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Seriously Misleading Results Using Inverse of Freeman-Tukey Double Arcsine Transformation in Meta-Analysis of Single Proportions

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Title: Seriously Misleading Results Using Inverse of Freeman-Tukey Double Arcsine Transformation in Meta-Analysis of Single Proportions
Language: English
Authors: Schwarzer, Guido (ORCID 0000-0001-6214-9087), Chemaitelly, Hiam (ORCID 0000-0002-8756-6968), Abu-Raddad, Laith J. (ORCID 0000-0003-0790-0506), Rücker, Gerta (ORCID 0000-0002-2192-2560)
Source: Research Synthesis Methods. Sep 2019 10(3):476-483.
Availability: Wiley-Blackwell. 350 Main Street, Malden, MA 02148. Tel: 800-835-6770; Tel: 781-388-8598; Fax: 781-388-8232; e-mail: cs-journals@wiley.com; Web site: http://www.wiley.com/WileyCDA
Peer Reviewed: Y
Page Count: 8
Publication Date: 2019
Document Type: Journal Articles
Reports - Research
Descriptors: Meta Analysis, Statistical Analysis, Research Problems
DOI: 10.1002/jrsm.1348
ISSN: 1759-2879
Abstract: Standard generic inverse variance methods for the combination of single proportions are based on transformed proportions using the logit, arcsine, and Freeman-Tukey double arcsine transformations. Generalized linear mixed models are another more elaborate approach. Irrespective of the approach, meta-analysis results are typically back-transformed to the original scale in order to ease interpretation. Whereas the back-transformation of meta-analysis results is straightforward for most transformations, this is not the case for the Freeman-Tukey double arcsine transformation, albeit possible. In this case study with five studies, we demonstrate how seriously misleading the back-transformation of the Freeman-Tukey double arcsine transformation can be. We conclude that this transformation should only be used with special caution for the meta-analysis of single proportions due to potential problems with the back-transformation. Generalized linear mixed models seem to be a promising alternative.
Abstractor: As Provided
Entry Date: 2020
Accession Number: EJ1255358
Database: ERIC
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  Value: <anid>AN0138519221;[bdct]01sep.19;2019Sep11.02:56;v2.2.500</anid> <title id="AN0138519221-1">Seriously misleading results using inverse of Freeman‐Tukey double arcsine transformation in meta‐analysis of single proportions </title> <p>Standard generic inverse variance methods for the combination of single proportions are based on transformed proportions using the logit, arcsine, and Freeman‐Tukey double arcsine transformations. Generalized linear mixed models are another more elaborate approach. Irrespective of the approach, meta‐analysis results are typically back‐transformed to the original scale in order to ease interpretation. Whereas the back‐transformation of meta‐analysis results is straightforward for most transformations, this is not the case for the Freeman‐Tukey double arcsine transformation, albeit possible. In this case study with five studies, we demonstrate how seriously misleading the back‐transformation of the Freeman‐Tukey double arcsine transformation can be. We conclude that this transformation should only be used with special caution for the meta‐analysis of single proportions due to potential problems with the back‐transformation. Generalized linear mixed models seem to be a promising alternative.</p> <p>Keywords: back‐transformation; generalized linear mixed model; harmonic mean; random intercept logistic regression; variance stabilization</p> <hd id="AN0138519221-2">1 INTRODUCTION</hd> <p>A key application of meta‐analytical methods is the pooling of proportions, such as prevalence of a specific infection or disease.([[<reflink idref="bib1" id="ref1">1</reflink>]]) Classic fixed‐effect and random‐effects meta‐analysis methods[<reflink idref="bib5" id="ref2">5</reflink>] are typically used to combine single proportions. In order to use these methods, proportions are generally transformed using either the log,[<reflink idref="bib6" id="ref3">6</reflink>] logit,[<reflink idref="bib7" id="ref4">7</reflink>] arcsine,[<reflink idref="bib8" id="ref5">8</reflink>] or Freeman‐Tukey double arcsine[<reflink idref="bib9" id="ref6">9</reflink>] transformations. These transformations are implemented for pure mathematical reasons, eg,  variance stabilization (details on the transformations are given in Appendix A and summarized in Table A1). For pooling, the transformed proportions and corresponding standard errors are used in the generic inverse variance method.[<reflink idref="bib5" id="ref7">5</reflink>] An alternative yet more elaborate approach based on the logit transformation are generalized linear mixed models (GLMMs),[<reflink idref="bib10" id="ref8">10</reflink>] which account for the binomial structure of the data and thus avoid the generic inverse variance method. Irrespective of the meta‐analysis method and transformation, results are usually presented on the original probability scale after using the corresponding back‐transformation.</p> <p>Whereas the back‐transformation of meta‐analysis results is straightforward for the log, logit, and arcsine transformations, this is not the case for the Freeman‐Tukey double arcsine transformation, albeit possible.[<reflink idref="bib11" id="ref9">11</reflink>] In order to calculate the inverse of the Freeman‐Tukey double arcsine transformation, a single sample size has to be specified. Accordingly, for a single study, a one to one relation exists between transformation and its inverse, however, in a meta‐analysis with different sample sizes the value of the back‐transformation depends on the specified sample size. Typically, the harmonic mean of sample sizes is used in the back‐transformation.[<reflink idref="bib11" id="ref10">11</reflink>]</p> <hd id="AN0138519221-3">2 CASE STUDY: META‐ANALYSIS ON PREVALENCE OF HEPATITIC C VIRUS INFECTIONS</hd> <p>We report results of meta‐analyses with five studies estimating the prevalence of hepatitis C virus (HCV) infections in the general population of Nepal, which constitute a subset of an unpublished dataset with 28 studies.