STATISTICAL METHOD
Bayesian Data Analysis
Statistical Consulting for Posterior Inference
We apply Bayesian data analysis within your study design, specifying priors that are checked against the outcome scale, confirming sampler convergence and predictive fit, and reporting posterior summaries with credible intervals in publication-ready form.
No Commitment Required
Strictly Confidential
In-House PhD-Level Statisticians
Fixed Project Fee
Same-Day Response
METHOD OVERVIEW
What Is Bayesian Analysis?
Bayesian analysis estimates unknown parameters by combining a prior distribution, which states what is plausible before the data are seen, with the likelihood of the observed data to give a posterior distribution. Results are summarized as posterior estimates with credible intervals and as probabilities of statements such as an effect exceeding a meaningful threshold. A 95% credible interval is a probability statement about the parameter given the model and prior, which a confidence interval is not. Posteriors are usually computed by Markov chain Monte Carlo, so convergence and predictive fit must be examined. OnlineSPSS, a specialist statistical consultancy for academic, clinical and professional research, provides Bayesian data analysis within your study design, with the relevant checks, effect estimates, uncertainty measures, interpretation and publication-ready reporting.
WHEN TO ENGAGE
When Is Bayesian Analysis Appropriate?
Bayesian inference suits questions where prior evidence, limited data or direct probability statements matter, and these are the situations in which researchers most often ask us for it.
Prior Evidence to Incorporate Earlier trials, pilot data or established effect sizes should inform the estimate, with the prior fixed and justified before the new data are analyzed.
Many Small Units to Pool Sites, clinics or species each contribute few observations, and a hierarchical model lets their estimates borrow strength, with the degree of pooling learned from the data.
Direct Probability Statements The question is how probable it is that an effect exceeds a meaningful threshold, which posterior probabilities answer directly and p-values do not.
Priors or Sampler Questioned A model was fitted with default priors and a reviewer queried them. Independent Statistical Review checks the priors, sampler and reporting.
ANALYTICAL APPROACH
How We Apply Bayesian Analysis
Within the family of Bayesian & Computational Methods, the priors, the sampler and the checks are specified together, not left to defaults.
Model and Prior Specification We state the likelihood and set priors on the outcome scale, then run prior predictive checks, which reveal implausible values before fitting. Defaults are justified, not assumed.
Convergence and Sampling A converged sampler is not enough. We inspect R-hat, effective sample size and trace behavior across several chains, and divergent transitions where Hamiltonian sampling is used.
Posterior Predictive Fit We simulate data from the fitted model and compare them with the observed data, to see which features that matter the model fails to reproduce, not only whether it ran.
Prior Sensitivity and Reporting We refit under alternative reasonable priors and report credible intervals directly, so readers see how much the prior, not the data, drives each result.
DELIVERABLES
What You Receive
Within our Quantitative Data Analysis service, the model is delivered with the priors, diagnostics and files needed to report and defend it.
Data & Variable Review Outcome coding, the unit of analysis, missing values and the scale of each predictor are reviewed, so that prior scales can be stated and no sentinel codes enter the model.
Method-Specific Checks Prior predictive checks, R-hat, effective sample size, trace plots and posterior predictive checks are run, with a prior sensitivity analysis under alternative reasonable priors.
Statistical Results Posterior estimates with credible intervals, posterior probabilities of meaningful effects, and Bayes factors only where prespecified, with their dependence on the prior stated.
Publication-Ready Tables & Figures Posterior density and interval plots, diagnostic figures and tables formatted to APA 7, the Bayesian Analysis Reporting Guidelines (BARG) or the journal's requirements.
Interpretation Findings are written as probability statements conditional on the model and the prior, within what the design supports, with the influence of the prior discussed openly.
Reproducible Analysis Files R or Python scripts, Stata do-files or SAS code are provided where applicable, with seeds and sampler settings, so every posterior summary can be regenerated.
ALTERNATIVE METHODS
When Another Method May Be More Appropriate
A prior earns its place only when it adds information or stability. In other situations, these methods answer the question with less modeling, or with a different safeguard.
01
Multiple Linear Regression
With ample data and no usable prior information, a conventional regression usually gives similar estimates and intervals and is simpler to explain and review.
02
Linear Mixed Models (Multilevel Modeling)
For clustered or repeated data with enough clusters, a frequentist mixed model estimated by restricted maximum likelihood often serves, without a prior to justify.
03
Meta-Analysis
When the earlier evidence is several published studies and the question is their combined effect, a random-effects meta-analysis pools them openly, not through a prior.
04
Independent Samples t-Test
A single two-group comparison with adequate data and no prior evidence is answered directly by Welch's t-test, with a mean difference and its confidence interval.
CASE EXAMPLE
How We Work in Practice
An illustration of how a Bayesian model moves from a prespecified prior to a reportable, defensible result.
ILLUSTRATIVE EXAMPLE · MEDICAL DEVICE CLINICAL TRIALS · UNITED STATES
A medical device sponsor's biostatistics group planned a single-arm study of a catheter and wanted its primary analysis to use data from the predecessor device.
