The Measurement and Analysis of Complex Traits

Everything you didn’t want to know about measuring behavioral and psychological constructs

Leuven Workshop August 2008

Overview

• SEM factor model basics

• Group differences: - practical

• Relative merits of factor scores & sum scores

• Test for normal distribution of factor

• Alternatives to the factor model

• Extensions for multivariate linkage & association

Structural Equation Model basics• Two kinds of relationships

– Linear regression X -> Y single-headed– Unspecified Covariance X<->Y double-headed

• Four kinds of variable– Squares – observed variables– Circles – latent, not observed variables

– Triangles – constant (zero variance) for specifying means– Diamonds -- observed variables used as moderators (on paths)

Single Factor Model

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Factor Model with Means

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Factor model essentials

• Diagram translates directly to algebraic formulae

• Factor typically assumed to be normally distributed: SEM

• Error variance is typically assumed to be normal as well

• May be applied to binary or ordinal data– Threshold model

What is the best way to measure factors?

• Use a sum score

• Use a factor score

• Use neither - model-fit

Factor Score Estimation

• Formulae for continuous case– Thompson 1951 (Regression method)– C = LL’ + V– f = (I+J)-1L’V-1x– Where J = L’V-1 L

Factor Score Estimation

• Formulae for continuous case

– Bartlett 1938

– C = LL’ + V

– fb = J-1L’V-1x

– where J = L’V-1 L

• Neither is suitable for ordinal data

Estimate factor score by ML

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Want ML estimate of this

ML Factor Score Estimation• Marginal approach

• L(f&x) = L(f)L(x|f) (1)• L(f) = pdf(f) • L(x|f) = pdf(x*) • x* ~ N(V,Lf)

• Maximize (1) with respect to f• Repeat for all subjects in sample

– Works for ordinal data too!

Multifactorial Threshold ModelNormal distribution of liability x. ‘Yes’ when liability x > t

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Item Response Theory - Factor model equivalence

• Normal Ogive IRT Model

• Normal Theory Threshold Factor Model

• Takane & DeLeeuw (1987 Psychometrika)– Same fit– Can transform parameters from one to

the other

Item Response ProbabilityExample item response probability shown in white

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Do groups differ on a measure?

• Observed– Function of observed categorical variable (sex)– Function of observed continuous variable (age)

• Latent– Function of unobserved variable– Usually categorical – Estimate of class membership probability

• Has statistical issues with LRT

Practical: Find the Difference(s)

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Variance

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Sequence of MNI testing1. Model fx of covariates on factor mean & variance

2. Model fx of covariates on factor loadings & thresholds

2. Identify which loadings & thresholds are non-invariant

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Yes Measurement invariance: Sum* or ML scores

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3. Revise scale

MNI: Compute ML factor scores using covariates

* If factor loadings equal

Continuous Age as a Moderator in the Factor Model

Stims Tranq MJ

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k - Item meansv - Item variances

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What is the best way to measure and model variation in my trait?

• Behavioral / Psychological characteristics usually Likert– Might use ipsative?

• What if Measurement Invariance does not hold?– How do we judge:

• Development• GxE interaction• Sex limitation

• Start simple: Finding group differences in mean

Simulation Study (MK)

• Generate True factor score f ~ N(0,1)• Generate Item Errors ej ~ N(0,1)

• Obtain vector of j item scores sj = L*fj + ej

• Repeat N times to obtain sample• Compute sum score• Estimate factor score by ML

Two measures of performance

• Reliability

QuickTime™ and aTIFF (LZW) decompressor

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Two measures of performance

QuickTime™ and aTIFF (LZW) decompressor

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• Validity

Simulation parameters

• 10 binary item scale

• Thresholds – [-1.8 -1.35 -0.9 -0.45 0.0 0.45 0.9 1.35 1.8]

• Factor Loadings– [.30 .80 .43 .74 .55 .68 .36 .61 .49]

Mess up measurement parameters

• Randomly reorder thresholds

• Randomly reorder factor loadings

• Blend reordered estimates with originals 0% - 100% ‘doses’

Measurement non-invariance

• Which works better: ML or Sum score?

• Three tests:– SEM - Likelihood ratio test difference in latent factor

mean– ML Factor score t-test– Sum score t-test

MNI figures

More Factors: Common Pathway Model

More Factors: Independent Pathway Model

= 1 = 1 = 1

Independent pathway model is submodel of 3 factor common pathway model

Example: Fat MZT MZA

Results of fitting twin

model

ML A, C, E or P Factor Scores• Compute joint likelihood of data and factor scores

– p(FS,Items) = p(Items|FS)*p(FS)– works for non-normal FS distribution

• Step 1: Estimate parameters of (CP/IP) (Moderated) Factor Model

• Step 2: Maximize likelihood of factor scores for each (family’s) vector of observed scores– Plug in estimates from Step 1

Business end of FS script

The guts of it

! Residuals only

Shell script to FS everyone

Central Limit TheoremAdditive effects of many small factors

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Measurement artifacts

• Few binary items• Most items rarely endorsed (floor effect)• Most items usually endorsed (ceiling effect)

• Items more sensitive at some parts of distribution

• Non-linear models of item-trait relationship

Assessing the distribution of latent trait• Schmitt et al 2006 MBR method

• N-variate binary item data have 2N possible patterns

• Normal theory factor model predicts pattern frequencies– E.g., high factor loadings but different thresholds– 0 0 0 0– 0 0 0 1 but 0 0 1 0 would be uncommon– 0 0 1 1– 0 1 1 1– 1 1 1 1 1 2 3 4

item threshold

Latent Trait (Factor) Model

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Difficulty

Use Gaussian quadrature weights to integrate over factor; then relax constraints on weights

Latent Trait (Factor) Model

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Difficulty

Difference in model fit: LRT~ 2

Chi-squared test for non-normality performs well

Detecting latent heterogeneityScatterplot of 2 classes

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Scatterplot of 2 classesCloser means

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Scatterplot of 2 classesLatent heterogeneity: Factors or classes?

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Latent Profile Model

Class 1: p

Class 2: (1-p)

ClassMembership probability

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Factor Mixture Model

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Class 1: p

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ClassMembership probability

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Classes or Traits? A Simulation Study

• Generate data under:– Latent class models– Latent trait models– Factor mixture models

• Fit above 3 models to find best-fitting model– Vary number of factors– Vary number of classes

• See Lubke & Neale Multiv Behav Res (2007 & In press)

What to do about conditional data

• Two things– Different base rates of “Stem” item– Different correlation between Stem and “Probe”

items

• Use data collected from relatives

Data from Relatives: Likely failure of conditional independence

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Can work with p-variate integration, best if p<m “Generalized MML” built into Mx

Dependence 1Did your use of it cause you physical problems or make you depressed or very nervous?

Consequence: physical & psychological

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Extensions to More Complex Applications

• Endophenotypes

• Linkage Analysis

• Association Analysis

Basic Linkage (QTL) Model

Q: QTL Additive Genetic F: Family Environment E: Random Environment3 estimated parameters: q, f and e Every sibship may have different model

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Measurement Linkage (QTL) Model

Q: QTL Additive Genetic F: Family Environment E: Random Environment3 estimated parameters: q, f and e Every sibship may have different model

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Fulker Association Model

Multilevel model for the means

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Measurement Fulker Association Model (SM)M

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Multivariate Linkage & Association Analyses

• Computationally burdensome• Distribution of test statistics questionable• Permutation testing possible

– Even heavier burden• Potential to refine both assessment and genetic models• Lots of long & wide datasets on the way

– Dense repeated measures EMA– fMRI– Need to improve software! Open source Mx