[<reflink idref="bib12" id="ref11">12</reflink>] This unpublished dataset comprises testing for a total of 972 123 individuals among whom 3696 were HCV antibody positive. The prevalence across studies ranged from 0% to 18.4% with a median of 0.5%. We restrict ourselves to the five‐study subset for didactic reasons; the same issues encountered in this subset also exist in the full dataset.</p> <p>We conducted classic meta‐analyses using the arcsine, Freeman‐Tukey double arcsine, and logit transformations, respectively. Furthermore, we fitted GLMMs implicitly using the logit transformation. Details on the statistical methods are provided in Appendix A. We used R function metaprop() from R package <bold>meta</bold>[<reflink idref="bib13" id="ref12">13</reflink>] (see Supporting Information). Results are summarized in Table.</p> <p>Estimates and 95% confidence intervals of HCV prevalence meta‐analyses using arcsine, Freeman‐Tukey double arcsine, and logit transformations, respectively</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left">Transformation</th><th align="left">Transformed</th><th align="left">HCV Infections</th></tr><tr><th align="left">(Meta‐analysis Model)</th><th align="left">Proportion</th><th align="left">per 1000 Observations</th></tr></thead><tbody valign="top"><tr><td align="left">Arcsine (fixed)</td><td align="left">  0.044 (0.042 to 0.046)</td><td align="left">1.94 (1.77 to 2.13)</td></tr><tr><td align="left">Double arcsine (fixed)</td><td align="left">  0.044 (0.042 to 0.046)</td><td align="left">0.00 (0.00 to 0.00)</td></tr><tr><td align="left">Logit (fixed)</td><td align="left">−6.231 (−6.323 to −6.139)</td><td align="left">1.96 (1.79 to 2.15)</td></tr><tr><td align="left">GLMM (fixed)</td><td align="left">−6.238 (−6.330 to −6.147)</td><td align="left">1.95 (1.78 to 2.14)</td></tr><tr><td align="left">Arcsine (random, <p><math display="inline" altimg="urn:x-wiley:jrsm:media:jrsm1348:jrsm1348-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true" xmlns=""><mi>τ</mi><mo>^</mo></mover><mo xmlns="">=</mo><mn xmlns="">0</mn><mo xmlns="">.</mo><mn xmlns="">0003</mn></math></p>)</td><td align="left">  0.044 (0.042 to 0.046)</td><td align="left">1.94 (1.76 to 2.13)</td></tr><tr><td align="left">Double arcsine (random, <p><math display="inline" altimg="urn:x-wiley:jrsm:media:jrsm1348:jrsm1348-math-0002" xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true" xmlns=""><mi>τ</mi><mo>^</mo></mover><mo xmlns="">=</mo><mn xmlns="">0</mn><mo xmlns="">.</mo><mn xmlns="">0020</mn></math></p>)</td><td align="left">  0.044 (0.041 to 0.048)</td><td align="left">0.00 (0.00 to 0.00)</td></tr><tr><td align="left">Logit (random, <p><math display="inline" altimg="urn:x-wiley:jrsm:media:jrsm1348:jrsm1348-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true" xmlns=""><mi>τ</mi><mo>^</mo></mover><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">.</mo><mn xmlns="">1758</mn></math></p>)</td><td align="left">−5.451 (−6.649 to −4.254)</td><td align="left">4.27 (1.29 to 14.01)</td></tr><tr><td align="left">GLMM (random, <p><math display="inline" altimg="urn:x-wiley:jrsm:media:jrsm1348:jrsm1348-math-0004" xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true" xmlns=""><mi>τ</mi><mo>^</mo></mover><mo xmlns="">=</mo><mn xmlns="">0</mn><mo xmlns="">.</mo><mn xmlns="">0000</mn></math></p>)</td><td align="left">−6.238 (−6.330 to −6.147)</td><td align="left">1.95 (1.78 to 2.14)</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note.</emph> GLMM (fixed) = logistic regression; GLMM (random) = random intercept logistic regression; between‐study variance estimate <ephtml> <math display="inline" altimg="urn:x-wiley:jrsm:media:jrsm1348:jrsm1348-math-0005" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mi>τ</mi><mo>^</mo></mover></mrow><mn>2</mn></msup></math> </ephtml> .</p> <p>2 Abbreviations: GLMM, generalized linear mixed model; HCV, hepatitis C virus.</p> <p>Under the fixed‐effect model, results depicted as transformed proportions (middle column in Table) are very similar for the two methods using the arcsine and logit transformations, respectively. Whereas the random‐effects estimates are also very similar with a slightly smaller confidence interval for the arcsine transformation, the results for the two logit methods are rather different due to a very different estimate for the between‐study variance.</p> <p>For easier interpretation, results are back‐transformed to the original scale. Due to the small prevalences, we express results as HCV infections per 1000 observations. In Table (right column), the results using the inverse of the Freeman‐Tukey double arcsine transformation based on the harmonic mean of 85 are highly irregular with HCV prevalences and confidence limits exactly equal to zero. Under the fixed‐effect model, all of the other three methods show very similar results. Conversely, under the random‐effects model, results for the classic meta‐analysis method using the logit transformation are very different from the other results.</p> <p>Looking at Figure , we see that the meta‐analysis estimators are reasonable summaries of transformed prevalences. On the other hand, back‐transformed meta‐analysis results are clearly off the mark in Figure with meta‐analysis estimators smaller than all individual study results. Note that the back‐transformation works as expected for individual study results, eg, the prevalence is 1/29 = 0.03448 for study 26, which corresponds to 34.48 HCV infections per 1000 observations.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01sep19/jrsm1348-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1348-fig-0001.jpg" title="Forest plot of hepatitis C virus (HCV) meta‐analysis with Freeman‐Tukey double arcsine transformation and without back‐transformation of results. PFT, Freeman‐Tukey double arcsine transformed proportion" /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01sep19/jrsm1348-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1348-fig-0002.jpg" title="Forest plot of hepatitis C virus (HCV) meta‐analysis with Freeman‐Tukey double arcsine transformation and back‐transformation according to Miller[11]" /> </p> <p></p> <p>The harmonic mean of 85 is obviously the wrong choice in this meta‐analysis with sample sizes ranging from 29 to more than 200 000. Figure shows the influence of sample size on meta‐analysis results (see also Table A2). For sample sizes between 10 and around 120, results are exactly zero for the back‐transformation of the Freeman‐Tukey double arcsine transformation. The number of HCV infections per 1000 observations then steeply increases up to a sample size of 500 when the effect of sample size starts to slowly level out.