Cluster-Bootstrapping a meta-analytic selection model

In this post, we will sketch out what we think is a promising and pragmatic method for examining selective reporting while also accounting for effect size dependency. The method is to use a cluster-level bootstrap, which involves re-sampling clusters of observations to approximate the sampling distribution of an estimator. To illustrate this technique, we will demonstrate how to bootstrap a Vevea-Hedges selection model.
bootstrap
dependent effect sizes
meta-analysis
publication bias
programming
Rstats
Authors

James E. Pustejovsky

Megha Joshi

Published

March 30, 2023

Important

The research reported here was supported, in whole or in part, by the Institute of Education Sciences, U.S. Department of Education, through grant R305D220026 to the American Institutes for Research. The opinions expressed are those of the authors and do not represent the views of the Institute or the U.S. Department of Education.

Selective reporting

Selective reporting of study results is a big concern for meta-analysts. By selective reporting, we mean the phenomenon where affirmative findings—that is, statistically significant findings in the theoretically expected direction—are more likely to be reported and more likely to be available for a systematic review compared to non-affirmative findings. Selective reporting arises due to biases in the publication process, on the part of journals, editors, and reviewers, as well as strategic decisions on part of the authors (Rothstein et al., 2006; Sutton, 2009). Research synthesists worry about selective reporting because it can distort the evidence base available for meta-analysis, almost like a fun-house mirror distorts your appearance, leading to inflation of average effect size estimates and biased estimates of heterogeneity.

If you read the meta-analysis methods literature, you will find scores of tools available to investigate and adjust for the biases created by selective reporting. Well known and widely used methods include:

However, nearly all of the statistical methods here have the limitation that they are premised on observing independent effect sizes. That presents a problem for meta-analyses in education, psychology, and other social science fields, where it is very common to have meta-analyses involving dependent effect sizes.

Dependent effects occur in meta-analyses of group comparisons when primary studies report effects for multiple correlated measures of an outcome, at multiple points in time, or for multiple treatment groups compared to the same control group (Becker, 2000). Dependent effects are also common in meta-analyses of correlational effect sizes, where primary studies report more than one relevant correlation coefficient based on the same sample of participants. Methods such as multi-level meta-analysis (Van den Noortgate et al., 2013) and robust variance estimation (Hedges et al., 2010) are available to accommodate dependent effects when summarizing findings across studies or investigating moderators of effect size using meta-regression, but these techniques have yet to be extended to methods for testing or correcting bias due to selective reporting. Consequently, it’s pretty common to see research synthesis papers that use very sophisticated models for part of the analysis, but then use kludgey, awkward, or hacky approaches when it comes time to investigating selective reporting (Rodgers & Pustejovsky, 2020).

Along with a group of our colleagues from the American Institutes for Research, we are currently working on a project to develop better methods for investigating selective reporting issues in meta-analyses of dependent effect sizes. In this post, we will share an early peek under the hood at one little piece of what we’re studying, by sketching out what we think is a promising and pragmatic method for examining selective reporting while also accounting for effect size dependency. The method is to use a cluster-level bootstrap, which involves re-sampling clusters of observations (i.e., the set of multiple effect size estimates reported within a given primary study) to approximate the sampling distribution of an estimator (Boos, 2003; Cameron et al., 2008). To illustrate this technique, we will demonstrate how to bootstrap a Vevea-Hedges selection model.

Selection models comprise a large class of models that have two parts: a model describing the evidence-generation process and a model describing the process by which evidence is reported (Hedges & Vevea, 2005). Vevea-Hedges selection models (Hedges, 1992; Hedges & Vevea, 1996; Vevea & Hedges, 1995) involve a random effects meta-regression model for the evidence-generation process and a step function for the reporting process. With a step function, we assume that the probability that an effect size estimate is observed depends on the range in which its p-value falls. For instance, effects with \(.01 < p \leq .05\) might have some probability \(\lambda_1\) of being reported, effects with \(.05 < p \leq .10\) might have some other probability \(\lambda_2\), and effects with \(.10 < p\) might have some other probability \(\lambda_3\).1 Because the Vevea-Hedges model and other selection models separate the data-generation process into these two distinct stages, their parameters have clear interpretations and they can be used to generate bias-adjusted estimates of the distribution of effect sizes and to test for selective reporting issues. The only problem is that available implementations of selection models do not account for effect size dependency—but that’s where cluster bootstrapping could potentially help.

Disclaimer

To be clear, this post is based on work in progress. The cluster-bootstrap selection model that we’re going to demonstrate is an experimental and exploratory technique. We’re currently studying its properties and performance using Monte Carlo simulations, but we don’t have formal results to share yet. In the spirit of open and collaborative science, we wrote this post to demonstrate our approach to coding the method, in case others would like to experiment with the technique. Given that there are so few methods available for investigating selective reporting in meta-analyses with dependent effect sizes, we think this method is worth playing with and investigating further, and we would be happy to have others try it out as well. But, if you do so, please treat the results as tentative until we learn more about when the methods work well enough to trust the results.

An Example

For demonstrating this method, we will use data from a recent meta-analysis by Lehmann and colleagues (2018) that examined the effects of the color red on attractiveness judgments. The data is available via the metadat package (White et al., 2022). The dataset includes 81 effect sizes from 41 unique studies. You can browse the data for yourself here:

Code
library(metadat)   # for the example dataset
library(tidyverse) # for tidying
library(janitor)   # for tidying variable names
library(metafor)   # for meta-analysis
library(boot)      # for bootstrapping
library(tictoc)    # for keeping time

