Meta-analysis integrates results from multiple independent studies with similar research objectives to provide a more reliable overall estimate of intervention effects. However, traditional one-time meta-analysis only provides a final pooled conclusion and makes it difficult to reveal the specific contribution of each individual study to the overall effect. To address this limitation, Cumulative Meta-analysis was proposed. By sequentially adding studies and repeatedly recalculating the pooled results, researchers can dynamically observe how intervention effects change with the number of studies and over time. Cumulative meta-analysis not only shows the influence of early studies on the overall conclusion but also evaluates the impact of small-sample or negative studies on the overall results, thereby improving the scientific rigor of evidence-based decision-making.

I. Understanding Cumulative Meta-Analysis

Meta-analysis essentially integrates results from multiple independent but similar studies for quantitative analysis. Through this approach, we can not only observe overall effects but also analyze differences between studies to arrive at a relatively reliable comprehensive conclusion. In medical research, Meta-analysis has become very common, and with advances in statistical methods and computational tools, it has become increasingly refined. Traditional Meta-analysis typically combines all studies at once to reach an overall conclusion. While this reflects the overall effect, it is difficult to see the specific contribution of each individual study to the comprehensive result. To solve this problem, scholars proposed the “cumulative Meta-analysis” method. The characteristic of cumulative Meta-analysis lies in treating study data as a continuous whole—each time a new study is completed, a Meta-analysis can be conducted. This not only inherits the advantages of traditional Meta-analysis but also allows dynamic observation of how research results change over time and the impact of each study on the comprehensive conclusion.

The operation of Cumulative Meta-analysis is actually not difficult. The first step is to determine the research objective, then collect relevant study materials as comprehensively as possible, including both published and unpublished studies, to reduce selection bias. Next, evaluate the quality of collected studies and exclude those that do not meet standards to ensure reliable analysis results. Then, unify the measurement indicators of study effects, such as Odds Ratio (OR), Relative Risk (RR), or Mean Difference, to ensure data from different studies can be combined. Arranging the order of studies is also important—they can be sorted by publication year, sample size, or intervention effect size. Finally, select appropriate statistical methods, generally either fixed-effect or random-effects models. Cumulative Meta-analysis repeats the analysis each time a new study is added, ultimately displaying trend changes in effects intuitively through charts while also showing the contribution of each study to the comprehensive conclusion.

A cardiovascular example makes this more intuitive. For instance, studying the effect of drug intervention on mortality in acute myocardial infarction patients. Traditional Meta-analysis shows that a certain antiplatelet drug can significantly reduce mortality. But if we use cumulative Meta-analysis, accumulating by publication year order, the effect value and confidence interval will gradually stabilize, and we can see which year and which study first showed statistically significant results. Through this approach, we can not only see the overall therapeutic effect but also understand the contribution of early studies to the conclusion, thereby avoiding excessive optimism or premature conclusions. Another approach is to conduct cumulative analysis by sample size.

Small-sample study results fluctuate greatly—if we rely only on these studies, we might overestimate therapeutic effects. But as large-sample studies are added, cumulative analysis can more accurately reflect true therapeutic effects while narrowing the confidence interval, making conclusions more reliable. Additionally, we can conduct cumulative analysis by the magnitude of therapeutic differences between treatment and control groups. This allows observation of how smaller therapeutic effects or negative results impact overall conclusions, providing a more comprehensive understanding of treatment effects.

Another advantage of cumulative Meta-analysis is that it can help determine when sufficient evidence already exists, avoiding unnecessary repeated experiments and thereby saving research resources and reducing patient risks. For example, some early cardiovascular drug studies showed significant effects, but subsequent studies found the therapeutic effects were not as pronounced as initially expected. Through cumulative Meta-analysis, we can detect this trend early and alert clinicians to exercise caution in medication use. At the same time, cumulative analysis can reveal the impact of small-sample or negative studies on comprehensive conclusions, thereby discovering potential biases and guiding subsequent research design and optimization of trial protocols.

Of course, cumulative Meta-analysis also has some issues that require attention. Since it is essentially continuous analysis, if significance levels are not adjusted, the risk of Type I error (false positive) may increase. Scholars have proposed several solutions, such as using stricter significance standards for each analysis or distributing the overall significance level across each analysis. However, some scholars believe Bayesian methods can be used to interpret cumulative analysis results without necessarily adjusting significance levels—this remains a methodological controversy. Another issue is whether different therapeutic indicators can be combined for analysis. Generally, as long as the research objectives are the same, they can be combined; otherwise, combination is not recommended. There is also debate about whether comprehensive conclusions can rely on small-sample studies. Overall, cumulative analysis can increase sample size and reduce random errors, but if the underlying study quality is poor or samples are too small, results may still be affected by bias, so conclusions must be interpreted cautiously.

