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Martel M, Negrín MA, Vázquez–Polo FJ. Bayesian heterogeneity in a meta-analysis with two studies and binary data. J Appl Stat 2022; 50:2760-2776. [PMID: 37720245 PMCID: PMC10503457 DOI: 10.1080/02664763.2022.2084719] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Accepted: 05/24/2022] [Indexed: 10/18/2022]
Abstract
The meta-analysis of two trials is valuable in many practical situations, such as studies of rare and/or orphan diseases focussed on a single intervention. In this context, additional concerns, like small sample size and/or heterogeneity in the results obtained, might make standard frequentist and Bayesian techniques inappropriate. In a meta-analysis, moreover, the presence of between-sample heterogeneity adds model uncertainty, which must be taken into consideration when drawing inferences. We suggest that the most appropriate way to measure this heterogeneity is by clustering the samples and then determining the posterior probability of the cluster models. The meta-inference is obtained as a mixture of all the meta-inferences for the cluster models, where the mixing distribution is the posterior model probability. We present a simple two-component form of Bayesian model averaging that is unaffected by characteristics such as small study size or zero-cell counts, and which is capable of incorporating uncertainties into the estimation process. Illustrative examples are given and analysed, using real sparse binomial data.
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Affiliation(s)
- M. Martel
- Dpt. of Quantitative Methods and TiDES Institute, U. of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Canary Islands, Spain
| | - M. A. Negrín
- Dpt. of Quantitative Methods and TiDES Institute, U. of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Canary Islands, Spain
| | - F. J. Vázquez–Polo
- Dpt. of Quantitative Methods and TiDES Institute, U. of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Canary Islands, Spain
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Jain S, Sharma SK, Jain K. Using Copulas for Bayesian Meta-analysis. STATISTICS IN BIOSCIENCES 2021. [DOI: 10.1007/s12561-021-09312-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Negrín-Hernández MA, Martel-Escobar M, Vázquez-Polo FJ. Bayesian Meta-Analysis for Binary Data and Prior Distribution on Models. INTERNATIONAL JOURNAL OF ENVIRONMENTAL RESEARCH AND PUBLIC HEALTH 2021; 18:809. [PMID: 33477861 PMCID: PMC7832911 DOI: 10.3390/ijerph18020809] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/05/2020] [Revised: 01/14/2021] [Accepted: 01/15/2021] [Indexed: 11/18/2022]
Abstract
In meta-analysis, the structure of the between-sample heterogeneity plays a crucial role in estimating the meta-parameter. A Bayesian meta-analysis for binary data has recently been proposed that measures this heterogeneity by clustering the samples and then determining the posterior probability of the cluster models through model selection. The meta-parameter is then estimated using Bayesian model averaging techniques. Although an objective Bayesian meta-analysis is proposed for each type of heterogeneity, we concentrate the attention of this paper on priors over the models. We consider four alternative priors which are motivated by reasonable but different assumptions. A frequentist validation with simulated data has been carried out to analyze the properties of each prior distribution for a set of different number of studies and sample sizes. The results show the importance of choosing an adequate model prior as the posterior probabilities for the models are very sensitive to it. The hierarchical Poisson prior and the hierarchical uniform prior show a good performance when the real model is the homogeneity, or when the sample sizes are high enough. However, the uniform prior can detect the true model when it is an intermediate model (neither homogeneity nor heterogeneity) even for small sample sizes and few studies. An illustrative example with real data is also given, showing the sensitivity of the estimation of the meta-parameter to the model prior.
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Affiliation(s)
- Miguel-Angel Negrín-Hernández
- Department of Quantitative Methods & TiDES Institute, University of Las Palmas de Gran Canaria, E-35017 Las Palmas de Gran Canaria, Spain; (M.M.-E.); (F.-J.V.-P.)
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Meta-Analysis with Few Studies and Binary Data: A Bayesian Model Averaging Approach. MATHEMATICS 2020. [DOI: 10.3390/math8122159] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/21/2023]
Abstract
In meta-analysis, the existence of between-sample heterogeneity introduces model uncertainty, which must be incorporated into the inference. We argue that an alternative way to measure this heterogeneity is by clustering the samples and then determining the posterior probability of the cluster models. The meta-inference is obtained as a mixture of all the meta-inferences for the cluster models, where the mixing distribution is the posterior model probabilities. When there are few studies, the number of cluster configurations is manageable, and the meta-inferences can be drawn with BMA techniques. Although this topic has been relatively neglected in the meta-analysis literature, the inference thus obtained accurately reflects the cluster structure of the samples used. In this paper, illustrative examples are given and analysed, using real binary data.