lehmann_dat <- 
  dat.lehmann2018 %>%
  clean_names() %>%
  mutate(study = str_split_fixed(short_title, pattern = "-", n = 2)[, 1]) %>%
  arrange(study) %>%
  select(study, presentation = stimuli_presentation, yi, vi, everything())
study presentation yi vi short_title full_citation year pr_publication source_type preregistered moderator_group context gender color_contrast color_form photo_type photo_similarity dv_type dv_items dv_scale dv_scale_bottom dv_scale_top location continent participants participant_notes design eth_majority eth_majority_detail eth_stim eth_match red_age control_age color_red color_control red_original color_match presentation_control red_n red_m red_sd control_n control_m control_sd sd_diff rm_r control_attractiveness notes total_sample_size pooled
Banas, 2014 Paper 0.06 0.10 Banas, 2014 - Exp 1 Banas, K. (2014, July 7). Replication of Elliot et al. (2010) for CREP at the University of Edinburgh. Retrieved from osf.io/cvdpw 2014.0 No CREP Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 Scotland Europe Students Undergrads Between Subjects White White Latino Mis-match 20.43 20.95 LCh(50.0, 59.6, 31.3) LCh(50.0, -, 69.1) Yes Yes Yes 20 6.05 1.59 19 5.96 1.49 NA NA 0.62 NA 39 NA
Berthold, 2013 Screen 0.55 0.06 Berthold, 2013 - Exp 1 - In Group Berthold, A. (2013). Unpublished data 2013.0 No Unpublished/Online Not Pre-Registered No Romantic Females Blue Background Bust Same between conditions Perceived attractiveness 4 1-7 1 7 Germany Europe Students Undergrads Between Subjects White White White Matched 25.00 25.00 No Data No Data No No Yes 36 2.31 1.29 33 1.73 0.65 NA NA 0.12 NA 69 NA
Bigelow et al., 2013 Screen 0.31 0.30 Bigelow et al., 2013 - Exp 1 Bigelow, M.G., Taylor, G. & Underwood, M. (2013). Context-moderated effect of color on physiological and self-report measures of emotional response. UNC Asheville Journal, Undergraduate Research Program, Asheville, NC 2013.0 Yes Journal Not Pre-Registered No Romantic Females Blue Background Head Shot Same between conditions Single item 1 1-9 1 9 USA North America Students Undergrads Between Subjects NA NA NA NA NA NA Lab(37.43/59.24/47.63) Lab(36.68/34.86/87.99) No No Yes 6 5.22 1.77 8 4.56 2.09 NA NA 0.44 NA 14 NA
Bigelow et al., 2013 Screen -0.73 0.53 Bigelow et al., 2013 - Exp 1 Bigelow, M.G., Taylor, G. & Underwood, M. (2013). Context-moderated effect of color on physiological and self-report measures of emotional response. UNC Asheville Journal, Undergraduate Research Program, Asheville, NC 2013.0 Yes Journal Not Pre-Registered No Romantic Males Blue Background Head Shot Same between conditions Single item 1 1-9 1 9 USA North America Students Undergrads Between Subjects NA NA NA NA NA NA Lab(37.43/59.24/47.63) Lab(36.68/34.86/87.99) No No Yes 4 4.65 2.05 4 6.28 1.81 NA NA 0.66 NA 8 NA
Blech, 2014 Screen 0.08 0.03 Blech, 2014 - Exp 1 Blech, C. (2014, August 4). Replication of Elliot et al. (2010). Red, rank, and romance in women viewing men. Retrieved from osf.io/tx2u5 2014.0 No Unpublished/Online Not Pre-Registered No Romantic Females White Background Head Shot Same between conditions Perceived attractiveness, German translation 4 1-9 1 9 Germany Europe Students Undergrads Between Subjects White White Latino Mis-match NA NA No Data No Data No No No 71 5.36 1.43 78 5.25 1.48 NA NA 0.53 Not part of CREP because Used white as control condition, dropped yellow as not an original control color, age not included because separated into categories (<25, 26-40, >=41) 149 NA
Blech, 2015 Screen -0.35 0.05 Blech, 2015 - Class Exp Blech, C. (2015). Unpublished data from a class experiment 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females White Clothing Bust Same between conditions Perceived attractiveness 4 1-9 1 9 Germany Europe Adults NA Between Subjects White White White Matched 31.40 31.40 NA NA No No No 37 3.95 1.29 37 4.45 1.50 NA NA 0.43 NA 74 NA
Boelk & Madden, 2014 Paper -0.27 0.06 Boelk & Madden, 2014 - Exp 1 Boelk, K., & Madden, W. (2014, August 5). Fork of Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Retrieved from osf.io/zf7c9 2014.0 No CREP Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 19.06 19.35 LCh(50.0, 59.6, 31.3) LCh(50.0, -, 69.1) Yes Yes Yes 34 6.03 1.24 34 6.40 1.49 NA NA 0.68 NA 68 NA
Buechner et al., 2015 Paper 0.68 0.09 Buechner et al., 2015 - Exp 1 - Prideful Pose Buechner, V. L., Maier, M. A., Lichtenfeld, S., & Elliot, A. J. (2015). Emotion Expression and Color: Their Joint Influence on Perceived Attractiveness and Social Position. Current Psychology, 34(2), 422-433. http://doi.org/10.1007/s12144-014-9266-x 2015.0 Yes Journal Not Pre-Registered No Romantic Females Blue Dot Bust Same between conditions Perceived attractiveness 3 1-9 1 9 Germany Europe Students High School Between Subjects White White White Matched 16.88 16.88 LCh(50.9, 59.7, 25.7) LCh(49.2, 60.2, 278.2) Yes Yes Yes 21 4.30 1.65 29 3.21 1.53 NA NA 0.28 NA 50 NA
Costello et al., 2017 Paper -0.12 0.03 Costello et al., 2017 - Exp 1 Costello J., Groeneboom L. & Pollet T. (2017). Romantic red: Do red products enhance the attractiveness of the consumer? Unpublished masters degree manuscript, University of Leiden 2017.0 No Unpublished/Online Not Pre-Registered No Romantic Males Blue Item Bust Same between conditions Single item perceived attractiveness 1 1-5 1 5 Netherlands Europe Students Undergrad Between Subjects White White Asian Mis-match 22.60 NA NA NA No No Yes 65 2.25 1.04 64 2.38 0.95 NA NA 0.34 NA 129 NA
Costello et al., 2017 Paper 0.08 0.02 Costello et al., 2017 - Exp 2 Costello J., Groeneboom L. & Pollet T. (2017). Romantic red: Do red products enhance the attractiveness of the consumer? Unpublished masters degree manuscript, University of Leiden 2017.0 No Unpublished/Online Not Pre-Registered No Romantic Females Blue/Green Item Head Shot Same between conditions Single item perceived attractiveness 1 1-5 1 5 Netherlands Europe Students Undergrad Between Subjects White White White Matched 21.32 21.06 No Data No Data No No Yes 67 2.60 1.37 140 2.49 1.38 NA NA 0.37 Same filters applied as in Exp 1 (excluded homosexual and preferred not to answer). Control combines blue and green 207 NA
Elliot & Maier, 2013 Paper 0.25 0.03 Elliot & Maier, 2013 - Exp 1 Elliot, A. J., & Maier, M. a. (2013). The red-attractiveness effect, applying the Ioannidis and Trikalinos (2007b) test, and the broader scientific context: a reply to Francis (2013). Journal of Experimental Psychology. General, 142(1), 297-300. http://doi.org/10.1037/a0029592 2013.0 Yes Journal Not Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 19.47 19.47 LCh(42.6, 45.2, 15.8) LCh(43.0, -, 296.7) No Yes Yes 75 6.29 1.35 69 5.93 1.49 NA NA 0.62 NA 144 NA
Elliot & Niesta, 2008 Paper 0.62 0.14 Elliot & Niesta, 2008 - Exp 4 Elliot, A. J., & Niesta, D. (2008). Romantic red: red enhances men's attraction to women. Journal of Personality and Social Psychology, 95(5), 1150-1164. http://doi.org/10.1037/0022-3514.95.5.1150 2008.0 Yes Journal Not Pre-Registered No Romantic Males Green Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.97 19.97 LCh(46.1, 51.2, 29.3) LCh(46.1, 51.0, 147.6) Yes Yes Yes 16 6.29 0.89 15 5.66 1.09 NA NA 0.58 NA 31 NA
Elliot & Niesta, 2008 Paper 0.67 0.11 Elliot & Niesta, 2008 - Exp 3 Elliot, A. J., & Niesta, D. (2008). Romantic red: red enhances men's attraction to women. Journal of Personality and Social Psychology, 95(5), 1150-1164. http://doi.org/10.1037/0022-3514.95.5.1150 2008.0 Yes Journal Not Pre-Registered No Romantic Males Gray Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 20.00 20.00 LCh(50.0, 58.7, 30.3) LCh(50.0, -, 52.6) Yes Yes Yes 20 6.65 1.10 17 5.91 1.07 NA NA 0.61 NA 37 NA
Elliot & Niesta, 2008 Paper 0.84 0.19 Elliot & Niesta, 2008 - Exp 5 Elliot, A. J., & Niesta, D. (2008). Romantic red: red enhances men's attraction to women. Journal of Personality and Social Psychology, 95(5), 1150-1164. http://doi.org/10.1037/0022-3514.95.5.1150 2008.0 Yes Journal Not Pre-Registered No Romantic Males Blue Clothing Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.77 19.77 LCh(45.9, 54.8, 32.5) LCh(46.0, 54.9, 283.0) Yes Yes Yes 12 7.21 1.05 11 6.09 1.49 NA NA 0.64 NA 23 NA
Elliot & Niesta, 2008 Paper 0.75 0.13 Elliot & Niesta, 2008 - Exp 2 Elliot, A. J., & Niesta, D. (2008). Romantic red: red enhances men's attraction to women. Journal of Personality and Social Psychology, 95(5), 1150-1164. http://doi.org/10.1037/0022-3514.95.5.1150 2008.0 Yes Journal Not Pre-Registered No Romantic Males White Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.27 19.27 LCh(55.5, 78.0, 28.0) No Data Yes No Yes 16 7.07 0.78 16 6.13 1.53 NA NA 0.64 Age includes both male and female participants. 32 NA
Elliot & Niesta, 2008 Paper 1.08 0.17 Elliot & Niesta, 2008 - Exp 1 Elliot, A. J., & Niesta, D. (2008). Romantic red: red enhances men's attraction to women. Journal of Personality and Social Psychology, 95(5), 1150-1164. http://doi.org/10.1037/0022-3514.95.5.1150 2008.0 Yes Journal Not Pre-Registered No Romantic Males White Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 20.52 20.52 LCh(50.3, 58.8, 29.9) No Data Yes No Yes 15 7.33 0.90 12 6.25 1.05 NA NA 0.66 NA 27 NA
Elliot et al., 2010 Paper 0.82 0.13 Elliot et al., 2010 - Exp 3 Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Red, rank, and romance in women viewing men. Journal of Experimental Psychology: General, 139(3), 399-417. http://doi.org/10.1037/a0019689 2010.0 Yes Journal Not Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 19.64 19.64 LCh(50.0, 59.6, 31.3) LCh(50.0, -, 69.1) Yes Yes Yes 16 6.69 1.22 17 5.27 2.04 NA NA 0.53 df doesn't match sample size 33 NA
Elliot et al., 2010 Paper 0.61 0.08 Elliot et al., 2010 - Exp 4 Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Red, rank, and romance in women viewing men. Journal of Experimental Psychology: General, 139(3), 399-417. http://doi.org/10.1037/a0019689 2010.0 Yes Journal Not Pre-Registered No Romantic Females Green Clothing Bust Same between conditions Mehrabian & Blum's Perceived attractiveness 4 1-9 1 9 China Asia Students Undergrads Between Subjects Chinese Chinese Chinese Matched 20.60 20.60 LCh(51.3, 51.7, 30.1) LCh(51.5, 51.6, 136.6) Yes Yes Yes 27 6.32 1.09 28 5.50 1.50 NA NA 0.56 NA 55 NA
Elliot et al., 2010 Paper 0.90 0.21 Elliot et al., 2010 - Exp 1 Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Red, rank, and romance in women viewing men. Journal of Experimental Psychology: General, 139(3), 399-417. http://doi.org/10.1037/a0019689 2010.0 Yes Journal Not Pre-Registered No Romantic Females White Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 20.19 20.19 LCh(49.6, 58.8, 30.4) No Data Yes No Yes 10 6.79 1.00 11 5.67 1.34 NA NA 0.58 df doesn't match sample size 21 NA
Elliot et al., 2010 Paper 0.82 0.16 Elliot et al., 2010 - Exp 7 Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Red, rank, and romance in women viewing men. Journal of Experimental Psychology: General, 139(3), 399-417. http://doi.org/10.1037/a0019689 2010.0 Yes Journal Not Pre-Registered No Romantic Females Blue Clothing Bust Same between conditions Maner et al perceived attractiveness 1 1-9 1 9 England Europe Students Undergrads Between Subjects White White White Matched 19.44 19.44 LCh(54.8, 43.2, 30.3) LCh(55.1, 43.7, 283.0) Yes Yes Yes 12 7.50 1.17 15 6.13 1.89 NA NA 0.64 NA 27 NA
