Methodological letter
Effect size in meta-analysis
El tamaño del efecto en el metaanálisis
Marina Iniesta-Sepúlvedaa,
, José Antonio Pereira-Rodríguezb
Corresponding author
miniesta@ucam.edu
Corresponding author at: UCAM, Campus de los Jerónimos, 30010 Guadalupe (Murcia), Spain.
Corresponding author at: UCAM, Campus de los Jerónimos, 30010 Guadalupe (Murcia), Spain.
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class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0005">Effect size indices</span><p id="par0005" class="elsevierStylePara elsevierViewall">The effect size is the statistical value that reports the magnitude at which a phenomenon occurs or exists in a given population, providing information on the practical or clinical significance of a result.<a class="elsevierStyleCrossRefs" href="#bib0005"><span class="elsevierStyleSup">1,2</span></a> In meta-analyses, effect size enables the results of different studies to be expressed in a common metric and presented as a global index. To this end, the effect size of an outcome of interest and its variance are calculated for each of the individual studies. The numerous effect size indices depend on the level of measurement of the outcome variables, the type of analysis, and the objectives of the study.<a class="elsevierStyleCrossRef" href="#bib0015"><span class="elsevierStyleSup">3</span></a> Some of the most commonly used indices in meta-analysis are shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>.</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><p id="par0010" class="elsevierStylePara elsevierViewall">For the comparison of groups in quantitative variables, the <span class="elsevierStyleItalic">d</span> family indices are most appropriate, as they are based on the difference between the means and allow for 2 groups to be compared with a quantitative variable (e.g., a 10-point pain scale). These can be used with both experimental and quasi-experimental designs by selecting the most appropriate index.<a class="elsevierStyleCrossRefs" href="#bib0020"><span class="elsevierStyleSup">4,5</span></a></p><p id="par0015" class="elsevierStylePara elsevierViewall">To compare 2 groups regarding a dichotomous variable, such as survival/mortality rate, indices based on hazard ratios are used. Hazard ratios refer to the probability that an event of interest (favorable or unfavorable) will occur in the presence or absence of a specific factor. The most common of these indices are the risk ratio (RR) and the advantage ratio or odds ratio (OR), along with their logarithmic transformations.<a class="elsevierStyleCrossRef" href="#bib0030"><span class="elsevierStyleSup">6</span></a></p><p id="par0020" class="elsevierStylePara elsevierViewall">Regarding effect size indices to evaluate the degree of association between variables, the most widely known is the Pearson correlation (<span class="elsevierStyleItalic">r</span>). In a meta-analysis of correlations, it is possible to use 2 different analytical strategies. In the first, the Pearson correlations would be combined directly, while in the second the Fisher <span class="elsevierStyleItalic">z</span> transformation would be applied first.<a class="elsevierStyleCrossRef" href="#bib0035"><span class="elsevierStyleSup">7</span></a></p><p id="par0025" class="elsevierStylePara elsevierViewall">When the study does not report the complete set of necessary statistics (a fairly common situation in practice), there are formulas to calculate effect sizes from partial data.<a class="elsevierStyleCrossRef" href="#bib0015"><span class="elsevierStyleSup">3</span></a></p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0010">Statistical models: Fixed effect vs. Random effects</span><p id="par0030" class="elsevierStylePara elsevierViewall">The main purpose of meta-analysis is usually to obtain an average or overall effect size. When conducting a meta-analysis, one of the decisions that researchers must make is the selection of the most appropriate statistical model, with important implications for both the analysis and interpretation of the results. From the fixed-effect model, it is assumed that all studies estimate a single true effect size in the population and that the differences between the effects observed in the studies are due to sampling errors. From the random-effect model, it is assumed that the different studies estimate parametric effect sizes belonging to different populations.<a class="elsevierStyleCrossRef" href="#bib0040"><span class="elsevierStyleSup">8</span></a></p><p id="par0035" class="elsevierStylePara elsevierViewall">The fixed-effect model is appropriate when the studies have been carried out following identical procedures and the participants belong to the same defined population (e.g., patients from the same hospital). On the other hand, the random-effect model is more appropriate when the studies are heterogeneous. This last situation is the most common in practice, especially in the context of bibliographic reviews, so the random-effect model is the appropriate choice in most cases.</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0015">Interpretation of effect size</span><p id="par0040" class="elsevierStylePara elsevierViewall">To interpret the magnitude of the effect, an effect can be classified as low, moderate or high in magnitude according to value ranges.<a class="elsevierStyleCrossRef" href="#bib0045"><span class="elsevierStyleSup">9</span></a> Despite the existence of these mathematical criteria, researchers must take into account the context and the nature of the phenomenon studied by explaining the meaning of the effect in the real world.</p><p id="par0045" class="elsevierStylePara elsevierViewall">Another factor to consider when interpreting the effect size is the accuracy of the estimate, which is determined by the confidence interval.<a class="elsevierStyleCrossRef" href="#bib0015"><span class="elsevierStyleSup">3</span></a> The range of this interval indicates the value range of the average effect size in a universe of comparable populations at a 95% confidence level.