@article{boos_monte_2000,
  title = {Monte {Carlo} evaluation of resampling-based hypothesis tests},
  volume = {95},
  issn = {0162-1459, 1537-274X},
  url = {http://www.tandfonline.com/doi/abs/10.1080/01621459.2000.10474226},
  doi = {10.1080/01621459.2000.10474226},
  language = {en},
  number = {450},
  urldate = {2025-10-29},
  journal = {Journal of the American Statistical Association},
  author = {Boos, Dennis D. and Zhang, Ji},
  month = jun,
  year = {2000},
  keywords = {Bootstrapping, Power, Simulation Tools},
  pages = {486--492},
}

@article{chalmers_solving_2024,
  title = {Solving variables with {Monte} {Carlo} simulation experiments: {A} stochastic root-solving approach},
  issn = {1939-1463},
  shorttitle = {Solving variables with {Monte} {Carlo} simulation experiments},
  doi = {10.1037/met0000689},
  abstract = {Despite their popularity and flexibility, questions remain regarding how to optimally solve particular unknown variables of interest using Monte Carlo simulation experiments. This article reviews two common approaches based on either performing deterministic iterative searches with noisy objective functions or by constructing interpolation estimates given fitted surrogate functions, highlighting the inefficiencies and inferential concerns of both methods. To address these limitations, and to fill a gap in existing Monte Carlo experimental methodology, a novel algorithm termed the probabilistic bisection algorithm with bolstering and interpolations (ProBABLI) is presented with the goal providing efficient, consistent, and unbiased estimates (with associated confidence intervals) for the stochastic root equations found in Monte Carlo simulation research. Properties of the ProBABLI approach are demonstrated using practical sample size planning applications for independent samples t tests and structural equation models given target power rates, precision criteria, and expected power functions that incorporate prior beliefs. (PsycInfo Database Record (c) 2024 APA, all rights reserved)},
  journal = {Psychological Methods},
  publisher = {American Psychological Association},
  author = {Chalmers, R. Philip},
  year = {2024},
  note = {Place: US},
  keywords = {Simulation Tools, Root Finding, Stochastic Root Finding, PBA (Probabilistic Bisection Algorithm)},
}

@article{waeber_bisection_2013,
  title = {Bisection search with noisy responses},
  volume = {51},
  issn = {0363-0129, 1095-7138},
  url = {http://epubs.siam.org/doi/10.1137/120861898},
  doi = {10.1137/120861898},
  language = {en},
  number = {3},
  urldate = {2026-02-13},
  journal = {SIAM Journal on Control and Optimization},
  author = {Waeber, Rolf and Frazier, Peter I. and Henderson, Shane G.},
  month = jan,
  year = {2013},
  keywords = {Power, Root Finding, Bisection Method, PBA (Probabilistic Bisection Algorithm)},
  pages = {2261--2279},
}



@article{bohrnstedt_exact_1969,
  title = {On the exact covariance of products of random variables},
  volume = {64},
  issn = {01621459},
  url = {http://www.jstor.org/stable/2286081?origin=crossref},
  doi = {10.2307/2286081},
  number = {328},
  journal = {Journal of the American Statistical Association},
  author = {Bohrnstedt, George W. and Goldberger, Arthur S.},
  month = dec,
  year = {1969},
  keywords = {Product of Random Variables},
  pages = {1439--1442},
}



@article{bodner_tumble_2016,
	title = {Tumble graphs: Avoiding misleading end point extrapolation when graphing interactions from a moderated multiple regression analysis},
	volume = {41},
	issn = {1076-9986},
	doi = {10.3102/1076998616657080},
	pages = {593--604},
	number = {6},
	journaltitle = {Journal of Educational and Behavioral Statistics},
	author = {Bodner, Todd E.},
	date = {2016},
	keywords = {Regression, Moderation (Interaction), Graphs, Tumble Graph},
}


@article{cheung_manymome_2024,
	title = {{\textit{manymome}}: {An} {R} package for computing the indirect effects, conditional effects, and conditional indirect effects, standardized or unstandardized, and their bootstrap confidence intervals, in many (though not all) models},
	volume = {56},
	issn = {1554-3528},
	shorttitle = {manymome},
	url = {https://link.springer.com/10.3758/s13428-023-02224-z},
	doi = {10.3758/s13428-023-02224-z},
	abstract = {Abstract

              Mediation, moderation, and moderated mediation are common in behavioral research models. Several tools are available for estimating indirect effects, conditional effects, and conditional indirect effects and forming their confidence intervals. However, there are no simple-to-use tools that can appropriately form the bootstrapping confidence interval for standardized conditional indirect effects. Moreover, some tools are restricted to a limited type of models. We developed an R package,
              manymome
              , which can be used to estimate and form confidence intervals for indirect effects, conditional effects, and conditional indirect effects, standardized or not, using a two-step approach: model parameters are estimated either by structural equation modeling using
              lavaan
              or by a set of linear regression models using
              lm
              , and then the coefficients are used to compute the requested effects and form confidence intervals. It can be used when there are missing data if the model is fitted by structural equation modeling. There are only a few limitations on some aspects of a model, and no inherent limitations on the number of predictors, the number of independent variables, or the number of moderators and mediators. The goal is to have a tool that allows researchers to focus on model fitting first and worry about estimating the effects later. The use of the model is illustrated using a few numerical examples, and the limitations of the package are discussed.},
	language = {en},
	number = {5},
	urldate = {2024-09-13},
	journal = {Behavior Research Methods},
	author = {Cheung, Shu Fai and Cheung, Sing-Hang},
	year = {2024},
	keywords = {Bootstrapping, Mediation, MI (Multiple Imputation), Moderation (Interaction), sfcheung, R Packages, Missing Data, sfcheung (First or Correspondence), Moderated-Mediation and Mediated-Moderation, manymome},
	pages = {4862--4882},
}

@article{hayes_index_2015,
	title = {An index and test of linear moderated mediation},
	volume = {50},
	issn = {0027-3171},
	url = {https://doi.org/10.1080/00273171.2014.962683},
	doi = {10.1080/00273171.2014.962683},
	abstract = {I describe a test of linear moderated mediation in path analysis based on an interval estimate of the parameter of a function linking the indirect effect to values of a moderator—a parameter that I call the index of moderated mediation. This test can be used for models that integrate moderation and mediation in which the relationship between the indirect effect and the moderator is estimated as linear, including many of the models described by Edwards and Lambert (2007) and Preacher, Rucker, and Hayes (2007) as well as extensions of these models to processes involving multiple mediators operating in parallel or in serial. Generalization of the method to latent variable models is straightforward. Three empirical examples describe the computation of the index and the test, and its implementation is illustrated using Mplus and the PROCESS macro for SPSS and SAS.},
	number = {1},
	urldate = {2022-05-23},
	journal = {Multivariate Behavioral Research},
	author = {Hayes, Andrew F.},
	month = jan,
	year = {2015},
	pmid = {26609740},
	note = {Publisher: Routledge
\_eprint: https://doi.org/10.1080/00273171.2014.962683},
	keywords = {Top Among Top, Effect Sizes, Moderation (Interaction), Moderated-Mediation and Mediated-Moderation, Index of Moderated Mediation},
	pages = {1--22},
}
