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Clinical Trials: the Statistician's Role
Published in Trevor F. Cox, Medical Statistics for Cancer Studies, 2022
We have to find the joint distribution of Z1 and Z2. Each has a normal distribution and so their joint distribution is a bivariate normal distribution (see Appendix for notes on the multivariate normal distribution). Now and . The covariance of Z1 and Z2 is found as follows. Let and . Then from Equations 6.8,
Complex Innovative Design
Published in Wei Zhang, Fangrong Yan, Feng Chen, Shein-Chung Chow, Advanced Statistics in Regulatory Critical Clinical Initiatives, 2022
Modeling the joint distribution of is more complicated because and are different types of variables, i.e., is a binary variable whereas is an ordinal variable, and they are correlated. To this end, we take the latent variable approach. Specifically, let and denote two continuous latent variables that are related to and , respectively, as follows,
Bayesian Methods for Meta-Analysis
Published in Christopher H. Schmid, Theo Stijnen, Ian R. White, Handbook of Meta-Analysis, 2020
Christopher H. Schmid, Bradley P. Carlin, Nicky J. Welton
Although MCMC encompasses a variety of algorithms, the Gibbs sampler is the simplest to understand and presents most of the key features. Consider a set of M model parameters . The sampler works by drawing in turn each parameter θi from its posterior distribution conditional on the most recent draws of all the other parameters θ−i. At iteration t in a normal likelihood model, we draw where indicates that we condition on the most recent draws from θ−i. Note the slight abuse of notation here because each newly drawn θi becomes part of the current set of parameters used to draw θi+1 and so some of the parameters in will actually be drawn from iteration t. These distributions are known as the full conditional distributions and are much easier to sample than the joint distribution, either because they are univariate distributions or because they are multivariate distributions such as of a vector of linear regression coefficients from which it is easy to sample.
Variability in in vitro biofilm production and antimicrobial sensitivity pattern among Pasteurella multocida strains
Published in Biofouling, 2020
Awadhesh Prajapati, Mohammed Mudassar Chanda, Arul Dhayalan, Revanaiah Yogisharadhya, Jitendra Kumar Chaudhary, Nihar Nalini Mohanty, Sathish Bhadravati Shivachandra
Statistical models were used to identify association/correlation factors with biofilm formation by the P. multocida strains. Generally, the correlation (r2) co-efficient can be estimated between two variables and in multivariate regression analysis it can be done between dependent and independent variables. In Joint distribution models, correlation between many dependent variables can be performed and are helpful in identifying correlation between multiple dependent variables by taking account of the effect of independent variables. Bayesian ordination and regression analysis allow such assessment. Correlation co-efficient values above zero were considered as ‘positive’ and below zero were considered as ‘negative’ correlations. Bayesian ordination and regression analysis with the BORAL package in R (Hui 2016) was used to correlate biofilm production with the capsule type, the host, the clinical condition and the presence of tadD. A total of six latent models were fit with biofilm production in different media and other factors (capsule type, host, clinical condition and the presence of the tadD gene). In addition, this method was used to correlate biofilm production in BHI medium with antimicrobial susceptibility.
Identifying mediating variables with graphical models: an application to the study of causal pathways in people living with HIV
Published in Journal of Applied Statistics, 2020
Adrian Dobra, Katherine Buhikire, Joachim G. Voss
One might ask whether it is possible to replace the determination of the graphical model from the first step of our proposed approach with the determination of relevant regression models that are best supported by the data corresponding with the three conditionals in Equations (1)–(3). This is a valid question since regression models are easier to understand than multivariate joint distributions represented as graphical models. However, the selection of regression models is not straightforward in this situation. First of all, the conditional distributions in Equations (1) and (2) are multivariate since there could exist one, two or more variables in G with three or more variables). Moreover, different types of regression models need to be considered for the three conditionals when the types of outcome variables vary (e.g. binary, ordinal or multinomial). Even if searches for regression models associated with 3], the existence of a valid joint distribution for
A Bayesian conditional model for bivariate mixed ordinal and skew continuous longitudinal responses using quantile regression
Published in Journal of Applied Statistics, 2018
S. Ghasemzadeh, M. Ganjali, T. Baghfalaki
Models for ordinal and continuous responses have been proposed in the previous subsections. Now, we can develop a conditional model based on these models. As we know, bivariate response variables have a joint distribution and each joint distribution leads to a joint model. Therefore, we have a joint model in which its joint distribution of mixed responses is factorized into a product of a marginal distribution of continuous variable and a conditional distribution of ordinal variable given continuous variable (that is 14) and model for latent variable, 10) with a difference of having one more term. In fact, the continuous response is considered as a predictor variable for ordinal model. The vectors of random effects γ is the associated parameter which causes inter-dependence between ordinal and continuous responses. It is worth noticing that γ is the only source of dependence between the two mixed ordinal and continuous responses. If γ is equal to zero then we will have two separate models.