Elliot et al., 2010 Paper 1.05 0.15 Elliot et al., 2010 - Exp 2 Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Red, rank, and romance in women viewing men. Journal of Experimental Psychology: General, 139(3), 399-417. http://doi.org/10.1037/a0019689 2010.0 Yes Journal Not Pre-Registered No Romantic Females White Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 20.46 20.46 LCh(49.6, 58.8, 30.4) No Data Yes No Yes 20 7.15 0.74 12 6.20 1.08 NA NA 0.65 Sample size taken from Francis, t(df) seems to be using ANOVA df for post-hoc, age data for both male and female participants (separate was not provided) 32 NA
Elliot et al., 2013 Paper 0.64 0.10 Elliot et al., 2013 - Exp 1 Elliot, A. J., Tracy, J. L., Pazda, A. D., & Beall, A. T. (2013). Red enhances women's attractiveness to men: First evidence suggesting universality. Journal of Experimental Social Psychology, 49(1), 165-168. http://doi.org/10.1016/j.jesp.2012.07.017 2013.0 Yes Journal Not Pre-Registered No Romantic Males Blue Background Head Shot Same between conditions Perceived attractiveness 3 1-5 1 5 Burkina Faso Africa Adults NA Between Subjects Black Black Black Matched 26.80 26.80 LCh(42.7, 51.5, 20.4) LCh(43.4, 51.5, 269.8) No Yes Yes 21 4.62 0.59 21 4.14 0.85 NA NA 0.78 NA 42 NA
Frazier, 2014 Paper 0.09 0.06 Frazier, 2014 - Exp 1 Frazier, A. (2014, November 13). Fork of Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Retrieved from osf.io/u0mig 2014.0 No CREP Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 18.90 18.90 LCh(50.0, 59.6, 31.3) LCh(50.0, -, 69.1) Yes Yes Yes 29 6.09 1.39 39 5.97 1.26 NA NA 0.62 NA 68 NA
Gilston & Privitera, 2016 Screen 1.92 0.04 Gilston & Privitera, 2016 - Exp 1 - Healthy Gilston, A., & Privitera, G. J. (2015). A 'Healthy' Color: Information About Healthy Eating Attenuates the 'Red Effect.' Global Journal of Health Science, 8(1), 56-61. https://doi.org/10.5539/gjhs.v8n1p56 2016.0 Yes Journal Not Pre-Registered No Romantic Males White Clothing Head Shot Same between conditions Perceived attractiveness 1 1-7 1 7 USA North America Students Undergrad Within Subjects NA NA White NA 19.85 19.85 No Data No Data No No Yes 54 5.65 1.51 54 3.09 1.09 NA 0.89 0.35 NA 54 1.32
Gueguen, 2012 Paper 0.74 0.05 Gueguen, 2012 - Exp 1 Gueguen, N. (2012). Color and Women Attractiveness: When Red Clothed Women Are Perceived to Have More Intense Sexual Intent. The Journal of Social Psychology, 152(3), 261-265. http://doi.org/10.1080/00224545.2011.605398 2012.0 Yes Journal Not Pre-Registered No Romantic Males Blue/Green/White Clothing Bust Same between conditions Perceived attractiveness 1 1-9 1 9 France Europe Students Undergrads Between Subjects NA NA NA NA 19.20 19.20 No Data No Data No No Yes 30 5.95 1.24 90 5.04 1.22 NA NA 0.50 Control is average of blue, white, and green 120 NA
Hesslinger et al. 2015 Paper 0.46 0.13 Hesslinger et al. 2015 - Exp 2 Hesslinger, V. M., Goldbach, L., Carbon, C.-C., Allgemeine, A., Psychologie, E., & Note, A. (2015). Men in red: A reexamination of the red-attractiveness effect. Psychonomic Bulletin & Review, 55(4), 1-6. http://doi.org/10.3758/s13423-015-0866-8 2015.0 Yes Journal Not Pre-Registered No Romantic Females White Background Full Body Same between conditions Perceived attractiveness, German translation 3 1-9 1 9 Germany Europe Students Undergrads Between Subjects White White White Matched 20.20 20.20 CIE-Lab(50, 51, 30) No Data Yes No Yes 16 4.98 0.70 16 4.63 0.79 NA NA 0.45 Age includes both male and female participants. 32 NA
Hesslinger et al. 2015 Paper 0.06 0.06 Hesslinger et al. 2015 - Exp 1 Hesslinger, V. M., Goldbach, L., Carbon, C.-C., Allgemeine, A., Psychologie, E., & Note, A. (2015). Men in red: A reexamination of the red-attractiveness effect. Psychonomic Bulletin & Review, 55(4), 1-6. http://doi.org/10.3758/s13423-015-0866-8 2015.0 Yes Journal Not Pre-Registered No Romantic Females White Background Bust Same between conditions Perceived attractiveness, German translation 3 1-9 1 9 Germany Europe Students Undergrads Between Subjects White White White Matched 21.20 22.60 CIE-Lab(50, 51, 30) No Data Yes No Yes 35 5.71 1.65 37 5.62 1.44 NA NA 0.58 Age includes both male and female participants. 72 NA
Johnson et al., 2015 Paper -0.01 0.05 Johnson et al., 2015 - Exp 1 Johnson, K., Meltzer, A., & Grahe, J. E. (2015, October 12). Fork of Elliot, A. J., Niesta Kayser, D., Greitemeyer, T., Lichtenfeld, S., Gramzow, R. H., Maier, M. A., & Liu, H. (2010). Retrieved from osf.io/ictud 2014.0 No CREP Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 18.94 18.92 LCh(50.0, 59.6, 31.3) LCh(50.0, -, 69.1) Yes Yes Yes 35 5.97 1.60 38 5.99 1.34 NA NA 0.62 Yellow group also run, but not included 73 NA
Khislavsky, 2016 Paper 0.06 0.02 Khislavsky, 2016 - Exp 1 Khislavsky, A. (2016, March 14). Replication of Elliot et al. (2010). Red, rank, and romance in women viewing men. Retrieved from osf.io/f2udj 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects Latino Latino Latino Matched 19.91 20.04 ~LCh(50.0, 59.6, 31.3) ~LCh(50.0, -, 69.1) Yes Yes Yes 95 5.73 1.76 92 5.61 1.84 NA NA 0.58 Not reviewed by CREP 187 NA
Kirsch, 2015 Screen -0.47 0.03 Kirsch, 2015 - Exp 1 - Heterosexual Kirsch, F. (2015). Wahrgenommene Attraktivitaet und sexuelle Orientierung: Die Wirkung von Rot und Farbpraeferenzen (Perceived attractiveness and sexual orientation: The effects of red and color preferences). Wiesbaden: Springer. doi: 10.1007/978-3-658-08405-9 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females White Clothing Bust Same between conditions Perceived attractiveness 2 1-9 1 9 Germany Europe Online NA Between Subjects NA NA White NA 22.45 22.45 No Data No Data No No No 76 5.33 1.83 85 6.13 1.54 NA NA 0.64 Cross-gender rating (females rating males). Age includes participants in all conditions. 161 NA
Kirsch, 2015 Screen 0.32 0.04 Kirsch, 2015 - Exp 1 - Heterosexual Kirsch, F. (2015). Wahrgenommene Attraktivitaet und sexuelle Orientierung: Die Wirkung von Rot und Farbpraeferenzen (Perceived attractiveness and sexual orientation: The effects of red and color preferences). Wiesbaden: Springer. doi: 10.1007/978-3-658-08405-9 2015.0 No Monograph Not Pre-Registered No Romantic Males White Clothing Bust Same between conditions Perceived attractiveness 2 1-9 1 9 Germany Europe Online NA Between Subjects NA NA White NA 22.45 22.45 No Data No Data No No No 57 6.85 1.44 48 6.42 1.22 NA NA 0.68 Cross-gender rating (males rating females). Age includes participants in all conditions. 105 NA
Kirsch, 2015 NA 0.03 0.07 Kirsch, 2015 - Exp 2 Kirsch, F. (2015). Wahrgenommene Attraktivitaet und sexuelle Orientierung: Die Wirkung von Rot und Farbpraeferenzen (Perceived attractiveness and sexual orientation: The effects of red and color preferences). Wiesbaden: Springer. doi: 10.1007/978-3-658-08405-9 2015.0 No Monograph Not Pre-Registered No Romantic Males White Clothing Bust Same between conditions Perceived attractiveness 2 1-9 1 9 Germany Europe Students Undergrads Between Subjects NA NA NA NA NA NA No Data No Data No No No 28 6.55 1.17 28 6.52 1.36 NA NA 0.69 Means are only for cross-gender rating (males rating females) 56 NA
Legate et al., 2015 Paper -0.35 0.08 Legate et al., 2015 - Exp 1 Legate, N., Baciu, C., Horne, L. M., Fiol, S., Paniagua, D., Muqeet, M., & Zachocki, E. (2015, June 27). Replication of Elliot et al. (2010) at IIT. Retrieved from osf.io/zih7c 2015.0 No CREP Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White (21 white, 16 asian) Latino Mis-match 20.73 20.25 ~LCh(50.0, 59.6, 31.3) ~LCh(50.0, -, 69.1) Yes Yes Yes 25 5.76 1.61 23 6.33 1.58 NA NA 0.67 NA 48 NA
Lehmann & Calin Screen 0.39 0.07 Lehmann & Calin-Jageman, 2015 - Class Exp Lehmann & Calin-Jageman (2017) Is red really romantic? Direct replications suggest little to no effect of the color red on perceived attractiveness for men and women. Social Psychology. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females White Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Online Mturk Between Subjects White White White Matched 31.70 36.15 No Data No Data No No No 29 4.34 1.63 28 3.69 1.65 NA NA 0.34 NA 57 NA
Lehmann & Calin Paper -0.11 0.03 Lehmann & Calin-Jageman, 2017 - Exp 1 Lehmann & Calin-Jageman (2017) Is red really romantic? Direct replications suggest little to no effect of the color red on perceived attractiveness for men and women. Social Psychology. 2017.0 Yes Journal Pre-Registered No Romantic Females White Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects Latino Mixed White/Latino Matched 20.41 21.22 LCh(51.3, 58.2, 29.5) No Data Yes No Yes 56 4.67 2.05 60 4.89 1.80 NA NA 0.49 NA 116 NA
Lehmann & Calin Screen -0.08 0.02 Lehmann & Calin-Jageman, 2017 - Exp 2 Lehmann & Calin-Jageman (2017) Is red really romantic? Direct replications suggest little to no effect of the color red on perceived attractiveness for men and women. Social Psychology. 2017.0 Yes Journal Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Online Mturk Between Subjects White White White/Latino Matched 36.05 38.32 No Data No Data No No No 114 5.24 1.58 130 5.38 1.85 NA NA 0.55 NA 244 NA
Lehmann & Calin Screen -0.23 0.04 Lehmann & Calin-Jageman, 2015 - Class Exp Lehmann & Calin-Jageman (2017) Is red really romantic? Direct replications suggest little to no effect of the color red on perceived attractiveness for men and women. Social Psychology. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Males White Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Online Mturk Between Subjects White White White Matched 33.15 33.88 No Data No Data No No No 56 5.24 1.59 46 5.59 1.34 NA NA 0.57 NA 102 NA
Lehmann & Calin Screen 0.10 0.02 Lehmann & Calin-Jageman, 2017 - Exp 2 Lehmann & Calin-Jageman (2017) Is red really romantic? Direct replications suggest little to no effect of the color red on perceived attractiveness for men and women. Social Psychology. 2017.0 Yes Journal Pre-Registered No Romantic Males Gray Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Online Mturk Between Subjects White White White Matched 35.78 35.85 No Data No Data No No No 104 6.51 1.32 106 6.38 1.32 NA NA 0.67 NA 210 NA
Lehmann & Calin Paper 0.00 0.14 Lehmann & Calin-Jageman, 2017 - Exp 1 Lehmann & Calin-Jageman (2017) Is red really romantic? Direct replications suggest little to no effect of the color red on perceived attractiveness for men and women. Social Psychology. 2017.0 Yes Journal Pre-Registered No Romantic Males White Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects Mixed Mixed White Mis-match 20.81 20.45 LCh(51.3, 58.2, 29.5) No Data Yes No Yes 21 6.52 1.16 11 6.52 1.56 NA NA 0.69 NA 32 NA
Lin, 2014 Paper 1.53 0.13 Lin, 2014 - Exp 1 Lin, H. (2014). Red-colored products enhance the attractiveness of women. Displays, 35(4), 202-205. http://doi.org/10.1016/j.displa.2014.05.009 2014.0 Yes Journal Not Pre-Registered No Romantic Males Blue Item Bust Same between conditions Single item perceived attractiveness 1 1-5 1 5 Taiwan Asia Students Undergrads Between Subjects Asian NA Asian Matched NA NA No Data No Data No No Yes 20 3.50 0.69 20 2.55 0.51 NA NA 0.39 Control condition is blue; silver condition is dropped as it is not an original control color 40 NA
Maves & Nadler, 2016 Paper -0.12 0.03 Maves & Nadler, 2016 - Exp 1 Maves, M., & Nadler, J. T. (2016, June 2). Data and Analysis. Retrieved from osf.io/9bm8v 2015.0 No CREP Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 21.86 22.48 ~LCh(50.0, 59.6, 31.3) ~LCh(50.0, -, 69.1) Yes Yes Yes 66 6.07 1.40 64 6.23 1.38 NA NA 0.65 NA 130 NA
