@InCollection{Chambers:1993,
  author = 	 {John M. Chambers},
  title = 	 {Linear Models},
  booktitle = 	 {Statistical Models in \proglang{S}},
  publisher =	 {Chapman \& Hall},
  year =	 1993,
  editor = 	 {John M. Chambers and Trevor J. Hastie},
  chapter =	 {4},
  pages = {95--144}}

                  
@Book{Rauden:Bryk:2002,
  author =	 {Stephen W. Raudenbush and Anthony S. Bryk},
  title = 	 {Hierarchical Linear Models:  Applications and Data
                  Analysis Methods},
  publisher = 	 {Sage},
  year = 	 2002,
  edition =	 {2nd},
  ISBN = 	 {0-7619-1904-X}
}

@Book{MLwiNUser:2000,
  author =	 {J. Rasbash and W. Browne and H. Goldstein and M. Yang and I. Plewis},
  title = 	 {A User's Guide to {MLwiN}},
  publisher = 	 {Multilevel Models Project, Institute of Education, University of London},
  year = 	 2000,
  address =	 {London}
}

@Book{davis06:csparse_book,
  author = 	 {Tim Davis},
  title = 	 {Direct Methods for Sparse Linear Systems},
  publisher = 	 {SIAM},
  address =	 {Philadelphia, PA},
  year = 	 2006}

@Article{laird_ware_1982,
  author = 	 {Nan M. Laird and James H. Ware},
  title = 	 {Random-Effects Models for Longitudinal Data},
  journal = 	 {Biometrics},
  year = 	 1982,
  volume = 	 38,
  pages = 	 {963--974}}

@Book{bateswatts88:_nonlin,
  author = 	 {Douglas M. Bates and Donald G. Watts},
  title = 	 {Nonlinear Regression Analysis and Its Applications},
  publisher = 	 {Wiley},
  year = 	 1988,
  ISBN = 	 {0-471-81643-4},
  address = 	 {Hoboken, NJ}}

@Book{R:Pinheiro+Bates:2000,
  author = {Jose C. Pinheiro and Douglas M. Bates},
  title = {Mixed-Effects Models in {S} and {S-Plus}},
  publisher = {Springer},
  year = 2000,
  isbn = {0-387-98957-0},
  publisherurl = {http://www.springeronline.com/sgw/cda/frontpage/0,11855,4-10129-22-2102822-0,00.html?changeHeader=true},
  abstract = {A comprehensive guide to the use of the `nlme' package
                  for linear and nonlinear mixed-effects models.},
  orderinfo = {springer.txt}
}

@article{bates04:_linear,
	Author = {Douglas M. Bates and Saikat DebRoy},
	Journal = {Journal of Multivariate Analysis},
	doi = {10.1016/j.jmva.2004.04.013},
	Number = 1,
	Pages = {1--17},
	Title = {Linear Mixed Models and Penalized Least Squares},
	Volume = 91,
	Year = 2004}

@article{gelman2005analysis,
  title={Analysis of Variance --- Why it is More Important than Ever},
  author={Gelman, Andrew},
  journal={The Annals of Statistics},
  volume={33},
  number={1},
  pages={1--53},
  year={2005},
  publisher={Institute of Mathematical Statistics}
}

@ARTICLE{1977EfronAndMorris,
   author = {{Efron}, B. and {Morris}, C.},
    title = "{Stein's Paradox in Statistics}",
  journal = {Scientific American},
     year = 1977,
    month = may,
   volume = 236,
    pages = {119-127},
      doi = {10.1038/scientificamerican0577-119},
   adsurl = {http://adsabs.harvard.edu/abs/1977SciAm.236e.119E},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{henderson_1982,
	Abstract = {The model generally considered in analysis of covariance has all levels of classification factors and interactions fixed, and also covariate regression coefficients fixed. Mixed models are more appropriate in most applications. A summary of estimation and hypothesis testing for analysis of covariance in the mixed model, including the case of random regression coefficients, is presented. Higher-level covariate regressions (i.e., regressions in which, for all levels of a factor or interaction, all observations on the same level have a common covariate value) are discussed. Nonestimability problems that result from defining such covariates at the levels of fixed effects are illustrated. The case of nonhomogeneous covariate regressions in the mixed model is considered in the context of interpreting predicted future differences among levels of a given factor or interaction. Nonhomogeneous regressions complicate interpretations only when they are associated with the contrast(s) of interest among fixed effects in the model. The question of whether the regressions are homogeneous is itself often of substantive interest. Different random regression coefficients associated with the levels of a random effect are also examined.},
	Author = {Henderson, Jr., Charles R.},
	Copyright = {Copyright {\copyright} 1982 International Biometric Society},
	Journal = {Biometrics},
	Jstor_Articletype = {research-article},
	Jstor_Formatteddate = {Sep., 1982},
	Jstor_Issuetitle = {Special Issue: Analysis of Covariance},
	Language = {English},
	Number = {3},
	Pages = {623--640},
	Publisher = {International Biometric Society},
	Title = {Analysis of Covariance in the Mixed Model: Higher-Level, Nonhomogeneous, and Random Regressions},
	Url = {http://www.jstor.org/stable/2530044},
	Volume = {38},
	Year = {1982},
	Bdsk-Url-1 = {http://www.jstor.org/stable/2530044}}

