Longitudinal Joint Modelling of Ordinal and Overdispersed Count Outcomes: A Bridge Distribution for the Ordinal Random Intercept

Associated longitudinal response variables are faced with variations caused by repeated measurements over time along with the association between the responses. To model a longitudinal ordinal outcome using generalized linear mixed models, integrating over a normally distributed random intercept in the proportional odds ordinal logistic regression does not yield a closed form. In this paper, we combined a longitudinal count and an ordinal response variable with Bridge distribution for the random intercept in the ordinal logistic regression submodel. We compared the results to that of a normal distribution. The two associated response variables are combined using correlated random intercepts. The random intercept in the count outcome submodel follows a normal distribution. The random intercept in the ordinal outcome submodel follows Bridge distribution. The estimations were carried out using a likelihood-based approach in direct and conditional joint modelling approaches. To illustrate the performance of the model, a simulation study was conducted. Based on the simulation results, assuming a Bridge distribution for the random intercept of ordinal logistic regression results in accurate estimation even if the random intercept is normally distributed. Moreover, considering the association between longitudinal count and ordinal responses resulted in estimation with lower standard error in comparison to univariate analysis. In addition to the same interpretation for the parameter in marginal and conditional estimates thanks to the assumption of a Bridge distribution for the random intercept of ordinal logistic regression, more efficient estimates were found compared to that of normal distribution.

Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic

We prove that the Behrens–Fisher statistic follows a Student bridge distribution, the mixing coefficient of which depends on the two sample variances only through their ratio. To this end, it is first shown that a weighted sum of two independent normalized chi-square distributed random variables is chi-square bridge distributed, and secondly that the Behrens–Fisher statistic is based on such a variable and a standard normally distributed one that is independent of the former. In case of a known variance ratio, exact standard statistical testing and confidence estimation methods apply without the need for any additional approximations. In addition, a three pillar bridges explanation is given for the choice of degrees of freedom in Welch’s approximation to the exact distribution of the Behrens–Fisher statistic.

Universal Properties in Filler-Loaded Rubbers

Stress-strain cycles in filler loaded rubbers can be described with the aid of the van der Waals-network model. Reinforcement comes about by drawing pairs of filler particles apart. Reinforcement is observed because the intrinsic strain within the rubber bridge which is located between the filler particles exceeds the macroscopic strain very much, so much that interfacial slippage is enforced. The rubbery intra-cluster bridge distribution is represented by three dominant filler particle distances. One of them describes direct filler-to-filler (FF-) contacts, the critical strength of which is different from the filler-to-matrix (FM-) contacts of the filler-to-filler chains which are located on the whole surface of the filler particles. Formation of clusters is described by a power law. Stress-strain experiments are described with the aid of this model for different filler-matrix combinations (NR, SBR, carbon blacks, silica). Many universal features are observed: The intra-cluster rubber bridges display the same mean thickness when being related to the radius of the primary filler particles. The exponent in the power law is always identical. The deformation mechanisms, including irreversible slippage, do not to depend on the type of strain (simple extension, uniaxial compression). Yet, the Einstein-Smallwood effect turns out to be anisotrop so far as quasipermanent filler-to-matrix interactions seem to be determined by normal forces in the particles surfaces only. Different filler and matrix combinations display different strengths of the FF- and FM-contacts independent of the type of strain.