Mixed-effects models (MEMs) and latent growth models (LGMs) are often considered interchangeable save the discipline-specific nomenclature. Software implementations of these models, however, are not interchangeable, particularly with small sample sizes. Restricted maximum likelihood estimation that mitigates small sample bias in MEMs has not been widely developed for LGMs, and fully Bayesian methods, while not dependent on asymptotics, can encounter issues because the choice for the factor covariance matrix prior distribution has substantial influence with small samples. This tutorial discusses differences between LGMs and MEMs and demonstrates how data-dependent priors, an established class of methods that blend frequentist and Bayesian paradigms, can be implemented within Mplus 7.1 to abate the small sample bias that is prevalent with LGM software while keeping additional programming to the bare minimum.
AllisonP. D. (2012, 4). Handling Missing data by maximum likelihood. Statistics and Data Analysis Keynote Presentation at the SAS Global Forum, Orlando, FL.
2.
AsparouhovT.MuthénB. (2010). Bayesian analysis of latent variable models using Mplus, version 2. Unpublished manuscript. Retrieved fromhttp://www.statmodel.com
3.
BaldwinS. A.FellinghamG. W. (2013). Bayesian methods for the analysis of small sample multilevel data with a complex variance structure. Psychological Methods, 18, 151–164.
4.
BarnardJ.McCullochR.MengX. L. (2000). Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Statistica Sinica, 10, 1281–1312.
5.
BartlettM. S. (1950). Tests of significance in factor analysis. British Journal of Statistical Psychology, 3, 77–85.
6.
BauerD. J.CurranP. J. (2003). Distributional assumptions of growth mixture models: Implications for overextraction of latent trajectory classes. Psychological Methods, 8, 338–363.
7.
BergerJ. (2006). The case for objective Bayesian analysis. Bayesian Analysis, 1, 385–402.
8.
BiesanzJ. C.Deeb-SossaN.PapadakisA. A.BollenK.CurranP. (2004). The role of coding time in estimating and interpreting growth curve models. Psychological Methods, 9, 30–52.
9.
BlozisS. A.CudeckR. (1999). Conditionally linear mixed-effects models with latent variable covariates. Journal of Educational and Behavioral Statistics, 24, 245–270.
10.
BrowneW. J.DraperD. (2006). A comparison of Bayesian and likelihood-based methods for fitting multilevel models. Bayesian Analysis, 1, 473–514.
11.
CarlinB. P.GelfandA. E. (1990). Approaches for empirical Bayes confidence intervals. Journal of the American Statistical Association, 85, 105–114.
12.
CarlinB. P.GelfandA. E. (1991). A sample reuse method for accurate parametric empirical Bayes confidence intervals. Journal of the Royal Statistical Society. Series B, 53, 189–200.
13.
CarlinB. P.LouisT. A. (2000). Empirical Bayes: Past, present and future. Journal of the American Statistical Association, 95, 1286–1289.
ChenF. (2011). The RANDOM statement and more: Moving on with PROC MCMC. In Proceedings of the SAS Global Forum 2011 Conference. Cary, NC: SAS Institute, Inc.
16.
CheungM. W. L. (2013). Implementing restricted maximum likelihood estimation in structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 20, 157–167.
17.
CurranP. J. (2003). Have multilevel models been structural equation models all along?Multivariate Behavioral Research, 38, 529–569.
18.
DarniederW. F. (2011). Bayesian methods for data-dependent priors (Doctoral dissertation). The Ohio State University, Columbus, OH.
19.
DepaoliS. (2014). The impact of inaccurate “informative” priors for growth parameters in Bayesian growth mixture modeling. Structural Equation Modeling: A Multidisciplinary Journal, 21, 239–252.
20.
de LeeuwJ.MeijerE.GoldsteinH. (2008). Handbook of multilevel analysis. New York, NY: Springer.
21.
DeelyJ. J.LindleyD. V. (1981). Bayes empirical Bayes. Journal of the American Statistical Association, 76, 833–841.
22.
DunsonD. B. (2000). Bayesian latent variable models for clustered mixed outcomes. Journal of the Royal Statistical Society: Series B, 62, 355–366.
23.
EfronB. (2010). Large-scale inference: empirical Bayes methods for estimation, testing, and prediction. New York, NY: Cambridge University Press.