@article{golub_pereyra_1973,
	Abstract = {For given data (ti, yi), i = 1, ⋯, m, we consider the least squares fit of nonlinear models of the form $\eta(\mathbf{a, \alpha};t) = \sum^n_{j = 1} a_j \varphi_j(\mathbf{\alpha};t),\quad \mathbf{a} \in \mathscr{R}^n, \mathbf{\alpha} \in \mathscr{R}^k.$ For this purpose we study the minimization of the nonlinear functional r(a, α) = ∑m i = 1 (yi - η(a, α, ti))2. It is shown that by defining the matrix {Φ(α)}i,j = φj(α; ti), and the modified functional r2(α) = |y - Φ(α)Φ+ (α)y|2 2, it is possible to optimize first with respect to the parameters α, and then to obtain, a posteriori, the optimal parameters $\hat{\mathbf{a}}$ . The matrix Φ+(α) is the Moore-Penrose generalized inverse of Φ(α). We develop formulas for the Frechet derivative of orthogonal projectors associated with Φ(α) and also for Φ+(α), under the hypothesis that Φ(α) is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik [20] and Guttman, Pereyra and Scolnik [9].},
	Author = {Golub, G. H. and Pereyra, V.},
	Copyright = {Copyright {\copyright} 1973 Society for Industrial and Applied Mathematics},
	Journal = {SIAM Journal on Numerical Analysis},
	Jstor_Articletype = {research-article},
	Jstor_Formatteddate = {Apr., 1973},
	Language = {English},
	Number = {2},
	Pages = {pp. 413-432},
	Publisher = {Society for Industrial and Applied Mathematics},
	Title = {The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate},
	Url = {http://www.jstor.org/stable/2156365},
	Volume = {10},
	Year = {1973},
	Bdsk-Url-1 = {http://www.jstor.org/stable/2156365}}

@article{sleepstudy,
	Author = {Gregory Belenky and Nancy J. Wesensten and David R. Thorne and Maria L. Thomas and Helen C. Sing and Daniel P. Redmond and Michael B. Russo and Thomas J. Balkin},
	Date-Modified = {2014-02-14 21:07:17 +0000},
	Journal = {Journal of Sleep Research},
	Pages = {pp. 1-12},
	Title = {Patterns of Performance Degradation and Restoration During Sleep Restriction and Subsequent Recovery: A Sleep Dose-Response Study},
	Volume = {12},
	Year = {2003}}

@article{Chen:2008:ACS:1391989.1391995,
 author = {Chen, Yanqing and Davis, Timothy A. and Hager, William W. and Rajamanickam, Sivasankaran},
 title = {Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate},
 journal = {ACM Trans. Math. Softw.},
 issue_date = {October 2008},
 volume = {35},
 number = {3},
 month = oct,
 year = {2008},
 pages = {22:1--22:14},
 articleno = {22},
 numpages = {14},
 url = {http://doi.acm.org/10.1145/1391989.1391995},
 doi = {10.1145/1391989.1391995},
 acmid = {1391995},
 publisher = {ACM},
 address = {New York, NY, USA},
 keywords = {Cholesky factorization, linear equations, sparse matrices},
} 


@article{kenward_small_1997,
	title = {Small Sample Inference for Fixed Effects from Restricted Maximum Likelihood},
	volume = {53},
	abstract = {Restricted maximum likelihood {(REML)} is now well established as a method for estimating the parameters of the general Gaussian linear model with a structured covariance matrix, in particular for mixed linear models. Conventionally, estimates of precision and inference for fixed effects are based on their asymptotic distribution, which is known to be inadequate for some small-sample problems. In this paper, we present a scaled Wald statistic, together with an F approximation to its sampling distribution, that is shown to perform well in a range of small sample settings. The statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias. This approach has the advantage that it reproduces both the statistics and F distributions in those settings where the latter is exact, namely for Hotelling T\${\textasciicircum}2\$ type statistics and for analysis of variance F-ratios. The performance of the modified statistics is assessed through simulation studies of four different {REML} analyses and the methods are illustrated using three examples.},
	number = {3},
	journal = {Biometrics},
	author = {M. G Kenward and J. H Roger},
	year = {1997},
	pages = {983--997}
}

@Article{Satterthwaite_1946,
  author = 	 {F. E. Satterthwaite},
  title = 	 {An Approximate Distribution of Estimates of Variance Components},
  journal = 	 {Biometrics Bulletin},
  year = 	 {1946},
  volume =	 {2},
  number =	 {6},
  pages =	 {110-114}
}

@Manual{gamm4,
    title = {\pkg{gamm4}: Generalized Additive Mixed Models using mgcv and lme4},
    author = {Simon Wood and Fabian Scheipl},
    year = {2013},
    note = {R package version 0.2-2},
    url = {http://CRAN.R-project.org/package=gamm4},
  }

@Manual{blme,
    title = {\pkg{blme}: {Bayesian} Linear Mixed-Effects Models},
    author = {Vincent Dorie},
    year = {2013},
    note = {R package version 1.0-1},
    url = {http://CRAN.R-project.org/package=blme},
  }

@article{doran2007estimating,
  title={Estimating the Multilevel {Rasch} Model: With the \pkg{lme4} Package},
  author={Doran, Harold and Bates, Douglas and Bliese, Paul and Dowling, Maritza},
  journal={Journal of Statistical Software},
  volume={20},
  number={2},
  pages={1--18},
  year={2007},
  publisher={American Statistical Association}
}



