This article develops a two-part finite mixture quantile regression model for semi-continuous longitudinal data. The proposed methodology allows heterogeneity sources that influence the model for the binary response variable to also influence the distribution of the positive outcomes. As is common in the quantile regression literature, estimation and inference on the model parameters are based on the asymmetric Laplace distribution. Maximum likelihood estimates are obtained through the EM algorithm without parametric assumptions on the random effects distribution. In addition, a penalized version of the EM algorithm is presented to tackle the problem of variable selection. The proposed statistical method is applied to the well-known RAND Health Insurance Experiment dataset which gives further insights on its empirical behaviour.
AitkinM, and AlfòM (1998) Regression models for binary longitudinal responses. Statistics and Computing, 8, 289–307.
2.
AitkinM, and AlfòM (2003) Longitudinal analysis of repeated binary data using autoregressive and random effect modelling. Statistical Modelling, 3, 291–303.
3.
AlfòM, and AitkinM (2000) Random coefficient models for binary longitudinal responses with attrition. Statistics and Computing, 10, 279–87.
4.
AlfòM, and MaruottiA (2010) Two-part regression models for longitudinal zero-inflated count data. Canadian Journal of Statistics, 38, 197–216.
5.
AlfòM, SalvatiN, and RanallliMG (2017) Finite mixtures of quantile and M-quantile regression models. Statistics and Computing, 27, 547–70.
Bagod'Uva T (2006) Latent class models for utilisation of health care. Health Economics, 15, 329–43.
8.
BassettGW, and ChenH-L (2002) Portfolio style: Return-based attribution using quantile regression, pages 293–305. Heidelberg: Physica-Verlag HD.
9.
BasuA, and ManningWG (2009) Issues for the next generation of health care cost analyses. Medical Care, 47, S109–14.
10.
BelascoEJ and GhoshSK (2012) Modelling semi-continuous data using mixture regression models with an application to cattle production yields. The Journal of Agricultural Science, 150, 109–21.
11.
BelottiF, DebP, and NortonEC (2015) Twopm: Two-part models. The Stata Journal, 15, 3–20.
12.
BernardiM, GayraudG, and PetrellaL (2015) Bayesian tail risk interdependence using quantile regression. Bayesian Analysis, 10, 553–603.
13.
BernardiM, BottoneM, and PetrellaL (2018) Bayesian quantile regression using the skew exponential power distribution. Computational Statistics & Data Analysis, 126, 92–111.
14.
BiswasJ, GhoshP, and DasK (2020) A semi-parametric quantile regression approach to zero-inflated and incomplete longitudinal outcomes. AStA Advances in Statistical Analysis, 1–23. doi:10.1007/s10182-020-00362-9
15.
ColeTJ, and GreenPJ (1992) Smoothing reference centile curves: The LMS method and penalized likelihood. Statistics in Medicine, 11, 1305–19.
16.
DebP, and TrivediPK (2002) The structure of demand for health care: Latent class versus two-part models. Journal of Health Economics, 21, 601–25.
17.
DiehrP, YanezD, AshA, HornbrookM, and LinD (1999) Methods for analyzing health care utilization and costs. Annual Review of Public Health, 20, 125–44.
18.
DuanN, ManningWG, MorrisCN, and NewhouseJP (1983) A comparison of alternative models for the demand for medical care. Journal of Business & Economic Statistics, 1, 115–26.
19.
FanJ, and LvJ (2010) A selective overview of variable selection in high dimensional feature space. Statistica Sinica, 20, 101.
20.
FarcomeniA, and VivianiS (2015) Longitudinal quantile regression in the presence of informative dropout through longitudinal-survival joint modelling. Statistics in Medicine, 34, 1199–1213.
21.
FarewellV, LongD, TomB, YiuS, and SuL (2017) Two-part and related regression models for longitudinal data. Annual Review of Statistics and Its Application, 4, 283–315.
22.
FriedmanJ, HastieT, and TibshiraniR (2010) Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33, 1.
