investigates several methods of dealing with ceiling effects through Monte Carlo simulations and empirical data analyses. Data were simulated based on a latent growth curve model with T = 5 occasions. The proportion of the ceiling data [10%–40%] was manipulated by using different thresholds, and estimated parameters were examined for R = 500 replications. The results showed that ceiling effects led to incorrect model selection and biased parameter estimation (shape of the curve and magnitude of the changes) when regular growth curve models were applied. The Tobit growth curve model, instead, performed very well in dealing with ceiling effects in longitudinal data analysis. The Tobit growth curve model was then applied in an empirical cognitive aging study and the results were discussed.

The effect of skewness on hypothesis tests for the existence of a mixture of univariate and bivariate normal distributions is examined through a Monte Carlo study. A likelihood ratio test based on results of the simultaneous estimation of skewness parameters, derived from power transformations, with mixture parameters is proposed. This procedure detects the difference between inherent distributional skewness and the apparent skewness which is a manifestation of the mixture of several distributions. The properties of this test are explored through a simulation study.

The Growth Curve Model (GCM) is a Generalized Multivariate Analysis of Variance (GMANOVA) model especially useful in the analysis of longitudinal data, growth curves as well as other response curves. The model is a natural extension of the classical Multivariate Analysis of Variance (MANOVA) model and among other assumptions, relies on the assumption that the mean for each group can be represented as a polynomial of degree q. The assumption that the mean over time for all groups follows a polynomial of the same degree is not always satisfied, since individuals across the different groups may respond differently. An excellent scenario is when we have clustered longitudinal data, where the response over time can be represented by polynomials of different degrees. In such situations, the natural extension is to use the Extended Growth Curve Model (EGCM), where one can assume different shapes to represent different groups or clusters. In this paper, we formulate hypotheses motivated by real life scenarios involving clustered longitudinal data, and propose tests that are motivated by residuals in the EGCM. We then mathematically derive the tests and evaluate performances using simulations. We provide real data examples as illustrations.

The heterogeneous growth curve model (HGM; Klein & Muthén, 2006) is a method for modeling heterogeneity of growth rates with a heteroscedastic residual structure for the slope factor. It has been developed as an extension of a conventional growth curve model and a complementary tool to growth curve mixture models. In this article, a robust version of the heterogeneous growth curve model (HGM-R) is presented that extends the original HGM with a mixture model to allow for an unbiased parameter estimation under the condition of nonnormal data. In two simulation studies, the performance of the method is examined under the condition of nonnormality and a misspecified heteroscedastic residual structure. The results of the simulation studies suggest an unbiased estimation of the heterogeneity by the HGM-R when sample size was large enough and a good approximation of the heteroscedastic residual structure even when the functional form of the heteroscedasticity was misspecified. The practical application of the approach is demonstrated for a data set from HIV-infected patients.

The finite mixture model is considered as an appropriate instrument for data clustering.Different parsimonious multivariate mixture distributions are introduced for skewed and/or heavy-tailed longitudinal data.The eigenvalue or modified Cholesky decomposition of covariance matrices develops the families of parsimonious mixture models.Thus, the finite mixture of matrix-variate t-distributions for clustering a three-way dataset with heavy-tailed or outlier observations (e.g., multivariate longitudinal data) is more appropriate compared to matrix-variate normal distributions.Accordingly, the present study considered a parsimonious family of the finite mixture of matrix-variate t-distributions using the eigenvalue and modified Cholesky decomposition for within and between covariance matrices, respectively.Finally, parameter estimates were calculated using the expectation-maximization algorithm, and simulations studies and real data analyses were conducted to confirm the obtained results.

Analysis of binary longitudinal data with time-varying effects

Unified computational methods for regression analysis of zero-inflated and bound-inflated data

The coding of time in growth curve models has important implications for the interpretation of the resulting model that are sometimes not transparent. The authors develop a general framework that includes predictors of growth curve components to illustrate how parameter estimates and their standard errors are exactly determined as a function of receding time in growth curve models. Linear and quadratic growth model examples are provided, and the interpretation of estimates given a particular coding of time is illustrated. How and why the precision and statistical power of predictors of lower order growth curve components changes over time is illustrated and discussed. Recommendations include coding time to produce readily interpretable estimates and graphing lower order effects across time with appropriate confidence intervals to help illustrate and understand the growth process.

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

The expectation maximisation (EM) algorithm is an iterative maximum likelihood procedure often used for estimating the parameters of a mixture model. Theoretically, increases in the likelihood function are guaranteed as the algorithm iteratively improves upon previously derived parameter estimates. The algorithm is considered to converge when all parameter estimates become stable and no further improvements can be made to the likelihood value. However, to reduce computational time, it is often common practice for the algorithm to be stopped before complete convergence using heuristic approaches. In this paper, we consider various stopping criteria and evaluate their effect on fitting Gaussian mixture models (GMMs) to patient length of stay (LOS) data. Although the GMM can be successfully fitted to positively skewed data such as LOS, the fitting procedure often requires many iterations of the EM algorithm. To our knowledge, no previous study has evaluated the effect of different stopping criteria on fitting GMMs to skewed distributions. Hence, the aim of this paper is to evaluate the effect of various stopping criteria in order to select and justify their use within a patient spell classification methodology. Results illustrate that criteria based on the difference in the likelihood value and on the GMM parameters may not always be a good indicator for stopping the algorithm. In fact we show that the values of the difference in the variance parameters should be used instead, as these parameters are the last to stabilise. In addition, we also specify threshold values for the other stopping criteria.