@TechReport{Powell_bobyqa,
  author = 	 {M. J. D. Powell},
  title = 	 {The {BOBYQA} Algorithm for Bound Constrained Optimization without Derivatives},
  institution =  {Centre for Mathematical Sciences, University of Cambridge},
  year = 	 {2009},
  number =	 {DAMTP 2009/NA06},
  address =	 {Cambridge, England},
  url =  {http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf}
}


@article{pinheiro_unconstrained_1996,
	title = {Unconstrained Parametrizations for Variance-Covariance Matrices},
	volume = {6},
	doi = {10.1007/BF00140873},
	abstract = {The estimation of variance-covariance matrices through optimization of an objective function, such as a log-likelihood function, is usually a difficult numerical problem. Since the estimates should be positive semi-definite matrices, we must use constrained optimization, or employ a parametrization that enforces this condition. We describe here five different parametrizations for variance-covariance matrices that ensure positive definiteness, thus leaving the estimation problem unconstrained. We compare the parametrizations based on their computational efficiency and statistical interpretability. The results described here are particularly useful in maximum likelihood and restricted maximum likelihood estimation in linear and non-linear mixed-effects models, but are also applicable to other areas of statistics.},
	number = {3},
	urldate = {2010-01-05},
	journal = {Statistics and Computing},
	author = {Pinheiro, José C. and Bates, Douglas M.},
	year = {1996},
	pages = {289--296}
}

@article{bolker_strategies_2013,
	title = {Strategies for Fitting Nonlinear Ecological Models in \proglang{R}, {AD} {Model} {Builder}, and \proglang{BUGS}},
	volume = {4},
	url = {http://doi.wiley.com/10.1111/2041-210X.12044},
	doi = {10.1111/2041-210X.12044},
	number = {6},
	urldate = {2013-06-11},
	journal = {Methods in Ecology and Evolution},
	author = {Bolker, Benjamin M. and Gardner, Beth and Maunder, Mark and Berg, Casper W. and Brooks, Mollie and Comita, Liza and Crone, Elizabeth and Cubaynes, Sarah and Davies, Trevor and de Valpine, Perry and Ford, Jessica and Gimenez, Olivier and Kéry, Marc and Kim, Eun Jung and Lennert-Cody, Cleridy and Magnusson, Arni and Martell, Steve and Nash, John and Nielsen, Anders and Regetz, Jim and Skaug, Hans and Zipkin, Elise},
	editor = {Ramula, Satu},
	month = jun,
	year = {2013},
	pages = {501--512},
        pdf={bbpapers/bolker_strategies_2013.pdf}
}


@Article{Klein_nelder_2013,
  author = 	 {Kyle Klein and Julian Neira},
  title = 	 {{Nelder-Mead} Simplex Optimization Routine for Large-Scale Problems: A Distributed Memory Implementation},
  journal = 	 {Computational Economics},
  year = 	 {2013},
doi =  {10.1007/s10614-013-9377-8}
}

@Misc{merBoot,
  author =	 {José A. Sánchez-Espigares and Jordi Ocaña},
  title =	 {An \proglang{R} Implementation of Bootstrap Procedures for Mixed Models},
  howpublished = {Conference presentation, useR!},
  month =	 {July},
  year =	 {2009},
  note =	 {accessed 25 May 2014},
  url = {http://www.r-project.org/conferences/useR-2009/slides/SanchezEspigares+Ocana.pdf}
}



@book{gelman_data_2006,
	address = {Cambridge, England},
	title = {Data Analysis Using Regression and {Multilevel/Hierarchical} Models},
	url = {http://www.stat.columbia.edu/~gelman/arm/},
	publisher = {Cambridge University Press},
	author = {Gelman, Andrew and Hill, Jennifer},
	year = {2006},
	keywords = {uploaded}
}

@article{khatri1968solutions,
  title={Solutions to Some Functional Equations and their Applications to Characterization of Probability Distributions},
  author={Khatri, CG and Rao, C Radhakrishna},
  journal={Sankhy{\=a}: The Indian Journal of Statistics, Series A},
  pages={167--180},
  year={1968},
  publisher={JSTOR}
}

@incollection{zhang2006schur,
   year={2005},
   booktitle={The Schur Complement and its Applications},
   volume={4},
   series={Numerical Methods and Algorithms},
   editor={Zhang, Fuzhen},
   title={Basic Properties of the Schur Complement},
   url={http://dx.doi.org/10.1007/0-387-24273-2_2},
   publisher={Springer-Verlag},
   author={Horn, RogerA. and Zhang, Fuzhen},
   pages={17-46}
}

@book{gelman2013bayesian,
  title={Bayesian Data Analysis},
  author={Gelman, Andrew and Carlin, John B and Stern, Hal S and Dunson, David B and Vehtari, Aki and Rubin, Donald B},
  year={2013},
  publisher={CRC press}
}

@Manual{Matrix_pkg,
    title = {\pkg{Matrix}: Sparse and Dense Matrix Classes and Methods},
    author = {Douglas Bates and Martin Maechler},
    year = {2014},
    note = {\proglang{R} package version 1.1-3},
    url = {http://CRAN.R-project.org/package=Matrix},
  }