O'Mara & Trujillo, 2015 Screen 0.97 0.10 O'Mara & Trujillo, 2015 - Exp 1 - Masculine Face O'Mara, E. M., & Trujillo, A. (2015), unpublished data. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females White Background Head Shot Same between conditions Perceived attractiveness 1 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.10 19.10 NA NA No No Yes 21 6.11 1.30 25 4.78 1.38 NA NA 0.47 NA 46 NA
O'Mara & Trujillo, 2015 Screen -0.20 0.09 O'Mara & Trujillo, 2015 - Exp 2 - Masculine Face O'Mara, E. M., & Trujillo, A. (2015), unpublished data. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females White Background Head Shot Same between conditions Perceived attractiveness 1 1-9 1 9 USA North America Online Mturk Between Subjects White White White Matched 28.23 28.23 NA NA No No No 24 4.88 1.84 21 5.23 1.61 NA NA 0.53 NA 45 NA
O'Mara et al., 2016 Screen 0.45 0.14 O'Mara et al., 2016 - Exp 2 - Long Shirt O'Mara, E. M., Kershaw, C., Receveur, A., Hunt, C., Askar, S., Ballas, T., Farmer, C., O'Koon, B., Stitzel, C., Vavro, C., & Wilhoit, S. (2016), unpublished data. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Males Blue Clothing Bust Different between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.26 19.26 NA NA No No Yes 20 4.42 1.37 12 3.79 1.41 NA NA 0.35 NA 32 NA
O'Mara et al., 2016 Screen -0.32 0.14 O'Mara et al., 2016 - Exp 2 - Tank Top O'Mara, E. M., Kershaw, C., Receveur, A., Hunt, C., Askar, S., Ballas, T., Farmer, C., O'Koon, B., Stitzel, C., Vavro, C., & Wilhoit, S. (2016), unpublished data. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Males Blue Clothing Bust Different between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.26 19.26 NA NA No No Yes 17 3.91 1.09 12 4.33 1.51 NA NA 0.42 NA 29 NA
O'Mara et al., 2016 Screen 0.47 0.07 O'Mara et al., 2016 - Exp 1 - Long Shirt O'Mara, E. M., Kershaw, C., Receveur, A., Hunt, C., Askar, S., Ballas, T., Farmer, C., O'Koon, B., Stitzel, C., Vavro, C., & Wilhoit, S. (2016), unpublished data. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Males White Clothing Bust Different between conditions Perceived attractiveness 2 1-9 1 9 USA North America Online Mturk Between Subjects White White White Matched 34.56 34.56 NA NA No No No 28 6.07 2.14 35 5.12 1.89 NA NA 0.52 NA 63 NA
O'Mara et al., 2016 Screen 0.10 0.06 O'Mara et al., 2016 - Exp 1 - Tank Top O'Mara, E. M., Kershaw, C., Receveur, A., Hunt, C., Askar, S., Ballas, T., Farmer, C., O'Koon, B., Stitzel, C., Vavro, C., & Wilhoit, S. (2016), unpublished data. 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Males White Clothing Bust Different between conditions Perceived attractiveness 2 1-9 1 9 USA North America Online Mturk Between Subjects White White White Matched 34.56 34.56 NA NA No No No 34 5.53 1.96 35 5.33 2.11 NA NA 0.54 NA 69 NA
O'Mara et al., 2016 Screen -0.12 0.04 O'Mara et al., 2016 - Exp 3 - Tank Top O'Mara, E. M., Kershaw, C., Receveur, A., Hunt, C., Askar, S., Ballas, T., Farmer, C., O'Koon, B., Stitzel, C., Vavro, C., & Wilhoit, S. (2016), unpublished data. 2016.0 No Unpublished/Online Not Pre-Registered No Romantic Males White Clothing Bust Different between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.54 19.54 NA NA No No Yes 51 6.70 1.14 55 6.83 1.00 NA NA 0.73 NA 106 NA
O'Mara et al., 2016 Screen -0.14 0.03 O'Mara et al., 2016 - Exp 3 - Long Shirt O'Mara, E. M., Kershaw, C., Receveur, A., Hunt, C., Askar, S., Ballas, T., Farmer, C., O'Koon, B., Stitzel, C., Vavro, C., & Wilhoit, S. (2016), unpublished data. 2016.0 No Unpublished/Online Not Pre-Registered No Romantic Males White Clothing Bust Different between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.54 19.54 NA NA No No Yes 63 6.97 1.27 57 7.12 0.93 NA NA 0.77 NA 120 NA
Pazda et al., 2012 Paper 0.63 0.09 Pazda et al., 2012 - Exp 2 Pazda, A. D., Elliot, A. J., & Greitemeyer, T. (2012). Sexy red: Perceived sexual receptivity mediates the red-attraction relation in men viewing woman. Journal of Experimental Social Psychology, 48(3), 787-790. http://doi.org/10.1016/j.jesp.2011.12.009 2012.0 Yes Journal Not Pre-Registered No Romantic Males Green Clothing Bust Same between conditions Perceived attractiveness 2 1-9 1 9 Austria Europe Students Undergrads Between Subjects White White White Matched 23.50 23.50 LCh(40.6, 40.4, 20.1) LCh(40.3, 41.2, 146.8) No Yes Yes 27 6.07 1.17 22 5.00 2.13 NA NA 0.50 NA 49 NA
Pazda et al., 2014 Screen 0.32 0.02 Pazda et al., 2014 - Exp 1 Pazda, A. D., Elliot, A. J., & Greitemeyer, T. (2014). Perceived sexual receptivity and fashionableness: Separate paths linking red and black to perceived attractiveness. Color Research & Application, 39(2), 208-212. http://doi.org/10.1002/col.21804 2014.0 Yes Journal Not Pre-Registered No Romantic Males White Clothing Full Body Same between conditions Perceived attractiveness 2 1-9 1 9 Not specified Unknown Online Mturk Between Subjects White White (170 white, 142 Asian) White Matched 24.50 24.50 No Data No Data No No No 109 6.03 1.78 125 5.46 1.77 NA NA 0.56 NA 234 NA
Pazda et al., 2017 Screen 0.17 0.00 Pazda et al., 2017 - Exp 1 Pazda, A., Thorstenson & Elliot, A. (2017). Unpublished data. 2017.0 No Unpublished/Online Not Pre-Registered No Romantic Males Green Clothing Full Body Same between conditions Single Item 1 1-9 1 9 USA North America Students Lab study Within Subjects NA NA White NA 19.76 NA red LCh(42.00, 57.92, 348.89), green LCh(41.50, 56.71, 92.17) NA No Yes Yes 115 7.81 1.21 115 7.58 1.41 NA 0.84 0.82 NA 115 1.31
Pazda et al., 2017 Screen 0.10 0.00 Pazda et al., 2017 - Exp 2 Pazda, A., Thorstenson & Elliot, A. (2017). Unpublished data. 2017.0 No Unpublished/Online Pre-Registered No Romantic Males Green Clothing Full Body Same between conditions Single Item 1 1-9 1 9 USA North America Online Mturk Within Subjects White NA White Matched 37.43 NA red LCh(42.00, 57.92, 348.89), green LCh(41.50, 56.71, 92.17) NA No No No 228 8.08 1.28 228 7.95 1.39 NA 0.80 0.87 NA 228 1.34
Peperkoorn et al., 2016 Paper -0.43 0.06 Peperkoorn et al., 2016 - Exp 1 - Short Term Peperkoorn, L. S., Roberts, S. C., & Pollet, T. V. (2016). Revisiting the Red Effect on Attractiveness and Sexual Receptivity: No Effect of the Color Red on Human Mate Preferences. Evolutionary Psychology, 14(4). http://doi.org/10.1177/1474704916673841 2016.0 Yes Journal Not Pre-Registered No Romantic Males White Clothing Bust Same between conditions Perceived attractiveness 2 1-11 2 22 Netherlands Europe Students Undergrads Between Subjects White White White Matched 23.67 23.67 NA NA No No Yes 34 14.26 3.31 34 15.47 2.18 NA NA 0.67 NA 68 NA
Peperkoorn et al., 2016 Screen -0.10 0.06 Peperkoorn et al., 2016 - Exp 2 - Short Term Peperkoorn, L. S., Roberts, S. C., & Pollet, T. V. (2016). Revisiting the Red Effect on Attractiveness and Sexual Receptivity: No Effect of the Color Red on Human Mate Preferences. Evolutionary Psychology, 14(4). http://doi.org/10.1177/1474704916673841 2016.0 Yes Journal Not Pre-Registered No Romantic Males White Clothing Bust Same between conditions Perceived attractiveness 2 1-11 2 22 USA North America Online Mturk Between Subjects White White White Matched 30.13 30.13 NA NA No No No 33 15.06 4.14 36 15.47 3.61 NA NA 0.67 NA 69 NA
Pollet, 2013 Screen 0.16 0.08 Pollet, 2013 - Exp 1 Pollet T. (2013). Unpublished data 2013.0 No Unpublished/Online Not Pre-Registered No Romantic Females Blue Clothing Full Body Same between conditions Single item 1 1-7 1 7 Netherlands Europe Students NA Between Subjects White Primarily dutch nationality White Matched 26.60 28.40 No Data No Data No No No 26 3.15 1.22 23 2.96 1.19 NA NA 0.33 Only blue control color used as other control colors were not in original 49 NA
Purdy, 2009 Screen 0.30 0.16 Purdy, 2009 - Exp 1 - High Arousal Purdy, M. A. (2009). The influence of the amygdala and color on judgments of attractiveness. UNC Asheville Journal, Undergraduate Research Program, Asheville, NC 2009.0 Yes Journal Not Pre-Registered No Romantic Males Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects NA NA White NA 21.00 21.00 RGB(185, 26, 23) RGB(216,216,216) No No Yes 13 7.26 3.27 13 6.24 3.27 NA NA 0.66 NA 26 NA
Purdy, 2009 Screen 0.01 0.15 Purdy, 2009 - Exp 1 - Low Arousal Purdy, M. A. (2009). The influence of the amygdala and color on judgments of attractiveness. UNC Asheville Journal, Undergraduate Research Program, Asheville, NC 2009.0 Yes Journal Not Pre-Registered No Romantic Males Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects NA NA White NA 21.00 21.00 RGB(185, 26, 23) RGB(216,216,216) No No Yes 13 6.37 3.31 13 6.33 3.27 NA NA 0.67 NA 26 NA
Roberts et al., 2010 Screen 0.33 0.05 Roberts et al., 2010 - Exp 2 Roberts, S. C., Owen, R. C., & Havlicek, J. (2010). Distinguishing between Perceiver and Wearer Effects in Clothing Color-Associated Attributions. Evolutionary Psychology, 8(3), 350-364. http://doi.org/10.1177/147470491000800304 2010.1 Yes Journal Not Pre-Registered No Romantic Females Blue/Green/White Clothing Bust Different between conditions Single item rating of attractiveness 1 1-10 1 10 England Europe Students Undergrads Within Subjects White Noted in manuscript all caucasian White Matched NA NA No Data No Data No No Yes 15 3.66 0.60 15 3.45 0.61 NA 0.66 0.27 Age range includes both male and female participants. 15 0.60
Roberts et al., 2010 Screen 0.40 0.02 Roberts et al., 2010 - Exp 2 Roberts, S. C., Owen, R. C., & Havlicek, J. (2010). Distinguishing between Perceiver and Wearer Effects in Clothing Color-Associated Attributions. Evolutionary Psychology, 8(3), 350-364. http://doi.org/10.1177/147470491000800304 2010.1 Yes Journal Not Pre-Registered No Romantic Males Blue/Green/White Clothing Bust Different between conditions Single item rating of attractiveness 1 1-10 1 10 England Europe Students Undergrads Within Subjects White Noted in manuscript all caucasian White Matched NA NA No Data No Data No No Yes 15 3.65 0.45 15 3.46 0.48 NA 0.92 0.27 Age range includes both male and female participants. 15 0.47
Roberts et al., 2010 Screen 0.19 0.02 Roberts et al., 2010 - Exp 1 Roberts, S. C., Owen, R. C., & Havlicek, J. (2010). Distinguishing between Perceiver and Wearer Effects in Clothing Color-Associated Attributions. Evolutionary Psychology, 8(3), 350-364. http://doi.org/10.1177/147470491000800304 2010.1 Yes Journal Not Pre-Registered No Romantic Females Blue/Green/White Clothing Bust Different between conditions Single item rating of attractiveness 1 1-10 1 10 England Europe Students Undergrads Within Subjects White Noted in manuscript all caucasian White Matched NA NA No Data No Data No No Yes 30 4.50 1.23 30 4.26 1.18 NA 0.66 0.36 Age range includes both male and female participants. 30 1.21
Roberts et al., 2010 Screen 0.19 0.01 Roberts et al., 2010 - Exp 1 Roberts, S. C., Owen, R. C., & Havlicek, J. (2010). Distinguishing between Perceiver and Wearer Effects in Clothing Color-Associated Attributions. Evolutionary Psychology, 8(3), 350-364. http://doi.org/10.1177/147470491000800304 2010.1 Yes Journal Not Pre-Registered No Romantic Males Blue/Green/White Clothing Bust Different between conditions Single item rating of attractiveness 1 1-10 1 10 England Europe Students Undergrads Within Subjects White Noted in manuscript all caucasian White Matched NA NA No Data No Data No No Yes 30 4.60 1.10 30 4.40 0.92 NA 0.92 0.38 Age range includes both male and female participants. 30 1.01
