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Standard Errors and Sample Sizes for Two-Level Research
Open accessAbstract
The hierarchical linear model approach to a two-level design is considered, some variables at the lower level having fixed and others having random regression coefficients. An approximation is derived to the covariance matrix of the estimators of the fixed regression coefficients (for variables at the lower and the higher level) under the assumption that the sample sizes at either level are large enough. This covariance matrix is expressed as a function of parameters occurring in the model. If a research planner can make a reasonable guess as to these parameters, this approximation can be used as a guide to the choice of sample sizes at either level.
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“Post hoc power analyses using the multilevel power analysis program PinT (Snijders & Bosker, 1993) confirmed that we had sufficient power (>.80) to detect a mediated pathway comprising paths that were of medium effect sizes and to detect a medium effect size for treatment condition as a moderator of the mediated pathways.”
From Respiratory and cognitive mediators of treatment change in panic disorder: Evidence for intervention specificity. (2010) · cites this paper as Snijders & Bosker, 1993
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In the citing paper
“Tyrer et al reported that brief CBT is no more effective than usual care when it comes to preventing repetition of self-harm, 15”
From Cognitive-behavioural intervention for self-harm: Randomised controlled trial (2008) · cites this paper as [15]
Experimental — accuracy not yet measuredThe check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“Furthermore, this study's findings that time-limited CBT decreases self-harm contrasts with the results reported by Tyrer et al . 15”
From Cognitive-behavioural intervention for self-harm: Randomised controlled trial (2008) · cites this paper as [15]
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Liu, X. (2003) · Journal of Educational and Behavioral Statistics · doi:10.3102/10769986028003231 · Cited by 85
“Snijders and Bosker (1993) derived approximate variance for testing parameters in a general two-level model with any number of covariates.”
BackgroundMinimum Sample Size Requirements for Mokken Scale Analysis
Straat, J. H., van der Ark, L. A., & Sijtsma, K. (2014) · Educational and Psychological Measurement · doi:10.1177/0013164414529793 · Cited by 87
“…Widaman, Preacher, & Hong, 2002; Mundfrom, Shaw, & Ke, 2005; Velicer & Fava, 1998), multilevel analysis (e.g., Cohen, 2005; Hedeker et al., 1999; Snijders & Bosker, 1993), structural equation modeling (e.g., Bentler & Yuan, 1999; Jackson, 2003; Muthén & Muthén, 2002), and item response theory…”
Family policy, family structure, and children’s educational achievement
Hampden‐Thompson, G. (2013) · Social Science Research · doi:10.1016/j.ssresearch.2013.01.005 · Cited by 78
Itzek‐Greulich, H., & Vollmer, C. (2016) · Journal of Research in Science Teaching · doi:10.1002/tea.21334 · Cited by 94
Mehrebenenanalysen in der psychologischen Forschung
Nezlek, J. B., Schröder-Abé, M., & Schütz, A. (2006) · Psychologische Rundschau · doi:10.1026/0033-3042.57.4.213 · Cited by 77
Browne, W. J., Lahi, M. G., & Parker, R. M. (2009) · Bristol Research (University of Bristol) · Cited by 76
Aarts, E., Dolan, C. V., Verhage, M., & van der Sluis, S. (2015) · BMC Neuroscience · doi:10.1186/s12868-015-0228-5 · Cited by 75
“The program PinT [22] can be used to evaluate how much a particular covariate increases power given the amount of explained variation in the experimental effect and the allocation of sample sizes.”
Oberle, E., Guhn, M., Gadermann, A., Thomson, K., & Schonert-Reichl, K. A. (2018) · Social Science & Medicine · doi:10.1016/j.socscimed.2018.06.041 · Cited by 96
Kim, H. J., Li, G., Gjertson, D. W., Elashoff, R. M., Shah, S. K., Ochs, R. A., Vasunilashorn, F., Abtin, F. G., Brown, M. S., & Goldin, J. G. (2008) · Academic Radiology · doi:10.1016/j.acra.2008.03.011 · Cited by 72
Fairclough, S. J., McGrane, B., Sanders, G., Taylor, S. L., Owen, M. B., & Curry, W. B. (2016) · BMC Public Health · doi:10.1186/s12889-016-3550-7 · Cited by 71
“12 statistical power analysis software [53].”
Does fear reactivity during exposure predict panic symptom reduction?