  @Manual{minqa_pkg,
    title = {\pkg{minqa}: Derivative-Free Optimization Algorithms by Quadratic
Approximation},
    author = {Douglas Bates and Katharine M. Mullen and John C. Nash and Ravi Varadhan},
    year = {2014},
    note = {\proglang{R} package version 1.2.3},
    url = {http://CRAN.R-project.org/package=minqa},
  }

  @Article{optimx_pkg,
    title = {Unifying Optimization Algorithms to Aid Software System Users: \pkg{optimx} for \proglang{R}},
    author = {John C. Nash and Ravi Varadhan},
    journal = {Journal of Statistical Software},
    year = {2011},
    volume = {43},
    number = {9},
    pages = {1--14},
    url = {http://www.jstatsoft.org/v43/i09/},
  }


@Misc{NLopt,
  author =	 {Steven G. Johnson},
  title =	 {The \pkg{NLopt} Nonlinear-Optimization Package},
  year = {2014},
  url = {http://ab-initio.mit.edu/nlopt}
}

  @Manual{nlme_pkg,
    title = {\pkg{nlme}: Linear and Nonlinear Mixed Effects Models},
    author = {Jose Pinheiro and Douglas Bates and Saikat DebRoy and Deepayan Sarkar and {\proglang{R} Core Team}},
    year = {2014},
    note = {\proglang{R} package version 3.1-117},
    url = {http://CRAN.R-project.org/package=nlme},
  }



  @Article{HLMdiag_pkg,
    title = {\pkg{HLMdiag}: A Suite of Diagnostics for Hierarchical Linear Models in \proglang{R}},
    author = {Adam Loy and Heike Hofmann},
    journal = {Journal of Statistical Software},
    year = {2014},
    volume = {56},
    number = {5},
    pages = {1--28},
    url = {http://www.jstatsoft.org/v56/i05/},
  }

  @Article{influenceME_pkg,
    title = {\proglang{Influence.ME}: Tools for Detecting Influential Data in Mixed Effects Models},
    author = {Rense Nieuwenhuis and Manfred {Te Grotenhuis} and Ben Pelzer},
    year = {2012},
    journal = {R Journal},
    volume = {4},
    number = {2},
    pages = {38-47},
  }


@Manual{boot_pkg,
  author = {  Angelo Canty and Brian Ripley},
   year = 2013,
   title = { \pkg{boot}: Bootstrap \proglang{R} (\proglang{S-PLUS}) Functions},
    note = {\proglang{R} package version 1.3-9},
    url = {http://CRAN.R-project.org/package=boot}
}


  @Book{DavisonHinkley1997,
  author = 	 {A. C. Davison and D. V. Hinkley},
  title = 	 { Bootstrap Methods and Their Applications},
  publisher = 	 {Cambridge University Press},
  year = 	 {1997},
  ISBN = {0-521-57391-2},
  address = {Cambridge, England}
}
  
@article{vaida2005conditional,
  title={Conditional Akaike Information for Mixed-Effects Models},
  author={Vaida, Florin and Blanchard, Suzette},
  journal={Biometrika},
  volume={92},
  number={2},
  pages={351--370},
  year={2005},
  publisher={Biometrika Trust}
}

@book{cook1982residuals,
  title={Residuals and Influence in Regression},
  author={Cook, R Dennis and Weisberg, Sanford},
  year={1982},
  publisher={New York: Chapman and Hall}
}

@article{lesnoff_within-herd_2004,
	title = {Within-herd spread of contagious bovine pleuropneumonia in Ethiopian highlands},
	volume = {64},
	doi = {10.1016/j.prevetmed.2004.03.005},
	abstract = {Contagious bovine pleuropneumonia ({CBPP)} is a major threat for cattle health and production in Africa. This disease is caused by the small-colony type of Mycoplasma mycoides subspecies mycoides ({MmmSC).} Transmission occurs from direct and repeated contacts between sick and healthy animals. Veterinary services recently reported a resurgence of {CBPP} in the province of West Wellega, in the Ethiopian highlands. A research program was set up to estimate the epidemiological parameters of the within-herd infection spread. A follow-up survey was implemented in 71 sampled herds of the Boji district (West Wellega province). Fifteen herds were classified as newly infected and used in a serological- and clinical-incidence study. The overall 16-month cumulative sero-incidence risk was 34\%. Clinical cases were recorded for 39\% of the seropositive cattle; case-fatality risk was 13\%. There was no evidence of benefit on infection spread of {CBPP-control} measures used locally by farmers (isolation or antibiotic treatments of sick animals). This might be related to a lack of power in the statistical analyses or to a quality problem for the medications used (and more generally, for health-care delivery in the Boji district).},
	number = {1},
	urldate = {2013-10-03},
	journal = {Preventive Veterinary Medicine},
	author = {Lesnoff, Matthieu and Laval, Géraud and Bonnet, Pascal and Abdicho, Sintayehu and Workalemahu, Asseguid and Kifle, Daniel and Peyraud, Armelle and Lancelot, Renaud and Thiaucourt, François},
	month = jun,
	year = {2004},
	keywords = {Clinical signs, Contagious bovine pleuropneumonia, Ethiopia, Herd monitoring, Within-herd incidence},
	pages = {27--40}
}

@book{harrell_regression_2001,
	title = {Regression Modeling Strategies},
	isbn = {0387952322},
	publisher = {Springer},
	author = {Harrell, Frank},
	year = {2001}
}