Roberts et al., 2010 Screen -0.08 0.01 Roberts et al., 2010 - Exp 3 Roberts, S. C., Owen, R. C., & Havlicek, J. (2010). Distinguishing between Perceiver and Wearer Effects in Clothing Color-Associated Attributions. Evolutionary Psychology, 8(3), 350-364. http://doi.org/10.1177/147470491000800304 2010.1 Yes Journal Not Pre-Registered No Romantic Males White Clothing Bust Same between conditions Single item rating of attractiveness 1 1-10 1 10 England Europe Students Undergrads Within Subjects White Noted in manuscript all caucasian White Matched NA NA No Data No Data No No Yes 25 4.82 0.90 25 4.88 0.85 0.36 0.92 0.43 Age range includes both male and female participants. 25 0.88
Roberts et al., 2010 Screen -0.05 0.03 Roberts et al., 2010 - Exp 3 Roberts, S. C., Owen, R. C., & Havlicek, J. (2010). Distinguishing between Perceiver and Wearer Effects in Clothing Color-Associated Attributions. Evolutionary Psychology, 8(3), 350-364. http://doi.org/10.1177/147470491000800304 2010.1 Yes Journal Not Pre-Registered No Romantic Females White Clothing Bust Same between conditions Single item rating of attractiveness 1 1-10 1 10 England Europe Students Undergrads Within Subjects White Noted in manuscript all caucasian White Matched NA NA No Data No Data No No Yes 23 4.87 0.75 23 4.90 0.73 0.61 0.66 0.43 Age range includes both male and female participants. 23 0.74
Schwarz & Singer, 2013 Paper 0.19 0.07 Schwarz & Singer, 2013 - Exp 1 - Adults Rate Young Schwarz, S., & Singer, M. (2013). Romantic red revisited: Red enhances men's attraction to young, but not menopausal women. Journal of Experimental Social Psychology, 49(1), 161-164. http://doi.org/10.1016/j.jesp.2012.08.004 2013.0 Yes Journal Not Pre-Registered No Romantic Males White Background Bust Same between conditions Perceived attractiveness 1 1-9 1 9 Germany Europe Adults NA Between Subjects NA NA White NA 53.47 53.47 Lum: 35.2, Chroma: 39.3, Hue no specified No Data No No Yes 30 6.53 1.96 30 6.13 2.17 NA NA 0.64 NA 60 NA
Schwarz & Singer, 2013 Paper -0.15 0.07 Schwarz & Singer, 2013 - Exp 1 - UGrads Rate Young Schwarz, S., & Singer, M. (2013). Romantic red revisited: Red enhances men's attraction to young, but not menopausal women. Journal of Experimental Social Psychology, 49(1), 161-164. http://doi.org/10.1016/j.jesp.2012.08.004 2013.0 Yes Journal Not Pre-Registered No Romantic Males White Background Bust Same between conditions Perceived attractiveness 1 1-9 1 9 Germany Europe Students Undergrads Between Subjects NA NA White NA 24.67 24.67 Lum: 35.2, Chroma: 39.3, Hue no specified No Data No No Yes 30 6.20 1.42 30 6.40 1.18 NA NA 0.68 NA 60 NA
Seibt & Klement, 2015 Screen 0.22 0.05 Seibt & Klement, 2015 - Exp 1 Seibt, T., & Klement, V. (2015). The Impact of the Colour Red on Attractiveness Perception. In 4th Advanced Research in Scientific Areas (pp. 20-24). http://doi.org/10.18638/arsa.2015.4.1.799 2015.0 No Conference Proceedings Not Pre-Registered No Romantic Females Green Background NA Same between conditions Perceived Attractiveness, subscale of Haselton und Gangestad (2006) NA 1-9 1 9 Germany Europe Students Undergrads and Grads Between Subjects NA NA NA NA 26.70 26.70 RGB(255.0.0) RGB(0.139.0) No No No 54 3.19 0.53 31 3.07 0.55 NA NA 0.26 NA 85 NA
Seibt & Klement, 2015 Screen 0.20 0.16 Seibt & Klement, 2015 - Exp 1 Seibt, T., & Klement, V. (2015). The Impact of the Colour Red on Attractiveness Perception. In 4th Advanced Research in Scientific Areas (pp. 20-24). http://doi.org/10.18638/arsa.2015.4.1.799 2015.0 No Conference Proceedings Not Pre-Registered No Romantic Males Green Background NA Same between conditions Perceived Attractiveness, subscale of Haselton und Gangestad (2006) NA 1-9 1 9 Germany Europe Students Undergrads and Grads Between Subjects NA NA NA NA 26.70 26.70 RGB(255.0.0) RGB(0.139.0) No No No 12 3.40 0.43 14 3.30 0.54 NA NA 0.29 NA 26 NA
Seibt, 2015 Screen 0.20 0.05 Seibt, 2015 - Exp 1 Seibt, T. (2015). Romantic Red Effect in the Attractiveness Perception. In Hassacc (pp. 31-34). http://doi.org/10.18638/hassacc.2015.3.1.186 2015.0 No Conference Proceedings Not Pre-Registered No Romantic Males Green Background NA Same between conditions Perceived Attractiveness NA 1-5 1 5 Germany Europe Students Undergrads and Grads Between Subjects NA NA NA NA 25.50 25.50 No Data No Data No No No 46 2.80 0.49 41 2.70 0.52 NA NA 0.42 NA 87 NA
Stefan & Gueguen, 2013 Paper 0.62 0.12 Stefan & Gueguen, 2013 - Exp 1 Stefan J. & Gueguen, N. (2013). Unpublished data. 2013.0 No Unpublished/Online Not Pre-Registered No Romantic Males White Clothing Full Body Same between conditions Single item 1 1-100 1 100 France Europe Students Undergrads and Grads Between Subjects White White White Matched 22.11 22.11 NA NA No No Yes 18 81.33 16.30 16 67.50 26.87 NA NA 0.67 Data provided by Elliot 34 NA
Sullivan et al., 2016 Screen 0.24 0.04 Sullivan et al., 2016 - Exp 1 Sullivan, J., Amaral Lavoie, E., Bays, R. B., Fontana, S., Goodkind, R., Johnson, R., ... Lavoie, M. (2016, April 4). Replication of Elliot et al., 2010: Red, Rank, and Romance. Retrieved from osf.io/pm7fx 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match 19.96 19.45 No Data No Data No No Yes 49 5.60 1.37 53 5.26 1.42 NA NA 0.53 Two-year class project posted on OSF 102 NA
Sullivan et al., 2016 Screen -0.34 0.06 Sullivan et al., 2016 - Exp 2 Sullivan, J., Amaral Lavoie, E., Bays, R. B., Fontana, S., Goodkind, R., Johnson, R., ... Lavoie, M. (2016, April 4). Replication of Elliot et al., 2010: Red, Rank, and Romance. Retrieved from osf.io/pm7fx 2016.0 No Unpublished/Online Not Pre-Registered No Romantic Females Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White Latino Mis-match NA NA No Data No Data No No Yes 38 4.85 1.65 27 5.37 1.38 NA NA 0.55 Age recorded as dichotomy (younger then 20, older than 20) so not included 65 NA
Sullivan et al., 2016 Screen 0.42 0.04 Sullivan et al., 2016 - Exp 1 Sullivan, J., Amaral Lavoie, E., Bays, R. B., Fontana, S., Goodkind, R., Johnson, R., ... Lavoie, M. (2016, April 4). Replication of Elliot et al., 2010: Red, Rank, and Romance. Retrieved from osf.io/pm7fx 2015.0 No Unpublished/Online Not Pre-Registered No Romantic Males Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched 19.62 19.75 No Data No Data No No Yes 45 6.21 1.38 48 5.58 1.56 NA NA 0.57 NA 93 NA
Sullivan et al., 2016 Screen 0.02 0.12 Sullivan et al., 2016 - Exp 2 Sullivan, J., Amaral Lavoie, E., Bays, R. B., Fontana, S., Goodkind, R., Johnson, R., ... Lavoie, M. (2016, April 4). Replication of Elliot et al., 2010: Red, Rank, and Romance. Retrieved from osf.io/pm7fx 2016.0 No Unpublished/Online Not Pre-Registered No Romantic Males Gray Background Bust Same between conditions Perceived attractiveness 3 1-9 1 9 USA North America Students Undergrads Between Subjects White White White Matched NA NA No Data No Data No No Yes 21 6.24 1.29 14 6.21 1.45 NA NA 0.65 Age recorded as dichotomy (younger then 20, older than 20) so not included 35 NA
Wartenberg et al., 2011 Screen 0.20 0.01 Wartenberg et al., 2011 - Exp 1 - In Group Wartenberg, W., Hoepfner, T., Potthast, P., & Mirau, A. (2011). If you wear red on a date, you will please your mate. Proceedings of Empiriepraktikumskongress, 6th, Aug. 7, pp 26-27. University of Jena, Germany. 2011.0 No Conference Proceedings Not Pre-Registered No Romantic Females Blue Background Bust Same between conditions Perceived attractiveness 4 1-7 1 7 Germany Europe Students Undergrads Within Subjects White White White Matched 21.47 21.47 No Data No Data No No Yes 39 3.00 1.18 39 2.76 1.13 0.63 0.86 0.29 Translation of summary provided by Elliot; within subjects info still needed 39 1.16
Wen et al., 2014 Paper 0.16 0.07 Wen et al., 2014 - Exp 1 - Feminine Females Wen, F., Zuo, B., Wu, Y., Sun, S., & Liu, K. (2014). Red is Romantic, but Only for Feminine Females: Sexual Dimorphism Moderates Red Effect on Sexual Attraction. Evolutionary Psychology, 12(4), 719-735. http://doi.org/10.1177/147470491401200404 2014.0 Yes Journal Not Pre-Registered No Romantic Males Blue/White Clothing Bust Same between conditions Perceived attractiveness, extracted factor 3 1-7 1 7 China Asia Students Undergrads Between Subjects Asian Asian Asian Matched 20.95 20.95 LCh(51.1, 57.7, 27.8) LCh(51.6, 57.6, 278.3) Yes Yes Yes 22 0.24 0.93 44 0.10 0.81 NA NA NA Control is average of blue and white conditions; only normalized scores provided. Age range includes participants in all conditions. 66 NA
Wen et al., 2014 Paper -0.08 0.07 Wen et al., 2014 - Exp 1 - Masculine Males Wen, F., Zuo, B., Wu, Y., Sun, S., & Liu, K. (2014). Red is Romantic, but Only for Feminine Females: Sexual Dimorphism Moderates Red Effect on Sexual Attraction. Evolutionary Psychology, 12(4), 719-735. http://doi.org/10.1177/147470491401200404 2014.0 Yes Journal Not Pre-Registered No Romantic Females Blue/White Clothing Bust Same between conditions Perceived attractiveness, extracted factor 3 1-7 1 7 China Asia Students Undergrads Between Subjects Asian Asian Asian Matched 20.95 20.95 LCh(51.1, 57.7, 27.8) LCh(51.6, 57.6, 278.3) Yes Yes Yes 21 -0.10 0.93 49 -0.03 0.85 NA NA NA Control is average of blue and white conditions; only normalized scores provided. Age range includes participants in all conditions. 70 NA
Williams & Neelon, 2013 Screen 0.58 0.13 Williams & Neelon, 2013 - Exp 1 - All Williams, C. L. & Neelon, M. (2013). Conditional beauty: The impact of emotionally linked images on the red effect in sexual attraction. Psi Chi Journal of Psychological Research, 18(1), 10-19. 2013.0 Yes Journal Not Pre-Registered No Romantic Males Blue Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White (noted by authors) White Matched 22.50 22.50 RGB(183,70,60) RGB(76,105,200) No No Yes 16 6.81 1.03 15 6.15 1.20 NA NA 0.64 Data provided by Elliot 31 NA
Williams & Neelon, 2013 Screen -0.13 0.08 Williams & Neelon, 2013 - Exp 2 - Positive and Neutral Williams, C. L. & Neelon, M. (2013). Conditional beauty: The impact of emotionally linked images on the red effect in sexual attraction. Psi Chi Journal of Psychological Research, 18(1), 10-19. 2013.0 Yes Journal Not Pre-Registered No Romantic Males Blue Background Bust Same between conditions Perceived attractiveness 2 1-9 1 9 USA North America Students Undergrads Between Subjects White White (noted by authors) White Matched 21.50 21.50 RGB(183,70,60) RGB(76,105,200) No No Yes 26 6.40 1.28 26 6.56 1.04 NA NA 0.69 Data provided by Elliot 52 NA
Young, 2015 Screen 0.14 0.00 Young, 2015 - Exp 1 - More Attractive Young, S. G. (2015). The effect of red on male perceptions of female attractiveness: Moderation by baseline attractiveness of female faces. European Journal of Social Psychology, 45(2), 146-151. http://doi.org/10.1002/ejsp.2098 2015.0 Yes Journal Not Pre-Registered No Romantic Males Gray Background Head Shot Same between conditions Single item 1 1-7 1 7 USA North America Students Undergrads Within Subjects Mixed Mixed White Mis-match 19.90 19.90 LCh(62.7, 84.6, 34.1) LCh(62.6, -, 265.5) No Yes Yes 19 4.29 0.86 19 4.17 0.81 0.23 0.96 0.53 NA 19 0.84
Young, 2015 Screen 0.10 0.00 Young, 2015 - Exp 2 - More Attractive Young, S. G. (2015). The effect of red on male perceptions of female attractiveness: Moderation by baseline attractiveness of female faces. European Journal of Social Psychology, 45(2), 146-151. http://doi.org/10.1002/ejsp.2098 2015.0 Yes Journal Not Pre-Registered No Romantic Males Blue/Gray Background Head Shot Same between conditions Single item 1 1-7 1 7 USA North America Students Undergrads Within Subjects Mixed Mixed White Mis-match 21.10 21.10 LCh(62.7, 84.6, 34.1) LCh(62.7, 84.6, 34.1) No Yes Yes 46 4.30 1.11 46 4.19 1.07 0.25 0.98 0.53 Blue and Gray ratings averaged, then compared to red; original data provided 46 1.09