Meuret, A. E., Seidel, A., Rosenfield, B., Hofmann, S. G., & Rosenfield, D. (2012) · Journal of Consulting and Clinical Psychology · doi:10.1037/a0028032 · Cited by 67
“Power analyses using the multilevel power analysis program PinT (Snijders & Bosker, 1993) confirmed that we had sufficient power (greater than .80) to detect a medium effect J Consult Clin Psychol.”
MethodStatistical power analysis for growth curve models using SAS
Zhang, Z. J., & Wang, L. (2009) · Behavior Research Methods · doi:10.3758/brm.41.4.1083 · Cited by 69
Kœrner, S. S., & Kenyon, D. B. (2006) · Family Relations · doi:10.1111/j.1741-3729.2007.00435.x · Cited by 67
Xie, M., & Baumer, E. P. (2019) · Criminology · doi:10.1111/1745-9125.12204 · Cited by 66
Concentrated Disadvantage and Beliefs about the Causes of Poverty: A Multi-Level Analysis
Merolla, D. M., Hunt, M. O., & Serpe, R. T. (2011) · Sociological Perspectives · doi:10.1525/sop.2011.54.2.205 · Cited by 61
Reducing care-resistant behaviors during oral hygiene in persons with dementia
Jablonski, R. A., Kolanowski, A. M., Therrien, B. A., Mahoney, E. K., Kassab, C. D., & LESLIE, D. L. (2011) · BMC Oral Health · doi:10.1186/1472-6831-11-30 · Cited by 62
“The power analyses were conducted using the software PINT [46], which is based on the work of Snijders and Bosker [47,48].”
Intraclass Correlations for Planning Group Randomized Experiments in Rural Education
Hedges, L. V., & Hedberg, E. C. (2007) · Journal of Research in Rural Education · Cited by 60
Daily mistrust: A resource perspective and its implications for work and home
Lanaj, K., Kim, P. H., Koopman, J., & Matta, F. K. (2018) · Personnel Psychology · doi:10.1111/peps.12268 · Cited by 67
Drinking to cope: a comparison of questionnaire and electronic diary reports.
Todd, M. W., Armeli, S., Tennen, H. A., Carney, M. A., Ball, S. A., Kranzler, H. R., & Affleck, G. G. (2005) · Journal of Studies on Alcohol · doi:10.15288/jsa.2005.66.121 · Cited by 60
de Jong, K., Moerbeek, M., & van der Leeden, R. (2009) · Psychotherapy Research · doi:10.1080/10503300903376320 · Cited by 59
“Although some literature (e.g., Raudenbusch, 1997; Snijders & Bosker, 1993) and software solutions (Bosker, Snijders, & Guldemond, 1996) are available to estimate the power for two-level models, there is little information about power analysis for three-level longitudinal models.”
BackgroundObesogenic environmental influences on young adults: Evidence from college dormitory assignments
Kapinos, K. A., Yakusheva, O., & Eisenberg, D. (2013) · Economics & Human Biology · doi:10.1016/j.ehb.2013.05.003 · Cited by 61
Social Identification in Multiteam Systems: The Role of Depletion and Task Complexity
Porck, J. P., Matta, F. K., Hollenbeck, J. R., Oh, J. K., Lanaj, K., & Lee, S. M. (2018) · Academy of Management Journal · doi:10.5465/amj.2017.0466 · Cited by 63
“To address this concern, we estimated the power of our tests in the primary model, using the program PinT (Power in Two-Level Designs) (based on calculations derived by Snijders & Bosker, 1993).”
MethodSample size issues in multilevel logistic regression models
Ali, A., Ali, S., Khan, S. A., Khan, D. M., Abbas, K., Khalil, A., Manzoor, S., & Khalil, U. (2019) · PLoS ONE · doi:10.1371/journal.pone.0225427 · Cited by 59
Twisk, J. W. R. (2019) · Cambridge University Press eBooks · doi:10.1017/9781108635660 · Cited by 59
Designing multi-level studies:
Stoker, L., & Bowers, J. (2002) · Electoral Studies · doi:10.1016/s0261-3794(01)00021-x · Cited by 59
Wang, P., Yang, L., Liu, C., Wei, X., Yang, X., Zhou, Y., Jiang, H., Lei, Z., Reinhardt, J. D., & He, C. (2015) · Clinical Rehabilitation · doi:10.1177/0269215515607970 · Cited by 58
Jouriles, E. N., McDonald, R., Rosenfield, D., Norwood, W. D., Spiller, L. C., Stephens, N. S., Corbitt‐Shindler, D., & Ehrensaft, M. K. (2010) · Journal of Family Psychology · doi:10.1037/a0019281 · Cited by 66
“A post hoc analysis (using Power in Two-Level Designs; PinT; Snijders & Bosker, 1993), indicated that the power to detect a medium effect was .80.”