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01sep19/jrsm1348-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1348-fig-0003.jpg" title="Influence of sample size on results of hepatitis C virus (HCV) meta‐analysis using inverse of Freeman‐Tukey double arcsine transformation according to Miller[11]" /> </p> <p></p> <p>As noted earlier, the results of the random‐effects model are very different for the two logit methods due to different between‐study variance estimates. This discrepancy can be explained by looking at the confidence intervals of individual studies in the corresponding forest plots (Figures and ). Confidence intervals, based on the normal approximation, are much narrower for the two smallest studies in the classic random‐effects meta‐analysis (Figure) than the confidence intervals, based on the Clopper‐Pearson method taking the binomial distribution into account,([[<reflink idref="bib14" id="ref13">14</reflink>]]) in the GLMM meta‐analysis (Figure). Apparently, in these two small studies with only 1 HCV infection and less than 50 observations, the assumption of a normally distributed logit transformed proportion is not fulfilled. With increasing numbers of infections and sample sizes, approximate and Clopper‐Pearson confidence intervals get closer to each other. Obviously, the very narrow confidence intervals of the two smallest studies result in an inflated between‐study variance estimate leading to a larger estimate for the pooled mean HCV prevalence and a much wider confidence interval for the pooled mean HCV prevalence.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01sep19/jrsm1348-fig-0004.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1348-fig-0004.jpg" title="Forest plot of hepatitis C virus (HCV) meta‐analysis using classic method and logit transformation. Confidence intervals for individual studies are based on normal approximation for logit transformed proportions" /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01sep19/jrsm1348-fig-0005.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1348-fig-0005.jpg" title="Forest plot of hepatitis C virus (HCV) meta‐analysis using generalized linear mixed model. Confidence intervals for individual studies are based on Clopper‐Pearson method([[14]])" /> </p> <p></p> <hd id="AN0138519221-9">3 DISCUSSION</hd> <p>Our case study shows that meta‐analysis results based on the back‐transformation of the Freeman‐Tukey double arcsine transformation[<reflink idref="bib11" id="ref14">11</reflink>] can be very misleading and even smaller than all individual study results. We observe similar undesirable results in a meta‐analysis using the complete dataset with 28 studies. To our knowledge, this is the first publication reporting such an anomaly and erratic results.</p> <p>In our view, the main reason for this unexpected behaviour is the very extreme pattern of sample sizes that range from 29 to more than 200 000. The harmonic mean of 85 is much smaller than 3 of the 5 sample sizes. For such highly skewed sample sizes, the harmonic mean is by definition rather small, which may result in nonsensical back‐transformed probabilities.</p> <p>In order to prevent misleading conclusions for the Freeman‐Tukey double arcsine transformation, several sample sizes could be used to evaluate the sensitivity of meta‐analysis results; however, this may lead to diverging meta‐analysis estimates. In our example, using the arithmetic or geometric mean in the back‐transformation (see Table) would result in random‐effects estimates of 1.96 and 1.59 HCV infections per 1000 observations, respectively. Here, results for the harmonic mean are obviously wrong; however, it is rather unclear whether to rely on the results for the arithmetic or geometric mean. All other transformations (arcsine, logit, and log) do not have this intrinsic problem in the presentation of meta‐analysis results.</p> <p>Overall, the arcsine transformation appears to be the best classic method for the meta‐analysis of single proportions. However, as application of GLMMs for meta‐analysis is nowadays straightforward due to its implementation in common software, there is neither a real reason nor a clear advantage for using an approximate method. Accordingly, we support the viewpoint of  previous works,([[<reflink idref="bib10" id="ref15">10</reflink>], [<reflink idref="bib16" id="ref16">16</reflink>]])  recommending the use of GLMMs for the meta‐analysis of single proportions. From our perspective, the only disadvantage of a GLMM is that individual study weights are not available, which we consider as a minor drawback; analysts seeing this differently should use the arcsine transformation.</p> <p>Our recommendation is purportedly in contrast to advice by Barendregt et al[<reflink idref="bib1" id="ref17">1</reflink>] promoting the use of the Freeman‐Tukey double arcsine transformation over the logit transformation. However, this publication only considered these transformations under the classic meta‐analysis model. We agree with Barendregt et al[<reflink idref="bib1" id="ref18">1</reflink>] that the use of the logit transformation is problematic in inverse variance meta‐analyses with small event numbers or sample sizes; this is also visible in our example. These problems with the logit transformation under the classic meta‐analysis do not translate to GLMMs. The classic meta‐analysis model assumes that treatment estimates of individual studies follow a normal distribution that is obviously critical in studies with small numbers of events and observations. The arcsine and Freeman‐Tukey double arcsine transformation are less affected by this normality assumption than the logit transformation. However, GLMMs taking into account the binomial structure of the data are not affected by this problem at all.