II. Running Cumulative Meta-Analysis in R

metacum.meta is a function provided by the meta package in R for conducting cumulative Meta-analysis. Simply put, it can accumulate similar studies in chronological or other order, calculating the comprehensive effect each time a new study is added, thereby dynamically observing trends in effect indicators. This analytical method is particularly suitable for clinical research or evidence-based medicine where understanding how intervention effects change over time is needed, or determining whether early studies have already provided sufficient evidence.

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Bayesian methods have become increasingly popular in systematic reviews and meta-analyses of diagnostic test accuracy because of their flexibility in handling small samples, sparse data, and high heterogeneity. Compared with classical (frequentist) approaches, Bayesian analysis not only provides point estimates but also fully characterizes uncertainty and its distribution.

In R, the bamdit package offers an integrated framework for Bayesian bivariate random-effects meta-analysis of diagnostic tests. It enables complex model fitting (via MCMC methods such as JAGS) and provides rich visualization tools that help researchers interpret model results, explore heterogeneity, and identify potential conflicting evidence.

Visualization in Bayesian Meta-Analysis

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In medical diagnostic test research, how to effectively synthesize evidence across multiple studies while accurately quantifying the uncertainty of sensitivity and specificity has always been a core statistical challenge. Compared with traditional Meta-analysis, the Bayesian framework can naturally handle uncertainty, flexibly adapt to different data structures, and maintain robust performance in settings with small sample sizes and high heterogeneity. This article demonstrates the full analytical process with code implementation.

Review of Bayesian Meta-Analysis

In the previous article, we discussed that Bayesian Meta-analysis is an advanced data synthesis method based on Bayesian statistical principles. It significantly extends and optimizes traditional Meta-analysis. Conventional Meta-analysis typically adopts fixed-effect or random-effects models, combining effect sizes from multiple independent studies through weighted aggregation to obtain an overall effect estimate.

The fixed-effect model assumes that all studies share the same true effect, which is suitable when no substantial heterogeneity exists across studies. In contrast, the random-effects model assumes that effect sizes vary between studies, making it more appropriate for highly heterogeneous research environments.

However, in diagnostic test Meta-analysis, sensitivity and specificity are often analyzed separately, ignoring their negative correlation and between-study heterogeneity. This may lead to underestimation of true uncertainty.

Bayesian Meta-analysis adopts a fundamentally different statistical framework: it treats both data and model parameters as random variables. By combining prior distributions with the likelihood derived from observed data, posterior distributions are generated to quantify parameter uncertainty. This approach allows incorporation of expert knowledge or prior research evidence into the analysis.

Beyond synthesizing evidence from different studies, Bayesian Meta-analysis provides more transparent, robust, and generalizable conclusions—particularly under small sample or sparse data conditions. Compared with traditional approaches, it offers substantial advantages in handling heterogeneity, quantifying parameter uncertainty, model flexibility, and extrapolation capability. It is widely used in complex hierarchical models, diagnostic bivariate analyses, and network Meta-analysis.

In practical applications, the R package bamdit provides a convenient and powerful tool for Bayesian diagnostic Meta-analysis. It is based on a scale-mixture bivariate random-effects model and can integrate research data of varying types and quality, including randomized controlled trials and observational studies. It accounts for both internal and external validity bias, improving robustness.

The core function metadiag() estimates sensitivity and specificity, supporting:

• Bivariate random-effects modeling

• Different link functions

• Evidence conflict handling

• Prediction for new studies

Computation is performed via JAGS using MCMC sampling, generating full posterior distributions of parameters. This enables researchers to quantify uncertainty under small samples or high heterogeneity, rather than relying solely on point estimates and confidence intervals.

Additionally, bamdit provides Bayesian summary ROC curves, AUC calculation, visualization tools, and prior specification functions, greatly simplifying traditional Bayesian diagnostic Meta-analysis workflows and enabling clinicians and statisticians to conduct complex analyses efficiently and robustly.

1. Data Preparation

The bamdit package includes the example dataset glas from Glas et al. (2003), covering detection data for multiple biomarkers. In actual analysis, researchers typically filter for markers of interest, such as Telomerase, to ensure targeted analysis. Filtering specific markers not only reduces heterogeneity interference but also facilitates accurate estimation of sensitivity and specificity in subsequent models.