MethodTreatment Of Missing Data At The Second Level Of Hierarchical Linear Models
Gibson, N. M., & Olejnik, S. F. (2003) · Educational and Psychological Measurement · doi:10.1177/0013164402250987 · Cited by 58
“Snijders and Bosker (1993) argued that for typical school effectiveness studies, optimal sample size for the number of schools is between 30 and 62, with between 11 and 28 students sampled in each school.”
BackgroundGaskin, C. J., & Happell, B. (2013) · International Journal of Nursing Studies · doi:10.1016/j.ijnurstu.2013.09.014 · Cited by 57
Brief acceptance and commitment therapy and exposure for panic disorder: A pilot study
Meuret, A. E., Twohig, M. P., Rosenfield, D., Hayes, S. C., & Craske, M. G. (2012) · Cognitive and Behavioral Practice · doi:10.1016/j.cbpra.2012.05.004 · Cited by 56
Zandberg, L. J., Rosenfield, D., McLean, C. P., Powers, M. B., Asnaani, A., & Foa, E. B. (2015) · Journal of Consulting and Clinical Psychology · doi:10.1037/ccp0000052 · Cited by 56
A Daily Diary Validity Test of Drinking to Cope Measures.
Todd, M. W., Armeli, S., Tennen, H. A., Carney, M. A., & Affleck, G. G. (2003) · Psychology of Addictive Behaviors · doi:10.1037/0893-164x.17.4.303 · Cited by 54
The Design of Cluster Randomized Crossover Trials
Rietbergen, C., & Moerbeek, M. (2011) · Journal of Educational and Behavioral Statistics · doi:10.3102/1076998610379136 · Cited by 53
“Equations for calculating optimal allocation of units given budget constraints for other designs can be found in for example Snijders and Bosker (1993), Moerbeek et al. (2000), and Liu (2003).”
BackgroundRodrigues, J., Müller, M., Mühlberger, A., & Hewig, J. S. (2017) · Psychophysiology · doi:10.1111/psyp.12908 · Cited by 61
Symptom Profile of Multiple Chemical Sensitivity in Actual Life
Saito, M., Kumano, H., Yoshiuchi, K., Kokubo, N., Ohashi, K., Yamamoto, Y., Shinohara, N., Yanagisawa, Y., Sakabe, K., 宮田, 幹., et al. (2005) · Psychosomatic Medicine · doi:10.1097/01.psy.0000155676.69030.28 · Cited by 53
“A power analysis for linear mixed models ( 18 ) was performed to ensure that nonsignificant results in the latter hypothesis testing were not the result of low statistical power.”
MethodSchmitt, A., Zacher, H., & de Lange, A. H. (2012) · European Journal of Work and Organizational Psychology · doi:10.1080/1359432x.2012.698055 · Cited by 57
“Also, the number of participants on the between-person level is well within the range of other recent diary studies (eg, Beal & Ghandour, 2011 ; Bissing-Olson, Iyer, Fielding, & Zacher, in press; Bledow, Schmitt, Frese, & Kuhnel, 2011 ) and this overall sample size has a stronger influence on the power than the number…”
BackgroundResultRural Neighborhood Context, Child Care Quality, and Relationship to Early Language Development
Marco, A. D., & Vernon‐Feagans, L. (2013) · Early Education and Development · doi:10.1080/10409289.2013.736036 · Cited by 51
“To utilize multi-level modeling it is suggested that neighborhood units contain at least 10 subjects (Snijders & Bosker, 1993).”
MethodCurci, A., Lanciano, T., & Soleti, E. (2014) · The American Journal of Psychology · doi:10.5406/amerjpsyc.127.4.0431 · Cited by 54
Modeling Levels and Time in Entrepreneurship Research
Holcomb, T. R., Combs, J. G., Sirmon, D. G., & Sexton, J. C. (2009) · Organizational Research Methods · doi:10.1177/1094428109338401 · Cited by 50
“Although some scholars suggest a group size of at least 5 with 20 or more groups for adequate power to detect a moderate effect size (Kreft & de Leeuw, 1998), others suggest that group sizes of at least 10 with at least 30 groups are needed (LaHuis & Ferguson, 2009; Snijders & Bosker, 1993).”
Zhang, Z., Zhang, L., Xiao, H., & Zheng, J. (2021) · Personality and Individual Differences · doi:10.1016/j.paid.2021.110774 · Cited by 49
“Following the research of Huang and Ryan (2011), the power analysis in two-level (PINT) program was used for the research design (Snijders & Bosker, 1993).”