@Article{steele_et_al_1996,
  author = 	 {F. Steele and I. Diamond and S. Amin},
  title = 	 {Immunization uptake in rural Bangladesh: a
                  multilevel analysis},
  journal = 	 {Journal of the Royal Statistical Society, Series A},
  year = 	 1996,
  volume = 	 159,
  pages = 	 {289--299}}

@book{huq1990bangladesh,
  title={Bangladesh fertility survey 1989},
  author={Huq, Najmul and Cleland, John},
  publisher={National Institute of Population Research and Training},
  address = {Dhaka},
  year={1990}
}

@BOOK{McCullaghNelder1989,
  title = {Generalized Linear Models},
  publisher = {Chapman and Hall},
  year = {1989},
  author = {P. McCullagh and J. A. Nelder},
  address = {London}
}

@article{zeger1991generalized,
  title={Generalized linear models with random effects: a Gibbs sampling approach},
  author={Zeger, Scott L and Karim, M Rezaul},
  journal={Journal of the American Statistical Association},
  volume={86},
  number={413},
  pages={79--86},
  year={1991},
  doi = {10.1080/01621459.1991.10475006}
}

@article{bates2015fitting,
  title={Fitting linear mixed-effects models using lme4},
  author={Bates, Douglas and M{\"a}chler, Martin and Bolker, Ben and Walker, Steve},
  journal={Journal of Statistical Software},
  doi = {10.18637/jss.v067.i01},
  volume={67},
  pages={1--48},
  year={2015}
}


@article{gelman2008scaling,
  title={Scaling regression inputs by dividing by two standard deviations},
  author={Gelman, Andrew},
  journal={Statistics in medicine},
  volume={27},
  number={15},
  pages={2865--2873},
  year={2008},
  doi = {10.1002/sim.3107}
}

@book{madsen2011,
  title = {Introduction to General and Generalized Linear Models},
  author = {Madsen, Henrik and Thyregod, Poul},
  year = 2011,
  publisher = {CRC Press},
  abstract = {Bridging the gap between theory and practice for modern statistical model building, Introduction to General and Generalized Linear Models presents likelihood-based techniques for statistical modelling using various types of data. Implementations using R are provided throughout the text, although other software packages are also discussed. Numerous examples show how the problems are solved with R.After describing the necessary likelihood theory, the book covers both general and generalized linear models using the same likelihood-based methods. It presents the corresponding/parallel results for the general linear models first, since they are easier to understand and often more well known. The authors then explore random effects and mixed effects in a Gaussian context. They also introduce non-Gaussian hierarchical models that are members of the exponential family of distributions. Each chapter contains examples and guidelines for solving the problems via R. Providing a flexible framework for data analysis and model building, this text focuses on the statistical methods and models that can help predict the expected value of an outcome, dependent, or response variable. It offers a sound introduction to general and generalized linear models using the popular and powerful likelihood techniques. Ancillary materials are available at www.imm.dtu.dk/\textasciitilde hm/GLM},
  isbn = {978-1-4200-9155-7},
  langid = {english},
  keywords = {Business & Economics / Statistics,Mathematics / Probability & Statistics / General}
}

@article{kristensenTMB2016,
  title = {{TMB} : Automatic Differentiation and {Laplace} Approximation},
  author = {Kristensen, Kasper and Nielsen, Anders and Berg, Casper W. and Skaug, Hans and Bell, Bradley M.},
  year = 2016,
  journal = {Journal of Statistical Software},
  volume = {70},
  number = {5},
  doi = {10.18637/jss.v070.i05},
  urldate = {2017-05-23},
  langid = {english}
}

  @article{debacker+1998,
  title = {Twelve weeks of continuous oral therapy for toenail onychomycosis caused by dermatophytes: a double-blind comparative trial of terbinafine 250 mg/day versus itraconazole 200 mg/day},
  author = {De Backer, M. and De Vroey, C. and Lesaffre, E. and Scheys, I. and De Keyser, P.},
  year = 1998,
  month = may,
  journal = {Journal of the American Academy of Dermatology},
  volume = {38},
  number = {5, Supplement 2},
  pages = {S57-S63},
  doi = {10.1016/S0190-9622(98)70486-4},
  urldate = {2015-03-28},
  abstract = {Background: Dermatophyte infections of the toenail have been difficult to treat, requiring long courses of therapy and having high recurrence rates. New oral antifungal agents with better outcomes and minimal adverse events are needed. Objective: The purpose of this study was to compare two newer antifungal compounds, terbinafine and itraconazole, for efficacy and safety in toenail onychomycosis caused by dermatophytes. Methods: The study was randomized and double-blind. It compared 12 weeks of continuous oral treatment with terbinafine 250 mg/day or itraconazole 200 mg/day for confirmed toenail dermatophyte onychomycosis. Clinical symptoms and mycologic outcome were assessed at weeks 4, 8, 12, 24, 36, and 48. A total of 372 patients (186 in each group) with dermatophyte infection confirmed by microscopy and culture were included in the intent-to-treat analysis. Results: At week 48, a statistically significantly greater percentage of the terbinafine group than itraconazole group showed negative mycology (73\% [119 of 163] vs 45.8\% [77 of 168]; p \&lt; 0.0001) (difference = 27.2\%; 95\% CI = [17.0\%, 37.3\%]). The difference was also confirmed clinically (p = 0.001) in the patients who were clinically cured or had only minimal symptoms at the end of the study (76.2\% [125 of 164] vs 58.1\% [100 of 172]) (difference = 18.1\%; 95\% CI = [8.24\%, 27.9\%]). The geometric mean length of healthy nail of the big toe was significantly greater in the terbinafine than itraconazole group (8.1 vs 6.4 mm; p = 0.026). Tolerability was good to very good in almost 90\% of patients in both groups, and all reported adverse events were known for these compounds. Conclusion: Terbinafine produced higher rates of clinical and mycologic cure at follow-up than did itraconazole. (J Am Acad Dermatol 1998;38:S57-63.)}
}