The core goal of data organization is to structure the 2×2 table information from each study into an R data frame, commonly with columns in the order: TP (True Positives), number of diseased patients (N1), FP (False Positives), and number of non-diseased patients (N2). This structure is the default reading format for the metadiag() function. During organization, attention must be paid to missing values, outliers, and sample size differences, as these factors directly affect MCMC sampling convergence and posterior distribution stability. After completing data organization, preliminary visualization is crucial. The plotdata() function can intuitively display the distribution of sensitivity and specificity for each study, helping researchers identify negative correlation patterns, outliers, and potential heterogeneity.

library(bamdit)
data("glas")

# Select data for Telomerase marker, keeping first four columns: tp, n1, fp, n2
glas.t <- glas[glas$marker == "Telomerase", 1:4]

# Visualize data distribution
plotdata(glas.t, two.by.two = FALSE)

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Meta-analysis is a statistical tool used to synthesize results from multiple independent studies in order to obtain more stable and reliable overall conclusions. Traditional meta-analysis commonly adopts either a fixed-effect model or a random-effects model, weighting and pooling study-specific effect sizes. However, in diagnostic test research, sensitivity and specificity are often analyzed separately, ignoring their negative correlation and between-study heterogeneity.

Bayesian meta-analysis adopts a different statistical framework, treating both data and model parameters as random variables. By combining prior distributions with the likelihood function derived from the data, it generates posterior distributions. This framework not only quantifies uncertainty but also allows the incorporation of subjective knowledge or prior research evidence when appropriate, leading to more precise and transparent analytical results.

I. Understanding Bayesian Meta-Analysis

Meta-analysis is a class of statistical tools capable of synthesizing results from multiple independent scientific studies. It allows researchers to obtain more stable overall conclusions from study data collected at different locations, using different methods, or even from different databases or literature. Meta-analysis is often considered a key component of systematic reviews, but it is important to emphasize that not all systematic reviews include meta-analysis; therefore, the two cannot be simply equated. Logically, meta-analysis is equivalent to “creating new data” under appropriate conditions because it weights and integrates effects from multiple studies and outputs a new, quantified uncertainty assessment of the overall effect.

Traditional meta-analysis primarily includes two types of statistical models: the fixed-effect model and the random-effects model. The former assumes that all studies share exactly the same true effect, suitable for situations where there is no statistical heterogeneity between studies and generalization of conclusions is not required. The latter posits that there is no single true effect, but rather effect values follow some distribution across studies, making it more appropriate when the number of studies is relatively large (usually ≥ 5), study differences are apparent, or generalization to broader populations is needed. Thus, the fixed-effect model estimates “the single effect shared by all studies,” while the random-effects model estimates “the mean of the effect distribution.” This difference lays the foundation for various extended meta-analysis methods.

However, in traditional diagnostic test meta-analysis, researchers typically summarize and analyze the two indicators—Sensitivity and Specificity—separately. The core assumption of this method is that the two are independent of each other. But in reality, sensitivity and specificity often exhibit statistical correlation—improving the sensitivity of a diagnostic tool (making it easier to detect patients) is likely to come at the cost of specificity (more false positives), and vice versa. More importantly, different study designs (e.g., prospective vs. retrospective studies), patient sources (multi-center vs. single-center, different regions), diagnostic tools and operational procedures, and data quality (sample size, missing data, risk of bias, etc.) can all greatly affect the performance of sensitivity and specificity. Therefore, pooling these two indicators separately not only ignores their potential negative correlation but also fails to reveal the complexity of data structures and methodological differences between studies. Such analysis results can easily underestimate true inter-study heterogeneity, leading to misleading clinical applications.

In recent years, methods adopting a completely different statistical framework—namely Bayesian meta-analysis—have also been growing rapidly. The fundamental difference between Bayesian meta-analysis and frequentist meta-analysis lies in: Bayesian methods treat both data and model parameters as random variables. Its main task is to calculate the probability distribution of data given parameters, and on this basis construct posterior distributions, under the premise that parameters are treated as random variables. Importantly, Bayesian methods allow researchers to incorporate subjective knowledge and prior beliefs—such as previous studies, expert judgments, prior data, etc.—into the construction of parameter distributions when appropriate. Although this characteristic of “introducing subjective information” is often criticized by opponents as “subjective statistics,” in reality, the posterior distribution formed by prior information and the likelihood function is precisely the core of Bayesian inference being more precise and controllable. The posterior distribution simultaneously reflects data evidence and argumentative assumptions; therefore, Bayesian meta-analysis often provides more accurate and transparent research synthesis results than frequentist methods.