There have been few Bayesian analyses of phase III sequential clinical trials that model divergent expert opinions in a single distribution. We used modeling of experts' opinions to perform additional Bayesian analyses of a randomized clinical trial (designed as a sequential trial), particularly when a bimodal shape is observed. We provide an illustrative example based on a randomized trial conducted in patients aged between 65 and 75 years with multiple myeloma as the case study. The main endpoint of the trial was overall survival (OS). Prior distribution of the log hazard ratio of death in the experimental versus the control arm ( $$\theta $$ ) was constructed based on elicitation of experts using a mixture of normal distributions estimated by the Expectation-Maximisation (EM) algorithm. At each interim and terminal analysis, the posterior probability of $$\theta $$ and the resulting increases in median OS in the experimental arm compared to the control were computed. The results were compared to results obtained using either skeptical, enthusiastic, or a mixture of those priors. Finally, we discuss our results in light of the frequentist approach originally designed for the trial. A total of 39 experts reported their opinion on the median OS in the experimental arm compared to the median control survival of 30 months. The resulting pooled distribution of the log hazard ratios exhibited a bimodal profile. When the prior mixture of the normal distribution was fitted to the data sets from the experts, 44% of the experts' opinions were optimistic and 56% were doubtful. At the final analysis, the percentage of doubting experts dropped to 18%. This corresponded to a posterior probability of an improved OS in the experimental arm compared to the control arm of at least 0.98, regardless of the prior. These findings are in agreement with the original conclusion of the trial regarding the beneficial effect of the experimental treatment in this population. Only 39 experts among the 120 questioned physicians responded to the inquiry. Our approach was hybrid because the prior mixture was estimated using the EM algorithm, and a full Bayesian approach may have been used. Bayesian inference allows the quantification of increased survival in terms of probability distributions and provides investigators with an additional tool in the analysis of a randomized phase III clinical trial. Using a mixture of densities appears to be a promising strategy for incorporating the bimodal profile of prior opinion, with actualization of the two components along the trial as an illustration of the evolution of opinions as data are accumulated.

We apply a machine learning model to determine the optimal strategy in an online auction for the rent of computing resources using the best-choice model. The best-choice model allows clients to minimize the expected cost of renting a computing resource based on the spot price distribution function. The spot price dynamics platform is investigated. The most suitable price distributions in an auction are the normal distribution and its mixtures. In this case, the problems of determining the number of components in the mixture and estimating its parameters arise. One of the well-known methods for determining the number of components in a mixture of normal distributions is the BIC criterion. The EM algorithm is a basic tool for estimating the parameters of a mixture of distributions if we know the number of components. However, parameter estimation by this method takes more time when both the sample size and the number of components of the mixture increase. To automate and expedite the process of determining the number of components for a mixture of normal distributions and estimating its parameters, a classification machine learning model based on a convolutional neural network is developed. The results of the model training and validation are presented. The suggested model is compared with other algorithms which do not use neural networks. The results show that the suggested model performs well in determining the most appropriate number of components for a mixture of normal distributions and in reducing the time spent on applying the EM algorithm to estimate its parameters. This model can be used in different arias, for example, in finance or for determination of the optimal strategy in an online auction for the rent of computing resources.

Introduction: In real-world datasets, outliers are a common occurrence that can have a significant impact on the accuracy and reliability of statistical analyses. Detecting these outliers and developing robust models to handle their presence is a crucial challenge in data analysis. For instance, natural images may have complex distributions of values due to environmental factors like noise and illumination, resulting in objects with overlapping regions and non-trivial contours that cannot be accurately described by Gaussian mixture models. In many real life applications, observed data always fall in bounded support regions. This leads to the idea of bounded support mixture models. Motivated by the aforementioned observations, we introduce a bounded multivariate cntaminated normal distribution for fitting data with non-Gaussian distributions, asymmetry, and bounded support which makes finite mixture models more robust to fitting, since rare observations are given less importance in calculations. Methods: A family of finite mixtures of bounded multivariate contaminated normal distributions is introduced. The model is well-suited for computer vision and pattern recognition problems due to its heavily-tailed and bounded nature, providing flexibility in modeling data in the presence of outliers. A feasible expectation- maximization algorithm is developed to compute the maximum likelihood estimates of the model parameters using a selection mechanism. Results: The proposed methodology is validated by conducting experiments on both simulated data and two real natural skin cancer images. We estimate the parameters by the proposed expectation-maximization algorithm. The obtained results shown that the proposed model has successfully enhanced accuracy in segmenting skin lesions. Conclusion: The reliable model-based clustering using finite mixtures of bounded multivariate contaminated normal distributions is introduced. An expectation-maximization algorithm was created to estimate parameters, with closed-form expressions utilized at the E-step. Practical tests on images for skin cancer detection showed enhanced accuracy in delineating skin lesions.

This paper introduces a mixture model that combines proportional hazards regression with logistic regression for the analysis of survival data, and describes its parameter estimation via an expectation maximization algorithm. The mixture model is then applied to analyze the determinants of the timing of intrauterine device (IUD) discontinuation and long-term IUD use, utilizing 14 639 instances of IUD use by Chinese women. The results show that socio-economic and demographic characteristics of women have different influences on the acceleration or deceleration of the timing of stopping IUD use and on the likelihood of long-term IUD use.