MethodAttention bias dynamics and symptom severity during and following CBT for social anxiety disorder.
Davis, M. L., Rosenfield, D., Bernstein, A., Zvielli, A., Reinecke, A., Beevers, C. G., Koster, E. H. W., & Smits, J. A. J. (2016) · Journal of Consulting and Clinical Psychology · doi:10.1037/ccp0000125 · Cited by 47
“Power analyses (PinT 2.12; Snijders & Bosker, 1993) indicated that we had sufficient power (> .80) to detect a medium effect size (d=.50) for our least powerful test (those involving treatment condition differences or the mean levels of TVPs).”
MethodInfluentialChallenging Conventional Wisdom for Multivariate Statistical Models With Small Samples
McNeish, D. (2017) · Review of Educational Research · doi:10.3102/0034654317727727 · Cited by 54
“…study to meet these requirements (the guidelines call for 25 observations per higher level unit), issues related to estimation and power with small sample sizes in multilevel models are almost exclusively tied to the higher level sample size (Scherbaum & Ferreter, 2009; Snijders & Bosker, 1993).”
BackgroundDeKay, M. L., Rubinchik, N., Li, Z., & de Boeck, P. A. L. (2022) · Perspectives on Psychological Science · doi:10.1177/17456916221079611 · Cited by 51
“, 2014) replication of Tversky and Kahneman’s (1981) risky-choice framing effect involved more than 6,000 participants, split between only two conditions. Shimizu and Udagawa’s (2011) two framing-effect studies were also large, with nearly 1,400 and 1,900 participants, respectively.”
BackgroundInfluentialYoon, J., & Kayes, D. C. (2016) · Journal of Organizational Behavior · doi:10.1002/job.2092 · Cited by 52
Dang, Q., Mazumdar, S., & Houck, P. R. (2008) · Computer Methods and Programs in Biomedicine · doi:10.1016/j.cmpb.2008.03.001 · Cited by 44
Reflecting on one's best possible self as a leader: Implications for professional employees at work
Jennings, R. E., Lanaj, K., Koopman, J., & McNamara, G. M. (2021) · Personnel Psychology · doi:10.1111/peps.12447 · Cited by 48
Hamilton, Z. K., Sullivan, C. J., Veysey, B. M., & Grillo, M. (2006) · Behavioral Sciences & the Law · doi:10.1002/bsl.720 · Cited by 44
“Although explicit rules of thumb with respect to power in HLM are not clear, reasonable estimates would suggest a need for more second-level observations (i.e. sites) (Kreft & DeLeeuw, 1998; Snijders & Bosker, 1993).”
BackgroundOPTIMAL EXPERIMENTAL DESIGNS FOR MULTILEVEL MODELS WITH COVARIATES
Moerbeek, M., van Breukelen, G., & Berger, M. P. F. (2001) · Communication in Statistics- Theory and Methods · doi:10.1081/sta-100108453 · Cited by 44
“Formulas for the optimal sample sizes in cases where all predictor variables are at the group level are given in [15], in which it is also shown how numerical results can be obtained for models including predictor variables at the individual level.”
BackgroundBoyce, J., & Bowers, A. J. (2018) · International Journal of Leadership in Education · doi:10.7916/d85t4xz3 · Cited by 43
Applications of multilevel modeling in sport injury rehabilitation research
Cornelius, A. E., Brewer, B. W., & Raalte, J. L. V. (2007) · International Journal of Sport and Exercise Psychology · doi:10.1080/1612197x.2007.9671843 · Cited by 42
Community-level prejudice and mortality among immigrant groups
Morey, B. N., Gee, G. C., Muennig, P. A., & Hatzenbuehler, M. L. (2017) · Social Science & Medicine · doi:10.1016/j.socscimed.2017.04.020 · Cited by 42
“To avoid spurious effects resulting from small samples and influential values, PSUs with fewer than 10 respondents providing antiimmigrant prejudice scores were excluded from analyses (Muth en and Asparouhov, 2011; Snijders and Bosker, 1993).”
BackgroundInternal consistency reliability in measurement: Aggregate and multilevel approaches
Sideridis, G., Saddaawi, A., & Al-Harbi, K. (2018) · Journal of Modern Applied Statistical Methods · doi:10.22237/jmasm/1530027194 · Cited by 44
Power and measures of effect size in analysis of variance with fixed versus random nested factors.