Systematic Comparison of Traditional Meta-Analysis (Fixed/Random Effects) and Bayesian Meta-Analysis

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In medical research, a single trial is often insufficient to comprehensively evaluate the effectiveness of an intervention or the progression pattern of a disease. This is because each study typically has a limited sample size, and differences in research design, intervention intensity, and measurement methods may lead to inconsistent results.

To obtain more reliable conclusions, researchers usually apply Meta-analysis to statistically synthesize results from multiple independent studies and infer an overall effect. However, traditional Meta-analysis typically focuses on a single outcome variable and ignores the dependence that may exist among multiple correlated effect sizes reported within the same study.

To address this limitation, Multivariate Meta-Analysis (MVMA) was developed. MVMA can simultaneously synthesize multiple correlated outcomes while accounting for between-study heterogeneity and within-study effect correlations. This provides more accurate and reliable overall effect estimates for clinical and public health research.

I. Understanding Multivariate Meta-Analysis

In medical and social science research, researchers frequently encounter a common yet complex phenomenon: multiple independent studies often exist for the same research topic. For example, when researchers aim to evaluate the efficacy of a particular drug for a specific disease, they may find dozens or even hundreds of clinical trials or observational studies with published results. Each study may differ in sample size, study design, measurement methods, intervention intensity, and follow-up duration, making direct comparison or simple aggregation of results difficult.

Traditional Meta-analysis methods typically treat these studies as independent units, using statistical methods such as weighted averaging to pool effect sizes and obtain an estimate of the overall effect. These analytical methods played important roles in early evidence-based medical research, helping researchers extract more reliable, generally applicable conclusions from scattered studies—for instance, clarifying a drug’s therapeutic effect on a disease or its degree of risk reduction. However, reality is often much more complex than such simple aggregation. Individual studies often contain multiple effect sizes, possibly because studies used multiple measurement tools to assess the same indicator, conducted measurements on different population subgroups, or performed repeated measurements at different time points. Each effect size contains certain information, but these effect sizes are not completely independent; instead, they exhibit correlations or dependencies. If this correlation is ignored, traditional Meta-analysis may underestimate standard errors or misjudge the significance of overall effects, leading to inaccurate or biased conclusions.

It is against this backdrop of complex data structures that scientists developed a new statistical method—Multivariate Meta-Analysis (MVMA)—to synthesize estimation results for multiple related parameters across different studies, thereby comprehensively characterizing the multidimensional associations between exposures and health outcomes. MVMA can not only synthesize results from multiple studies but also handle multiple effect sizes within individual studies and their dependency issues, providing researchers with more refined and reliable analytical tools. Through multivariate Meta-analysis, researchers can obtain more accurate estimates of overall effect sizes while accounting for dependencies among effect values, and can also explore potential moderating factors and sources of heterogeneity, thereby advancing Meta-analysis from simple aggregation to complex data parsing and further enhancing the quality and scientific rigor of evidence-based research.

The core concept of multivariate Meta-analysis is “hierarchical” or “nested” structures. From a statistical perspective, data from each study is not a single plane but a complex structure composed of multiple levels. Taking the random-effects model as an example, it is essentially a multilevel model: the first level consists of participants or individual samples, and the second level consists of the studies themselves.

At the first level, each participant’s measurements produce certain random errors due to individual differences; at the second level, differences between different studies introduce between-study heterogeneity. Random-effects models can estimate overall effect sizes and their uncertainty by simultaneously considering these two sources of error. This means that even when we use only summary averages from each study in Meta-analysis, the model implicitly reflects the multilevel structure. If we further complicate this, when a study contributes multiple effect sizes—for example, results obtained from different populations or different measurement tools within the same study—these effect sizes are nested within the same study, forming a third level. The Three-Level Model is specifically designed to handle such situations. By introducing a third level, the model can not only reflect differences between studies but also capture correlations among effect sizes within studies, making the overall analysis closer to the true structure of the data.

Specifically, variance at the first level reflects sampling error at the participant level, variance at the second level measures differences among effect sizes within studies, and the third level quantifies between-study heterogeneity. Such hierarchical structures enable the model to clearly identify the contribution of different levels to total variation while estimating overall effects, avoiding biases that arise from ignoring dependencies.

