Abstract
Latent class (LC) analysis is used to construct empirical evidence on the existence of latent subgroups based on the associations among a set of observed discrete variables. One of the tests used to infer about the number of underlying subgroups is the bootstrap likelihood ratio test (BLRT). Although power analysis is rarely conducted for this test, it is important to identify, clarify, and specify the design issues that influence the statistical inference on the number of latent classes based on the BLRT. This paper proposes a computationally efficient ‘short-cut’ method to evaluate the power of the BLRT, as well as presents a procedure to determine a required sample size to attain a specific power level. Results of our numerical study showed that this short-cut method yields reliable estimates of the power of the BLRT. The numerical study also showed that the sample size required to achieve a specified power level depends on various factors of which the class separation plays a dominant role. In some situations, a sample size of 200 may be enough, while in others 2000 or more subjects are required to achieve the required power.
Similar content being viewed by others
References
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723
Bock HH (1996) Probabilistic models in cluster analysis. Comput Stat Data Anal 23:6–28
Cohen J (1988) Statistical power analysis for the behavioral sciences. Lawrence Erlbaum, New Jersey
Collins LM, Lanza ST (2010) Latent class and latent transition analysis: with applications in the social, behavioral, and health sciences. Wiley, New Jersey
Davidson R, MacKinnon JG (2006) The power of bootstrap and asymptotic tests. J Econom 133:421–441
Dias JG, Vermunt JK (2007) Latent class modeling of website users’ search patterns: implications for online market segmentation. J Retail Consum Serv 14:359–368
Everitt BS (1981) A Monte Carlo investigation of the likelihood ratio test for the number of components in a mixture of normal distributions. Multivar Behav Res 16:171–180
Genge E (2014) A latent class analysis of the public attitude towards the euro adoption in Poland. Adv Data Anal Classif 8:427–442
Hartigan JA (1977) Distribution problems in clustering. In: Ryzin JV (ed) Classification and Clustering. Academic Press, New York, pp 45–72
Holt JA, Macready GB (1989) A simulation study of the difference Chi-square statistic for comparing latent class models under violation of regularity conditions. Appl Psychol Meas 13:221–231
Jeffries NO (2003) A note on ’testing the number of components in a normal mixture. Biometrika 90:991–994
Johnson VE, Rossell D (2010) On the use of non-local prior densities in Bayesian hypothesis tests. J R Stat Soc 27:143–170
Langeheine R, Pannekoek J, van de Pol F (1996) Bootstrapping goodness-of-fit measures in categorical data analysis. Sociol Methods Res 24:492–616
Lazarsfeld PF, Henry NW (1968) Latent Structure Analysis. Houghton Mifflin, Boston
Leask SJ, Vermunt JK, Done DJ, Crowd TJ, Blows M, Boks MP (2009) Beyond symptom dimensions: Schizophrenia risk factors for patient groups derived by latent class analysis. Schizophr Res 115:346–350
Lo YT, Mendell NR, Rubin DB (2001) Testing the number of components in a normal mixture. Biometrika 88:767–778
Magidson J, Vermunt JK (2004) Latent class models. In: Kaplan D (ed) The sage handbook of quantitative methodology for the social sciences. Sage Publications, Thousand Oakes, pp 175–198
McLachlan G (1987) On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Appl Stat J R Stat Soc 36:318–324
McLachlan G, Basford K (1988) Mixture models: inference and applications to clustering. Marcel Dekker, New York
McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York
Muthén LK, Muthén BO (1998–2010) Mplus User’s Guide. Sixth Edition, Muthén & Muthén, Los Angeles, CA
Nylund KL, Muthen M, Muthen BO (2007) Deciding on the number of classes in latent class analysis and growth mixture modeling: a monte carlo simulation study. Struct Equ Model 14:535–569
Oberski D (2015) Beyond the number of classes: separating substantive from non-substantive dependence in latent class analysis. Adv Data Anal Classif. doi:10.1007/s11634-015-0211-0
Rindskopf D (2002) The use of latent class analysis in medical diagnosis. Proceedings of the Annual Meeting of the American Statistical Association, American Statistical Association, Alexandria VA, pp 2912–2916
Rubin DB (1981) The Bayesian bootstrap. Ann Stat 9(1):130–134
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464
Self SG, Mauritsen RH, Ohara J (1992) Power calculations for likelihood ratio tests in generalized linear models. Biometrics 48:31–39
Shapiro A (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72:133–144
Steiger JH, Shapiro A, Browne MW (1985) On the multivariate asymptotic distribution of sequential Chi-square statistics. Psychometrika 50:253–263
Takane Y, van der Heijden PGM, Browne MW (2003) On likelihood ratio tests for dimensionality selection. In: Higuchi T, Iba Y, Ishiguro M (eds) Proceedings of Science of Modeling: The 30th Anniversary Meeting of the Information Criterion (AIC). Report on Research and Education 17. The Institute of Statistical Mathematics, Tokyo, pp 348–349
Tekle FB, Tan FEE, Berger MPF (2008) Maximin D-optimal designs for binary longitudinal responses. Comput Stat Data Anal 52(12):5253–5262
Tollenaar N, Mooijaart A (2003) Type I errors and power of the parametric bootstrap goodness-of-fit test: full and limited information. Br J Math Stat Psychol 56:271–288
Van der Heijden PGM, HitHart H, Dessens JAG (1997) A parametric bootstrap procedure to perform statistical tests in latent class analysis. In: Rost J, Langeheine R (eds) Applications of latent trait and latent class models in the social sciences. Waxman Muenster, New York, pp 190–202
Vermunt JK (2010) Latent class modeling with covariates: two improved three-step approaches. Political Anal 18:450–469
Vermunt JK, Magidson J (2008) Manual for latent GOLD 4.5 syntax module. Statistical Innovations Inc, Belmont, MA
Vermunt JK, Magidson J (2013) Latent GOLD 5.0 Upgrade Manual. Statistical Innovations Inc, Belmont, MA
Wolfe JH (1970) Pattern clustering by multivariate mixture analysis. Multivar Behav Res 5:329–350
Zenor MJ, Srivastava RK (1993) Inferring market structure with aggregate data: a latent segment logit approach. J Mark Res 25:369–379
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tekle, F.B., Gudicha, D.W. & Vermunt, J.K. Power analysis for the bootstrap likelihood ratio test for the number of classes in latent class models. Adv Data Anal Classif 10, 209–224 (2016). https://doi.org/10.1007/s11634-016-0251-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11634-016-0251-0