Siemer, M., & Joormann, J. (2003) · Psychological methods · doi:10.1037/1082-989x.8.4.497 · Cited by 43
Jouriles, E. N., Rosenfield, D., McDonald, R., Vu, N. L., Rancher, C. E., & Mueller, V. (2016) · Journal of Clinical Child & Adolescent Psychology · doi:10.1080/15374416.2016.1163706 · Cited by 42
“Power analyses using the MLM power analysis program PinT 2.12 (Power in Two-Level Models; Snijders & Bosker, 1993) indicated that power exceeded .90 for detecting a medium effect (d = .50).”
MethodInfluentialSocietal Level of Religiosity and Religious Identity Expression in Europe
van der Noll, J., Rohmann, A., & Saroglou, V. (2018) · Journal of Cross-Cultural Psychology · doi:10.1177/0022022117737302 · Cited by 42
“Power analyses with PinT version 2.12 (Snijders & Bosker, 1993; Snijders, Bosker, & Guldemond, 2007) suggested that the analyses had sufficient power (1 – β = .80) to detect a small direct effect of a society-level variable at α = .95.”
MethodInfluentialPowerful and Cost-Efficient Designs for Longitudinal Intervention Studies With Two Treatment Groups
Moerbeek, M. (2007) · Journal of Educational and Behavioral Statistics · doi:10.3102/1076998607302630 · Cited by 41
Harbourne, R. T., Dusing, S. C., Lobo, M. A., McCoy, S. W., Bovaird, J. A., Sheridan, S. M., Galloway, J. C., Ch'ang, H.-J., Hsu, L., Koziol, N. A., et al. (2018) · Physical Therapy · doi:10.1093/ptj/pzy033 · Cited by 40
Daily Spousal Responsiveness Predicts Longer-Term Trajectories of Patients’ Physical Function
Wilson, S. J., Martire, L. M., & Sliwinski, M. J. (2017) · Psychological Science · doi:10.1177/0956797617697444 · Cited by 38
“We used PINT Version 2.11 (Snijders & Bosker, 1993) for the parent study’s multilevel power calculations (described in greater detail in the Supplemental Material available online).”
MethodInfluentialThe Power of the Test for Treatment Effects in Three-Level Block Randomized Designs
Konstantopoulos, S. (2008) · Journal of Research on Educational Effectiveness · doi:10.1080/19345740802328216 · Cited by 42
“…for computing power in two-level designs has been well documented and statistical software for two-level balanced designs is currently available (e.g., Hedges & Hedberg, 2007; Raudenbush, 1997; Raudenbush & Liu, 2000, 2001; Raudenbush, Spybrook, Liu, & Congdon, 2006; Snijders & Bosker, 1993, 1999).”
MethodVascak, M., Sun, J., Baer, M. L., Jacobs, K. M., & Povlishock, J. T. (2017) · Frontiers in Cellular Neuroscience · doi:10.3389/fncel.2017.00157 · Cited by 39
“Thus, failure to account for such dependent data can inflate the Type I error rate (α) above the standard nominal value of 0.05 (Snijders and Bosker, 1993).”
BackgroundInfluentialLofgreen, A. M., Mattson, R. E., Wagner, S. A., Ortiz, E. G., & Johnson, M. D. (2017) · Journal of Interpersonal Violence · doi:10.1177/0886260517738777 · Cited by 38
“This is because the literature on calculating statistical power in multilevel models is still developing, and the existing software (e.g., PINT, Snijders & Bosker, 1993) to guide sample size estimation does not readily accommodate more complex designs like the one presently used (Scherbaum &…”
MethodPersonality Consistency and Situational Influences on Behavior
Green, J. P., Dalal, R. S., Swigart, K. L., Bleiberg, M. A., Wallace, D. M., & Hargrove, A. K. (2018) · Journal of Management · doi:10.1177/0149206318781950 · Cited by 40
“Our reasoning for this, based on prior research (Snijders & Bosker, 1993), was that 10 data points would permit a somewhat stable estimate of within-person variability (specifically, the standard deviation) needed for the operationalization of consistency across situations.”
BackgroundMethodHorn, M. L. V., Karlin, E., Ramey, S. L., Aldridge, J. T., & Snyder, S. W. (2005) · The Elementary School Journal · doi:10.1086/429946 · Cited by 38
“Not only are multilevel analyses typically called for, but use of this data structure also results in reduced power, requiring greater sample sizes to detect the same size effect as when more traditional analytic methods are used (Murray, 1998; Snijders & Bosker, 1993, 1999).”
