Clinician's Approach to Advanced Statistical Methods: Win Ratios, Restricted Mean Survival Time, Responder Analyses, and Standardized Mean Differences.

Novel statistical methods have emerged in recent medical literature, which clinicians must understand to properly appraise and integrate evidence into their practice. Some of these key concepts include win ratios, restricted mean survival time, responder analyses, and standardized mean difference. This article offers guidance to busy clinicians on the comprehension and practical applicability of the results to patients. Win ratios provide an alternative method to analyze composite outcomes by prioritizing individual components of the composite; prioritization of the outcomes should be evidence-based, pre-specified, and patient-centered. Restricted mean survival time presents a method to analyze Kaplan-Meier curves when assumptions required for Cox proportional hazards analysis are not met. As it only considers outcomes that occur within a specific timeframe, the duration of follow-up must be appropriately defined and based on prior epidemiologic and mechanistic evidence. Researchers can analyze continuous outcomes with responder analyses, in which participants are dichotomized into "responders" or "non-responders." While clinicians and patients may more easily grasp outcomes analyzed in this way, they should be aware of the loss of information and resulting imprecision, as well as potential to manipulate data presentation. When meta-analyzing continuous outcomes, point estimates can be converted to standardized mean differences to facilitate the combination of data utilizing various outcome measures. However, clinicians may find it challenging to grasp the clinical meaningfulness of a standardized mean difference, and may benefit from converting it to well-known outcomes. By providing the background knowledge of these statistical methods, along with practical applicability, benefits, and inevitable limitations, this article aims to provide clinicians with an approach to appraise the literature and apply the results in clinical practice.

MMP12 is a Potential Predictive and Prognostic Biomarker of Various Cancers Including Lung Adenocarcinoma.

OBJECTIVE: This study sought to explore the clinical value of matrix metalloproteinases 12 (MMP12) in multiple cancers, including lung adenocarcinoma (LUAD). METHODS: Using >10,000 samples, this retrospective study demonstrated the first pan-cancer analysis of MMP12. The expression of MMP12 between cancer groups and their control groups was analyzed using Wilcoxon rank-sum tests. The clinical significance of MMP12 expression in multiple cancers was assessed using receiver operating characteristic curves, Kaplan-Meier curves, and univariate Cox analysis. A further LUAD-related analysis based on 4565 multi-center and in-house samples was performed to verify the findings regarding MMP12 in pan-cancer analysis partly. RESULTS: MMP12 mRNA is highly expressed in 13 cancers compared to their controls, and the MMP12 protein level is elevated in some of these cancers (e.g., colon adenocarcinoma) (P < .05). MMP12 expression makes it feasible to distinguish 21 cancer tissues from normal tissues (AUC = 0.86). A high MMP12 expression is a prognosis risk factor in eight cancers, such as adrenocortical carcinoma (hazard ratio >1, P < .05). The elevated MMP12 expression is also a prognosis protective factor in breast-invasive carcinoma and colon adenocarcinoma (hazard ratio <1, P < .05). Some pan-cancer findings regarding MMP12 are verified in LUAD-MMP12 expression is upregulated in LUAD at both the mRNA and protein levels (P < .05), has the potential to distinguish LUAD with considerable accuracy (AUC = .91), and plays a risk prognosis factor for patients with the disease (P < .05). CONCLUSIONS: MMP12 is highly expressed in most cancers and may serve as a novel biomarker for the prediction and prognosis of numerous cancers.

Standardization and other approaches to meta-analyze differences in means.

Meta-analysts often use standardized mean differences (SMD) to combine mean effects from studies in which the dependent variable has been measured with different instruments or scales. In this tutorial we show how the SMD is properly calculated as the difference in means divided by a between-subject reference-group, control-group, or pooled pre-intervention SD, usually free of measurement error. When combining mean effects from controlled trials and crossovers, most meta-analysts have divided by either the pooled SD of change scores, the pooled SD of post-intervention scores, or the pooled SD of pre- and post-intervention scores, resulting in SMDs that are biased and difficult to interpret. The frequent use of such inappropriate standardizing SDs by meta-analysts in three medical journals we surveyed is due to misleading advice in peer-reviewed publications and meta-analysis packages. Even with an appropriate standardizing SD, meta-analysis of SMDs increases heterogeneity artifactually via differences in the standardizing SD between settings. Furthermore, the usual magnitude thresholds for standardized mean effects are not thresholds for clinically important differences. We therefore explain how to use other approaches to combining mean effects of disparate measures: log transformation of factor effects (response ratios) and of percent effects converted to factors; rescaling of psychometrics to percent of maximum range; and rescaling with minimum clinically important differences. In the absence of clinically important differences, we explain how standardization after meta-analysis with appropriately transformed or rescaled pre-intervention SDs can be used to assess magnitudes of a meta-analyzed mean effect in different settings.

