Doubly bounded continuous data are common in the social and behavioral sciences. Examples include judged probabilities, confidence ratings, derived proportions such as percent time on task, and bounded scale scores. Dependent variables of this kind are often difficult to analyze using normal theory models because their distributions may be quite poorly modeled by the normal distribution. The authors extend the beta-distributed generalized linear model (GLM) proposed in Smithson and Verkuilen (2006) to discrete and continuous mixtures of beta distributions, which enables modeling dependent data structures commonly found in real settings. The authors discuss estimation using both deterministic marginal maximum likelihood and stochastic Markov chain Monte Carlo (MCMC) methods. The results are illustrated using three data sets from cognitive psychology experiments.
AitchisonJ. (2003). The statistical analysis of compositional dataLondon, England: The Blackburn Press.
2.
AlickeM. D.GovorunO. (2005). The better-than-average effect. In AlickeM. D.DunningD. A.KruegerJ. I. (Eds.), The self in social judgmentNew York, NY: Psychology Press, 85–108.
3.
AtkinsonA. C. (1994). Fast very robust methods for the detection of multiple outliers. Journal of the American Statistical Association, 89, 1329–1339.
4.
Barndorff-NielsenO. E.JorgensenB. (1991). Some parametric models on the simplex. Journal of Multivariate Analysis, 39, 106–116.
5.
BrehmJ.GatesS. (1993). Donut shops and speed traps: Evaluating models of supervision on police behavior. American Journal of Political Science, 37, 555–581.
6.
BrowneW. J.DraperD. (2006). A comparison of Bayesian and likelihood-based methods for fitting multilevel models. Bayesian Analysis, 1, 473–514.
7.
BrowneW. J.SubramanianS. V.JonesK.GoldsteinH. (2005). Variance partitioning in multilevel logistic models that exhibit over-dispersion. Journal of the Royal Statistical Society, Series A, 168, 599–613.
8.
BuckleyJ. (2002). Estimation of models with beta-distributed dependent variables: A replication and extension of Paolino (2001). Political Analysis, 11, 1–12.
9.
CarrollR. J. (2003). Variances are not always nuisance parameters. Biometrics, 59, 211–220.
10.
ChenM.-H.IbrahimJ. G.ShaoQ.-M.WeissR. E. (2003). Prior elicitation for model selection and estimation in generalized linear mixed models. Journal of Statistical Planning and Inference, 111, 57–76.
CoxC. (1996). Nonlinear quasi-likelihood models: Applications to continuous proportions. Computational Statistics and Data Analysis, 21, 449–461.
13.
CrowderM. J. (1978). Beta-binomial ANOVA for proportions. Applied Statistics, 27, 34–37.
14.
DolnicarS.GrunB. (2007). Cross-cultural differences in survey response patterns. International Marketing Review, 24, 127–143.
15.
EpsinheiraP. L.FerrariS. L. P.Cribari-NetoF. (2008a). On beta regression residuals. Journal of Applied Statistics, 35, 407–419.
16.
EpsinheiraP. L.FerrariS. L. P.Cribari-NetoF. (2008b). Influence diagnostics in beta regression. Computational Statistics and Data Analysis, 52, 4417–4431.
17.
FahrmeirL.TutzG. (2001). Multivariate statistical modelling based on generalized linear modelsNew York, NY: Springer.
18.
FerrariS. L. P.Cribari-NetoF. (2004). Beta regression for modeling rates and proportions. Journal of Applied Statistics, 31, 799–815.
19.
FoxC. R.RottenstreichY. (2003). Partition priming in judgment under uncertainty. Psychological Science, 14, 195–200.
20.
GelmanA. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1, 516–534.
21.
GelmanA.RubinD. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511.
GuptaA. K.NadarajahS. (Eds.). (2004). Handbook of beta distribution and its applicationsNew York, NY: Marcel Dekker.
24.
GurrM. (2009). Partition dependence: Investigating the principle of insufficient reason, uncertainty and dispositional predictors. Unpublished Honours thesis, The Australian National University, Canberra, Australia.
25.
JohnsonT. R. (2003). On the use of heterogeneous thresholds ordinal regression models to account for individual differences in response style. Psychometrika, 68, 563–583.
26.
KieschnickR.McCulloghB. D. (2003). Regression analysis of variates observed on (0,1): Percentages, proportions, and fractions. Statistical Modelling, 3, 193–213.
27.
KumaraswamyP. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46, 79–88.
28.
LongfordN. T. (2001). Simulation-based diagnostics in random-coefficient models. Journal of the Royal Statistical Society, Series A, 164, 259–273.