<a class="elsevierStyleCrossRef" href="#bib0050"><span class="elsevierStyleSup">10</span></a> The confidence interval also provides information about the statistical significance of the effect size. When this does not include the null value (e.g., 0 in the case of the standardized mean), the effect is statistically significant.</p><p id="par0050" class="elsevierStylePara elsevierViewall">The confidence interval should not be confused with the prediction interval, which provides information about the dispersion of the population effects, indicating the minimum and maximum values between which 95% of the population effect sizes would be found.<a class="elsevierStyleCrossRef" href="#bib0050"><span class="elsevierStyleSup">10</span></a> This heterogeneity of effect sizes also has important implications for the interpretation of the results. If there is little dispersion between the true effects of the studies, the overall effect size will be a good estimator of the average effect of each of the different populations. In contrast, if the dispersion is high, the meta-analysis should focus on the factors responsible for this variability.</p></span></span>" "textoCompletoSecciones" => array:1 [ "secciones" => array:4 [ 0 => array:2 [ "identificador" => "sec0005" "titulo" => "Effect size indices" ] 1 => array:2 [ "identificador" => "sec0010" "titulo" => "Statistical models: Fixed effect vs. Random effects" ] 2 => array:2 [ "identificador" => "sec0015" "titulo" => "Interpretation of effect size" ] 3 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "multimedia" => array:1 [ 0 => array:8 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "detalles" => array:1 [ 0 => array:3 [ "identificador" => "at0045" "detalle" => "Table " "rol" => "short" ] ] "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="\n \t\t\t\t\ttable-head\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t" scope="col" style="border-bottom: 2px solid black">Index \t\t\t\t\t\t\n \t\t\t\t\t\t</th><th class="td" title="\n \t\t\t\t\ttable-head\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t" scope="col" style="border-bottom: 2px solid black">Example of use \t\t\t\t\t\t\n \t\t\t\t\t\t</th><th class="td" title="\n \t\t\t\t\ttable-head\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t" scope="col" style="border-bottom: 2px solid black">Measurement of result \t\t\t\t\t\t\n \t\t\t\t\t\t</th><th class="td" title="\n \t\t\t\t\ttable-head\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t" scope="col" style="border-bottom: 2px solid black">Statistic for calculation \t\t\t\t\t\t\n \t\t\t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td-with-role" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t ; entry_with_role_rowhead " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Standardized mean difference \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Magnitude of the difference between the means of 2 groups on a pain scale (1–10 points) after a surgical procedure. \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Quantitative \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">n, means and standard deviation post-test in each group \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t ; entry_with_role_rowhead " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Standardized mean change \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Magnitude of change pre-test to post-test on a pain scale (1–10 points) in a single group after a surgical procedure \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Quantitative \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">n, means and standard deviation pre-test and post-test \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t ; entry_with_role_rowhead " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Standardized mean difference \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Magnitude of the difference between the pre-test and post-test changes of 2 groups on a pain scale (1–10 points) after a surgical procedure \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Quantitative \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">n, means and standard deviations pre-test and post-test in each group \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t ; entry_with_role_rowhead " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Relative risk \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Magnitude of the difference between mortality rates in 2 groups \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Dichotomous \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Total n and n with the event of interest in each group \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t ; entry_with_role_rowhead " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Odds ratio \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Ratio between the mortality advantages in 2 groups \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Dichotomous \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Total n and n with the event of interest in each group \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t ; entry_with_role_rowhead " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Pearson correlation coefficient \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Magnitude of the correlation between the pain scale score (1–10 points) and the time since the surgical intervention \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">Quantitative \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="\n \t\t\t\t\ttable-entry\n \t\t\t\t " align="left" valign="\n \t\t\t\t\ttop\n \t\t\t\t">n and correlation \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab3593935.png" ] ] ] ] "descripcion" => array:1 [ "en" => "<p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Effect size indices and examples of their use.</p>" ] ] ] "bibliografia" => array:2 [ "titulo" => "References" "seccion" => array:1 [ 0 => array:2 [ "identificador" => "bibs0005" "bibliografiaReferencia" => array:10 [ 0 => array:3 [ "identificador" => "bib0005" "etiqueta" => "1" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Understanding the new statistics: effect sizes, confidence intervals, and meta-analysis" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "G. 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