Preliminary Analysis

As a little warm-up exercise, here is a basic random effects meta-analysis of these data, fit via the metafor package (Viechtbauer, 2010). We use cluster-robust standard errors to account for effect size dependency.

Code
# Estimate random effects model
RE_mod <- rma.uni(yi, vi = vi, data = lehmann_dat, method = "ML")

# Calculate cluster-robust standard errors
RE_robust <- robust(RE_mod, cluster = study, clubSandwich = TRUE)
RE_robust

Random-Effects Model (k = 81; tau^2 estimator: ML)

tau^2 (estimated amount of total heterogeneity): 0.1009 (SE = 0.0245)
tau (square root of estimated tau^2 value):      0.3176
I^2 (total heterogeneity / total variability):   81.54%
H^2 (total variability / sampling variability):  5.42

Test for Heterogeneity:
Q(df = 80) = 246.9683, p-val < .0001

Number of estimates:   81
Number of clusters:    41
Estimates per cluster: 1-6 (mean: 1.98, median: 1)

Model Results:

estimate      se¹    tval¹     df¹    pval¹   ci.lb¹   ci.ub¹     
  0.2069  0.0569   3.6331   23.41   0.0014   0.0892   0.3245   ** 

---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

1) results based on cluster-robust inference (var-cov estimator: CR2,
   approx t-test and confidence interval, df: Satterthwaite approx)

The random effects model indicates an average effect size of about 0.21 standard deviations and substantial heterogeneity, with \(\hat\tau = 0.32\). The cluster-robust standard error is about 27% bigger than the model-based standard error (not shown) because the latter does not account for dependent effect sizes.

Here is a contour-enhanced funnel plot of the data:

Code
funnel(RE_mod, refline = 0, level=c(90, 95, 99), shade=c("white", "gray55", "gray75"))

The funnel plot shows some asymmetry, suggesting that there is reason to be concerned about selective reporting bias in these data.

A Selection Model

For starters, we will fit a very simple selection model, with a single step in the probability of reporting at \(\alpha = .025\). This is what’s come to be called the three-parameter selection model (McShane et al., 2016). The step is defined in terms of a one-sided p-value, so \(\alpha = .025\) corresponds to the point where an effect size estimate in the theoretically expected direction would have a regular, two-sided p-value of .05, right at the mystical threshold of statistical significance. We fit the model using metafor’s selmodel() function:

Code
RE_sel <- selmodel(RE_mod, type = "stepfun", steps = .025)
RE_sel

Random-Effects Model (k = 81; tau^2 estimator: ML)

tau^2 (estimated amount of total heterogeneity): 0.0811 (SE = 0.0261)
tau (square root of estimated tau^2 value):      0.2848

Test for Heterogeneity:
LRT(df = 1) = 37.3674, p-val < .0001

Model Results:

estimate      se    zval    pval   ci.lb   ci.ub    
  0.1328  0.0655  2.0267  0.0427  0.0044  0.2612  * 

Test for Selection Model Parameters:
LRT(df = 1) = 1.7646, p-val = 0.1840

Selection Model Results:

                     k  estimate      se     zval    pval   ci.lb   ci.ub    
0     < p <= 0.025  25    1.0000     ---      ---     ---     ---     ---    
0.025 < p <= 1      56    0.5485  0.2495  -1.8097  0.0703  0.0594  1.0375  . 

---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The selection parameter represents the probability that an effect size estimate that is not in the theoretically expected direction or not statistically significant at the conventional level would be included in the synthesis, relative to the probability that an affirmative, statistically significant effect size estimate would be included. In this example, the probability of selection is estimated as 0.55, meaning only about 55% of the non-significant results that we would expect were generated are actually reported. Adjusting for this selection bias, the model estimates an overall average effect size of 0.13 SD—smaller than the unadjusted random effects estimate—with heterogeneity of \(\hat\tau = 0.28\).

The problem with this analysis is that the selection model is set up under the assumption that the effect size estimates are all independent. As a result, the reported standard errors are probably smaller than they should be and the confidence intervals are narrower than they should be. We’ll use cluster bootstrapping to get standard errors and confidence intervals that should better account for effect size dependency.