  @article{scandola2024,
  title = {Reliability and Feasibility of Linear Mixed Models in Fully Crossed Experimental Designs},
  author = {Scandola, Michele and Tidoni, Emmanuele},
  year = 2024,
  month = jan,
  journal = {Advances in Methods and Practices in Psychological Science},
  volume = {7},
  number = {1},
  pages = {25152459231214454},
  publisher = {SAGE Publications Inc},
  doi = {10.1177/25152459231214454},
  urldate = {2024-12-24},
  abstract = {The use of linear mixed models (LMMs) is increasing in psychology and neuroscience research In this article, we focus on the implementation of LMMs in fully crossed experimental designs. A key aspect of LMMs is choosing a random-effects structure according to the experimental needs. To date, opposite suggestions are present in the literature, spanning from keeping all random effects (maximal models), which produces several singularity and convergence issues, to removing random effects until the best fit is found, with the risk of inflating Type I error (reduced models). However, defining the random structure to fit a nonsingular and convergent model is not straightforward. Moreover, the lack of a standard approach may lead the researcher to make decisions that potentially inflate Type I errors. After reviewing LMMs, we introduce a step-by-step approach to avoid convergence and singularity issues and control for Type I error inflation during model reduction of fully crossed experimental designs. Specifically, we propose the use of complex random intercepts (CRIs) when maximal models are overparametrized. CRIs are multiple random intercepts that represent the residual variance of categorical fixed effects within a given grouping factor. We validated CRIs and the proposed procedure by extensive simulations and a real-case application. We demonstrate that CRIs can produce reliable results and require less computational resources. Moreover, we outline a few criteria and recommendations on how and when scholars should reduce overparametrized models. Overall, the proposed procedure provides clear solutions to avoid overinflated results using LMMs in psychology and neuroscience.}
}

@article{matuschekBalancingTypeError2017,
  title = {Balancing {{Type I}} Error and Power in Linear Mixed Models},
  author = {Matuschek, Hannes and Kliegl, Reinhold and Vasishth, Shravan and Baayen, Harald and Bates, Douglas},
  year = 2017,
  month = jun,
  journal = {Journal of Memory and Language},
  volume = {94},
  pages = {305--315},
  doi = {10.1016/j.jml.2017.01.001},
  urldate = {2021-05-08},
  abstract = {Linear mixed-effects models have increasingly replaced mixed-model analyses of variance for statistical inference in factorial psycholinguistic experiments. Although LMMs have many advantages over ANOVA, like ANOVAs, setting them up for data analysis also requires some care. One simple option, when numerically possible, is to fit the full variance-covariance structure of random effects (the maximal model; Barr, Levy, Scheepers \& Tily, 2013), presumably to keep Type I error down to the nominal {$\alpha$} in the presence of random effects. Although it is true that fitting a model with only random intercepts may lead to higher Type I error, fitting a maximal model also has a cost: it can lead to a significant loss of power. We demonstrate this with simulations and suggest that for typical psychological and psycholinguistic data, higher power is achieved without inflating Type I error rate if a model selection criterion is used to select a random effect structure that is supported by the data.},
  langid = {english},
  keywords = {Hypothesis testing,Linear mixed effect model,Power}
}

@article{barrRandomEffectsStructure2013,
  title = {Random Effects Structure for Confirmatory Hypothesis Testing: {{Keep}} It Maximal},
  shorttitle = {Random Effects Structure for Confirmatory Hypothesis Testing},
  author = {Barr, Dale J. and Levy, Roger and Scheepers, Christoph and Tily, Harry J.},
  year = 2013,
  month = apr,
  journal = {Journal of Memory and Language},
  volume = {68},
  number = {3},
  pages = {255--278},
  doi = {10.1016/j.jml.2012.11.001},
  urldate = {2016-02-20},
  langid = {english}
}