([[<reflink idref="bib10" id="ref19">10</reflink>], [<reflink idref="bib16" id="ref20">16</reflink>]])</p> <hd id="AN0138519221-10">4 CONCLUSIONS</hd> <p>Our case study shows that the Freeman‐Tukey double arcsine transformation should only be used with special caution for the meta‐analysis of single proportions due to potential problems in the back‐transformation of meta‐analysis results. In our view, a sensitivity analysis using other sample sizes is mandatory for this transformation. GLMMs seem to be a promising alternative which is nowadays available in common meta‐analysis software.</p> <hd id="AN0138519221-11">ACKNOWLEDGEMENTS</hd> <p>H.C. and L.J.A. acknowledge support by NPRP grant number 9‐040‐3‐008  from the Qatar National Research Fund (a member of Qatar Foundation), and support provided by the Biostatistics, Epidemiology, and Biomathematics Research Core at Weill Cornell Medicine in Qatar. G.R. acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG), grant number RU1747/1‐2. The findings achieved herein are solely the responsibility of the authors.</p> <hd id="AN0138519221-12">CONFLICT OF INTEREST</hd> <p>The author reported no conflict of interest.</p> <p>A APPENDIX STATISTICAL METHODS</p> <p>We consider a meta‐analysis of <emph>K</emph> studies where each study reports the number of events, <emph>a</emph><subs><emph>k</emph></subs>, and the number of observations <emph>n</emph><subs><emph>k</emph></subs>, <emph>k</emph> = 1,...,<emph>K</emph>. We assume that the number of events follows a binomial distribution. Specifically, cell count <emph>a</emph><subs><emph>k</emph></subs>∼Binomial(<emph>n</emph><subs><emph>k</emph></subs>,<emph>p</emph><subs><emph>k</emph></subs>), where <emph>p</emph><subs><emph>k</emph></subs> denotes the probability of the event in study <emph>k</emph>. These probabilities are estimated from the observed number of events and sample sizes by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>p</mi><mo>^</mo></mover></mrow><mi>k</mi></msub><mo>=</mo><msub><mi>a</mi><mi>k</mi></msub><mo stretchy="false">/</mo><msub><mi>n</mi><mi>k</mi></msub></math> </ephtml> .</p> <p>A.1 Transformations</p> <p>In this subsection, we briefly introduce the arcsine, Freeman‐Tukey double arcsine, and logit transformations in the context of a single study. In the next subsection, the use of these transformations in meta‐analyses will be described.</p> <p>A.1.1 Arcsine transformation</p> <p>The arcsine‐transformed event probability <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> [<reflink idref="bib8" id="ref21">8</reflink>] is defined as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo>=</mo><mi>arcsin</mi><msqrt><mrow><msub><mi>p</mi><mi>k</mi></msub></mrow></msqrt><mspace width="0.1em" /><mo>.</mo></mrow></math> </ephtml> </p> <p>An estimate of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> is given by replacing <emph>p</emph><subs><emph>k</emph></subs> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>p</mi><mo>^</mo></mover></mrow><mi>k</mi></msub></math> </ephtml> . The main advantage of this transformation is the property of variance stabilization. The approximate variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> is calculated using</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>Var</mi><mo>^</mo></mover><mspace width="0.1em" /><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow><mn>4</mn><mspace width="0.1em" /><msub><mi>n</mi><mi>k</mi></msub></mrow><mo>,</mo></mrow></math> </ephtml> </p> <p>where the approximation improves as <emph>n</emph><subs><emph>k</emph></subs> increases. Notice that the approximate variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> only depends on the sample size. A confidence interval for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> can be constructed as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mspace width="3.0235pt" /><mo>±</mo><mspace width="2.0235pt" /><msub><mi>z</mi><mrow><mn>1</mn><mo>−</mo><mi>α</mi><mn>2</mn></mrow></msub><mspace width="3.0235pt" /><mtext>S.E.</mtext><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> </ephtml> </p> <p>with standard error <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S.E.</mtext><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mover accent="true"><mi>Var</mi><mo>^</mo></mover><mspace width="0.1em" /><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></msqrt></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mrow><mn>1</mn><mo>−</mo><mi>α</mi><mn>2</mn></mrow></msub></math> </ephtml> denoting the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><mi>α</mi><mn>2</mn></math> </ephtml> quantile of the standard normal distribution.</p> <p>A.1.2 Freeman‐Tukey double arcsine transformation</p> <p>The Freeman‐Tukey double arcsine‐transformed event probability <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> [<reflink idref="bib9" id="ref22">9</reflink>] is an average of two arcsine‐transformed probabilities. Its estimate is given by</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mfenced separators="" open="(" close=")"><mrow><mi>arcsin</mi><msqrt><mrow><mrow><msub><mi>a</mi><mi>k</mi></msub></mrow><mrow><msub><mi>n</mi><mi>k</mi></msub><mo>+</mo><mn>1</mn></mrow></mrow></msqrt><mo>+</mo><mi>arcsin</mi><msqrt><mrow><mrow><msub><mi>a</mi><mi>k</mi></msub><mo>+</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mi>k</mi></msub><mo>+</mo><mn>1</mn></mrow></mrow></msqrt></mrow></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>The Freeman‐Tukey double arcsine transformation was introduced in order to improve on the variance stabilizing property of the arcsine transformation. The approximate variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> is</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>Var</mi><mo>^</mo></mover><mspace width="0.1em" /><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow><mn>4</mn><mspace width="0.1em" /><msub><mi>n</mi><mi>k</mi></msub><mo>+</mo><mn>2</mn></mrow><mo>,</mo></mrow></math> </ephtml> </p> <p>where the approximation—again—improves as <emph>n</emph><subs><emph>k</emph></subs> increases. A confidence interval for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> can be constructed following the same methodology for that of the arcsine transformed probability described above.