24.
EnderP. (2011, 7). xtmixed and denominator degrees of freedom: Myth or magic. Stata Conference, Chicago, IL.
25.
EndersC. K. (2010). Applied missing data analysis. New York, NY: Guilford Publications.
26.
FloraD. B.CurranP. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9, 466–491.
27.
FouladiR. T. (2000). Performance of modified test statistics in covariance and correlation structure analysis under conditions of multivariate nonnormality. Structural Equation Modeling, 7, 356–410.
28.
GelmanA. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1, 515–534.
29.
GelmanA.RubinD. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–472.
30.
GhoshJ. K.MukerjeeR. (1992). Non-informative priors (with discussion). In BernardoJ. M.BergerJ. O.DawidA. P.SmithA. F. M. (Eds.), Bayesian statistics4 (pp. 195–2010). Oxford, England: University Press.
HancockG. R.KuoW. L.LawrenceF. R. (2001). An illustration of second-order latent growth models. Structural Equation Modeling, 8, 470–489.
34.
HerzogW.BoomsmaA. (2009). Small-sample robust estimators of noncentrality-based and incremental model fit. Structural Equation Modeling: A Multidisciplinary Journal, 16, 1–27.
35.
HooglandJ. J.BoomsmaA. (1998). Robustness studies in covariance structure modeling: An overview and a meta-analysis. Sociological Methods & Research, 26, 329–367.
36.
HoxJ. (2010). Multilevel analysis: Techniques and applications. Mahwah, NJ: Routledge.
37.
HoxJ.van de SchootR.MatthijsseS. (2012). How few countries will do? Comparative survey analysis from a Bayesian perspective. Survey Research Methods, 6, 87–93.
38.
KennyD. A.McCoachD. B. (2003). Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling, 10, 333–351.
39.
KenwardM. G.RogerJ. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53, 983–997.
40.
KenwardM. G.RogerJ. H. (2009). An improved approximation to the precision of fixed effects from restricted maximum likelihood. Computational Statistics & Data Analysis, 53, 2583–2595.
41.
KruschkeJ. K.AguinisH.JooH. (2012). The time has come Bayesian methods for data analysis in the organizational sciences. Organizational Research Methods, 15, 722–752.
42.
LairdN. M.LouisT. A. (1987). Empirical Bayes confidence intervals based on bootstrap samples. Journal of the American Statistical Association, 82, 739–750.
43.
LairdN. M.WareJ. H. (1982). Random-effects models for longitudinal data. Biometrics, 38, 963–974.
44.
LambertP. C.SuttonA. J.BurtonP. R.AbramsK. R.JonesD. R. (2005). How vague is vague? A simulation study of the impact of the use of vague prior distributions in MCMC using WinBUGS. Statistics in Medicine, 24, 2401–2428.
45.
LeeS. Y.SongX. Y. (2004). Evaluation of the Bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes. Multivariate Behavioral Research, 39, 653–686.
46.
LindleyD. V. (1969). Discussion of compound decisions and empirical Bayes, CopasJ. B.. Journal of the Royal Statistical Society, Series B, 31, 419–421.
47.
LunnD.JacksonC.BestN.ThomasA.SpiegelhalterD. (2013). The BUGS book. Boca Raton, FL: CRC Press.
48.
McArdleJ. J. (1988). Dynamic but structural equation modeling of repeated measures data. In NesselroadeJ. R.CattellR. B. (Eds.), The handbook of multivariate experimental psychology (pp. 561–614). New York, NY: Plenum Press.
McNeishD. M.StapletonL. M. (2014). The effect of small sample size on two level model estimates: A review and illustration. Educational Psychology Review. doi:10.1007/s10648-014-9287-x
MuthénB. (2010). Bayesian analysis in Mplus: A brief introduction. Unpublished manuscript. Retrieved fromhttp://www.statmodel.com
53.
MuthénB. O. (2001). Second-generation structural equation modeling with a combination for categorical and continuous latent variables: New opportunities for latent class/latent curve modeling. In SayersA.CollinsL. (Eds.), New methods for the analysis of change (pp. 291–322). Washington, DC: American Psychological Association.
54.
MuthénL. K.MuthénB. O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling: A Multidisciplinary Journal, 9, 599–620.