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Wang Y, Gao L, Lang W, Li H, Cui P, Zhang N, Jiang W. Serum Calcium Levels and Parkinson's Disease: A Mendelian Randomization Study. Front Genet 2020; 11:824. [PMID: 32849817 PMCID: PMC7431982 DOI: 10.3389/fgene.2020.00824] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2019] [Accepted: 07/08/2020] [Indexed: 12/12/2022] Open
Abstract
BACKGROUND Though increasing epidemiological studies have evaluated the correlation between serum calcium contents and Parkinson's disease (PD), the results are inconsistent. At present, whether there is a causal association between serum calcium content and PD remains undetermined. OBJECTIVE AND METHODS This study was designed to explore the relationship between increased serum calcium contents and PD risk. In this present study, a Mendelian randomization trial was carried out using a large-scale serum calcium genome-wide association study (GWAS) dataset (N = 61,079, Europeans) and a large-scale PD GWAS dataset (N = 8,477, Europeans including 4,238 PD patients and 4,239 controls). Here, a total of four Mendelian randomization methods comprising weighted median, inverse-variance weighted meta-analysis (IVW), MR-Egger, and MR-PRESSO were used. RESULTS Our data concluded that genetically higher serum calcium contents were not significantly related to PD. CONCLUSION In conclusion, we provided genetic evidence that there was no direct causal relationship between serum calcium contents and PD. Hence, calcium supplementation may not result in reduced PD risk.
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Affiliation(s)
- Yanchao Wang
- Department of Neurology, Tianjin Neurological Institute, Tianjin Medical University General Hospital, Tianjin, China
- Department of Neurology, Affiliated Hospital of Chifeng University, Chifeng, China
| | - Luyan Gao
- Department of Neurology, The Fourth Central Clinical College of Tianjin Medical University, Tianjin, China
| | - Wenjing Lang
- Department of Neurology, Tianjin Neurological Institute, Tianjin Medical University General Hospital, Tianjin, China
| | - He Li
- Department of Neurology, Tianjin Neurological Institute, Tianjin Medical University General Hospital, Tianjin, China
| | - Pan Cui
- Department of Neurology, Tianjin Neurological Institute, Tianjin Medical University General Hospital, Tianjin, China
| | - Nan Zhang
- Department of Neurology, Tianjin Neurological Institute, Tianjin Medical University General Hospital, Tianjin, China
| | - Wei Jiang
- Department of Neurology, Tianjin Neurological Institute, Tianjin Medical University General Hospital, Tianjin, China
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Pateras K, Nikolakopoulos S, Roes KCB. Prior distributions for variance parameters in a sparse-event meta-analysis of a few small trials. Pharm Stat 2020; 20:39-54. [PMID: 32767452 PMCID: PMC7818503 DOI: 10.1002/pst.2053] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2019] [Revised: 02/19/2020] [Accepted: 06/24/2020] [Indexed: 11/08/2022]
Abstract
In rare diseases, typically only a small number of patients are available for a randomized clinical trial. Nevertheless, it is not uncommon that more than one study is performed to evaluate a (new) treatment. Scarcity of available evidence makes it particularly valuable to pool the data in a meta-analysis. When the primary outcome is binary, the small sample sizes increase the chance of observing zero events. The frequentist random-effects model is known to induce bias and to result in improper interval estimation of the overall treatment effect in a meta-analysis with zero events. Bayesian hierarchical modeling could be a promising alternative. Bayesian models are known for being sensitive to the choice of prior distributions for between-study variance (heterogeneity) in sparse settings. In a rare disease setting, only limited data will be available to base the prior on, therefore, robustness of estimation is desirable. We performed an extensive and diverse simulation study, aiming to provide practitioners with advice on the choice of a sufficiently robust prior distribution shape for the heterogeneity parameter. Our results show that priors that place some concentrated mass on small τ values but do not restrict the density for example, the Uniform(-10, 10) heterogeneity prior on the log(τ2 ) scale, show robust 95% coverage combined with less overestimation of the overall treatment effect, across varying degrees of heterogeneity. We illustrate the results with meta-analyzes of a few small trials.
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Affiliation(s)
- Konstantinos Pateras
- Department of Biostatistics and Research Support, Julius Center for Health Sciences and Primary Care, University medical center Utrecht, Utrecht, The Netherlands
| | - Stavros Nikolakopoulos
- Department of Biostatistics and Research Support, Julius Center for Health Sciences and Primary Care, University medical center Utrecht, Utrecht, The Netherlands
| | - Kit C B Roes
- Department of Health Evidence, Section Biostatistics, Radboud University Medical Centre, Nijmegen, The Netherlands
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Moreno E, Vázquez-Polo FJ, Negrín MA. Bayesian meta-analysis: The role of the between-sample heterogeneity. Stat Methods Med Res 2017; 27:3643-3657. [PMID: 28511616 DOI: 10.1177/0962280217709837] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
The random effect approach for meta-analysis was motivated by a lack of consistent assessment of homogeneity of treatment effect before pooling. The random effect model assumes that the distribution of the treatment effect is fully heterogenous across the experiments. However, other models arising by grouping some of the experiments are plausible. We illustrate on simulated binary experiments that the fully heterogenous model gives a poor meta-inference when fully heterogeneity is not the true model and that the knowledge of the true cluster model considerably improves the inference. We propose the use of a Bayesian model selection procedure for estimating the true cluster model, and Bayesian model averaging to incorporate into the meta-analysis the clustering estimation. A well-known meta-analysis for six major multicentre trials to assess the efficacy of a given dose of aspirin in post-myocardial infarction patients is reanalysed.
