Bayesian Mendelian randomization with an interval causal null hypothesis: ternary decision rules and loss function calibration.

We enhance the Bayesian Mendelian Randomization (MR) framework of Berzuini et al. (Biostatistics 21(1):86-101, 2018) by allowing for interval null causal hypotheses, where values of the causal effect parameter that fall within a user-specified interval of "practical equivalence" (ROPE) (Kruschke, Adv Methods Pract Psychol Sci 1(2):270-80, 2018) are regarded as equivalent to "no effect". We motivate this move in the context of MR analysis. In this approach, the decision over the hypothesis test is taken on the basis of the Bayesian posterior odds for the causal effect parameter falling within the ROPE. We allow the causal effect parameter to have a mixture prior, with components corresponding to the null and the alternative hypothesis. Inference is performed via Markov chain Monte Carlo (MCMC) methods. We speed up the calculations by fitting to the data a simpler model than the intended, "true", one. We recover a set of samples from the "true" posterior distribution by weighted importance resampling of the MCMC-generated samples. From the final samples we obtain a simulation consistent estimate of the desired posterior odds, and ultimately of the Bayes factor for the interval-valued null hypothesis, [Formula: see text], vs [Formula: see text]. In those situations where the posterior odds is neither large nor small enough, we allow for an uncertain outcome of the test decision, thereby moving to a ternary decision logic. Finally, we present an approach to calibration of the proposed method via loss function. We illustrate the method with the aid of a study of the causal effect of obesity on risk of juvenile myocardial infarction based on a unique prospective dataset.

Bayesian mendelian randomization with study heterogeneity and data partitioning for large studies.

BACKGROUND: Mendelian randomization (MR) is a useful approach to causal inference from observational studies when randomised controlled trials are not feasible. However, study heterogeneity of two association studies required in MR is often overlooked. When dealing with large studies, recently developed Bayesian MR can be computationally challenging, and sometimes even prohibitive. METHODS: We addressed study heterogeneity by proposing a random effect Bayesian MR model with multiple exposures and outcomes. For large studies, we adopted a subset posterior aggregation method to overcome the problem of computational expensiveness of Markov chain Monte Carlo. In particular, we divided data into subsets and combined estimated causal effects obtained from the subsets. The performance of our method was evaluated by a number of simulations, in which exposure data was partly missing. RESULTS: Random effect Bayesian MR outperformed conventional inverse-variance weighted estimation, whether the true causal effects were zero or non-zero. Data partitioning of large studies had little impact on variations of the estimated causal effects, whereas it notably affected unbiasedness of the estimates with weak instruments and high missing rate of data. For the cases being simulated in our study, the results have indicated that the "divide (data) and combine (estimated subset causal effects)" can help improve computational efficiency, for an acceptable cost in terms of bias in the causal effect estimates, as long as the size of the subsets is reasonably large. CONCLUSIONS: We further elaborated our Bayesian MR method to explicitly account for study heterogeneity. We also adopted a subset posterior aggregation method to ease computational burden, which is important especially when dealing with large studies. Despite the simplicity of the model we have used in the simulations, we hope the present work would effectively point to MR studies that allow modelling flexibility, especially in relation to the integration of heterogeneous studies and computational practicality.

Overlapping-sample Mendelian randomisation with multiple exposures: a Bayesian approach.

BACKGROUND: Mendelian randomization (MR) has been widely applied to causal inference in medical research. It uses genetic variants as instrumental variables (IVs) to investigate putative causal relationship between an exposure and an outcome. Traditional MR methods have mainly focussed on a two-sample setting in which IV-exposure association study and IV-outcome association study are independent. However, it is not uncommon that participants from the two studies fully overlap (one-sample) or partly overlap (overlapping-sample). METHODS: We proposed a Bayesian method that is applicable to all the three sample settings. In essence, we converted a two- or overlapping- sample MR to a one-sample MR where data were partly unmeasured. Assume that all study individuals were drawn from the same population and unmeasured data were missing at random. Then the missing data were treated au pair with the model parameters as unknown quantities, and thus, were imputed iteratively conditioning on the observed data and estimated parameters using Markov chain Monte Carlo. We generalised our model to allow for pleiotropy and multiple exposures and assessed its performance by a number of simulations using four metrics: mean, standard deviation, coverage and power. We also compared our method with classic MR methods. RESULTS: In our proposed method, higher sample overlapping rate and instrument strength led to more precise estimated causal effects with higher power. Pleiotropy had a notably negative impact on the estimates. Nevertheless, the coverages were high and our model performed well in all the sample settings overall. In comparison with classic MR, our method provided estimates with higher precision. When the true causal effects were non-zero, power of their estimates was consistently higher from our method. The performance of our method was similar to classic MR in terms of coverage. CONCLUSIONS: Our model offers the flexibility of being applicable to any of the sample settings. It is an important addition to the MR literature which has restricted to one- or two- sample scenarios. Given the nature of Bayesian inference, it can be easily extended to more complex MR analysis in medical research.

Synthesis of magnetic ZnO/ZnFe2O4 by a microwave combustion method, and its high rate of adsorption of methylene blue.

The magnetic ZnO/ZnFe2O4 particles have been synthesized by a microwave combustion method using NaAc as fuel. The as-obtained ZnO/ZnFe2O4 was characterized and applied for the removal of methylene blue (MB) from aqueous solution in the batch system. The ZnO/ZnFe2O4 particles display larger S(BET) and smaller size with increase of NaAc dosage. Because a certain amount of gas is generated during NaAc decomposing and the gas prevent the particles from growing larger. More interestingly, even at neutral pH value, the ZnO/ZnFe2O4 obtained with 24 mL NaAc shows high-rate adsorption properties with the MB removal efficiency up to 90% in 0.5 min and a maximum adsorption capacity of 37.27 mg/g.

Synthesis of high surface area, mesoporous MgO nanosheets with excellent adsorption capability for Ni(II) via a distillation treating.

Hexagonal mesoporous MgO nanosheets with a side length of 250 nm and specific surface area of 181.692 m(2)/g were fabricated by a three-step process. Firstly, MgO powders were obtained by sintered Mg5(OH)2(CO3)4⋅4H2O, which was synthesized by a wet precipitation process using ammonium hydrogen carbonate as precipitants. Secondly, the above-MgO were distilled 2 h in a three-necked bottle with condenser device. Lastly, we annealed the distilled-MgO at 500-800 °C to form mesoporous MgO nanosheets. We found the pore size distribution and the thicknesses of nanosheets were determined by the distillation process in step 2 and annealed temperature in step 3. By optimizing the experimental parameters, the mesoporous dis-MgO annealed at 600 °C displayed uniform hexagonal structure with the largest pore volume (0.875 cm(3)/g) and highest BET surface area (181.692 m(2)/g), as well as the maximum adsorption capability of 1684.25 mg/g for Ni(II).