MethodTime to the doctorate: Multilevel discrete-time hazard analysis
Wao, H. (2010) · Educational Assessment Evaluation and Accountability · doi:10.1007/s11092-010-9099-6 · Cited by 42
Optimal Sample Sizes in Experimental Designs With Individuals Nested Within Clusters
Moerbeek, M., van Breukelen, G., Berger, M. P. F., & Ausems, M. (2003) · Understanding Statistics · doi:10.1207/s15328031us0203_01 · Cited by 37
“…also possible and sometimes more realistic to calculate optimal sample sizes at both levels (clusters and individuals) given a certain budget for sampling and measuring individuals and clusters, as has been done for continuous outcome variables by Raudenbush (1997) and Snijders and Bosker (1993).”
Estimation Methods for Mixed Logistic Models with Few Clusters
McNeish, D. (2016) · Multivariate Behavioral Research · doi:10.1080/00273171.2016.1236237 · Cited by 37
“As discussed in the previous studies, once cluster sizes are about 30, increasing cluster size has almost no discernible effect on properties of the estimates (Scherbaum & Ferreter, 2009; Snijders & Bosker, 1993; Stegmueller, 2013).”
BackgroundDetecting Consensus Emergence in Organizational Multilevel Data: Power Simulations
Lang, J. W. B., Bliese, P. D., & Runge, J. M. (2019) · Organizational Research Methods · doi:10.1177/1094428119873950 · Cited by 37
Power Analysis in Two-Level Unbalanced Designs
Konstantopoulos, S. (2010) · The Journal of Experimental Education · doi:10.1080/00220970903292876 · Cited by 36
“Power analysis in unbalanced designs has been discussed previously, and general methods for computing standard errors of regression estimates expressed in matrix notation have been provided (e.g., Longford, 1993; Snijders & Bosker, 1993, 1999).”
MethodBuffering and direct effect of posttraumatic growth in predicting distress following cancer.
Wang, A. W.-T., Chang, C., Chen, S.-T., Chen, D., Fan, F., Carver, C. S., & Hsu, W. (2017) · Health Psychology · doi:10.1037/hea0000490 · Cited by 36
Goedert, K. M., Boston, R. C., & Barrett, A. M. (2013) · Frontiers in Human Neuroscience · doi:10.3389/fnhum.2013.00211 · Cited by 35
“Rather,the F distribution”
Raising positive expectations helps patients with minor ailments: A cross-sectional study
Fassaert, T. J. L., van Dulmen, S., Schellevis, F., van der Jagt, L., & Bensing, J. M. (2008) · BMC Family Practice · doi:10.1186/1471-2296-9-38 · Cited by 34
“90, but fewer observations (either groups or individuals) show a rapid decline of power for the detection of cross-level interactions [49].”
BackgroundWang, S., Hu, B. Y., LoCasale‐Crouch, J., & Li, J. (2021) · Journal of Applied Developmental Psychology · doi:10.1016/j.appdev.2021.101331 · Cited by 37
Raudenbush, S. W. (2007) · Handbook of Multilevel Analysis · doi:10.1007/978-0-387-73186-5_5 · Cited by 51
“Chapter 4 in this volume; also [7, 26, 30, 38]).”
Optimal Experimental Designs for Multilevel Logistic Models with Two Binary Predictors
Moerbeek, M., & Maas, C. J. M. (2005) · Communication in Statistics- Theory and Methods · doi:10.1081/sta-200056839 · Cited by 33
“Optimal sample sizes for linear multilevel models with covariates are also available (Moerbeek et al., 2001b; Raudenbush, 1997; Raudenbush and Liu, 2000; Snijders and Bosker, 1993).”
MethodVacher, P., Nicolas, M., Martinent, G., & Mourot, L. (2017) · Frontiers in Psychology · doi:10.3389/fpsyg.2017.01043 · Cited by 40
“…was performed using Power Frontiers in Psychology | www.frontiersin.org 3 June 2017 | Volume 8 | Article 1043 IN Two-level designs software which is designed to estimate standard errors of regression coefficients in hierarchical linear models for power calculations (Snijders and Bosker, 1993).”