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Affiliation(s)
- Elías Moreno
- 1 Department of Statistics, University of Granada, Granada, Spain
| | - Francisco-José Vázquez-Polo
- 2 Department of Quantitative Methods and TiDES Institute, University of Las Palmas de Gran Canaria, Las Palmas, Spain
| | - Miguel A Negrín
- 2 Department of Quantitative Methods and TiDES Institute, University of Las Palmas de Gran Canaria, Las Palmas, Spain
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Li D, Sivaganesan S. An objective Bayesian analysis of a crossover design via model selection and model averaging. Stat Med 2016; 35:4509-4527. [PMID: 27364101 DOI: 10.1002/sim.7015] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2015] [Revised: 02/02/2016] [Accepted: 05/16/2016] [Indexed: 11/12/2022]
Abstract
Inference about the treatment effect in a crossover design has received much attention over time owing to the uncertainty in the existence of the carryover effect and its impact on the estimation of the treatment effect. Adding to this uncertainty is that the existence of the carryover effect and its size may depend on the presence of the treatment effect and its size. We consider estimation and testing hypothesis about the treatment effect in a two-period crossover design, assuming normally distributed response variable, and use an objective Bayesian approach to test the hypothesis about the treatment effect and to estimate its size when it exists while accounting for the uncertainty about the presence of the carryover effect as well as the treatment and period effects. We evaluate and compare the performance of the proposed approach with a standard frequentist approach using simulated data, and real data. Copyright © 2016 John Wiley & Sons, Ltd.
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Affiliation(s)
- Dandan Li
- Department of Mathematical Sciences, University of Cincinnati, Cincinnati, 45221, OH, U.S.A
| | - Siva Sivaganesan
- Department of Mathematical Sciences, University of Cincinnati, Cincinnati, 45221, OH, U.S.A..
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Vázquez FJ, Moreno E, Negrín MA, Martel M. Bayesian robustness in meta-analysis for studies with zero responses. Pharm Stat 2016; 15:230-7. [DOI: 10.1002/pst.1741] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2015] [Revised: 10/13/2015] [Accepted: 01/14/2016] [Indexed: 11/10/2022]
Affiliation(s)
- F. J. Vázquez
- Department of Quantitative Methods and TiDES Institute; University of Las Palmas de GC; Las Palmas de Gran Canaria 35017 Spain
| | - E. Moreno
- Department of Statistics and O.R.; University of Granada; Granada 18071 Spain
| | - M. A. Negrín
- Department of Quantitative Methods and TiDES Institute; University of Las Palmas de GC; Las Palmas de Gran Canaria 35017 Spain
| | - M. Martel
- Department of Quantitative Methods and TiDES Institute; University of Las Palmas de GC; Las Palmas de Gran Canaria 35017 Spain
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Vázquez-Polo FJ, Moreno E, Negrín MA, Martel M. A Bayesian sensitivity study of risk difference in the meta-analysis of binary outcomes from sparse data. Expert Rev Pharmacoecon Outcomes Res 2015; 15:317-22. [PMID: 25673174 DOI: 10.1586/14737167.2015.1011131] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
In most cases, including those of discrete random variables, statistical meta-analysis is carried out using the normal random effect model. The authors argue that normal approximation does not always properly reflect the underlying uncertainty of the original discrete data. Furthermore, in the presence of rare events the results from this approximation can be very poor. This review proposes a Bayesian meta-analysis to address binary outcomes from sparse data and also introduces a simple way to examine the sensitivity of the quantities of interest in the meta-analysis with respect to the structure dependence selected. The findings suggest that for binary outcomes data it is possible to develop a Bayesian procedure, which can be directly applied to sparse data without ad hoc corrections. By choosing a suitable class of linking distributions, the authors found that a Bayesian robustness study can be easily implemented. For illustrative purposes, an example with real data is analyzed using the proposed Bayesian meta-analysis for binomial sparse data.
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Affiliation(s)
- Francisco-Jose Vázquez-Polo
- Department of Quantitative Methods and TiDES Institute, University of Las Palmas de GC, 35017-Las Palmas de Gran Canaria, Spain
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