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In medical research, the relationship between dose and health risk is often influenced by differences in study design, dose levels, and population characteristics, making results from individual studies difficult to compare or synthesize directly. For example, the impact of alcohol intake on cardiovascular disease, body mass index (BMI) on diabetes risk, and vitamin intake on cancer risk all involve complex dose–response relationships. In this context, dose-response meta-analysis provides an effective approach to integrate multiple studies and quantify the impact of dose changes on risk.

Within the random-effects model framework, BLUP (Best Linear Unbiased Prediction) enables robust prediction of study-specific effects. By borrowing strength from the overall distribution, BLUP “shrinks” small-sample or highly variable studies toward the overall mean, reducing random noise and improving the reliability of study-specific estimates.

I. Understanding Best Linear Unbiased Prediction in Dose-Response Meta-Analysis

What is BLUP?

In medical research, the dosage-risk relationship is a core research question—such as the association between alcohol intake and cardiovascular disease risk, BMI and diabetes incidence, and vitamin intake and cancer risk. Variations in dosage levels and study populations across different studies hinder the direct comparison and synthesis of individual study results. In such cases, Dose-response Meta-analysis (DRMA) offers an effective methodological framework: by integrating dosage and effect data from multiple studies, it enables a more comprehensive and quantitative assessment of the specific impact of dosage changes on health risks.

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In epidemiology and clinical research, understanding the relationship between exposure dose and disease risk is a fundamental basis for developing public health strategies and clinical intervention plans. Traditional case–control or cohort studies often compare only the high-exposure group with the reference group, ignoring the continuous influence of different dose levels on disease risk. As a result, detailed quantification of risk changes is limited.

A typical fixed-effect dose–response analysis provides a methodological solution by integrating risk estimates at each dose level into a continuous dose–risk trend using regression models. This approach quantifies how much risk changes with each unit increase in dose. Under this framework, the dose–effect relationship within the study is assumed to be unique and fixed, without between-study heterogeneity. Thus, model parameters accurately reflect the average effect across dose groups within that study. This method fully utilizes information from all exposure categories, improves statistical efficiency and trend detection, and lays the foundation for subsequent dose–response meta-analysis.

I. Understanding Classical Fixed-Effect Dose–Response Analysis

Fixed-effect dose–response analysis and dose–response meta-analysis are methodologically related but differ in scope and purpose. Fixed-effect analysis typically applies to grouped data within a single study, fitting the relationship between continuous dose and outcome through regression modeling. It assumes that the internal dose–effect relationship is true and unique, with no heterogeneity.

For example, in a case–control study, log odds ratios (log OR) or log relative risks (log RR) can serve as dependent variables, while dose level serves as the independent variable. Linear or nonlinear trend models quantify how risk changes per unit increase in dose. Unlike simple high-versus-low comparisons, this method uses information across all exposure groups, improving statistical efficiency and trend detection.

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Dose–response meta-analysis (DRMA) is an advanced statistical method used to quantitatively evaluate the relationship between exposure dose and outcome risk. Compared with traditional meta-analyses of binary or continuous data, DRMA can handle data from three or more exposure levels, allowing direct estimation of the relationship between dose and disease risk. By synthesizing multiple original dose–response studies, researchers can obtain a unified effect estimate and construct linear or nonlinear curves, providing more precise and reliable evidence for scientific decision-making.

I. Understanding Dose–Response Meta-analysis

Meta-analysis (MA) is a class of statistical methods for quantitatively synthesizing multiple study results, dating back to the 1970s and 1980s. Initially, meta-analysis was defined as “a quantitative analytical method for synthesizing multiple study results,” a concept proposed by L. V. Hedges in 1985. In 1991, Fleiss provided a more rigorous and precise definition, considering meta-analysis as a class of statistical methods for comparing and synthesizing research results addressing the same scientific question, with the validity of conclusions depending on whether included studies met certain conditions. Over time, institutions and scholars such as the Cochrane Collaboration, the U.S. National Library of Medicine (NLM), the Himmelfarb Health Sciences Library, and Gene Glass have proposed their own definitions. Thus, meta-analysis is fundamentally a statistical method that can combine different study results to obtain more precise and statistically powerful conclusions. However, as the theoretical framework and methods of meta-analysis have continuously improved, new meta-analysis methods have emerged, among which dose-response meta-analysis (DRMA) is a typical representative.