MethodIlieș, R., Aw, S. S. Y., & Lim, V. K. G. (2016) · Applied Psychology · doi:10.1111/apps.12069 · Cited by 38
Reducing Approach Bias to Achieve Smoking Cessation: A Pilot Randomized Placebo-Controlled Trial
Baird, S. O., Rinck, M., Rosenfield, D., Davis, M. L., Fisher, J. R., Becker, E. S., Powers, M. B., & Smits, J. A. J. (2017) · Cognitive Therapy and Research · doi:10.1007/s10608-017-9835-z · Cited by 33
“Post-hoc power analyses for the MLM models, performed using the MLM power analysis program PinT 2.12 (Snijders and Bosker 1993), indicated that we had greater than 0.90 power to detect a medium effect size ( d = 0.50) for hypotheses 1 and 2.”
InfluentialKueny, C. R., & Shoss, M. K. (2020) · Journal of Business and Psychology · doi:10.1007/s10869-020-09686-4 · Cited by 38
Cho, S., Lee, M., & Lee, S. M. (2022) · Psychological Reports · doi:10.1177/00332941211054776 · Cited by 40
Olatunji, B. O., Rosenfield, D., Monzani, B. A., Krebs, G., Heyman, I., Turner, C. M., Isomura, K., & Mataix‐Cols, D. (2015) · Depression and Anxiety · doi:10.1002/da.22423 · Cited by 35
“12[52] indicated that we had greater than .”
BackgroundMultilevel Modeling and its Application in Counseling Psychology Research
Reise, S. P., & Duan, N. (1999) · The Counseling Psychologist · doi:10.1177/0011000099274003 · Cited by 31
“However, Snijders and Bosker (1993) provide equations for computing the optimal number of groups and participants within groups in order to achieve desired standard errors for the participant-level and group-level regression coefficients in MLM. Barcikowski (1981) provides numerous tables that…”
BackgroundMeuret, A. E., Chmielewski, M. S., Steele, A. M., Rosenfield, D., Petersen, S., Smits, J. A. J., Simon, N. M., Otto, M. W., Marques, L. M., Pollack, M. H., et al. (2016) · Behaviour Research and Therapy · doi:10.1016/j.brat.2016.03.008 · Cited by 36
“Thus, although we had quite a few predictors in the MLM models, power analyses, conducted using the MLM power analysis program PinT 2.12 (Snijders & Bosker,1993), indicated greater than 0.95 power to detect a medium effect size for the social identification predictors.”
InfluentialNicolas, M., Martinent, G., Millet, G. Y., Bagneux, V., & Gaudino, M. (2019) · Journal of Sports Sciences · doi:10.1080/02640414.2019.1597827 · Cited by 36
“A priori power analysis was performed using Power IN Twolevel designs software which is designed to estimate standard errors of regression coefficients in hierarchical linear models for power calculations (Snijders & Bosker, 1993).”
MethodWilliams, D. R., Martin, S. R., & Rast, P. (2021) · Behavior Research Methods · doi:10.3758/s13428-021-01646-x · Cited by 31
“This can be accomplished either within an ANOVA framework (Shieh, 2012), or relatedly, within an unconditional hierarchical mixed-effects model with only random intercepts (i.e., “multilevel” models; Snijders & Bosker, 1993).”
MethodEmployee perception of corporate social responsibility authenticity: A multilevel approach
Kim, H. O., & Lee, M. (2022) · Frontiers in Psychology · doi:10.3389/fpsyg.2022.948363 · Cited by 47
Contextual Predictors of Cumulative Biological Risk: Segregation and Allostatic Load
Bellatorre, A., Finch, B. K., Phuong, D., Bird, C. E., & Beck, A. N. (2011) · Social Science Quarterly · doi:10.1111/j.1540-6237.2011.00821.x · Cited by 31
Kingston, S. (2013) · American Journal of Community Psychology · doi:10.1007/s10464-013-9601-7 · Cited by 30
Emotion regulation as a determinant of recovery experiences and well-being: A day-level study.
Schraub, E. M., Turgut, S., Clavairoly, V. C. S., & Sonntag, K. (2013) · International Journal of Stress Management · doi:10.1037/a0034483 · Cited by 32
Jacquart, J., Roquet, R. F., Papini, S., Powers, M. B., Rosenfield, D., Smits, J. A. J., & Monfils, M. (2017) · Journal of Anxiety Disorders · doi:10.1016/j.janxdis.2017.05.010 · Cited by 35
Affective influences on within-person changes in work performance as mediated by attentional focus
Merlo, K. L., Shaughnessy, S. P., & Weiss, H. M. (2017) · European Journal of Work and Organizational Psychology · doi:10.1080/1359432x.2017.1417258 · Cited by 29
“Based on a power analysis assuming normal distribution of the dependent variables, a moderate effect size (d = .50), and standard significance and power levels (α = .05, β = 1 – .80), we set a sample size target of 70 participants and 35 within-person observations (Snijders & Bosker, 1993).”