Propensity score analysis for health care disparities: a deweighting approach.

BACKGROUND: Propensity score weighting is a useful tool to make causal or unconfounded comparisons between groups. According to the definition by the Institute of Medicine (IOM), estimates of health care disparities should be adjusted for health-status factors but not for socioeconomic status (SES) variables. There have been attempts to use propensity score weighting to generate estimates that are concordant with IOM's definition. However, the existing propensity score methods do not preserve SES distributions in minority and majority groups unless SES variables are independent of health status variables. METHODS: The present study introduces a deweighting method that uses two types of propensity scores. One is a function of all covariates of health status and SES variables and is used to weight study subjects to adjust for them. The other is a function of only the SES variables and is used to deweight the subjects to preserve the original SES distributions. RESULTS: The procedure of deweighting is illustrated using a dataset from a right heart catheterization (RHC) study, where it was used to examine whether there was a disparity between black and white patients in receiving RHC. The empirical example provided promising evidence that the deweighting method successfully preserved the marginal SES distributions for both racial groups but balanced the conditional distributions of health status given SES. CONCLUSIONS: Deweighting is a promising tool for implementing the IOM-definition of health care disparities. The method is expected to be broadly applied to quantitative research on health care disparities.

Associations Between BLG, CSN3, DGAT1, GH, PIT1, and PRL Gene Polymorphisms and Milk Production Traits in Holstein Dairy Cows: A Meta-Analysis.

The current meta-analysis was employed to combine the results of multiple studies into a single estimate related to B-LG, CSN3, DGAT1, PRL, GH, and PIT1 gene polymorphisms and their effects on milk production traits. The purpose of this meta-analysis was to investigate associations between B-LG, CSN3, DGAT1, GH, PIT1, and PRLgene polymorphisms with milk production traits in Holstein dairy cows. An extensive search was done from 2002 to 2022 year. Statistical analyses were performed by using Stata 11.2 software. Genetic models viz codominant (AA vs. AB, AA vs. BB, and AB vs. BB), dominant (AA + AB vs. BB), recessive (AA vs. AB + BB), and completely over-dominant (AA + BB vs. AB) were applied. The results of meta-analysis of association between B-LG genotypes with milk yield where found a significant (P < 0.05) and with fat and protein contents (P < 0.01). In CSN3 polymorphisms of A/A and A/B genotypes had a significant effect on fat yield (P < 0.05) and protein content (P < 0.01). DGAT1 polymorphisms had a significant effect on milk yield, fat yield, protein yield (P < 0.05), with fat and protein contents showed high effect (P < 0.01). No significant association was detected between GH and milk traits (P > 0.05). PIT1 genotype polymorphisms had a significant effect on milk yield (P < 0.05) and protein content (P < 0.01). PRL genotype polymorphisms were significantly associated with milk yield (P < 0.05), fat content and protein yield (P < 0.01). The B-LG, DGAT1,CSN3 and PRL gene polymorphisms could be utilized as good markers to improve milk production traits in the Holstein cattle breed.

Examining the normality assumption of a design-comparable effect size in single-case designs.

What Works Clearinghouse (WWC, 2022) recommends a design-comparable effect size (D-CES; i.e., gAB) to gauge an intervention in single-case experimental design (SCED) studies, or to synthesize findings in meta-analysis. So far, no research has examined gAB's performance under non-normal distributions. This study expanded Pustejovsky et al. (2014) to investigate the impact of data distributions, number of cases (m), number of measurements (N), within-case reliability or intra-class correlation (ρ), ratio of variance components (λ), and autocorrelation (ϕ) on gAB in multiple-baseline (MB) design. The performance of gAB was assessed by relative bias (RB), relative bias of variance (RBV), MSE, and coverage rate of 95% CIs (CR). Findings revealed that gAB was unbiased even under non-normal distributions. gAB's variance was generally overestimated, and its 95% CI was over-covered, especially when distributions were normal or nearly normal combined with small m and N. Large imprecision of gAB occurred when m was small and ρ was large. According to the ANOVA results, data distributions contributed to approximately 49% of variance in RB and 25% of variance in both RBV and CR. m and ρ each contributed to 34% of variance in MSE. We recommend gAB for MB studies and meta-analysis with N ≥ 16 and when either (1) data distributions are normal or nearly normal, m = 6, and ρ = 0.6 or 0.8, or (2) data distributions are mildly or moderately non-normal, m ≥ 4, and ρ = 0.2, 0.4, or 0.6. The paper concludes with a discussion of gAB's applicability and design-comparability, and sound reporting practices of ES indices.

