Random effect models have been popularly used as a mainstream statistical technique over several decades; and the same can be said for response transformation models such as the Box–Cox transformation. The latter aims at ensuring that the assumptions of normality and of homoscedasticity of the response distribution are fulfilled, which are essential conditions for inference based on a linear model or a linear mixed model. However, methodology for response transformation and simultaneous inclusion of random effects has been developed and implemented only scarcely, and is so far restricted to Gaussian random effects. We develop such methodology, thereby not requiring parametric assumptions on the distribution of the random effects. This is achieved by extending the ‘Nonparametric Maximum Likelihood’ towards a ‘Nonparametric profile maximum likelihood’ technique, allowing to deal with overdispersion as well as two-level data scenarios.
AgrestiA, CaffoB and Ohman-StricklandP (2004) Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies. Computational Statistics & Data Analysis, 47, 639–53.
2.
AitkinM (1996) A general maximum likelihood analysis of overdispersion in generalized linear models. Statistics and Computing, 6, 251–262.
3.
AitkinM A, FrancisB, HindeJ and DarnellR (2009) Statistical Modelling in R. Oxford:Oxford University Press.
4.
AkaikeH (1998) Information theory and an extension of the maximum likelihood principle. In Selected Papers of Hirotugu Akaike, edited by E Parzen, K Tanabe and G Kitagawa, pages 199–213. New York, NY: Springer.
5.
AlmohaimeedA and EinbeckJ (2020) boxcoxmix: Box–Cox-type transformations for linear and logistic models with random effects. R package version 0.28.
6.
AsarÖIlk O and DagO (2017) Estimating Box–Cox power transformation parameter via goodness-of-fit tests. Communications in Statistics — Simulation and Compu- tation, 46, 91–105.
7.
BhatH S and KumarN (2010) On the Derivation of the Bayesian Information Criterion. Merced, CA: School of Natural Sciences, University of California.
8.
BockR D and AitkinM (1981) Marginal maximum likelihood estimation of item parameters: application of an EM Algorithm. Psychometrika, 46, 443–459.
9.
BonatePeter L. (2011). Pharmacokinetic-Pha- rmacodynamic Modeling and Simulation. 2nd Edition, Springer.
10.
BoxG. E. and CoxD. R. (1964) An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26, 211–252.
11.
BöhningD, KuhnertR and RattanasiriS (2008) Meta-analysis of Binary Data using Prole Likelihood. Boca Raton, FL:
12.
BöhningD (2000) Computer-assisted Analysis of Mixtures and Applications: Metaanalysis, Disease Mapping and Others. London: Chapman & Hall.
13.
ButlerS M and LouisTA (1992) Random effects models with non-parametric priors. Statistics in Medicine, 11, 1981–2000.
14.
CarrollR J (1982) Prediction and power transformations when the choice of power is restricted to a finite set. Journal of the American Statistical Association, 77, 908–915.
15.
CeleuxG,Fr–hwirth-SchnatterS and RobertC (2018) Handbook of Mixture Analysis: Model Selection for Mixture Models: Perspectives and Strategies. Boca Raton, FL: CRC Press.
16.
ChangyongF, HongyueW, NaijiL, TianC, HuaH, YingL and XinM TU (2014) Log-transformation and its implications for data analysis. Shanghai Archives of Psychiatry, 26, 105–09.
17.
ClarkT S and LinzerDA (2015) Should I use fixed or random effects? Political Science Research and Methods, 3, 399–408.
18.
DaviesR (1987) Mass point methods for dealing with nuisance parameters in longitudinal studies. In Longitudinal Data Analysis, edited by R Crouchley, pages 88–109. Aldershot: Avebury.
19.
DempsterA P, LairdNM and RubinDB (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B, 39, 1–38.
20.
EinbeckJ and HindeJ (2006) A note on NPML estimation for exponential family regression models with unspecified dispersion parameter. Austrian Journal of Statistics, 35, 233–43.