Dose-response meta-analysis has become an indispensable method in modern epidemiology and clinical research primarily because it can quantitatively describe the relationship between exposure levels and disease risk, compensating for the limitations of traditional binary or simple continuous variable meta-analyses. Traditional meta-analyses typically only compare effect differences between exposed and non-exposed groups, unable to reveal the specific impact of dose changes on risk, and difficult to answer the core question of “how much risk change will increasing or decreasing exposure bring.” In real-world research, many exposure factors (such as smoking amount, drug dosage, nutrient intake, etc.) affect health outcomes not simply by presence or absence, but rather change linearly or nonlinearly with increasing dose. Therefore, relying solely on binary analysis tends to underestimate or overestimate the true impact of exposure, or even miss critical dose thresholds or saturation points. Through dose-response meta-analysis, researchers can integrate data from different dose groups across multiple studies, quantitatively estimate risk at each dose level, and plot dose-response curves through linear or nonlinear models, thereby revealing potential dose-dependent effects. This analytical method not only improves statistical power, enabling comprehensive utilization of results from small-sample studies, but can also explore nonlinear relationships and dose thresholds, providing scientific evidence for developing public health guidelines, drug dosage regimens, and clinical intervention strategies. Furthermore, dose-response meta-analysis can identify heterogeneity between studies, assess the impact of confounding factors, and improve the robustness of conclusions through sensitivity and subgroup analyses, thereby playing an important role in evidence-based medicine and policy-making.

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With the continuous development of medical research, traditional meta-analysis can no longer fully meet the needs of clinical decision-making. Network Meta-Analysis (NMA), as an emerging statistical method, enables comprehensive comparisons among multiple treatment options and provides more reliable evidence for researchers and clinicians. By simultaneously incorporating both direct and indirect comparisons, NMA makes full use of existing study data and supports the identification of optimal treatment strategies for patients. This article introduces the fundamental concepts, statistical methods, practical applications, and the significance of NMA in healthcare decision-making.

I. Understanding Network Meta-Analysis

1. Basic Concept of Network Meta-Analysis

Network Meta-Analysis (NMA) is an advanced statistical technique designed to evaluate the effectiveness of multiple treatment options simultaneously. Unlike traditional meta-analysis, which typically compares only two interventions, NMA allows comparisons across multiple treatments within a single analytical framework. This method was developed to address the challenge clinicians face when choosing among several available therapies.

In many clinical scenarios, physicians must select among several effective treatments. Traditional meta-analysis provides limited information because it cannot integrate all possible treatment options. NMA overcomes this limitation by synthesizing results from multiple studies to identify the most effective intervention and support evidence-based clinical decision-making.

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In today’s medical and clinical research, meta-analysis, as a method for synthesizing results from multiple studies, is receiving increasing attention. It not only improves statistical significance but also provides a more comprehensive perspective, helping decision-makers understand the effects of specific treatments or interventions.

Single-rate meta-analysis focuses specifically on the proportion of a particular event, such as the cure rate in a treatment group or the incidence of adverse events. A single rate refers to the proportion of a specific event occurring within a given sample. This analytical approach is widely used in epidemiology, clinical trials, and public health research because it effectively integrates event data from different studies and provides reliable statistical inference for broader populations.

I. Understanding Single-Rate Meta-Analysis

A single rate usually refers to the probability of a specific event (e.g., disease occurrence, treatment response) within a particular sample. For example, if 25 out of 100 patients develop a certain disease, the incidence rate is 25%.

The main purpose of single-rate meta-analysis is to summarize the incidence of the same event across multiple studies. It is widely used in epidemiology and clinical research. For instance, analyzing disease incidence across different regions can help inform public health policy and improve disease prevention strategies.

Advantages:

• Data integration: Effectively combines data from different studies to provide more accurate estimates.

• Increased sample size: Pooling multiple studies significantly increases the total sample size and statistical power.

• Heterogeneity analysis: Differences may exist between studies; heterogeneity analysis helps explore the sources of variation.

II. Conducting Single-Rate Meta-Analysis in R

This article uses data from Ford et al.’s paper:

Systematic Review and Meta-analysis of the Prevalence of Irritable Bowel Syndrome in Individuals With Dyspepsia.

The study background indicates that dyspepsia and irritable bowel syndrome (IBS) are common and may coexist. Through systematic review and meta-analysis, the authors estimated the prevalence of IBS among patients with dyspepsia.

Relevant literature was retrieved from MEDLINE and EMBASE. Ultimately, 19 studies were included from 239 screened papers, covering 18,173 participants.

Results:

• Prevalence of dyspepsia: 27%

• Prevalence of IBS among dyspepsia patients: 37%

• Prevalence of IBS in patients without dyspepsia: 7%

This indicates that IBS occurs eight times more frequently in dyspepsia patients than in the general population, suggesting a possible shared pathogenic mechanism. The study therefore recommends routine IBS assessment in patients with dyspepsia.

