The Bayes Estimators of the Variance and Scale Parameters of the Normal Model With a Known Mean for the Conjugate and Noninformative Priors Under Stein’s Loss

For the normal model with a known mean, the Bayes estimation of the variance parameter under the conjugate prior is studied in Lehmann and Casella (1998) and Mao and Tang (2012). However, they only calculate the Bayes estimator with respect to a conjugate prior under the squared error loss function. Zhang (2017) calculates the Bayes estimator of the variance parameter of the normal model with a known mean with respect to the conjugate prior under Stein’s loss function which penalizes gross overestimation and gross underestimation equally, and the corresponding Posterior Expected Stein’s Loss (PESL). Motivated by their works, we have calculated the Bayes estimators of the variance parameter with respect to the noninformative (Jeffreys’s, reference, and matching) priors under Stein’s loss function, and the corresponding PESLs. Moreover, we have calculated the Bayes estimators of the scale parameter with respect to the conjugate and noninformative priors under Stein’s loss function, and the corresponding PESLs. The quantities (prior, posterior, three posterior expectations, two Bayes estimators, and two PESLs) and expressions of the variance and scale parameters of the model for the conjugate and noninformative priors are summarized in two tables. After that, the numerical simulations are carried out to exemplify the theoretical findings. Finally, we calculate the Bayes estimators and the PESLs of the variance and scale parameters of the S&P 500 monthly simple returns for the conjugate and noninformative priors.

Modeling across-trial variability in the Wald drift rate parameter

The shifted-Wald model is a popular analysis tool for one-choice reaction-time tasks. In its simplest version, the shifted-Wald model assumes a constant trial-independent drift rate parameter. However, the presence of endogenous processes—fluctuation in attention and motivation, fatigue and boredom—suggest that drift rate might vary across experimental trials. Here we show how across-trial variability in drift rate can be accounted for by assuming a trial-specific drift rate parameter that is governed by a positive-valued distribution. We consider two candidate distributions: the truncated normal distribution and the gamma distribution. For the resulting distributions of first-arrival times, we derive analytical and sampling-based solutions, and implement the models in a Bayesian framework. Recovery studies and an application to a data set comprised of 1469 participants suggest that (1) both mixture distributions yield similar results; (2) all model parameters can be recovered accurately except for the drift variance parameter; (3) despite poor recovery, the presence of the drift variance parameter facilitates accurate recovery of the remaining parameters; (4) shift, threshold, and drift mean parameters are correlated.

On the variance parameter estimator in general linear models

In the present note we consider general linear models where the covariates may be both random and non-random, and where the only restrictions on the error terms are that they are independent and have finite fourth moments. For this class of models we analyse the variance parameter estimator. In particular we obtain finite sample size bounds for the variance of the variance parameter estimator which are independent of covariate information regardless of whether the covariates are random or not. For the case with random covariates this immediately yields bounds on the unconditional variance of the variance estimator—a situation which in general is analytically intractable. The situation with random covariates is illustrated in an example where a certain vector autoregressive model which appears naturally within the area of insurance mathematics is analysed. Further, the obtained bounds are sharp in the sense that both the lower and upper bound will converge to the same asymptotic limit when scaled with the sample size. By using the derived bounds it is simple to show convergence in mean square of the variance parameter estimator for both random and non-random covariates. Moreover, the derivation of the bounds for the above general linear model is based on a lemma which applies in greater generality. This is illustrated by applying the used techniques to a class of mixed effects models.

Identifying and quantifying variation between healthcare organisations and geographical regions: using mixed-effects models

When the degree of variation between healthcare organisations or geographical regions is quantified, there is often a failure to account for the role of chance, which can lead to an overestimation of the true variation. Mixed-effects models account for the role of chance and estimate the true/underlying variation between organisations or regions. In this paper, we explore how a random intercept model can be applied to rate or proportion indicators and how to interpret the estimated variance parameter.

Beyond the disconnectivity hypothesis of schizophrenia

To go beyond the disconnectivity hypothesis of schizophrenia, directed (effective) connectivity was measured between 94 brain regions, to provide evidence on the source of the changes in schizophrenia and a mechanistic model. Effective connectivity (EC) was measured in 180 participants with schizophrenia and 208 controls. For the significantly different effective connectivities in schizophrenia, on average the forward (stronger) effective connectivities were smaller, whereas the backward connectivities tended to be larger. Further, higher EC in schizophrenia was found from the precuneus and posterior cingulate cortex (PCC) to areas such as the parahippocampal, hippocampal, temporal, fusiform, and occipital cortices. These are backward effective connectivities and were positively correlated with the positive symptoms of schizophrenia. Lower effective connectivities were found from temporal and other regions and were negatively correlated with the symptoms, especially the negative and general symptoms. Further, a signal variance parameter was increased for areas that included the parahippocampal gyrus and hippocampus, consistent with the hypothesis that hippocampal overactivity is involved in schizophrenia. This investigation goes beyond the disconnectivity hypothesis by drawing attention to differences in schizophrenia between backprojections and forward connections, with the backward connections from the precuneus and PCC implicated in memory stronger in schizophrenia.

Variance prior specification for a basket trial design using Bayesian hierarchical modeling

Background: In the era of targeted therapies, clinical trials in oncology are rapidly evolving, wherein patients from multiple diseases are now enrolled and treated according to their genomic mutation(s). In such trials, known as basket trials, the different disease cohorts form the different baskets for inference. Several approaches have been proposed in the literature to efficiently use information from all baskets while simultaneously screening to find individual baskets where the drug works. Most proposed methods are developed in a Bayesian paradigm that requires specifying a prior distribution for a variance parameter, which controls the degree to which information is shared across baskets. Methods: A common approach used to capture the correlated binary endpoints across baskets is Bayesian hierarchical modeling. We evaluate a Bayesian adaptive design in the context of a non-randomized basket trial and investigate three popular prior specifications: an inverse-gamma prior on the basket-level variance, a uniform prior and half-t prior on the basket-level standard deviation. Results: From our simulation study, we can see that the inverse-gamma prior is highly sensitive to the input hyperparameters. When the prior mean value of the variance parameter is set to be near zero [Formula: see text], this can lead to unacceptably high false-positive rates [Formula: see text] in some scenarios. Thus, use of this prior requires a fully comprehensive sensitivity analysis before implementation. Alternatively, we see that a prior that places sufficient mass in the tail, such as the uniform or half-t prior, displays desirable and robust operating characteristics over a wide range of prior specifications, with the caveat that the upper bound of the uniform prior and the scale parameter of the half-t prior must be larger than 1. Conclusion: Based on the simulation results, we recommend that those involved in designing basket trials that implement hierarchical modeling avoid using a prior distribution that places a majority of the density mass near zero for the variance parameter. Priors with this property force the model to share information regardless of the true efficacy configuration of the baskets. Many commonly used inverse-gamma prior specifications have this undesirable property. We recommend to instead consider the more robust uniform prior or half-t prior on the standard deviation.