</p> <p>A.1.3 Logit transformation</p> <p>The logit transformation is another classic transformation[<reflink idref="bib7" id="ref23">7</reflink>] defined as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo>=</mo><mi>log</mi><mfenced separators="" open="(" close=")"><mrow><mrow><msub><mi>p</mi><mi>k</mi></msub></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>p</mi><mi>k</mi></msub></mrow></mrow></mfenced><mspace width="0.1em" /><mo>.</mo></mrow></math> </ephtml> </p> <p>Again, an estimate of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> is given by replacing <emph>p</emph><subs><emph>k</emph></subs> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>p</mi><mo>^</mo></mover></mrow><mi>k</mi></msub></math> </ephtml> . The approximate variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> is</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>Var</mi><mo>^</mo></mover><mspace width="0.1em" /><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow><msub><mi>a</mi><mi>k</mi></msub></mrow><mo>+</mo><mn>1</mn><mrow><msub><mi>n</mi><mi>k</mi></msub><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub></mrow><mspace width="0.1em" /><mo>.</mo></mrow></math> </ephtml> </p> <p>It is clear from this variance formula that the approximate variance of a logit transformed proportion can become infinite if the number of events is zero or equal to the sample size. Typically, in this situation, a small increment is added to each denominator in order to yield a finite variance estimate.</p> <p>A confidence interval for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> can be constructed following the same methodology for that of the arcsine transformed probability described earlier.</p> <p>A.2 Meta‐analysis of single proportions</p> <p>We briefly describe both the classic meta‐analysis method assuming approximate normally distributed study effects (ie, prevalence measures) as well as the generalized linear mixed model taking the binary structure of the data into account.</p> <p>All methods are available in R function metaprop() from R package <bold>meta</bold>.[<reflink idref="bib13" id="ref24">13</reflink>]</p> <p>A.2.1 Classic random‐effects model</p> <p>Classic fixed‐effect and random‐effects meta‐analysis methods using the inverse variance method[<reflink idref="bib5" id="ref25">5</reflink>] can be implemented to combine single proportions. As the random‐effects model is a generalization of the fixed‐effect model, we only introduce the random‐effects model, which is defined as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub><mo>=</mo><mspace width="3.0235pt" /></mtd><mtd columnalign="left"><msub><mi>θ</mi><mi>k</mi></msub><mo>+</mo><msub><mi>ϵ</mi><mi>k</mi></msub><mo>,</mo><mspace width="1em" /><msub><mi>ϵ</mi><mi>k</mi></msub><mover accent="true"><mo>∼</mo><mtext>i.i.d.</mtext></mover><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>σ</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mtd><mtd columnalign="right" /></mtr><mtr><mtd columnalign="right"><msub><mi>θ</mi><mi>k</mi></msub><mo>=</mo><mspace width="3.0235pt" /></mtd><mtd columnalign="left"><mi>θ</mi><mo>+</mo><msub><mi>u</mi><mi>k</mi></msub><mo>,</mo><mspace width="1em" /><msub><mi>u</mi><mi>k</mi></msub><mover accent="true"><mo>∼</mo><mtext>i.i.d.</mtext></mover><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msup><mi>τ</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mspace width="1em" /><mo>,</mo></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>where the <emph>ϵ</emph>'s and <emph>u</emph>'s are independent. This model contains two sources of variation: the within‐study variances <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math> </ephtml> , <emph>k</emph> = 1,...,<emph>K</emph>, and the between‐study variance <emph>τ</emph><sups>2</sups>. The classic meta‐analysis methods assume that the variances <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math> </ephtml> are estimated without error by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math> </ephtml> . The estimated effects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub></math> </ephtml> and corresponding standard errors <emph>σ</emph><subs><emph>k</emph></subs> (which are assumed known) are used to estimate <emph>τ</emph><sups>2</sups> with the restricted maximum likelihood method.[<reflink idref="bib19" id="ref26">19</reflink>] Results are very similar using the classic DerSimonian and Laird estimator, which is still the default in most statistical software for meta‐analysis. The fixed‐effect model is a special case when <emph>τ</emph><sups>2</sups> = 0. Accordingly, results of fixed‐effect and random‐effects meta‐analysis are identical if the estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mi>τ</mi><mo>^</mo></mover></mrow><mn>2</mn></msup></math> </ephtml> equals zero.</p> <p>Given estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>σ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> the random‐effects estimate of <emph>θ</emph>, denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub></math> </ephtml> , is</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover></mstyle><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>^</mo></mover></mrow><mi>k</mi><mn>2</mn></msubsup><mo>+</mo><msup><mrow><mover accent="true"><mi>τ</mi><mo>^</mo></mover></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover></mstyle><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>^</mo></mover></mrow><mi>k</mi><mn>2</mn></msubsup><mo>+</mo><msup><mrow><mover accent="true"><mi>τ</mi><mo>^</mo></mover></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mspace width="0.1em" /><mo>,</mo></mrow></math> </ephtml> </p> <p>which is a weighted average of the individual effect estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub></math> </ephtml> with weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msup><mrow><mover accent="true"><mi>τ</mi><mo>^</mo></mover></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>Definition and properties of prevalence transformations with number of events a and total sample size n</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left" /><th align="left">Approximate</th><th align="left" /></tr><tr><th align="left">Transformation</th><th align="left">Estimate</th><th align="left">Variance</th><th align="left">Comments</th></tr></thead><tbody valign="top"><tr><td align="left">log<xref