23.
GeraciM, and BottaiM (2014) Linear quantile mixed models. Statistics and Computing, 24, 461–79.
24.
GreenPJ (1990) On use of the EM algorithm for penalized likelihood estimation. Journal of the Royal Statistical Society: Series B (Methodological), 52, 443–52.
25.
GrilliL, RampichiniC, and VarrialeR (2016) Statistical modelling of gained university credits to evaluate the role of pre-enrolment assessment tests: An approach based on quantile regression for counts. Statistical Modelling, 16, 47–66.
26.
HendricksW, and KoenkerR (1992) Hierarchical spline models for conditional quantiles and the demand for electricity. Journal of the American Statistical Association, 87, 58–68.
27.
HerasA, MorenoI, and Vilar-ZanonJL (2018) An application of two-stage quantile regression to insurance ratemaking. Scandinavian Actuarial Journal, 2018, 753–69.
28.
IversenT, AasE, RosenqvistG, HakkinenU and on behalf of the EuroHOPE study group (2015) Comparative analysis of treatment costs in EUROHOPE. Health Economics, 24, 5–22.
29.
KoenkerR (2005) Quantile Regression. Cambridge: Cambridge University Press.
30.
KoenkerR, and BassettG (1978) Regression quantiles. Econometrica: Journal of the Econometric Society, 46, 33–50.
31.
KoenkerR, ChernozhukovV, HeX, and PengL (2017) Handbook of Quantile Rregression. Boca Raton, FL: CRC Press.
32.
KoenkerR, and MizeraI (2014) Convex optimization in R. Journal of Statistical Software, 60, 1–23.
33.
KozumiH, and KobayashiG (2011) Gibbs sampling methods for Bayesian quantile regression. Journal of Statistical Computation and Simulation, 81, 1565–78.
34.
LairdN (1978) Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73, 805–11.
35.
LaportaAG, MerloL, and PetrellaL (2018) Selection of value at risk models for energy commodities. Energy Economics, 74, 628–43.
36.
LiuL (2009) Joint modelling longitudinal semi-continuous data and survival, with application to longitudinal medical cost data. Statistics in Medicine, 28, 972–86.
37.
ManningWG, NewhouseJP, DuanN, KeelerEB, and LeibowitzA (1987) Health insurance and the demand for medical care: Evidence from a randomized experiment. The American Economic Review, 77, 251–77.
38.
MarinoMF, and FarcomeniA (2015) Linear quantile regression models for longitudinal experiments: an overview. Metron, 73, 229–47.
39.
MarinoMF, TzavidisN, and AlfòM (2018) Mixed hidden Markov quantile regression models for longitudinal data with possibly incomplete sequences. Statistical Methods in Medical Research, 27, 2231–46.
40.
MaruottiA, and RaponiV (2014) On baseline conditions for zero-inflated longitudinal count data. Communications in Statistics: Simulation and Computation, 43, 743–60.
41.
MaruottiA, RaponiV, and LagonaF (2016) Handling endogeneity and nonnegativity in correlated random effects models: Evidence from ambulatory expenditure. Biometrical Journal, 58, 280–302.
42.
MerloL, PetrellaL, and RaponiV (2020) Sectoral decomposition of CO2 world emissions: A joint quantile regression approach. International Review of Environmental and Resource Economics, 14, 197–239.
43.
MihaylovaB, BriggsA, O'HaganA, and ThompsonSG (2011) Review of statistical methods for analysing healthcare resources and costs. Health Economics, 20, 897–916.
44.
MinY, and AgrestiA (2002) Modelling nonnegative data with clumping at zero: A survey. Journal of the Iranian Statistical Society, 1, 7–33.
45.
MinY, and AgrestiA (2005) Random effect models for repeated measures of zero-inflated count data. Statistical Modelling, 5, 1–19.
46.
MullahyJ (1986) Specification and testing of some modified count data models. Journal of Econometrics, 33, 341–65.
47.