Atypical hemispheric lateralization of brain function and structure in autism: a comprehensive meta-analysis study.

BACKGROUND: Characteristic changes in the asymmetric nature of the human brain are associated with neurodevelopmental differences related to autism. In people with autism, these differences are thought to affect brain structure and function, although the structural and functional bases of these defects are yet to be fully characterized. METHODS: We applied a comprehensive meta-analysis to resting-state functional and structural magnetic resonance imaging datasets from 370 people with autism and 498 non-autistic controls using seven datasets of the Autism Brain Imaging Data Exchange Project. We studied the meta-effect sizes based on standardized mean differences and standard deviations (s.d.) for lateralization of gray matter volume (GMV), fractional amplitude of low-frequency fluctuation (fALFF), and regional homogeneity (ReHo). We examined the functional correlates of atypical laterality through an indirect annotation approach followed by a direct correlation analysis with symptom scores. RESULTS: In people with autism, 85, 51, and 51% of brain regions showed a significant diagnostic effect for lateralization in GMV, fALFF, and ReHo, respectively. Among these regions, 35.7% showed overlapping differences in lateralization in GMV, fALFF, and ReHo, particularly in regions with functional annotations for language, motor, and perceptual functions. These differences were associated with clinical measures of reciprocal social interaction, communication, and repetitive behaviors. A meta-analysis based on s.d. showed that people with autism had lower variability in structural lateralization but higher variability in functional lateralization. CONCLUSION: These findings highlight that atypical hemispheric lateralization is a consistent feature in autism across different sites and may be used as a neurobiological marker for autism.

Validity of the food frequency questionnaire for adults in nutritional epidemiological studies: A systematic review and meta-analysis.

As the most widely used tool for assessing dietary intake, the validity of food frequency questionnaires (FFQs) should be evaluated before application. A comprehensive search of the PubMed and Web of Science databases was conducted for publications from January 2000 to April 1, 2020. Pooled estimates were calculated for correlation coefficients and mean differences for energy and 61 nutrients between FFQs and standard methods. The literature search identified 130 articles that included 21,494 participants. Subgroup analyses according to the number of administrations of the reference method, sample size, administration methods, FFQ items, reference periods, quality of the studies, gender, and regions were also performed. We conducted a meta-analysis by summarizing the available evidence to comprehensively assess the validity of FFQs stratified by the reference method type (24-hour recall (24HRs) and food records (FRs). We also performed subgroup analyses to examine the impact on the final summary estimates. After a meta-analysis of the FFQs' validity correlation coefficients of the included studies, this study showed that the range (median) of the validity coefficients of the 24HRs as reference methods was 0.220-0.770 (0.416), and for the FRs, it was 0.173-0.735 (0.373), which indicated that FFQs were suitable to assess the overall dietary intake in nutritional epidemiological studies. The results of the subgroup analysis showed that the number of administrations of the reference method, administration mode, number of items, reference periods, sample size, and gender mainly affected the validity correlation of FFQs.Supplemental data for this article is available online at https://doi.org/10.1080/10408398.2021.1966737 .

A Bayesian model for combining standardized mean differences and odds ratios in the same meta-analysis.

In meta-analysis practice, researchers frequently face studies that report the same outcome differently, such as a continuous variable (e.g., scores for rating depression) or a binary variable (e.g., counts of patients with depression dichotomized by certain latent and unreported depression scores). For combining these two types of studies in the same analysis, a simple conversion method has been widely used to handle standardized mean differences (SMDs) and odds ratios (ORs). This conventional method uses a linear function connecting the SMD and log OR; it assumes logistic distributions for (latent) continuous measures. However, the normality assumption is more commonly used for continuous measures, and the conventional method may be inaccurate when effect sizes are large or cutoff values for dichotomizing binary events are extreme (leading to rare events). This article proposes a Bayesian hierarchical model to synthesize SMDs and ORs without using the conventional conversion method. This model assumes exact likelihoods for continuous and binary outcome measures, which account for full uncertainties in the synthesized results. We performed simulation studies to compare the performance of the conventional and Bayesian methods in various settings. The Bayesian method generally produced less biased results with smaller mean squared errors and higher coverage probabilities than the conventional method in most cases. Nevertheless, this superior performance depended on the normality assumption for continuous measures; the Bayesian method could lead to nonignorable biases for non-normal data. In addition, we used two case studies to illustrate the proposed Bayesian method in real-world settings.