Cluster Bootstrapping a Selection Model

In R, the boot package provides tools for running a variety of different bootstrap techniques and obtaining confidence intervals based on bootstrap distributions (Canty & Ripley, 2021). It’s been around for ages and has some very nice features, but it requires a bit of trickery to use it for cluster bootstrapping. The main challenge is that the package functionality is set up under the assumption that every row of the dataset should be treated as an independent observation. To make it work for cluster bootstrapping, we will need a function to fit the selection model, which takes in a dataset with one row per cluster and returns a vector of parameter estimates. The function also has to have an index argument which is a vector of row indexes used to create the bootstrap sample. Here is a skeleton for such a function:

Code
fit_selmodel <- function(
    dat,   # dataset with one row per cluster
    index, # vector of indexes used to create the bootstrap sample
    ...    # any further arguments
) { 
  
  # take subset of data
  boot_dat <- dat[index,]
  
  # fit selection model
  
  # compile parameter estimates into a vector
  
}

Clustering and unclustering

The Lehmann dataset has one row per effect size, sometimes with multiple rows per study, so we need to modify the data to have one row per study. There are at least two ways to accomplish this. One option is to create a dataset consisting only of study-level IDs, then merge it back on to the full data to get the effect-size level data:

Code
# Make a dataset of study IDs
cluster_IDs <- data.frame(study = unique(lehmann_dat$study))

glimpse(cluster_IDs)
Rows: 41
Columns: 1
$ study <chr> "Banas, 2014 ", "Berthold, 2013 ", "Bigelow et al., 2013 ", "Ble…
Code
# Merge with full data
full_dat <- merge(cluster_IDs, lehmann_dat, by = "study")

full_dat %>% select(study, yi, vi) %>% glimpse()
Rows: 81
Columns: 3
$ study <chr> "Banas, 2014 ", "Berthold, 2013 ", "Bigelow et al., 2013 ", "Big…
$ yi    <dbl> 0.05716727, 0.55411916, 0.31467980, -0.73259462, 0.07921700, -0.…
$ vi    <dbl> 0.10267348, 0.06030579, 0.29520322, 0.53354343, 0.02692608, 0.05…

Another option is to use the dplyr::nest_by() function to nest the data by cluster (Wickham et al., 2019). Then, we can use tidyr::unnest() to recover the effect size level data (Wickham et al., 2019). Like so:

Code
# Nest the data for each study
lehmann_nested <- nest_by(lehmann_dat, study, .key = "data")

glimpse(lehmann_nested)
Rows: 41
Columns: 2
Rowwise: study
$ study <chr> "Banas, 2014 ", "Berthold, 2013 ", "Bigelow et al., 2013 ", "Ble…
$ data  <list<tibble[,49]>> [<tbl_df[1 x 49]>], [<tbl_df[1 x 49]>], [<tbl_df[2…
Code
# Recover the full dataset
full_dat <-
  lehmann_nested %>%
  unnest(data)

full_dat %>% select(study, yi, vi) %>% glimpse()
Rows: 81
Columns: 3
Groups: study [41]
$ study <chr> "Banas, 2014 ", "Berthold, 2013 ", "Bigelow et al., 2013 ", "Big…
$ yi    <dbl> 0.05716727, 0.55411916, 0.31467980, -0.73259462, 0.07921700, -0.…
$ vi    <dbl> 0.10267348, 0.06030579, 0.29520322, 0.53354343, 0.02692608, 0.05…

We will follow the latter strategy for the remainder of our example.

With this nest-and-unnest approach, we can fill in a little bit more of our function skeleton:

Code
fit_selmodel <- function(
    dat,   # dataset with one row per cluster
    index, # vector of indexes used to create the bootstrap sample
    ...    # any further arguments
) { 
  
  # take subset of data
  boot_dat_cluster <- dat[index, ]
  
  # expand to one row per effect size
  boot_dat <- tidyr::unnest(boot_dat_cluster, data)
  
  # fit selection model
  
  # compile parameter estimates into a vector
  
}

Selection model function

Next, we need to complete the function by writing code to fit the selection model. This is a little bit involved because of the way the metafor package implements the Vevea-Hedges selection model. We first need to fit a regular random effects model using metafor::rma.uni(), then pass the result to the metafor::selmodel() function to fit a selection model, and then pull out the parameter estimates as a vector. To make the code clearer, we will move this step out into its own function:

Code
run_sel_model <- function(dat, steps = .025) {
  
  # initial random effects model
  RE_mod <- metafor::rma.uni(
      yi = yi, vi = vi, data = dat, method = "ML"
  )
  
  # fit selection model
  res <- metafor::selmodel(
    RE_mod, type = "stepfun", steps = steps,
    skiphes = TRUE, # turn off SE calculation
    skiphet = TRUE # turn off heterogeneity test
  )
  
  # compile parameter estimates into a vector
  c(beta = res$beta[,1], tau = sqrt(res$tau2), delta = res$delta[-1])
  
}

Note the use of skiphes and skiphet arguments in the selmodel() call, which skip the calculation of standard errors and skip the calculation of the test for heterogeneity. We don’t need the standard errors here because we’re going to use bootstrapping instead, and we’re not interested in the heterogeneity test. Turning off these calculations saves computational time.

A further complication with fitting a selection model is that the parameter estimates are obtained by maximum likelihood, using an iterative optimization algorithm that sometimes fails to converge. To handle non-convergence, we will pass our function through purrr::possibly() so that errors are suppressed, rather than causing everything to grind to a halt (Wickham et al., 2019). We set the otherwise argument so that non-convergent results are returned as NA values:

Code
run_sel_model <- purrr::possibly(run_sel_model, otherwise = rep(NA_real_, 3))

A first bootstrap

Here is the completed fitting function called fit_selmodel():

Code
fit_selmodel <- function(dat, 
                         index = 1:nrow(dat), 
                         steps = 0.025) {
  
  # take subset of data
  boot_dat_cluster <- dat[index, ]
  
  # expand to one row per effect size
  boot_dat <- tidyr::unnest(boot_dat_cluster, data)
  
  # fit selection model, return vector
  run_sel_model(boot_dat, steps = steps)
  
}

First, we take a subset of the data based on the index argument. This generates a bootstrap sample from the original data based on re-sampled clusters. We then use tidyr::unnest() to get the effect size level data for those re-sampled clusters. We then re-fit the model using our run_sel_model() function.

Now let’s apply our function to the Lehmann dataset. We will first need to create a nested version of the dataset, with one row per study:

Code
lehmann_nested <- nest_by(lehmann_dat, study, .key = "data")

Now we can fit the selection model using fit_sel_model():

Code
fit_selmodel(lehmann_nested)
beta.intrcpt          tau        delta 
   0.1327996    0.2848304    0.5484541 

The results reproduce what we saw earlier when we estimated the three parameter selection model.

Now we can bootstrap using the boot::boot() function. The inputs to boot() are the nested dataset, the function to fit the selection model, and then any additional arguments passed to fit_selmodel()—here we include an argument for steps—and finally, the number of bootstrap replications:

Code
# Generate bootstrap
set.seed(20230321)

tic()

boots <- boot(
  data = lehmann_nested,            # nested dataset
  statistic = fit_selmodel,         # function for fitting selection model
  steps = .025,                     # further arguments to the fitting function
  R = 1999                          # number of bootstraps
)

time_seq <- toc()
114.99 sec elapsed

This code takes a while to run, but we can speed it up with parallel processing.

Parallel processing

The boot package has some handy parallel processing features, but they can be a bit finicky to use here because of how R manages environments across multiple processes. With the above code, we can’t simply turn on parallel processing because the worker processes won’t know where to find the run_sel_model() function that gets called inside fit_selmodel(). To fix this, we include the run_sel_model() function inside our fitting function, as follows:

Code
fit_selmodel <- function(dat, 
                         index = 1:nrow(dat), 
                         steps = 0.025) {
  
  # take subset of data
  boot_dat_cluster <- dat[index, ]
  
  # expand to one row per effect size
  boot_dat <- tidyr::unnest(boot_dat_cluster, data)
  
  # build run_selmodel
  run_sel_model <- function(dat, steps = .025) {
  
    # initial random effects model
    RE_mod <- metafor::rma.uni(
      yi = yi, vi = vi, data = dat, method = "ML"
    )
    
    # fit selection model
    res <- metafor::selmodel(
      RE_mod, type = "stepfun", steps = steps,
      skiphes = TRUE, # turn off SE calculation
      skiphet = TRUE # turn off heterogeneity test
    )
    
    # compile parameter estimates into a vector
    c(beta = res$beta[,1], tau = sqrt(res$tau2), delta = res$delta[-1])
    
  }
  
  p <- 2L + length(steps)  # calculate total number of model parameters
  
  # error handling for run_sel_model
  run_sel_model <- purrr::possibly(run_sel_model, otherwise = rep(NA_real_, p))
  
  # fit selection model, return vector of parameter estimates
  run_sel_model(boot_dat, steps = steps)
  
}

Now we can call boot() with options for parallel processing. The machine we used to compile this post has 12 cores. We will use half of the available cores for parallel processing. The configuration of parallel processing will depend on your operating system (different options are available for Mac), so you may need to adapt this code a bit.

Code
ncpus <- parallel::detectCores() / 2

# Generate bootstrap
set.seed(20230321)

tic()

boots <- boot(
  data = lehmann_nested,
  statistic = fit_selmodel, steps = .025,
  R = 1999,
  parallel = "snow", ncpus = ncpus # parallel processing options
)

time_par <- toc()
35.97 sec elapsed

Parallel processing is really helpful here. We get 2000 bootstraps in 36 seconds, 3.2 times faster than sequential processing.

Standard Errors

The standard deviations of the bootstrapped parameter estimates can be interpreted as standard errors for the parameter estimates that take into account the dependence structure of the effect size estimates. Here is a table comparing the cluster-bootstrapped standard errors to the model-based standard errors generated by selmodel():

Code
est <- boots$t0 # original parameter estimates

# calculate bootstrap SEs
boot_SE <- apply(boots$t, 2, sd, na.rm = TRUE)  

# calculate model-based SEs
model_SE <- with(RE_sel, c(se, se.tau2 / (2 * sqrt(tau2)), se.delta[-1]))

# make a table
res <- tibble(
  Parameter = names(est),
  Est = est,
  `SE(bootstrap)` = boot_SE,
  `SE(model)` = model_SE,
  `SE(bootstrap) / SE(model)` = boot_SE / model_SE
)
Parameter Est SE(bootstrap) SE(model) SE(bootstrap) / SE(model)
beta.intrcpt 0.133 0.113 0.066 1.732
tau 0.285 0.148 0.046 3.234
delta 0.548 0.780 0.250 3.127

The standard errors based on cluster bootstrapping are all substantially larger than the model-based standard errors, which don’t account for dependence.