Before the new data were analyzed, we specified a prior for the success rate from the earlier device data, with a robust mixture component so that a conflict between old and new data would reduce the borrowing, and fixed the posterior probability needed to meet the performance goal. Prior predictive checks showed which success rates the prior implied, and we reviewed R-hat, effective sample size and posterior predictive fit. Because the earlier population differed somewhat, we also refitted with weaker borrowing and with a vague prior, and reported how far the posterior moved.
Outcome: The group received the prior and its justification, the posterior success rate with its 95% credible interval, the probability of meeting the goal, the diagnostics, the sensitivity results and the code.
This example does not describe a real client or project. We never share details of actual client work, in line with the confidentiality we promise every researcher. This illustrative example is included to show the type, scope, and standard of work involved in this kind of analysis.
SOFTWARE
Statistical Software for Bayesian Models
Bayesian models are run in R and Python samplers, Stata's bayes prefix or SAS PROC MCMC, with code and seeds provided where applicable.
RESEARCH DISCIPLINES
Bayesian Analysis Across Research Fields
Bayesian methods appear across research fields, each with its own sources of prior information, study designs and reporting conventions.
A Defined Process for Your Analysis
We review your dataset, research question and design before any work begins, so the scope, deliverables, timeline and fixed project fee are clear from the outset.
01
Share Your Research Materials
Send your dataset, research question, study design and any existing analysis plan or output. We will let you know if anything else is needed to assess the project.
02
Receive Your Assessment
Your statistically trained Research Coordinator reviews the project and confirms the required analysis, deliverables, timeline and fixed project price before work begins.
03
Analysis & Delivery
Once you confirm the scope, an in-house statistician completes the agreed analysis and delivers the outputs, interpretation and reporting files.
RESEARCHER VOICE
What Researchers Say
Feedback from a researcher who needed the priors and posterior summaries justified and reported in a form usable in a manuscript and in reviewer responses.
“The difficult part was not pressing the button for Bayesian Analysis; it was how the priors and posterior summaries should be justified and interpreted. They caught the problem early, showed us the defensible approach, and kept the reporting focused on the estimates that mattered. The final reporting was much easier to use in the manuscript and in responding to reviewer comments.”
RESEARCH RESPONSIBILITY
Statistical Support Without Taking Over Your Research
We provide the statistical work for your Bayesian model, and the study, its meaning and its authorship stay entirely with you.
Agreed Statistical Support We provide the agreed methodological, analytical and statistical reporting support for your Bayesian analysis, within the scope confirmed before work begins.
No Intellectual Contribution OnlineSPSS claims no intellectual contribution to your research. Our work is statistical support, and authorship remains yours.
Your Research Decisions The research question, the substantive meaning of your priors, interpretation and conclusions remain with you, and our statistical interpretation stays within what the posterior and the design support.
Confidential Handling Research data and project details are handled confidentially and are not published. An NDA is available where required, and can be arranged before sensitive materials are shared.
FREQUENTLY ASKED QUESTIONS
Questions About Bayesian Analysis
Short answers to what researchers ask before a Bayesian project, from credible intervals and priors to confidentiality and the assessment process.
Is a credible interval the same as a confidence interval?
No. A 95% credible interval says that, given the model, the prior and the data, the parameter lies in the interval with probability 0.95. A 95% confidence interval describes how often the procedure would capture the parameter over repeated samples. The two can be numerically close when the prior is weak, but they answer different questions, so we state which one is reported.
How should the priors in a Bayesian analysis be chosen?
Priors come from earlier studies, subject-matter knowledge or weakly informative defaults that regularize without dominating, and they are fixed on the scale of the outcome before the data are analyzed. Prior predictive checks show what a prior implies, and a sensitivity analysis under alternative reasonable priors shows how much the conclusions depend on the choice.
Is Bayesian analysis or a conventional model better for my study?
It depends on what the analysis must deliver. A Bayesian model is worth its extra specification when earlier evidence should formally inform the estimate, when many small units need partial pooling, or when a decision rests on the probability that an effect exceeds a threshold. When none of these applies, a frequentist analysis usually reaches comparable conclusions with fewer choices for reviewers to question, and we say so at the assessment.
Can you review an analysis that has already been completed?
Yes, we can review a completed Bayesian analysis before submission or a response to reviewers. We check how the priors were justified, the convergence diagnostics, the posterior predictive checks, the prior sensitivity and the reporting of credible intervals, and confirm what stands or needs to change.
How do you protect my dataset and research confidentiality?
Your dataset, model code and any protocol or analysis plan are handled confidentially, and project information is not published. An NDA is available where required and can be arranged before sensitive materials are shared. Read more about confidentiality and data security.
What happens when I request an assessment, and how long does it take?
You share your research question, design and relevant materials, and a statistically trained Research Coordinator reviews the request and responds the same day. The analysis, deliverables, timeline and one fixed project fee are then confirmed before any work begins, so the timeline follows from the agreed scope.
Still have a question? Find more answers in our full FAQ.
NEXT STEP
Need Support With
Bayesian Analysis?
Share your research question, study design and data. We assess whether a Bayesian model fits your question and confirm the approach before any work begins.
No Commitment Required
Strictly Confidential
Fixed Project Fee
In-House PhD-Level Statisticians
Same-Day Response
Serving Researchers Since 2014