NeelonB, O'MalleyAJ, and SmithVA (2016a) Modelling zero-modified count and semicontinuous data in health services research part 1: Background and overview. Statistics in Medicine, 35, 5070–93.
48.
NeelonB, O'MalleyAJ, and SmithVA (2016b) Modelling zero-modified count and semicontinuous data in health services research part 2: Case studies. Statistics in Medicine, 35, 5094–5112.
49.
PanW, and ShenX (2007) Penalized model-based clustering with application to variable selection. Journal of Machine Learning Research, 8, 1145–64.
50.
PandeyGR and Nguyen V-T-V (1999) A com- parative study of regression based methods in regional flood frequency analysis. Journal of Hydrology, 225, 92–101.
51.
PetrellaL, LaportaAG, and MerloL (2018) Cross-country assessment of systemic risk in the European Stock Market: Evidence from a CoVaR analysis. Social Indicators Research, 146, 1–18.
52.
PetrellaL, and RaponiV (2019) Joint estimation of conditional quantiles in multivariate linear regression models with an application to financial distress. Journal of Multivariate Analysis, 173, 70–84.
53.
PortnoyS, and KoenkerR (1997) The Gaussian hare and the Laplacian tortoise: Computability of squared-error versus absolute-error estimators. Statistical Science, 12, 279–300.
54.
Rabe-HeskethS, SkrondalA, and PicklesA (2005) Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128, 301–23.
55.
ReichBJ, FuentesM, and DunsonDB (2011) Bayesian spatial quantile regression. Journal of the American Statistical Association, 106, 6–20.
56.
RigbyRA, and StasinopoulosDM (2005) Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54, 507–54.
57.
RoystonP, and AltmanDG (1994) Regression using fractional polynomials of continuous covariates: Parsimonious parametric modelling. Journal of the Royal Statistical Society: Series C (Applied Statistics), 43, 429–53.
58.
SauzetO, RazumO, WideraT, and BrzoskaP (2019) Two-part models and quantile regression for the analysis of survey data with a spike. The example of satisfaction with health care. Frontiers in Public Health, 7, 146.
59.
SchwarzG (1978) Estimating the dimension of a model. The Annals of Statistics, 6, 461–64.
60.
SottileG, FrumentoP, ChiodiM, and BottaiM (2020) A penalized approach to covariate selection through quantile regression coefficient models. Statistical Modelling, 20, 369–85.
61.
StädlerN, BühlmannP and Van De GeerS (2010) 1-penalization for mixture regression models. Test, 19, 209–56.
62.
TianF, GaoJ, and YangK (2018). A quantile regression approach to panel data analysis of health-care expenditure in organisation for economic co-operation and development countries. Health Economics, 27, 1921–44.
63.
TianY, TangM, and TianM (2016) A class of finite mixture of quantile regressions with its applications. Journal of Applied Statistics, 43, 1240–52.
64.
TianY, TianM, and ZhuQ (2014) Linear quantile regression based on EM algorithm. Communications in Statistics-Theory and Methods, 43, 3464–84.
65.
TibshiraniR (1996) Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society. Series B, 58, 267–88.
66.
WaldmannE (2018) Quantile regression: A short story on how and why. Statistical Modelling, 18, 203–18.
67.
WassermanL, and RoederK (2009) High dimensional variable selection. Annals of Statistics, 37, 2178.
68.
WinkelmannR (2004) Health care reform and the number of doctor visits: An econometric analysis. Journal of Applied Econometrics, 19, 455–72.
69.
YiC, and HuangJ (2017). Semismooth Newton coordinate descent algorithm for elastic-net penalized Huber loss regression and quantile regression. Journal of Computational and Graphical Statistics, 26, 547–57.
70.
YuK, and MoyeedRA (2001) Bayesian quantile regression. Statistics & Probability Letters, 54, 437–47.
71.
ZhouX-H (2002) Inferences about population means of health care costs. Statistical Methods in Medical Research, 11, 327–39.