Confidence Intervals

For reporting results from this sort of analysis, it is useful to provide confidence intervals along with the model parameter estimates and standard errors. These can be calculated using the boot::boot_ci() function. This function provides several different types of confidence intervals; for illustration, we will stick with simple percentile confidence intervals, which are calculated by taking percentiles of the bootstrap distribution of each parameter estimate. To use the function, we’ll specify the type of confidence interval and the index of the parameter we want. An index of 1 is for the overall average effect size:

Code
boot.ci(boots, type = "perc", index = 1) # For overall average ES
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 1995 bootstrap replicates

CALL : 
boot.ci(boot.out = boots, type = "perc", index = 1)

Intervals : 
Level     Percentile     
95%   (-0.0038,  0.4171 )  
Calculations and Intervals on Original Scale

Here is the confidence interval for between-study heterogeneity:

Code
boot.ci(boots, type = "perc", index = 2) # For heterogeneity
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 1995 bootstrap replicates

CALL : 
boot.ci(boot.out = boots, type = "perc", index = 2)

Intervals : 
Level     Percentile     
95%   ( 0.001,  0.489 )  
Calculations and Intervals on Original Scale

And for the selection weight:

Code
boot.ci(boots, type = "perc", index = 3) # For selection weight
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 1995 bootstrap replicates

CALL : 
boot.ci(boot.out = boots, type = "perc", index = 3)

Intervals : 
Level     Percentile     
95%   ( 0.0574,  2.7536 )  
Calculations and Intervals on Original Scale

Percentile confidence intervals can be asymmetric, and here the confidence interval for the selection weight parameter is notably asymmetric. The end-points of the confidence interval range from 0.057 (which represents very strong selective reporting, with only 6% of non-significant results reported) to 2.754 (which represents very strong selection against affirmative results).2 So, overall we can’t really draw any strong conclusions about the strength of selective reporting.

Conclusion

In this post, we’ve demonstrated how to code a cluster-level bootstrap for a three parameter version of the Vevea-Hedges selection model. We think this cluster-bootstrapping technique is interesting and promising because it can be applied with a very broad swath of models and methods to investigate selective reporting. For instance, the code we’ve demonstrated could be modified by:

  • using a meta-regression model instead of just a summary meta-analysis;
  • using a different form of selection function such as the beta-weight model proposed by Citkowicz and Vevea (2017) or a more elaborate step function with multiple steps; or
  • using a step function model applied across subsets of effect sizes, as in Coburn & Vevea (2015).

The metafor package implements an expansive set of selection models with the selmodel function, so one could really just swap specifications in and out. In principle, the cluster-level bootstrap could also be used in combination with other forms of selective reporting analysis such as PET-PEESE (although with such regression adjustments, cluster-robust variance estimation is also an option, see Rodgers & Pustejovsky, 2020) or Copas-style sensitivity analyses (Copas & Shi, 2001).

We are currently studying the performance of bootstrapping a three parameter selection model in some big Monte Carlo simulations. Based on some very preliminary results, it looks like the cluster bootstrapped selection model provides confidence intervals with reasonable coverage levels. We have more to do before we share these results, so again we want to emphasize that what we have demonstrated in this post is experimental and exploratory.

Further questions we need to investigate are how things work if we include covariates in the selection model, whether there are better variations of the bootstrap than what we have demonstrated here (e.g., the fractionally weighted bootstrapping, Xu et al., 2020), and the limits of this method in terms of the number of studies needed for adequate performance. If things pan out, we also plan to turn the workflow we’ve demonstrated here into some more user-friendly functions, perhaps as part of the wildmeta package.

Back to top

References

Aert, R. C. M. van, Wicherts, J. M., & Assen, M. A. L. M. van. (2016). Conducting meta-analyses based on p values: Reservations and recommendations for applying p -uniform and p -curve. Perspectives on Psychological Science, 11(5), 713–729. https://doi.org/10.1177/1745691616650874
Assen, M. A. L. M. van, Van Aert, R. C. M., & Wicherts, J. M. (2015). Meta-analysis using effect size distributions of only statistically significant studies. Psychological Methods, 20(3), 293–309. https://doi.org/http://dx.doi.org/10.1037/met0000025
Becker, B. J. (2000). Multivariate Meta-analysis. In S. D. Brown & H. E. A. Tinsley (Eds.), Handbook of applied multivariate statistics and mathematical modeling (pp. 499–525). Academic Press. https://doi.org/10.1016/B978-012691360-6/50018-5
Begg, C. B., & Mazumdar, M. (1994). Operating characteristics of a rank correlation test for publication bias. Biometrics, 50(4), 1088. https://doi.org/10.2307/2533446
Boos, D. D. (2003). Introduction to the bootstrap world. Statistical Science, 18(2), 168–174.
Cameron, A. C., Gelbach, J. B., & Miller, D. L. (2008). Bootstrap-based improvements for inference with clustered errors. The Review of Economics and Statistics, 90(3), 414–427.
Canty, A., & Ripley, B. D. (2021). Boot: Bootstrap r (s-plus) functions.
Citkowicz, M., & Vevea, J. L. (2017). A parsimonious weight function for modeling publication bias. Psychological Methods, 22(1), 28–41. https://doi.org/10.1037/met0000119
Coburn, K. M., & Vevea, J. L. (2015). Publication bias as a function of study characteristics. Psychological Methods, 20(3), 310–330. https://doi.org/10.1037/met0000046
Copas, J., & Shi, J. Q. (2001). A sensitivity analysis for publication bias in systematic reviews. Statistical Methods in Medical Research, 10, 251–265.
Duval, S., & Tweedie, R. (2000). A nonparametric "trim and fill" method of accounting for publication bias in meta-analysis. Journal of the American Statistical Association, 95(449), 89–98. https://doi.org/10.2307/2669529
Egger, M., Smith, G. D., Schneider, M., & Minder, C. (1997). Bias in meta-analysis detected by a simple, graphical test. BMJ, 315(7109), 629–634.
Hedges, L. V. (1992). Modeling publication selection effects in meta-analysis. Statistical Science, 7(2), 246–255.
Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1(1), 39–65. https://doi.org/10.1002/jrsm.5
Hedges, L. V., & Vevea, J. L. (1996). Estimating effect size under publication bias: Small sample properties and robustness of a random effects selection model. Journal of Educational and Behavioral Statistics, 21(4), 299. https://doi.org/10.3102/10769986021004299
Hedges, L. V., & Vevea, J. L. (2005). Selection method approaches. In Publication bias in meta-analysis: Prevention, assessment, and adjustments (pp. 145–174). John Wiley & Sons.
Ioannidis, J. P. A., & Trikalinos, T. A. (2007). An exploratory test for an excess of significant findings. Clinical Trials, 4(3), 245–253. https://doi.org/10.1177/1740774507079441
Lehmann, G. K., Elliot, A. J., & Calin-Jageman, R. J. (2018). Meta-analysis of the effect of red on perceived attractiveness. Evolutionary Psychology, 16(4), 147470491880241. https://doi.org/10.1177/1474704918802412
Mathur, M. B., & VanderWeele, T. J. (2020). Sensitivity analysis for publication bias in meta‐analyses. Journal of the Royal Statistical Society: Series C (Applied Statistics), 69(5), 1091–1119. https://doi.org/10.1111/rssc.12440
McShane, B. B., Böckenholt, U., & Hansen, K. T. (2016). Adjusting for publication bias in meta-analysis an evaluation of selection methods and some cautionary notes. Perspectives on Psychological Science, 11(5), 730–749. http://pps.sagepub.com/content/11/5/730.short
Rodgers, M. A., & Pustejovsky, J. E. (2020). Evaluating meta-analytic methods to detect selective reporting in the presence of dependent effect sizes. Psychological Methods. https://doi.org/10.1037/met0000300
Rothstein, H. R., Sutton, A. J., & Borenstein, M. (2006). Publication bias in meta-analysis: Prevention, assessment and adjustments. John Wiley & Sons.
Simonsohn, U., Nelson, L. D., & Simmons, J. P. (2014). P-curve and effect size: Correcting for publication bias using only significant results. Perspectives on Psychological Science, 9(6), 666–681. https://doi.org/10.1177/1745691614553988
Stanley, T. D. (2008). Meta-regression methods for detecting and estimating empirical effects in the presence of publication selection*. Oxford Bulletin of Economics and Statistics, 70(1), 103–127. https://doi.org/10.1111/j.1468-0084.2007.00487.x
Stanley, T. D., & Doucouliagos, H. (2014). Meta-regression approximations to reduce publication selection bias. Research Synthesis Methods, 5(1), 60–78.
Sterne, J. A. C., & Egger, M. (2001). Funnel plots for detecting bias in meta-analysis: Guidelines on choice of axis. Journal of Clinical Epidemiology, 54(10), 1046–1055.
Sterne, J. A. C., Sutton, A. J., Ioannidis, J. P. A., Terrin, N., Jones, D. R., Lau, J., Carpenter, J., Rücker, G., Harbord, R. M., Schmid, C. H., Tetzlaff, J., Deeks, J. J., Peters, J. L., Macaskill, P., Schwarzer, G., Duval, S., Altman, D. G., Moher, D., & Higgins, J. P. T. (2011). Recommendations for examining and interpreting funnel plot asymmetry in meta-analyses of randomised controlled trials. BMJ, 343, d4002. https://doi.org/10.1136/bmj.d4002
Sutton, A. (2009). Publication bias. In The handbook of research synthesis and meta-analysis (pp. 435–445). Russell Sage Foundation.
Van den Noortgate, W., López-López, J. A., Marín-Martínez, F., & Sánchez-Meca, J. (2013). Three-level meta-analysis of dependent effect sizes. Behavior Research Methods, 45(2), 576–594. https://doi.org/10.3758/s13428-012-0261-6
Vevea, J. L., & Hedges, L. V. (1995). A general linear model for estimating effect size in the presence of publication bias. Psychometrika, 60(3), 419–435. https://doi.org/10.1007/BF02294384
Vevea, J. L., & Woods, C. M. (2005). Publication bias in research synthesis: Sensitivity analysis using a priori weight functions. Psychological Methods, 10(4), 428–443. https://doi.org/10.1037/1082-989X.10.4.428
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48.
White, T., Noble, D., Senior, A., Hamilton, W. K., & Viechtbauer, W. (2022). Metadat: Meta-analysis datasets. https://CRAN.R-project.org/package=metadat
Wickham, H., Averick, M., Bryan, J., Chang, W., McGowan, L. D., François, R., Grolemund, G., Hayes, A., Henry, L., Hester, J., Kuhn, M., Pedersen, T. L., Miller, E., Bache, S. M., Müller, K., Ooms, J., Robinson, D., Seidel, D. P., Spinu, V., … Yutani, H. (2019). Welcome to the tidyverse. Journal of Open Source Software, 4(43), 1686. https://doi.org/10.21105/joss.01686
Xu, L., Gotwalt, C., Hong, Y., King, C. B., & Meeker, W. Q. (2020). Applications of the fractional-random-weight bootstrap. The American Statistician, 74(4), 345–358. https://doi.org/10.1080/00031305.2020.1731599

Footnotes

  1. Technically, these parameters \(\lambda_1,\lambda_2,\lambda_3\) are not absolute probabilities but instead relative risks of being reported, compared to a reference range of \(p\)-values. In the above example, they would be defined relative to the probability of being reported for an effect size estimate with \(p \leq .01\).↩︎

  2. Compare this to the model-based confidence interval of \([0.059, 1.037]\).↩︎