55.
MuthénB.SheddenK. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463–469.
56.
NaginD. S. (1999). Analyzing developmental trajectories: a semiparametric, group-based approach. Psychological Methods, 4, 139–157.
57.
NevittJ.HancockG. R. (2004). Evaluating small sample approaches for model test statistics in structural equation modeling. Multivariate Behavioral Research, 39, 439–478.
58.
MalleyA. J.ZaslavskyA. M. (2005). Variance-covariance functions for domain means of ordinal survey items. Survey Methodology, 31, 169–182.
59.
PeughJ. L.EndersC. K. (2005). Using the SPSS mixed procedure to fit cross-sectional and longitudinal multilevel models. Educational and Psychological Measurement, 65, 717–741.
60.
PotthoffR. F.RoyS. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51, 313–326.
61.
PriceL. R. (2012). Small sample properties of Bayesian multivariate autoregressive time series models. Structural Equation Modeling: A Multidisciplinary Journal, 19, 51–64.
62.
RaudenbushS. W.BrykA. S. (2002). Hierarchical linear models: Applications and data analysis methods. Thousand Oaks, CA: Sage.
63.
RobbinsH. (1955). An empirical Bayes approach to statistics. In NeymanJ.. (Ed.), Proceedings of the third Berkeley symposium on mathematical statistics and probability (pp. 157–164). Berkeley: University of California Press.
64.
RobertC. (2007). The Bayesian choice: A decision-theoretic motivation. New York, NY: Springer.
65.
SAS Institute, Inc. (2008). SAS/STAT® 9.2 user’s guide. Cary, NC: Author.
66.
SatterthwaiteF. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2, 110–114.
67.
SavaleiV. (2010). Small sample statistics for incomplete nonnormal data: Extensions of complete data formulae and a Monte Carlo comparison. Structural Equation Modeling, 17, 241–264.
68.
SavaleiV.KolenikovS. (2008). Constrained versus unconstrained estimation in structural equation modeling. Psychological Methods, 13, 150–170.
69.
SchaaljeG. B.McBrideJ. B.FellinghamG. W. (2002). Adequacy of approximations to distributions of test statistics in complex mixed linear models. Journal of Agricultural, Biological, and Environmental Statistics, 7, 512–524.
70.
ScheinesR.HoijtinkH.BoomsmaA. (1999). Bayesian estimation and testing of structural equation models. Psychometrika, 64, 37–52.
71.
SoaresT. M.GonçalvesF. B.GamermanD. (2009). An integrated Bayesian model for DIF analysis. Journal of Educational and Behavioral Statistics, 34, 348–377.
72.
StataCorp. (2013). Stata 13 base reference manual. College Station, TX: Stata Press.
73.
StegmuellerD. (2013). How many countries for multilevel modeling? A comparison of frequentist and Bayesian approaches. American Journal of Political Science, 57, 748–761.
74.
SwainA. J. (1975). Analysis of parametric structures for variance matrices (Unpublished doctoral dissertation). Department of Statistics, University of Adelaide, Australia.
75.
van de SchootR.BroereJ. J.PerryckK. H.Zondervan-ZwijnenburgM.Van LoeyN. E. (2015). Analyzing small data sets using Bayesian estimation: The case of posttraumatic stress symptoms following mechanical ventilation in burn survivors. European Journal of Psychotraumatology, 6, 25216.
76.
van de SchootR.KaplanD.DenissenJ.AsendorpfJ. B.NeyerF. J.AkenM. A. (2014). A gentle introduction to Bayesian analysis: Applications to developmental research. Child Development, 85, 842–860.
77.
WakefieldJ. (2013). Bayesian and frequentist regression methods. New York, NY: Springer.
78.
WuW.WestS. G.TaylorA. B. (2009). Evaluating model fit for growth curve models: Integration of fit indices from SEM and MLM frameworks. Psychological Methods, 14, 183–201.
79.
YuanK.-H. (2005). Fit indices versus test statistics. Multivariate Behavioral Research, 40, 115–148.
80.
ZyphurM. J.OswaldF. L.RuppD. E. (2015). Rendezvous Overdue Bayes analysis meets organizational research. Journal of Management, 41, 387–389.
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