Parametric and nonparametric methods for confidence intervals and sample size planning for win probability in parallel-group randomized trials with Likert item and Likert scale data.

Data on the Likert scale are ubiquitous in medical research, including randomized trials. Statistical analysis of such data may be conducted using the means of raw scores or the rank information of the scores. In the context of parallel-group randomized trials, we quantify treatment effects by the probability that a subject in the treatment group has a better score than (or a win over) a subject in the control group. Asymptotic parametric and nonparametric confidence intervals for this win probability and associated sample size formulas are derived for studies with only follow-up scores, and those with both baseline and follow-up measurements. We assessed the performance of both the parametric and nonparametric approaches using simulation studies based on real studies with Likert item and Likert scale data. The simulation results demonstrate that even without baseline adjustment, the parametric methods did not perform well, in terms of bias, interval coverage percentage, balance of tail error, and assurance of achieving a pre-specified precision. In contrast, the nonparametric approach performed very well for both the unadjusted and adjusted win probability. We illustrate the methods with two examples: one using Likert item data and the other using Like scale data. We conclude that non-parametric methods are preferable for two-group randomization trials with Likert data. Illustrative SAS code for the nonparametric approach using existing procedures is provided.

Clinical assessment and molecular mechanism of the upregulation of Toll-like receptor 2 (TLR2) in myocardial infarction.

OBJECTIVE: The prevalence and mortality of cardiovascular diseases remain ranked first worldwide. Myocardial infarction (MI) is the central cause of death from cardiovascular diseases, seriously endangering human health. The clinical implication of toll-like receptor 2 (TLR2) remains contradictory, and its mechanism is still unknown. Hence, the objective of this study was to elucidate the clinical value and molecular mechanism of TLR2 in MI. METHODS: All high-throughput datasets and eligible literature were screened, and the expression levels of TLR2 were collected from the MI. The integrated expression level of TLR2 was displayed by calculating the standardized mean difference (SMD) and the area under the curve (AUC) of the summary receiver operating characteristic curve (sROC). The related TLR2 genes were sent for pathway analyses by gene ontology (GO), Kyoto encyclopedia of genes and genome (KEGG), and disease ontology (DO). Single-cell RNA-seq was applied to ascertain the molecular mechanism of TLR2 in MI. RESULTS: Nine microarrays and four reported data were available to calculate the comprehensive expression level of TLR2 in MI, including 325 cases of MI and 306 cases of controls. The SMD was 2.55 (95% CI = 1.35-3.75), and the AUC was 0.76 (95% CI = 0.72-0.79), indicating the upregulation of TLR2 in MI. The related TLR2 genes were primarily enriched in the pathways of atherosclerosis, arteriosclerotic cardiovascular disease, and arteriosclerosis, suggesting the clinical role of TLR2 in the progression of MI. Afterward, TLR2 was upregulated in myeloid cells in MI. CONCLUSIONS: TLR2 may have a crucial role in progressing from coronary atherosclerosis to MI. The upregulation of TLR2 may have a favorable screening value for MI.

The clinical significance of integrin subunit alpha V in cancers: from small cell lung carcinoma to pan-cancer.

BACKGROUND: Little is known about the relationship between integrin subunit alpha V (ITGAV) and cancers, including small cell lung cancer (SCLC). METHODS: Using large sample size from multiple sources, the clinical roles of ITGAV expression in SCLC were explored using differential expression analysis, receiver operating characteristic curves, Kaplan-Meier curves, etc. RESULTS: Decreased mRNA (SMD = - 1.05) and increased protein levels of ITGAV were detected in SCLC (n = 865). Transcription factors-ZEB2, IK2F1, and EGR2-may regulate ITGAV expression in SCLC, as they had ChIP-Seq (chromatin immunoprecipitation followed by sequencing) peaks upstream of the transcription start site of ITGAV. ITGAV expression made it feasible to distinguish SCLC from non-SCLC (AUC = 0.88, sensitivity = 0.78, specificity = 0.84), and represented a risk role in the prognosis of SCLC (p < 0.05). ITGAV may play a role in cancers by influencing several immunity-related signaling pathways and immune cells. Further, the extensive pan-cancer analysis verified the differential expression of ITGAV and its clinical significance in multiple cancers. CONCLUSION: ITGAV served as a potential marker for prognosis and identification of cancers including SCLC.

