The meta suite now supports meta-analysis (MA) of a correlation coefficient. All standard meta-analysis features, such as forest plots and subgroup analysis, are supported. This feature is part of StataNow™.

Traditionally, MA focuses on two-sample binary or continuous data, where the outcome of interest is measured across two groups often labeled as the treatment and control groups. For example, an MA may compare the efficacy of a new drug versus a placebo or the impact of two different educational interventions on student performance.

Sometimes, we may want to investigate the strength and direction of relationships between variables across multiple studies. This is where the MA of correlations comes into play. For example, a researcher might be interested in synthesizing findings from various studies to understand the correlation between digital device usage and sleep quality. Or an economist might conduct an MA to analyze the relationship between market volatility and investor behavior across countries from different studies. MA can be used in these cases to synthesize correlation coefficients from different studies.

Let's see it work

Example dataset: Correlation between legume consumption and weight loss

The meeting with your future in-laws was a resounding success, thanks in part to your savvy MA of the proportions of vegetarians in the United States. Riding high on this triumph, your future mother-in-law, now brimming with entrepreneurial spirit, is curious about expanding her online restaurant's menu. This time, she is pondering whether the amount of legume consumption correlates with weight loss. As the resident statistician, you are up to bat again.

You propose conducting an MA of correlations to explore the relationship between legume consumption and weight loss across the United States. With these insights, the online restaurant could spice up its menu to include legume-based recipes to promote a healthier lifestyle, potentially boosting its appeal. Assume you have identified 13 studies.

. describe

Contains data from legumes.dta
 Observations:            13                  Fictional data of correlations
                                                between legume consumption and
                                                weight loss
    Variables:             4                  10 Apr 2024 14:09

 
Variable      Storage   Display    Value
    name         type    format    label      Variable label                   
 
studylbl        str22   %22s                  Study label
corr            float   %9.0g                 Correlation value
ntotal          int     %9.0g                 Study sample size
pulse           byte    %9.0g      pulsetype  Pulse type
 
Sorted by:

Meta-analysis of correlation data

Variables corr and ntotal represent the correlation and the total number of subjects in each study, respectively. We use meta esize to compute the Fisher's \(z\)-transformed correlation for each study. This Fisher's \(z\) transformation is a variance-stabilizing transformation and is particularly preferable when the correlations are close to −1 or 1.

Declare your data as meta data via meta esize

. meta esize corr ntotal, fisherz studylabel(studylbl)

Meta-analysis setting information

 Study information
    No. of studies: 13
       Study label: studylbl
        Study size: _meta_studysize
      Summary data: corr ntotal

       Effect size
              Type: fisherz
             Label: Fisher's z
          Variable: _meta_es

         Precision
         Std. err.: _meta_se
                CI: [_meta_cil, _meta_ciu]
          CI level: 95%

  Model and method
             Model: Random effects
            Method: REML

You may instead specify the untransformed (raw) correlation as the effect size using the correlation option. Because the variance of the untransformed correlation depends on the correlation itself, an MA of this effect size tends to assign artificially large weights for studies with correlations close to −1 or 1.

. meta update, correlation
-> meta esize corr ntotal , correlation studylabel(studylbl)

Meta-analysis setting information from meta esize

 Study information
    No. of studies: 13
       Study label: studylbl
        Study size: _meta_studysize
      Summary data: corr ntotal

       Effect size
              Type: correlation
             Label: Correlation
          Variable: _meta_es

         Precision
         Std. err.: _meta_se
                CI: [_meta_cil, _meta_ciu]
          CI level: 95%

  Model and method
             Model: Random effects
            Method: REML

Forest plots and other meta-analysis techniques

Let's continue with the first specification of meta esize. After computing the effect size of interest and declaring your data as meta data, you may use any MA technique as usual. For example, to construct a forest plot, we type

. meta forestplot, correlation

  Effect-size label: Correlation
        Effect size: _meta_es
          Std. err.: _meta_se
        Study label: studylbl

The correlation option specifies that the results be reported as correlations instead of Fisher's \(z\)-values. This is equivalent to applying the hyperbolic tangent transformation using the transform("Correlation": tanh) option. The overall (mean) correlation between legume consumption and weight loss is 0.70 with a confidence interval (CI) of [0.63, 0.76].

You may conduct a subgroup analysis to investigate whether the correlations differ significantly among the pulse groups:

. meta forestplot, subgroup(pulse) esrefline insidemarker correlation

  Effect-size label: Correlation
        Effect size: _meta_es
          Std. err.: _meta_se
        Study label: studylbl

The above forest plots reveal substantive differences within each pulse category. Intriguingly, within each pulse, certain studies show stronger correlations, possibly associated with a supplemental physical activity program alongside the dietary changes. We do not have any evidence that there is a difference between the subgroup correlations (\(Q_b(2) = 1.00\), \(p = 0.61)\).

Confident in their potential health benefits, you may advise your mother-in-law to diversify the menu with an array of legume-based dishes without the need to prioritize one pulse type over another in the restaurant recipes.

See more examples in [META] meta esize and [META] meta summarize.