Latent Variable Modeling Summary / Final Thoughts Karen Bandeen-Roche Qian-Li Xue October 28, 2016

Objectives What is a latent variable (LV)? What are some common LV models? What are major features of LV modeling? Hierarchical: structural and measurement components Fitting Evaluating fit Predictions Identifiability Why should I consider using—or decide against using—LV models?

Objectives What is a latent variable (LV)?

“LATENT” “…concepts in their purest form… unobserved or unmeasured … hypothetical” Bollen KA, Structural Equations with Latent Variables, p. 11, 1989 “…in principle or practice, cannot be observed” Bartholomew DJ, The Statistical Approach to Social Measurement, p. 12 “Underlying: not directly measurable. Existing in hidden form but capable of being measured indirectly by observables.” Bandeen-Roche K, Synthesis, 2006

Objectives What is a latent variable (LV)? Measurement is strongest when model linking observables to underlying variables is informed by scientific theory

Objectives What are some common LV models?

Well-used latent variable models Latent variable scale Observed variable scale Continuous Discrete Factor analysis LISREL Discrete FA IRT (item response) Latent profile Growth mixture Latent class analysis, regression General software: MPlus, Latent Gold, WinBugs (Bayesian), NLMIXED (SAS) gllamm (Stata)

Well-used latent variable models Latent variable scale Observed variable scale Continuous Discrete Factor analysis LISREL Discrete FA IRT (item response) Latent profile Growth mixture Latent class analysis, regression

Objectives What are major features of LV modeling? Hierarchical: structural and measurement components Fitting Evaluating fit Predictions Identifiability

Objectives Why should I consider using—or decide against using—LV models? Perhaps the highest reason: measurement properties (reliability, validity)

Advanced topics More models Many of them! Hybrids (ex/ Factor mixture model) Lubke & Muthen, Psych Methods, 2005 Specialties (ex/ latent class logit model for discrete choice data) Greene & Henscher, Transportation Res B, 2003 Scientifically relevant models

Advanced topics Differential measurement Implications for scoring / prediction Study designs Ramifications for risk factor analysis Translation of findings into improved measurement strategies

Advanced topics Novel fitting methods Big data Flexible models Penalized models Houseman, Coull, Betensky, Biometrics, 2006 Leoutsakos et al., Statist Med, 2011 Flexible models Methods that merge model based (latent variable) and data descriptive (robust) features

Advanced topics Novel scoring methods Latent class outcome scoring “Error” correction Croon, Lat Var & Lat Struct Model, 2002 Bartlett-like method Petersen et al., Psychometrika, 2012 Estimating equations approaches Sanchez et al., Ann Appl Stat, 2009 Vermunt, Political Analysis, 2010

Advanced topics Beyond model checking / identifiability Characterization of model family consistent with one’s data Sensitivity analysis

Closing thought: Philosophy Why? To operationalize / test theory To learn about measurement errors, differential reporting They summarize multiple measures parsimoniously To describe population heterogeneity Popperian learning Why not? Their modeling assumptions may determine scientific conclusions Their interpretation may be ambiguous Nature of latent variables? Uniqueness (identifiability) What if very different models fit comparably? (estimability) Seeing is believing Import: They are widely used

Proper use of latent variable models? The complexity of my problem demands it NIH wants me to be sophisticated Reveal underlying truth Operationalize and test theory Model checking is crucial Sensitivity analyses Acknowledge, study issues with measurement; correct attenuation; etc.

Thank you!

Three Excellent Textbooks Bartholomew D, Knott M & Moustaki I. Latent Variable Models and Factor Analysis: A Unified Approach, 3d. Edition. Wiley: London, 2011. Bollen KA. Structural Equations with Latent Variables. Wiley: New York, 1989. Skrondal A, Rabe-Hesketh S. Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equations Models. Chapman & Hall: Boca Raton, 2004.