ref-type="bibr" rid="bibr6">6</xref></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">log</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">a</mi><mo stretchy="false" xmlns="">/</mo><mi xmlns="">n</mi><mo stretchy="false" xmlns="">)</mo></math></p></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn xmlns="">1</mn><mi xmlns="">a</mi><mo xmlns="">−</mo><mn xmlns="">1</mn><mi xmlns="">n</mi></math></p></td><td align="left">Infinite estimate and variance for zero events</td></tr><tr><td align="left">logit<xref ref-type="bibr" rid="bibr7">7</xref></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">log</mi><mfenced separators="" open="(" close=")" xmlns=""><mrow><mrow><mi>a</mi><mo stretchy="false">/</mo><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>a</mi><mo stretchy="false">/</mo><mi>n</mi></mrow></mrow></mfenced></math></p></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn xmlns="">1</mn><mi xmlns="">a</mi><mo xmlns="">+</mo><mn xmlns="">1</mn><mrow xmlns=""><mi>n</mi><mo>−</mo><mi>a</mi></mrow></math></p></td><td align="left">Infinite estimate and variance for zero or all events</td></tr><tr><td align="left">arcsine<xref ref-type="bibr" rid="bibr8">8</xref></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">arcsin</mi><msqrt xmlns=""><mrow><mi>a</mi><mo stretchy="false">/</mo><mi>n</mi></mrow></msqrt></math></p></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn xmlns="">1</mn><mrow xmlns=""><mn>4</mn><mspace width="0.1em" /><mi>n</mi></mrow></math></p></td><td align="left">Variance stabilizing; defined for zero events</td></tr><tr><td align="left">Double arcsine<xref ref-type="bibr" rid="bibr9">9</xref></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn xmlns="">0</mn><mo xmlns="">.</mo><mn xmlns="">5</mn><mspace width="0.1em" xmlns="" /><mfenced separators="" open="(" close="" xmlns=""><mrow><mi>arcsin</mi><msqrt><mrow><mi>a</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msqrt><mo>+</mo></mrow></mfenced></math></p></td><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn xmlns="">1</mn><mrow xmlns=""><mn>4</mn><mspace width="0.1em" /><mi>n</mi><mo>+</mo><mn>2</mn></mrow></math></p></td><td align="left">Outperforms arcsine for small prevalences;</td></tr><tr><td align="left" /><td align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="" open="" close=")" xmlns=""><mrow><mi>arcsin</mi><msqrt><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msqrt></mrow></mfenced></math></p></td><td align="left" /><td align="left">sample size needed in back‐transformation<xref ref-type="bibr" rid="bibr11">11</xref></td></tr></tbody></table> </ephtml> </p> <p>Estimated number of HCV infections per 1000 observations for additional sample sizes in fixed‐effect and random‐effects meta‐analyses using the back‐transformation of the Freeman‐Tukey double arcsine method</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">HCV Infections per 1000 Observations</th><th align="left" /></tr><tr><th align="left">Sample Size</th><th align="left">Fixed Effect</th><th align="left">Random Effects</th><th align="left">Mean</th></tr></thead><tbody valign="top"><tr><td align="left">85</td><td align="left">0.000</td><td align="left">0.000</td><td align="left">Harmonic</td></tr><tr><td align="left">500</td><td align="left">1.083</td><td align="left">1.097</td><td align="left" /></tr><tr><td align="left">1000</td><td align="left">1.486</td><td align="left">1.500</td><td align="left" /></tr><tr><td align="left">1254</td><td align="left">1.575</td><td align="left">1.590</td><td align="left">Geometric</td></tr><tr><td align="left">10 000</td><td align="left">1.902</td><td align="left">1.917</td><td align="left" /></tr><tr><td align="left">46 892</td><td align="left">1.941</td><td align="left">1.956</td><td align="left">Arithmetic</td></tr><tr><td align="left">100 000</td><td align="left">1.947</td><td align="left">1.962</td><td align="left" /></tr><tr><td align="left">1 000 000</td><td align="left">1.951</td><td align="left">1.966</td><td align="left" /></tr></tbody></table> </ephtml> </p> <p>The variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub></math> </ephtml> is estimated by</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>Var</mi><mo>^</mo></mover><mspace width="0.1em" /><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow><mstyle displaystyle="true"><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover></mstyle><msub><mi>w</mi><mi>k</mi></msub></mrow></mrow></math> </ephtml> </p> <p>and a (1‐<emph>α</emph>) confidence interval for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub></math> </ephtml> can be calculated using</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub><mo>±</mo><msub><mi>z</mi><mrow><mn>1</mn><mo>−</mo><mi>α</mi><mn>2</mn></mrow></msub><mspace width="3.0235pt" /><mtext>S.E.</mtext><mspace width="3.0235pt" /><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> </p> <p>with standard error <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S.E.</mtext><mspace width="3.0235pt" /><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mover accent="true"><mi>Var</mi><mo>^</mo></mover><mspace width="0.1em" /><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>R</mi></msub><mo stretchy="false">)</mo></mrow></msqrt></math> </ephtml> .</p> <p>A fixed‐effect meta‐analysis can be conducted by assuming a between‐study variance <emph>τ</emph><sups>2</sups> = 0 resulting in a fixed‐effect estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>F</mi></msub></math> </ephtml> .</p> <p>Instead of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mi>σ</mi><mo>^</mo></mover></mrow><mi>k</mi></msub></math> </ephtml> , we use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S.E.</mtext><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo></math> </ephtml> for the arcsine method, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S.E.</mtext><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo></math> </ephtml> for the Freeman‐Tukey double arcsine method, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S.E.