21.
EinbeckJ, HindeJ and DarnellR (2007) A new package for fitting random effect models: The npmlreg package. R News, 7, 26–30.
22.
GurkaM J (2004) The Box–Cox transformation in the general linear mixed model for longitudinal data. PhD thesis, The University of North Carolina, Chapel Hill, North Carolina.
23.
GurkaMJ, EdwardsLJ, MullerKE and KupperLL (2006) Extending the Box–Cox transformation to the linear mixed model. Journal of the Royal Statistical Society: Series A (Statistics in Society), 169, 273–88.
24.
HeckmanJ and SingerB (1984) A method for minimizing the impact of distributional assumptions in econometric models for duration aata. Econometrica: Journal of the Econometric Society, 52, 271–320.
25.
HindeJ and DemetrioC (1998) Overdispersion: Models and estimation. Computational Statistics & Data Analysis, 27, 151–70.
26.
KeeneON (1995) The log transformation is special. Statistics in Medicine, 14, 811–19.
27.
LairdN (1978) Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73, 805–11.
28.
LerouxB G and PutermanML (1992) Maximum-penalized-likelihood estimation for independent and Markov-dependent mixture models. Biometrics, 48, 545–58.
29.
LindsayB G (1983) The geometry of mixture likelihoods: A general theory. The Annals of Statistics, 11, 86–94.
30.
LubkeG H and MuthenB (2005) Investigating population heterogeneity with factor mixture models. Psychological Methods, 10, 21–39.
31.
LukoĉieneO (2010) Latent class models for categorical data with a multilevel structure. PhD thesis, Tilburg University, Tilburg, Netherlands.
32.
LukoĉieneO and VermuntJK (2009) Determining the number of components in mixture models for hierarchical data. Advances in Data Analysis, Data Handling and Business Intelligence–Proceedings of the 32nd Annual Conference of the GfKl e.V., Hamburg, 16–18 July 2008, pages 241–49.
33.
MaruoK, YamaguchiY, NomaH and GoshoM (2017) Interpretable inference on the mixed effect model with the Box-Cox transformation. Statistics in Medicine, 36, 2420–34.
NaikPA, ShiP and TsaiC-L (2007) Extending the Akaike Information Criterion to mixture regression models. Journal of the American Statistical Association, 102, 244–54.
OsborneJW (2013) Best Practices in Data Cleaning: A Complete Guide to Everything You Need to Do Before and After Collecting Your Data. Washington, DC:SAGE.
38.
PearsonK (1895) Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Philosophical Transactions of the Royal Society of London, 186, 343–414.
39.
PiephoH-P and McCullochCE (2004) Transformations in mixed models: Application to risk analysis for a multi- environment trial. Journal of Agricultural, Biological, and Environmental Statistics, 9, 123–37.
40.
PinheiroJ, BatesD, DebRoyS, Sarkar D and R Core Team (2016) nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-128.
41.
QarmalahN M, EinbeckJ and CoolenFPA (2018) k-Boxplots for mixture data. Statistical Papers, 59, 513–28.
42.
RCore Team: R (2016) A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing.
43.
SakiaR (1992) The Box–Cox transformation technique: A review. The Statistician, 41, 169–78.
44.
SchwarzG (1978) Estimating the dimension of a model. The Annals of Statistics, 6, 461–64.
45.
SteeleR J and RafteryA (2010) Performance of Bayesian model selection criteria for Gaussian mixture models. In Frontiers of Statistical Decision Making and Bayesian Analysis, edited by M-H Chen, D Dey, P Muller, D Sun and K Ye, pages 113–30. New York: Springer.
46.
WangL (2004) Parameter estimation for mixtures of generalized linear mixed-effects models. PhD thesis, The University of Georgia, Athens. Georgia.
47.
WangP, TsaiG-f and QuA (2012) Conditional inference functions for mixed effects models with unspecified random-effects distribution. Journal of the American Statistical Association, 107, 725–36.