1. Dataset Construction

When performing a single-rate meta-analysis, the first step is to construct the dataset. In this study, the dataset ford.ibs is structured as follows:

study <- c("Talley", "Holtmann", "Holtmann", "Talley", "Agreus", "Schlemper", "Schlemper", "Kennedy", "Talley", "Caballero-Plasencia", "Shah", "Curioso", "Hu", "Locke", "Lu", "Papatheoridis", "Perona", "Minocha", "Bolling-Sternevald")
years <- c("1992", "1994", "1994", "1994", "1995", "1995", "1995", "1998", "1998", "1999", "2001", "2002", "2002", "2005", "2005", "2005", "2005", "2006", "2008")
event.e <- as.numeric(c(213, 44, 124, 200, 372, 46, 80, 833, 92, 63, 774, 87, 304, 90, 561, 339, 47, 247, 397))
event.n <- as.numeric(c(835, 180, 423, 919, 1154, 175, 473, 3169, 730, 264, 2549, 231, 1649, 643, 2018, 700, 70, 990, 1001))
ford.ibs <- data.frame(study, years, event.e, event.n)

ford.ibs

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Meta-analysis is a method for synthesizing results from multiple studies and is commonly used to evaluate the overall effect of an intervention or treatment. In many scientific studies, meta-analysis helps assess the robustness and generalizability of experimental data. Especially in the fields of medicine and psychology, meta-analysis can effectively integrate heterogeneity across studies, improving statistical power and the accuracy of results.

This article focuses on meta-analysis of continuous variables, particularly using the meta extension package in R. The analysis is demonstrated using the Fleiss93cont dataset.

I. The Fleiss93cont Dataset

The Fleiss93cont dataset in the R meta extension package originates from a meta-analysis published by E. Mumford et al. (1984) on the impact of mental health treatment on medical service utilization. The dataset contains basic information from five studies, including study year, sample sizes, means, and standard deviations for both treatment and control groups.

library(meta)
data(Fleiss93cont)
Fleiss93cont

The contents of the Fleiss93cont dataset are as follows:

    study   year n.e mean.e sd.e n.c mean.c sd.c
1   Davis  1973  13    5.0  4.70  13   6.50  3.80
2 Florell  1971  30    4.9  1.71  50   6.10  2.30
3   Gruen  1975  35   22.5  3.44  35  24.90 10.65
4    Hart  1975  20   12.5  1.47  20  12.30  1.66
5  Wilson  1977   8    6.5  0.76   8   7.38  1.41

• study: study label

• year: year of the study

• n.e: total number of participants in the treatment group

• mean.e: mean outcome in the treatment group

• sd.e: standard deviation in the treatment group

• n.c: total number of participants in the control group

• mean.c: mean outcome in the control group

• sd.c: standard deviation in the control group

The dataset contains 5 studies, primarily used to analyze the impact of mental health treatment on medical service utilization.

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Sensitivity analysis and subgroup analysis play a crucial role in Meta-analysis. They help evaluate the robustness of study results and identify potential sources of heterogeneity. This article provides a detailed discussion of these two analytical methods, illustrating how to perform them using the metainf() and metabin() functions in R, supported by practical dataset examples.

I. Review of Previous Content

1. Formula Construction

The Fleiss93 dataset comes from the meta extension package and includes seven clinical trials conducted in the 1970s–1980s on aspirin for preventing death after myocardial infarction.

library(meta)
data(Fleiss93)

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I. Review of Previous Content: Formula Construction and Result Interpretation

The Fleiss93 dataset comes from the meta extension package and includes seven clinical trials conducted in the 1970s–1980s on aspirin for the prevention of mortality after myocardial infarction.

Sec.2-Ch.1-Subsec.3：Fixed-Effect Dose–Response Modeling in R

This article is based on the Fleiss93 dataset and uses the meta extension package to conduct a meta-analysis of binary outcomes, with a detailed discussion of the relevant statistical methods and interpretation of heterogeneity and effect sizes.

library(meta)
data(Fleiss93)
str(Fleiss93)
head(Fleiss93)

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In epidemiology and clinical research, understanding the relationship between exposure dose and disease risk is a fundamental basis for developing public health strategies and clinical interventions. Traditional case–control or cohort studies often compare only the risk difference between a high-exposure group and a reference group, while ignoring the continuous effect of different dose levels on disease risk. This makes it difficult to quantify the fine-grained details of risk variation.

