Michael Hallquist, Thomas M. Olino, Paul A. Pilkonis University of Pittsburgh

Transcription

1 Comparing the evidence for categorical versus dimensional representations of psychiatric disorders in the presence of noisy observations: a Monte Carlo study of the Bayesian Information Criterion and Akaike Information Criterion in latent variable models. Michael Hallquist, Thomas M. Olino, & Paul A. Pilkonis University of Pittsburgh

2 Introduction Considerable debate has centered on whether psychopathology is best represented by continuous dimensions or categorical subtypes (First, 2005; Kramer, Noda, & O Hara, 2004). Prior statistical research has focused on the ability of model selection criteria (e.g., the Bayesian Information Criterion) to adjudicate the latent structure of psychometric indicators.

3 Introduction Psychological Methods 2006, Vol. 11, No. 3, Copyright 2006 by the American Psychological Association X/06/$12.00 DOI: / X Information-Theoretic Latent Distribution Modeling: Distinguishing Discrete and Continuous Latent Variable Models Kristian E. Markon and Robert F. Krueger University of Minnesota, Twin Cities Campus Distinguishing between discrete and continuous latent variable distributions has become Markon & Krueger (2006): selection increasingly important in numerous domains of behavioral science. Here, the authors explore an information-theoretic approach to latent distribution modeling, in which the ability of latent distribution models to represent statistical information in observed data is emphasized. criteria (esp. BIC) can be used to probe The authors conclude that loss of statistical information with a decrease in the number of latent values provides an attractive basis for comparing discrete and continuous latent variable models. Theoretical considerations as well as the results of 2 Monte Carlo simulations indicate that information theory provides a sound basis modeling latent distributions and distinguishing between discrete and continuous latent variable models particular. the loss of statistical information resulting Keywords: information theory, latent class, latent trait, model selection from discretizing an underlying dimension. Methods for distinguishing between discrete and continvariable models. The approach has a number of benefits, including explicit formulation and comparison of competing e.g., if latent structure models, is quantification dimensional, of model fit, and situation within a larger latent variable modeling framework. We first review relationships between discrete and continuous latent distribution models and theory regarding in- estimating a few points along a latent formation-theoretic comparison of latent distribution models more generally. may Using be simulations, less we explore the distribution (oligovalued) behavior of information-theoretic criteria in comparisons between discrete and continuous latent distribution models, informative than a parametric under conditions in which the models are correctly and Here, we explore a novel approach to distinguishing be- incorrectly specified. We conclude that information-theoretic approaches provide approximating the dimension (e.g., an attractive basis normal). for comparisons uous latent distributions have become increasingly important in a variety of areas of behavioral science. For example, interest in these methods has grown greatly in clinical psychology, because of the importance of distinguishing between discrete versus continuous phenomena in etiology, assessment, and treatment of psychological disorders (e.g., Cole, 2004). Such methods have also become increasingly important in other areas of psychology and are being applied in other fields of behavioral science as well, such as in the study of language impairment (e.g., Dollaghan, 2004). tween discrete and continuous latent structure. This approach is based on information-theoretic comparison of discrete and continuous latent distribution models within a logistic latent variable model framework. We make a dis- between latent distribution models, including comparisons between discrete and continuous latent distribution models.

4 Introduction Factor mixture models that combine categorical and dimensional structure can often be resolved by model selection criteria (Lubke & Neale, 2005). Resolution of dimensional, categorical, and hybrid latent structure is clearest when 1. indicators are well separated across latent classes 2. there are more indicators 3. there is little unmodeled within-class residual association among indicators

5 Introduction Yet simulation studies have only tested multivariate normal (or skew-normal) data with little attention to the influence of scatter/outliers on latent structure decisions. There has been recent interest in the robustness of clustering methods to data that include scatter (Maitra & Ramler, 2008). In the case of empirical studies of mental disorders, data are rarely normal and heterogeneity sometimes manifests as indicators that are difficult to cluster.

6 Scientific question If mental disorders in the population were sorted by severity (e.g., low, medium, high), would we be able to resolve categorical versus dimensional structure when some observations are noisy? Parameterization: latent profile model (LPM) with diagonal covariance matrices.

7 Model selection criteria Generally: Model log-likelihood Penalty coefficient Number of parameters Akaike Information Criterion (AIC): α = 2 Bayesian Information Criterion (BIC): α = log(n) Finite-sample adjusted AIC (AIC C ) α = 2*[n/(n-κ-1)]

8 Asymptotic properties of AIC and BIC AIC Efficient: tends to select the model that minimizes prediction error as sample size increases. Primarily concerned with minimizing the relative Kullback-Leibler (K-L) distance between a statistical model and the unknown mechanisms giving rise to the observed data. BIC Consistent: true model will be selected with increasing probability as sample size increases. Vrieze 2012; Yang 2005

9 Simulation Conditions Number of latent classes: 3 (equal prob.) Number of indicators: 5 Sample size: 45, 90, 180, 360, 720, 1440 Mean separation:.25,.5, 1.0, 1.5, 2.0, 3.0 Proportion of noisy observations added: 0%, 2.5%, 5%, 10%, 15%, 20%, 25%, 30% Number of replications: 200/condition

10 Simulation methods Within-class multivariate normal data generated using the MixSim package in R. Noise observations added by randomly sampling observations that were outside the 99% ellipsoids of concentration for all clusters (Maitra & Ramler, 2009). Noise observations were uniformly distributed with bounds -1.5*IQR < x < 1.5*IQR

11 Simulation Pipeline For each replication, simulate mvnorm Gaussian mixtures using MixSim in R Add some proportion (0 0.3) of scatter observations outside of 99% ellipsoid bounds Analyze simulated data in Mplus 7.0 using LPM and CFA Summarize and visualize results across conditions

12 Example of clean latent structure n = 720, LC separation = 2.0 SD, 0% Noise

13 Example of noisy latent structure n = 720, LC separation = 2.0 SD, 10% Noise

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22 Discussion When indicators were separated by 1SD or less, scatter observations (esp. > 10%) seriously degraded latent structure. For models where indicators were well separated (2 3 SDs), parameter coverage for LPMs was relatively robust to scatter. Adding scatter to data separated by < 1SD shifted evidence toward categorical model. For indicators separated by > 1SD, adding scatter weakened evidence for categorical model. Greater latent separation was more important than sample size in determining coverage of latent means.

23 Discussion For moderate sample sizes (n = 360 or less) and low latent separation (0.75SD or less) BIC tends to select dimensional model, whereas AIC prefers categorical model regardless of scatter. AIC C fell in the middle, but has a steeper penalty than BIC at small sample sizes.

24 Future directions Exploring how scatter observations are clustered (normalized mutual information). New approaches to scatter: corruption of selected observations by adding random deviate. Testing clustering algorithms (e.g., k-clips; Maitral & Ramler, 2009) that identify scatter and remove these observations from substantive clusters. Testing performance of model selection criteria for dimensional data with scatter.