Appropriately estimating the standardized average treatment effect with missing data: A simulation and primer.

Reporting standardized effects in randomized treatment studies aids interpretation and facilitates future meta-analyses and policy considerations. However, when outcome data are missing, achieving an unbiased, accurate estimate of the standardized average treatment effect, sATE, can pose challenges even for those with general knowledge of missing data handling, given that the sATE is a ratio of a mean difference to a (within-group) standard deviation. Under both homogeneity and heterogeneity of variance, a Monte Carlo simulation study was conducted to compare missing data handling strategies in terms of bias and accuracy in the sATE, under specific missingness patterns plausible for randomized pretest posttest studies. Within two broad missing data handling approaches, maximum likelihood and multiple imputation, modeling choices were thoroughly investigated including the analysis model, variance estimator, imputation algorithm, and method of pooling results across imputed datasets. Results demonstrated that although the sATE can be estimated with little bias using either maximum likelihood or multiple imputation, particular attention should be paid to the model and variance estimator, especially at smaller sample sizes (i.e., N = 50). Differences in accuracy were driven by differences in bias. To improve estimation of the sATE in practice, recommendations and a software demonstration are provided. Moreover, a pedagogical explanation of the causes of bias, described separately for the numerator and denominator of the sATE is provided, demonstrating visually how and why bias occurs with certain methods.

Phosphodiesterase inhibitor for heart failure with preserved ejection fraction: A systematic review and meta-analysis.

Background: Although heart failure with preserved ejection fraction (HFpEF) is a serious disease, only limited options are available for its treatment. Recent studies have analyzed the effects of phosphodiesterase (PDE) inhibitors, especially PDE5 and PDE3 inhibitors, in patients with HFpEF, with mixed outcomes. Methods: We searched PUBMED and EMBASE databases up to August 2021. Randomized controlled trials (RCTs) and clinical trials that tested the effects of PDE inhibitors on patients with HFpEF were included as eligible studies. Indicators of left ventricular (LV) function, pulmonary arterial pressure (PAP), right ventricular (RV) function, exercise capacity, and quality of life (QOL) were used to evaluate the efficacy of PDE inhibitors in HFpEF. Results: Six RCTs that reported in 7 studies were included to evaluate the efficiency of PDE inhibitors on HFpEF patients. In the pooled analysis, PDE inhibitors showed insignificant changes in the ratio of early diastolic mitral inflow to annular velocities, left atrial volume index, pulmonary artery systolic pressure (PASP), pulmonary vascular resistance (PVR), peak oxygen uptake, 6-minute walking test distance, as well as Kansas City Cardiomyopathy Questionnaire score. However, substantial improvement was observed in the tricuspid annular plane systolic excursion (TAPSE). Additionally, the regression analysis showed that PDE inhibitor administration time is a critical factor for the decrease in PASP. Conclusions: PDE inhibitors did not effectively improve LV function, PAP, exercise capacity, and QOL in patients with HFpEF. However, they improved RV function with significant difference, suggesting that PDE inhibitors might be a promising option for HFpEF patients with RV dysfunction.

Evaluation of various estimators for standardized mean difference in meta-analysis.

Meta-analyses of a treatment's effect compared with a control frequently calculate the meta-effect from standardized mean differences (SMDs). SMDs are usually estimated by Cohen's d or Hedges' g. Cohen's d divides the difference between sample means of a continuous response by the pooled standard deviation, but is subject to nonnegligible bias for small sample sizes. Hedges' g removes this bias with a correction factor. The current literature (including meta-analysis books and software packages) is confusingly inconsistent about methods for synthesizing SMDs, potentially making reproducibility a problem. Using conventional methods, the variance estimate of SMD is associated with the point estimate of SMD, so Hedges' g is not guaranteed to be unbiased in meta-analyses. This article comprehensively reviews and evaluates available methods for synthesizing SMDs. Their performance is compared using extensive simulation studies and analyses of actual datasets. We find that because of the intrinsic association between point estimates and standard errors, the usual version of Hedges' g can result in more biased meta-estimation than Cohen's d. We recommend using average-adjusted variance estimators to obtain an unbiased meta-estimate, and the Hartung-Knapp-Sidik-Jonkman method for accurate estimation of its confidence interval.