A typical fixed-effect dose–response analysis provides a methodological approach to address this limitation. By using regression models, the risk values corresponding to each dose level are integrated into a continuous dose–risk trend, enabling quantification of “the change in risk associated with each unit increase in dose.” In this type of analysis, it is assumed that the dose–effect relationship within the study is unique and fixed, with no interference from between-study heterogeneity. As a result, the model parameters accurately reflect the average effect of each dose group within that study.

This approach not only fully utilizes information from all dose groups—thereby improving statistical efficiency and trend detection—but also lays the foundation for subsequent dose–response meta-analyses.

I. Understanding Typical Fixed-Effect Dose–Response Analysis

Typical fixed-effect dose–response analysis and dose–response meta-analysis are closely related methodologically, yet differ clearly in scope and purpose.

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Meta-analysis is a method that systematically summarizes and quantitatively synthesizes results from multiple independent studies. Its goal is to evaluate the overall effect of a specific research question using statistical techniques. As an important clinical research tool, meta-analysis helps integrate existing evidence, improve statistical power, and derive more reliable conclusions.

This article provides an in-depth explanation of the seven major steps of meta-analysis and their key indicators, helping readers understand and master the core principles of this methodology.

Step 1: Research Background and Objectives

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Meta-analysis, as an essential component of evidence-based medicine, provides scientifically reliable evidence support through systematic reviews and quantitative synthesis of multiple study results. Although meta-analysis faces numerous challenges in practical applications, its advantages in improving statistical power and identifying heterogeneity and bias make it indispensable in medical research and clinical decision-making. With advancements in technology and data processing capabilities, the application prospects of meta-analysis will continue to expand.

I. Understanding Evidence-Based Medicine and Meta-Analysis

1. Evidence-Based Medicine

Evidence-Based Medicine (EBM) is a medical practice approach that emphasizes optimizing decision-making and clinical practice through the application of well-designed and well-executed research evidence.

In 1992, JAMA first introduced the concept of evidence-based medicine, emphasizing that physicians should master the skills of searching for, understanding, and applying scientific research reports.

In 1996, BMJ published a new definition of evidence-based medicine, acknowledging that previous medical practice was also evidence-based and that clinical experience is an indispensable source of evidence. Since then, evidence-based medicine has gradually evolved and gained widespread recognition.

Currently, selecting the best available evidence according to real-world circumstances is the core principle of evidence-based medicine.

2. Meta-Analysis

Meta-analysis is the core component of a Systematic Review. It uses statistical methods to quantitatively synthesize results from multiple independent studies, thereby drawing a more comprehensive conclusion.

The emergence of meta-analysis stems from the diversity and complexity of scientific research findings. Even within the same field, different studies often reach different conclusions due to variations in study design, data collection, and sample size.

Through meta-analysis, we can overcome the limitations of single-study conclusions, improve statistical power, and provide more reliable evidence for clinical decision-making.

A Brief History of Meta-Analysis

• 17th century: The foundations of meta-analysis can be traced back to mathematicians such as Blaise Pascal, who developed statistical methods for analyzing probabilities in games of chance. This marked the beginning of combining quantitative methods with observation, later influencing astronomy and other fields.

• 18th–19th centuries: Mathematicians such as Gauss and Laplace further advanced these methods, although distinctions between individual study results and aggregated results were not yet clearly defined.

• Mid-20th century: With the rapid growth in the number of studies, systematic methods for synthesizing research results became increasingly necessary. In 1940, researchers in psychology first began using quantitative synthesis, particularly in studies of extrasensory perception.

• 1976: Gene Glass first proposed the term “meta-analysis” and defined it as the statistical analysis of results from multiple individual studies to integrate their findings. During this period, meta-analysis was widely adopted in many fields, especially in social sciences and education.

• After the 1970s, medical researchers began to use meta-analysis more extensively. An early influential example was the work of Peter Elwood and Archie Cochrane, who evaluated the effect of aspirin in reducing recurrent heart disease. Their synthesis of trial results provided strong evidence of aspirin’s benefits, and the findings were published in The Lancet in 1980.

Meta-analysis was initially applied to Randomized Controlled Trials (RCTs) to evaluate treatment effects. Its application later expanded to observational studies to investigate disease incidence, prevalence, risk factors, and prognosis.

With advances in statistics and research methodology, meta-analysis techniques have further evolved, including Cumulative Meta-Analysis (CMA), Indirect Comparison, Network Meta-Analysis (NMA), and Trial Sequential Analysis (TSA).

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