@article{stringerAsymptoticsNumericalIntegration2022,
  title = {Asymptotics of Numerical Integration for Two-Level Mixed Models},
  author = {Stringer, Alex and Bilodeau, Blair and Tang, Yanbo},
  year = 2026,
  journal = {Bernoulli},
  note = {(forthcoming)},
  abstract = {We study mixed models with a single grouping factor, where inference about unknown parameters requires optimizing a marginal likelihood defined by an intractable integral. Low-dimensional numerical integration techniques are regularly used to approximate these integrals, with inferences about parameters based on the resulting approximate marginal likelihood. For a generic class of mixed models that satisfy explicit regularity conditions, we derive the stochastic relative error rate incurred for both the likelihood and maximum likelihood estimator when adaptive numerical integration is used to approximate the marginal likelihood. We then specialize the analysis to well-specified generalized linear mixed models having exponential family response and multivariate Gaussian random effects, verifying that the regularity conditions hold, and hence that the convergence rates apply. We also prove that for models with likelihoods satisfying very weak concentration conditions that the maximum likelihood estimators from non-adaptive numerical integration approximations of the marginal likelihood are not consistent, further motivating adaptive numerical integration as the preferred tool for inference in mixed models. Code to reproduce the simulations in this paper is provided at https://github.com/awstringer1/aq-theory-paper-code.},
  keywords = {Statistics - Applications,Statistics - Methodology},
  annotation = {ADS Bibcode: 2022arXiv220207864S},
  file = {/home/bolker/Documents/zotero_new/storage/4DSMQGKX/Stringer et al. - 2022 - Asymptotics of numerical integration for two-level mixed models.pdf}
}

@article{stringerExactGradientEvaluation2024,
  title = {Exact Gradient Evaluation for Adaptive Quadrature Approximate Marginal Likelihood in Mixed Models for Grouped Data},
  author = {Stringer, Alex},
  year = 2024,
  month = nov,
  journal = {Statistics and Computing},
  volume = {35},
  number = {1},
  pages = {4},
  issn = {1573-1375},
  doi = {10.1007/s11222-024-10536-z},
  urldate = {2025-08-24},
  abstract = {A method is introduced for approximate marginal likelihood inference via adaptive Gaussian quadrature in mixed models with a single grouping factor. The core technical contribution is an algorithm for computing the exact gradient of the approximate log-marginal likelihood. This leads to efficient maximum likelihood via quasi-Newton optimization that is demonstrated to be faster than existing approaches based on finite-differenced gradients or derivative-free optimization. The method is specialized to Bernoulli mixed models with multivariate, correlated Gaussian random effects; here computations are performed using an inverse log-Cholesky parameterization of the Gaussian density that involves no matrix decomposition during model fitting, while Wald confidence intervals are provided for variance parameters on the original scale. Simulations give evidence of these intervals attaining nominal coverage if enough quadrature points are used, for data comprised of a large number of very small groups exhibiting large between-group heterogeneity. The Laplace approximation is well-known to give especially poor coverage and high bias for data comprised of a large number of small groups. Adaptive quadrature mitigates this, and the methods in this paper improve the computational feasibility of this more accurate method. All results may be reproduced using code available at https://github.com/awstringer1/aghmm-paper-code.},
  langid = {english},
  keywords = {Adaptive quadrature,Approximate inference,Grouped data,Longitudinal data,Mixed models},
  file = {/home/bolker/Documents/zotero_new/storage/ZMVMUHFS/Stringer - 2024 - Exact gradient evaluation for adaptive quadrature approximate marginal likelihood in mixed models fo.pdf}
}

@incollection{ripleySelectingAmongstLarge2004a,
  title = {Selecting amongst Large Classes of Models},
  author = {Ripley, Brian D.},
  year = 2004,
  booktitle = {Methods and models in statistics: In honor of Professor John Nelder, FRS},
  editor = {Adams, Niall M. and Crowder, Martin and Hand, D.J. and Stephens, Dave},
  place = London,
  publisher = {Imperial College Press},
  pages = {155--170},
}

@misc{stringerFittingGeneralizedLinear2022b,
  title = {Fitting {{Generalized Linear Mixed Models}} Using {{Adaptive Quadrature}}},
  author = {Stringer, Alex and Bilodeau, Blair},
  year = 2022,
  month = feb,
  number = {arXiv:2202.07864},
  eprint = {2202.07864},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2202.07864},
  urldate = {2024-11-14},
  abstract = {We describe how to approximate the intractable marginal likelihood that arises when fitting generalized linear mixed models. We prove that non-adaptive quadrature approximations yield high error asymptotically in every statistical model satisfying weak regularity conditions. We derive the rate of error incurred when using adaptive quadrature to approximate the marginal likelihood in a broad class of generalized linear mixed models, which includes non-exponential family response and non-Gaussian random effects distributions. We provide an explicit recommendation for how many quadrature points to use, and show that this recommendation recovers and explains many empirical results from published simulation studies and data analyses. Particular attention is paid to models for dependent binary and survival/time-to-event observations. Code to reproduce results in the manuscript is found at https://github.com/awstringer1/glmm-aq-paper-code.},
  archiveprefix = {arXiv},
  keywords = {Statistics - Applications,Statistics - Methodology},
  file = {/home/bolker/Documents/zotero_new/storage/CCAGE8L6/Stringer and Bilodeau - 2022 - Fitting Generalized Linear Mixed Models using Adaptive Quadrature.pdf;/home/bolker/Documents/zotero_new/storage/99P5FKXQ/2202.html}
}