</mtext><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></math> </ephtml> for the logit method. We denote the corresponding fixed‐effect and random‐effects estimates as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>F</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>F</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>R</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>F</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>R</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> , respectively.</p> <p>A.2.2 Generalized linear mixed model</p> <p>An excellent tutorial[<reflink idref="bib10" id="ref27">10</reflink>] describes how generalized linear mixed models can be utilized in the meta‐analysis of event outcomes. One special case considered in the paper is the meta‐analysis of single proportions, which—like the classic meta‐analysis model—assumes a normal distribution for the effect size (ie, transformed proportion) across studies. However, a binomial distribution is assumed for the number of events within a study, ie, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>k</mi></msub><mo>∼</mo><mtext>Binomial</mtext><mfenced separators="" open="(" close=")"><mrow><msub><mi>n</mi><mi>k</mi></msub><mo>,</mo><msub><mi>p</mi><mi>k</mi></msub></mrow></mfenced></math> </ephtml> . Using the above defined logit transformed proportion <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></math> </ephtml> , this relation can be re‐expressed in the following way to define the random‐effects model</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><msub><mi>a</mi><mi>k</mi></msub></mtd><mtd columnalign="left"><mover accent="true"><mo>∼</mo><mtext>i.i.d.</mtext></mover><mspace width="1em" /><mtext>Binomial</mtext><mfenced separators="" open="(" close=")"><mrow><msub><mi>n</mi><mi>k</mi></msub><mo>,</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="right" /></mtr><mtr><mtd columnalign="right"><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup></mtd><mtd columnalign="left"><mover accent="true"><mo>∼</mo><mtext>i.i.d.</mtext></mover><mspace width="1em" /><mi>θ</mi><mo>+</mo><msub><mi>u</mi><mi>k</mi></msub><mspace width="1em" /><msub><mi>u</mi><mi>k</mi></msub><mover accent="true"><mo>∼</mo><mtext>i.i.d.</mtext></mover><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msup><mi>τ</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>This model uses the binomial likelihood <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>exp</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><msub><mi>a</mi><mi>k</mi></msub></mrow></msup><mfenced separators="" open="/" close=""><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>exp</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><msub><mi>n</mi><mi>k</mi></msub></mrow></msup></mrow></mfenced></math> </ephtml> instead of the likelihood from the normal distribution[<reflink idref="bib10" id="ref28">10</reflink>] and is also known as a random intercept logistic regression model that implicitly uses the logit transformation. Accordingly, the GLMM estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>F</mi></mrow><mrow><mi>G</mi><mi>L</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>R</mi></mrow><mrow><mi>G</mi><mi>L</mi></mrow></msubsup></math> </ephtml> correspond to the logit transformed probabilities in the fixed‐effect and random‐effects model, respectively.</p> <p>Estimation of GLMMs for meta‐analysis of single proportions is straightforward with R function metaprop() by specifying argument method = "GLMM".</p> <p>In principle, individual study weights could be derived from the likelihood contribution of each individual study; however, this information is at the moment not available in the utilized R software. Alternatively, the width of the Clopper‐Pearson confidence intervals that also takes the binomial data structure into account([[<reflink idref="bib14" id="ref29">14</reflink>]]) could be used to get approximate study weights.</p> <p>A.3 Back‐transformations</p> <p>For a single study, several statistical methods exist to calculate a confidence interval for a single proportion.([[<reflink idref="bib14" id="ref30">14</reflink>]]) These methods do not use the arcsine or the Freeman‐Tukey double arcsine transformations, and therefore, the back‐transformation is not strictly relevant for individual study results. However, in a meta‐analysis context, the back‐transformation of the (double) arcsine as well as the logit transformation is essential to report results on the original scale, ie, as proportions.</p> <p>A.3.1 Arcsine back‐transformation</p> <p>The back‐transformation/inverse of the arcsine transformation is defined as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo>=</mo><mi>sin</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>A</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mspace width="0.1em" /><mo>.</mo></mrow></math> </ephtml> </p> <p>This back‐transformation can be used for a single study as well as the result of a meta‐analysis, eg, for the random‐effects estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>S</mi></mrow></msubsup></math> </ephtml> and its lower and upper confidence limits.</p> <p>A.3.2 Inverse of Freeman‐Tukey double arcsine transformation</p> <p>Miller[<reflink idref="bib11" id="ref31">11</reflink>] introduced the back‐transformation of the Freeman‐Tukey double arcsine transformation that was published almost 30  years after the initial publication.[<reflink idref="bib9" id="ref32">9</reflink>] For study <emph>k</emph>, the back‐transformation is defined as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mspace width="0.1em" /><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>−</mo><mtext>sgn</mtext><mo stretchy="false">(</mo><mi>cos</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mspace width="0.1em" /><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mfenced separators="" open="[" close="]"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mspace width="0.1em" /><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">/</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mspace width="0.1em" /><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>F</mi><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced><mo stretchy="false">/</mo><msub><mi>n</mi><mi>k</mi></msub></mrow></mfenced></mrow><mn>2</mn></msup></mrow></msqrt></mrow></mfenced><mspace width="0.1em" /><mo>.