MethodGaffney, H., Mansell, W., & Tai, S. (2020) · Digital Health · doi:10.1177/2055207620911580 · Cited by 26
“from previous studies).(64) We did not aim to demonstrate efficacy, as a much larger sample, a longer period with the intervention, and a control condition would be required to investigate this.”
ResultOptimal Designs for Multilevel Studies
Moerbeek, M., Breukelen, G. J. P. V., & Berger, M. P. F. (2007) · Handbook of Multilevel Analysis · doi:10.1007/978-0-387-73186-5_4 · Cited by 28
Power analysis software for educational researchers
Peng, C.-Y. J., Long, H., & Abaci, S. (2012) · The Journal of Experimental Education · doi:10.1080/00220973.2011.647115 · Cited by 59
“The free software PinT (stands for Power in Two-level designs) considers budget constraints while estimating sample sizes for two-level HLM models (Bosker, Snijders, & Guldemond, 1996; Snijders & Bosker, 1993).”
MethodHow the Religious Cleavages of Civil Society Shape National Identity
Schnabel, A., & Hjerm, M. (2014) · SAGE Open · doi:10.1177/2158244014525417 · Cited by 27
“If a single-level model is performed, the assumption of independency of the error terms is violated because observations within a group can be expected to be more similar than between groups, and standard errors are underestimated (Hox, 2002; Snijders & Bosker, 1993).”
BackgroundRapa, L. J., Bolding, C. W., & Jamil, F. M. (2020) · Applied Developmental Science · doi:10.1080/10888691.2020.1861946 · Cited by 27
“Others have demonstrated that 10 or fewer level 2 units is certainly too low (Maas & Hox, 2005; Snijders & Bosker, 1993), with less than 20 being suboptimal (Snijders & 1Tables 1-3 are reprinted/adapted from Journal of Applied Developmental Psychology, 70, Rapa et al., Development and intitial…”
BackgroundPoulou, M. S., Garner, P. W., & Bassett, H. H. (2021) · Psychology in the Schools · doi:10.1002/pits.22631 · Cited by 34
Hofmann, S. G., Carpenter, J. K., Otto, M. W., Rosenfield, D., Smits, J. A. J., & Pollack, M. H. (2015) · Contemporary Clinical Trials · doi:10.1016/j.cct.2015.06.015 · Cited by 27
“12 (Power in Two Level Models) [57].”
Shingleton, R. M., Pratt, E. M., Gorman, B. S., Barlow, D. H., Palfai, T. P., & Thompson‐Brenner, H. (2016) · Behavior Therapy · doi:10.1016/j.beth.2016.01.005 · Cited by 27
On determination of sample size in hierarchical binomial models
Zou, K. H., & Normand, S. T. (2001) · Statistics in Medicine · doi:10.1002/sim.855 · Cited by 27
“Two important extensions involve use of other sampling models, such as the Poisson or Normal and inclusion of covariates in both the sampling model and in the prior distribution [25; 26].”
MethodStein, M. S., Kilbride, C., & Reynolds, F. A. (2015) · Disability and Rehabilitation · doi:10.3109/09638288.2015.1037865 · Cited by 26
Metadata
- Authors
- Snijders, T. A. B., & Bosker, R. J.
- Affiliations
- University of Groningen (NL)
- University of Twente (NL)
- Venue
- Journal of Educational Statistics
- Published
- 1993-09
- Volume
- 18
- Issue
- 3
- Pages
- 237-259
- ISSN
- 0362-9791 (ISSN-L)
- OpenAlex
- W2097507923
- Cited by
- 487
- References
- 17
- Citations by year
- 2026: 11 · 2025: 13 · 2024: 14 · 2023: 14 · 2022: 14 · 2021: 27 · 2020: 26 · 2019: 34 · 2018: 20 · 2017: 28 · 2016: 22 · 2015: 25 · 2014: 24 · 2013: 35 · 2012: 21
- FWCI
- 1.01
- Type
- journal article
- Language
- en
- Open access
- green
- www.jstor.org
- Locations
- Journal of Educational Statistics (publishedVersion)
- University of Twente Research Information (submittedVersion, other-oa)
- RePEc: Research Papers in Economics (submittedVersion)
- University of Groningen research database (University of Groningen / Centre for Information Technology) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Survey Sampling and Estimation Techniques
- Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
- Keywords
- two-level designs, multilevel analysis, standard errors, sample size determination, covariance matrix approximation, multilevel data
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