Analysis of microtubule dynamics using growth curve models

Microtubules are part of the structural network within a cell's cytoplasm, providing structural support as well as taking part in many of the cellular processes. A large body of data provide evidence that dynamics of microtubules in a cell is reponsible for the performance of many critical cellular functions such as cell division. In this article, we study the effect of four different isoforms of a protein tau on microtubule dynamics using growth curve models. The results show that a linear growth curve model is sufficient to explain the data. Moreover, we find that a mutated version of a 3-repeat tau protein has a similar effect as a 4-repeat tau protein on microtubule dynamics. The latter findings conform with the biological understanding of the effect of the protein tau on microtubule dynamics.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

Approximating the risk score for disease diagnosis using MARS

In disease screening and diagnosis, often multiple markers are measured and combined to improve the accuracy of diagnosis. McIntosh and Pepe [Combining several screening tests: optimality of the risk score, Biometrics 58 (2002), pp. 657–664] showed that the risk score, defined as the probability of disease conditional on multiple markers, is the optimal function for classification based on the Neyman–Pearson lemma. They proposed a two-step procedure to approximate the risk score. However, the resulting receiver operating characteristic (ROC) curve is only defined in a subrange (L, h) of false-positive rates in (0,1) and the determination of the lower limit L needs extra prior information. In practice, most diagnostic tests are not perfect, and it is usually rare that a single marker is uniformly better than the other tests. Using simulation, I show that multivariate adaptive regression spline is a useful tool to approximate the risk score when combining multiple markers, especially when ROC curves from multiple tests cross. The resulting ROC is defined in the whole range of (0,1) and is easy to implement and has intuitive interpretation. The sample code of the application is shown in the appendix.     

On the asymptotic distribution of Cook's distance in logistic regression models

It sometimes occurs that one or more components of the data exert a disproportionate influence on the model estimation. We need a reliable tool for identifying such troublesome cases in order to decide either eliminate from the sample, when the data collect was badly realized, or otherwise take care on the use of the model because the results could be affected by such components. Since a measure for detecting influential cases in linear regression setting was proposed by Cook [Detection of influential observations in linear regression, Technometrics 19 (1977), pp. 15–18.], apart from the same measure for other models, several new measures have been suggested as single-case diagnostics. For most of them some cutoff values have been recommended (see [D.A. Belsley, E. Kuh, and R.E. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, 2nd ed., John Wiley & Sons, New York, Chichester, Brisban, (2004).], for instance), however the lack of a quantile type cutoff for Cook's statistics has induced the analyst to deal only with index plots as worthy diagnostic tools. Focussed on logistic regression, the aim of this paper is to provide the asymptotic distribution of Cook's distance in order to look for a meaningful cutoff point for detecting influential and leverage observations.     

The generalized two-sided power distribution

The generalized standard two-sided power (GTSP) distribution was mentioned only in passing by Kotz and van Dorp Beyond Beta, Other Continuous Families of Distributions with Bounded Support and Applications, World Scientific Press, Singapore, 2004. In this paper, we shall further investigate this three-parameter distribution by presenting some novel properties and use its more general form to contrast the chronology of developments of various authors on the two-parameter TSP distribution since its initial introduction. GTSP distributions allow for J-shaped forms of its pdf, whereas TSP distributions are limited to U-shaped and unimodal forms. Hence, GTSP distributions possess the same three distributional shapes as the classical beta distributions. A novel method and algorithm for the indirect elicitation of the two-power parameters of the GTSP distribution is developed. We present a Project Evaluation Review Technique example that utilizes this algorithm and demonstrates the benefit of separate powers for the two branches of activity GTSP distributions for project completion time uncertainty estimation.     

Modelling udder infection data using copula models for quadruples

We study copula models for correlated infection times in the four udder quarters of dairy cows. Both a semi-parametric and a nonparametric approach are considered to estimate the marginal survival functions, taking into account the effect of a binary udder quarter level covariate. We use a two-stage estimation approach and we briefly discuss the asymptotic behaviour of the estimators obtained in the first and the second stage of the estimation. A pseudo-likelihood ratio test is used to select an appropriate copula from the power variance copula family that describes the association between the outcomes in a cluster. We propose a new bootstrap algorithm to obtain the p-value for this test. This bootstrap algorithm also provides estimates for the standard errors of the estimated parameters in the copula. The proposed methods are applied to the udder infection data. A small simulation study for a setting similar to the setting of the udder infection data gives evidence that the proposed method provides a valid approach to select an appropriate copula within the power variance copula family.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

Shrinkage inverse regression estimation for model-free variable selection

Summary.  The family of inverse regression estimators that was recently proposed by Cook and Ni has proven effective in dimension reduction by transforming the high dimensional predictor vector to its low dimensional projections. We propose a general shrinkage estimation strategy for the entire inverse regression estimation family that is capable of simultaneous dimension reduction and variable selection. We demonstrate that the new estimators achieve consistency in variable selection without requiring any traditional model, meanwhile retaining the root n estimation consistency of the dimension reduction basis. We also show the effectiveness of the new estimators through both simulation and real data analysis.     

On distribution-weighted partial least squares with diverging number of highly correlated predictors

Summary.  Because highly correlated data arise from many scientific fields, we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated. For this, we first develop a distribution-weighted least squares estimator that can recover directions in the central subspace, then use the distribution-weighted least squares estimator as a seed vector and project it onto a Krylov space by partial least squares to avoid computing the inverse of the covariance of predictors. Thus, distrbution-weighted partial least squares can handle the cases with high dimensional and highly correlated predictors. Furthermore, we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares. For theoretical investigation, we obtain strong consistency and asymptotic normality when the dimension p of predictors is of convergence rate O { n ^1/2/ log ( n )} and o ( n ^1/3) respectively where n is the sample size. When there are no other constraints on the covariance of predictors, the rates n ^1/2 and n ^1/3 are optimal. We also propose a Bayesian information criterion type of criterion to estimate the dimension of the Krylov space in the partial least squares procedure. Illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non-ellipticity and works well even in 'small n –large p ' problems.