@article{harrisonComparisonObservationlevelRandom2015,
  title = {A Comparison of Observation-Level Random Effect and Beta-Binomial Models for Modelling Overdispersion in Binomial Data in Ecology and Evolution},
  author = {Harrison, Xavier A.},
  year = 2015,
  month = jul,
  journal = {PeerJ},
  volume = {3},
  pages = {e1114},
  publisher = {PeerJ Inc.},
  issn = {2167-8359},
  doi = {10.7717/peerj.1114},
  urldate = {2023-06-20},
  abstract = {Overdispersion is a common feature of models of biological data, but researchers often fail to model the excess variation driving the overdispersion, resulting in biased parameter estimates and standard errors. Quantifying and modeling overdispersion when it is present is therefore critical for robust biological inference. One means to account for overdispersion is to add an observation-level random effect (OLRE) to a model, where each data point receives a unique level of a random effect that can absorb the extra-parametric variation in the data. Although some studies have investigated the utility of OLRE to model overdispersion in Poisson count data, studies doing so for Binomial proportion data are scarce. Here I use a simulation approach to investigate the ability of both OLRE models and Beta-Binomial models to recover unbiased parameter estimates in mixed effects models of Binomial data under various degrees of overdispersion. In addition, as ecologists often fit random intercept terms to models when the random effect sample size is low ({$<$}5 levels), I investigate the performance of both model types under a range of random effect sample sizes when overdispersion is present. Simulation results revealed that the efficacy of OLRE depends on the process that generated the overdispersion; OLRE failed to cope with overdispersion generated from a Beta-Binomial mixture model, leading to biased slope and intercept estimates, but performed well for overdispersion generated by adding random noise to the linear predictor. Comparison of parameter estimates from an OLRE model with those from its corresponding Beta-Binomial model readily identified when OLRE were performing poorly due to disagreement between effect sizes, and this strategy should be employed whenever OLRE are used for Binomial data to assess their reliability. Beta-Binomial models performed well across all contexts, but showed a tendency to underestimate effect sizes when modelling non-Beta-Binomial data. Finally, both OLRE and Beta-Binomial models performed poorly when models contained {$<$}5 levels of the random intercept term, especially for estimating variance components, and this effect appeared independent of total sample size. These results suggest that OLRE are a useful tool for modelling overdispersion in Binomial data, but that they do not perform well in all circumstances and researchers should take care to verify the robustness of parameter estimates of OLRE models.},
  langid = {english},
  file = {/home/bolker/Documents/zotero_new/storage/5W9YL625/Harrison - 2015 - A comparison of observation-level random effect an.pdf}
}

@article{rabe-heskethMaximumLikelihoodEstimation2005b,
  title = {Maximum Likelihood Estimation of Limited and Discrete Dependent Variable Models with Nested Random Effects},
  author = {{Rabe-Hesketh}, Sophia and Skrondal, Anders and Pickles, Andrew},
  year = 2005,
  month = oct,
  journal = {Journal of Econometrics},
  volume = {128},
  number = {2},
  pages = {301--323},
  issn = {0304-4076},
  doi = {10.1016/j.jeconom.2004.08.017},
  urldate = {2026-06-20},
  abstract = {Gauss--Hermite quadrature is often used to evaluate and maximize the likelihood for random component probit models. Unfortunately, the estimates are biased for large cluster sizes and/or intraclass correlations. We show that adaptive quadrature largely overcomes these problems. We then extend the adaptive quadrature approach to general random coefficient models with limited and discrete dependent variables. The models can include several nested random effects (intercepts and coefficients) representing unobserved heterogeneity at different levels of a hierarchical dataset. The required multivariate integrals are evaluated efficiently using spherical quadrature rules. Simulations show that adaptive quadrature performs well in a wide range of situations.},
  keywords = {Adaptive quadrature,GLLAMM,Hierarchical models,Multilevel models,Numerical integration,Random coefficients,Random effects,Spherical quadrature rules},
  file = {/home/bolker/Documents/zotero_new/storage/V84YE3JC/Rabe-Hesketh et al. - 2005 - Maximum likelihood estimation of limited and discrete dependent variable models with nested random e.pdf;/home/bolker/Documents/zotero_new/storage/3TFPFBID/S0304407604001599.html}
}

@article{tuerlinckxStatisticalInferenceGeneralized2006a,
  title = {Statistical Inference in Generalized Linear Mixed Models: {{A}} Review},
  shorttitle = {Statistical Inference in Generalized Linear Mixed Models},
  author = {Tuerlinckx, Francis and Rijmen, Frank and Verbeke, Geert and De Boeck, Paul},
  year = 2006,
  journal = {British Journal of Mathematical and Statistical Psychology},
  volume = {59},
  number = {2},
  pages = {225--255},
  issn = {2044-8317},
  doi = {10.1348/000711005X79857},
  urldate = {2026-06-24},
  abstract = {We present a review of statistical inference in generalized linear mixed models (GLMMs). GLMMs are an extension of generalized linear models and are suitable for the analysis of non-normal data with a clustered structure. A GLMM contains parameters common to all clusters (fixed regression effects and variance components) and cluster-specific parameters. The latter parameters are assumed to be randomly drawn from a population distribution. The parameters of this population distribution (the variance components) have to be estimated together with the fixed effects. We focus on the case in which the cluster-specific parameters are normally distributed. The cluster-specific effects are integrated out of the likelihood so that the fixed effects and variance components can be estimated. Unfortunately, the integral over the cluster-specific effects is intractable for most GLMMs with a normal mixing distribution. Within a classical statistical framework, we distinguish between two broad classes of methods to handle this intractable integral: methods that rely on a numerical approximation to the integral and methods that use an analytical approximation to the integrand. Finally, we present an overview of available methods for testing hypotheses about the parameters of GLMMs.},
  copyright = {2006 The British Psychological Society},
  langid = {english}
}