Bootstrap Confidence Intervals for Multilevel Standardized Effect Size.

Although many methodologists and professional organizations have urged applied researchers to compute and report effect size measures accompanying tests of statistical significance, discussions on obtaining confidence intervals (CIs) for effect size with clustered/multilevel data have been scarce. In this paper, I explore the bootstrap as a viable and accessible alternative for obtaining CIs for multilevel standardized mean difference effect size for cluster-randomized trials. A simulation was carried out to compare 17 analytic and bootstrap procedures for constructing CIs for multilevel effect size, in terms of empirical coverage rate and width, for both normal and nonnormal data. Results showed that, overall, the residual bootstrap with studentized CI had the best coverage rates (94.75% on average), whereas the residual bootstrap with basic CI had better coverage in small samples. These two procedures for constructing CIs showed better coverage than using analytic methods for both normal and nonnormal data. In addition, I provide an illustrative example showing how bootstrap CIs for multilevel effect size can be easily obtained using the statistical software R and the R package bootmlm. I strongly encourage applied researchers to report CIs to adequately convey the uncertainty of their effect size estimates.

Estimation in meta-analyses of mean difference and standardized mean difference.

Methods for random-effects meta-analysis require an estimate of the between-study variance, τ2 . The performance of estimators of τ2 (measured by bias and coverage) affects their usefulness in assessing heterogeneity of study-level effects and also the performance of related estimators of the overall effect. However, as we show, the performance of the methods varies widely among effect measures. For the effect measures mean difference (MD) and standardized MD (SMD), we use improved effect-measure-specific approximations to the expected value of Q for both MD and SMD to introduce two new methods of point estimation of τ2 for MD (Welch-type and corrected DerSimonian-Laird) and one WT interval method. We also introduce one point estimator and one interval estimator for τ2 in SMD. Extensive simulations compare our methods with four point estimators of τ2 (the popular methods of DerSimonian-Laird, restricted maximum likelihood, and Mandel and Paule, and the less-familiar method of Jackson) and four interval estimators for τ2 (profile likelihood, Q-profile, Biggerstaff and Jackson, and Jackson). We also study related point and interval estimators of the overall effect, including an estimator whose weights use only study-level sample sizes. We provide measure-specific recommendations from our comprehensive simulation study and discuss an example.

The influence of nonnormality from primary studies on the standardized mean difference in meta-analysis.

In this study we investigated the influence of data nonnormality in the primary studies on meta-analysis of the standardized mean difference (SMD) for a two-independent-group design. The bias, mean squared error, and confidence interval coverage probability of the mean effect sizes under different types of population distributions were compared. Also, the performance of the Q test was examined. The results showed that oppositely skewed distributions (i.e., distributions skewed in different directions) showed poor performance for point and interval estimates of mean effect sizes in meta-analysis, especially when the tails were pointing toward each other. The previously found adverse impacts due to nonnormality in primary studies do not disappear when primary studies with nonnormal data are meta-analyzed, even when the average sample size and number of studies are large. The results also showed that, when the tails were pointing toward each other, the Type I error rates of the Q test were inflated. We suggest that the impact of violating the assumption of normality should not be ignored in meta-analysis.

A parametric meta-analysis.

In a meta-analysis, we assemble a sample of independent, nonidentically distributed p-values. The Fisher's combination procedure provides a chi-squared test of whether the p-values were sampled from the null uniform distribution. After rejecting the null uniform hypothesis, we are faced with the problem of how to combine the assembled p-values. We first derive a distribution for the p-values. The distribution is parameterized by the standardized mean difference (SMD) and the sample size. It includes the uniform as a special case. The maximum likelihood estimate (MLE) of the SMD can then be obtained from the independent, nonidentically distributed p-values. The MLE can be interpreted as a weighted average of the study-specific estimate of the effect size with a shrinkage. The method is broadly applicable to p-values obtained in the maximum likelihood framework. Simulation studies show that our method can effectively estimate the effect size with as few as 6 p-values in the meta-analyses. We also present a Bayes estimator for SMD and a method to account for publication bias. We demonstrate our methods on several meta-analyses that assess the potential benefits of citicoline for patients with memory disorders or patients recovering from ischemic stroke.