</mo></mrow></math> </ephtml> </p> <p>This rather complex back‐transformation arises from using an average of two arcsine transformed proportions. The sample size <emph>n</emph><subs><emph>k</emph></subs> is included in the back‐transformation, which is no problem for a single study. However, in a meta‐analysis with different sample sizes, a single sample size has to be specified to apply the back‐transformation. Miller[<reflink idref="bib11" id="ref33">11</reflink>] suggested to use the harmonic mean of the sample sizes, ie, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ñ</mi><mo>=</mo><mi>K</mi><mfenced separators="" open="/" close=""><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover></mstyle><mn>1</mn><mrow><msub><mi>n</mi><mi>k</mi></msub></mrow></mrow></mfenced></math> </ephtml> . Accordingly, this harmonic mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ñ</mi></math> </ephtml> and the meta‐analysis estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>F</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><mrow><mi>R</mi></mrow><mrow><mi>F</mi><mi>T</mi></mrow></msubsup></math> </ephtml> are used in the back‐transformation.</p> <p>A.3.3 Inverse of logit transformation</p> <p>The inverse of the logit transformation is defined as</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo>=</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mi>L</mi><mi>O</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mspace width="0.1em" /><mo>.</mo></mrow></math> </ephtml> </p> <p>This well‐known back‐transformation can be used both for a single study and in a meta‐analysis setting (classic method or GLMM).</p> <p>GRAPH: Supporting info item</p> <p>GRAPH: Supporting info item</p> <ref id="AN0138519221-13"> <title> REFERENCES </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Barendregt J, Doi S, Lee Y, Norman R, Vos T. 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  Data: Seriously Misleading Results Using Inverse of Freeman-Tukey Double Arcsine Transformation in Meta-Analysis of Single Proportions
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  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Schwarzer%2C+Guido%22">Schwarzer, Guido</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-6214-9087">0000-0001-6214-9087</externalLink>)<br /><searchLink fieldCode="AR" term="%22Chemaitelly%2C+Hiam%22">Chemaitelly, Hiam</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-8756-6968">0000-0002-8756-6968</externalLink>)<br /><searchLink fieldCode="AR" term="%22Abu-Raddad%2C+Laith+J%2E%22">Abu-Raddad, Laith J.</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-0790-0506">0000-0003-0790-0506</externalLink>)<br /><searchLink fieldCode="AR" term="%22Rücker%2C+Gerta%22">Rücker, Gerta</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-2192-2560">0000-0002-2192-2560</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Research+Synthesis+Methods%22"><i>Research Synthesis Methods</i></searchLink>. Sep 2019 10(3):476-483.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Wiley-Blackwell. 350 Main Street, Malden, MA 02148. Tel: 800-835-6770; Tel: 781-388-8598; Fax: 781-388-8232; e-mail: cs-journals@wiley.com; Web site: http://www.wiley.com/WileyCDA
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 8
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2019
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Meta+Analysis%22">Meta Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Research+Problems%22">Research Problems</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1002/jrsm.1348
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1759-2879
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Standard generic inverse variance methods for the combination of single proportions are based on transformed proportions using the logit, arcsine, and Freeman-Tukey double arcsine transformations. Generalized linear mixed models are another more elaborate approach. Irrespective of the approach, meta-analysis results are typically back-transformed to the original scale in order to ease interpretation. Whereas the back-transformation of meta-analysis results is straightforward for most transformations, this is not the case for the Freeman-Tukey double arcsine transformation, albeit possible. In this case study with five studies, we demonstrate how seriously misleading the back-transformation of the Freeman-Tukey double arcsine transformation can be. We conclude that this transformation should only be used with special caution for the meta-analysis of single proportions due to potential problems with the back-transformation. Generalized linear mixed models seem to be a promising alternative.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2020
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1255358
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1255358
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  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1002/jrsm.1348
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 8
        StartPage: 476
    Subjects:
      – SubjectFull: Meta Analysis
        Type: general
      – SubjectFull: Statistical Analysis
        Type: general
      – SubjectFull: Research Problems
        Type: general
    Titles:
      – TitleFull: Seriously Misleading Results Using Inverse of Freeman-Tukey Double Arcsine Transformation in Meta-Analysis of Single Proportions
        Type: main
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      – PersonEntity:
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            NameFull: Schwarzer, Guido
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            NameFull: Chemaitelly, Hiam
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            NameFull: Abu-Raddad, Laith J.
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            NameFull: Rücker, Gerta
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          Dates:
            – D: 01
              M: 09
              Type: published
              Y: 2019
          Identifiers:
            – Type: issn-print
              Value: 1759-2879
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              Value: 10
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Research Synthesis Methods
              Type: main
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