29.
McAuliffeJ. D.BleiD. M.JordanM. I. (2006). Nonparametric empirical Bayes for the Dirichlet process mixture model. Statistics and Computing, 16, 5–14.
30.
McCullaghP.NelderJ. A. (1989). Generalized linear models2nd ed.London, England: Chapman and Hall.
31.
McCullochC. E.SearleS. R.NeuhausJ. M. (2008). Generalized, linear, and mixed models2nd ed.New York, NY: Wiley.
32.
MerkleE.SmithsonM.VerkuilenJ. (2009). Hierarchical Bayesian beta models for subjective probabilities. Paper presented at the Joint Annual Convention for the Society of Mathematical Psychology and the European Mathematical Psychology Group, Amsterdam, August 1–4.
33.
NoëlY. (2008August). When extreme responses are substantial: A generalized beta response model of behavior change. Paper presented at the Joint Annual Convention for the Society of Mathematical Psychology and the European Mathematical Psychology Group, Amsterdam, the Netherlands.
34.
NoëlY.DauvierB. (2007). A beta item response model for continuous bounded responses. Applied Psychological Measurement, 31, 47–73.
35.
NtzoufrasI. (2009). Bayesian modeling using WinBUGSHoboken, NJ: Wiley.
36.
OspinaR.Cribari-NetoF.VasconcellosK. L. P. (2006). Improved point and interval estimates for a beta regression model. Computational Statistics and Data Analysis, 51, 960–981.
37.
PaolinoP. (2001). Maximum likelihood estimation of models with beta-distributed dependent variables. Political Analysis, 9, 325–346.
38.
PapkeL.WooldridgeJ. (1996). Econometric methods for fractional response variables with an application to 401(K) plan participation rates. Journal of Applied Econometrics, 11, 619–632.
39.
PawitanY. (2001). In all likelihood: Statistical modeling and inference using likelihood. Oxford, UK: Clarendon Press.
40.
PinheiroJ. C.ChaoE. C. (2006). Efficient Laplacian and adaptive Gaussian quadrature algorithms for multilevel generalized linear models. Journal of Computational and Graphical Statistics, 15, 58–81.
41.
QiuZ.SongP. X. K.TanM. (2008). Simplex mixed-effects models for longitudinal proportional data. Scandinavian Journal of Statistics, 35, 577–596.
42.
RoyM.LierschM. (2009). Most people are above average, sometimes. Unpublished manuscript, Elizabethtown College, Elizabethtown, PA.
43.
SAS 9.1.3. (2006). Cary, NC, U.S.A.: SAS Institute, Inc.
44.
SAS 9.2 (2008). Cary, NC, U.S.A.: SAS Institute, Inc.
45.
SeeK. E.FoxC. R.RottenstreichY. S. (2006). Between ignorance and truth: Partition dependence and learning in judgment under uncertainty. Journal of Experimental Psychology: Learning, Memory and Cognition, 32, 1385–1402.
46.
SmithsonM. J.MerkleE.VerkuilenJ. (2009August). Beta regression finite mixture models of polarization and priming. Paper presented at the Joint Annual Convention for the Society of Mathematical Psychology and the European Mathematical Psychology Group, Amsterdam, the Netherlands.
47.
SmithsonM. J.SegaleC. (2009). Partition priming in judgments of imprecise probabilities. Journal of Statistical Theory and Practice, 3, 169–182.
48.
SmithsonM. J.VerkuilenJ. (2006). A better lemon squeezer? Maximum likelihood regression with beta-distributed dependent variables. Psychological Methods, 11, 54–71.
49.
SmythG. K. (1989). Generalized linear models with varying dispersion. Journal of the Royal Statistical Society, Series B, 51, 47–60.
50.
SpiegelhalterD.ThomasA.BestN.LunnD. (2004). WinBUGS Version 1.4.3. Computer software Cambridge, England: Medical Research Council, Biostatistics Unit.
51.
SteigerJ. H. (2001). Driving fast in reverse: The relationship between software development, theory and education in structural equation modeling. Journal of the American Statistical Association, 96, 331–338.
52.
TadikamallaP. R.JohnsonN. L. (1982). Systems of frequency curves generated by transformation of logistic variables. Biometrika, 69, 461–465.
53.
VasconcellosK. L. P.Cribari-NetoF. (2005). Improved maximum likelihood estimation in a new class of beta regression models. Brazilian Journal of Probability and Statistics, 19, 13–31.
54.
WeissR. E.ChoM. (1998). Bayesian marginal influence assessment. Journal of Statistical Planning and Inference, 71, 163–177.
