Transcript leuvenmeasurement2008 - Institute for Behavioral Genetics

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
MF
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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
Want ML estimate of this
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ML Factor Score Estimation
• Marginal approach
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L(f&x) = L(f)L(x|f)
L(f) = pdf(f)
L(x|f) = pdf(x*)
x* ~ N(V,Lf)
(1)
• Maximize (1) with respect to f
• Repeat for all subjects in sample
– Works for ordinal data too!
Multifactorial Threshold Model
Normal distribution of liability x. ‘Yes’ when liability x > t
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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 Probability
Example item response probability shown in white
Response
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Probability
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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
Mean
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Mean
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Factor
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Item 1
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Item 2
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Factor
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Item 1
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Sequence of MNI testing
1. Model fx of covariates
on factor mean & variance
2. Model fx of covariates on
factor loadings & thresholds
* If factor loadings equal
3. Revise scale
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beats
2?
No
Yes
2. Identify which loadings &
thresholds are non-invariant
Measurement
invariance: Sum*
or ML scores
MNI: Compute
ML factor scores
using covariates
Continuous Age as a Moderator in the Factor Model
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b - Factor Variance
d - Factor Mean
d
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L5
b
Age
j - Factor Loadings
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Factor
k - Item means
v - Item variances
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[dk and bj confound]
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Stims
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Age
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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)
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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 a
TIFF (LZW) decompressor
are needed to see this picture.
Two measures of performance
• Validity
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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
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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 Theorem
Additive effects of many small factors
1 Gene
2 Genes
3 Genes
4 Genes
 3 Genotypes
 3 Phenotypes
 9 Genotypes
 5 Phenotypes
 27 Genotypes
 7 Phenotypes
 81 Genotypes
 9 Phenotypes
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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
– 0000
– 0001
but 0 0 1 0 would be uncommon
– 0011
– 0111
– 1111
1 234
item threshold
Latent Trait (Factor) Model
F1
Use Gaussian quadrature
weights to integrate over factor;
then relax constraints on
weights
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Discrimination
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Difficulty
Latent Trait (Factor) Model
Difference in model fit:
LRT~ 2
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Discrimination
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Difficulty
Chi-squared test for non-normality performs well
Detecting latent heterogeneity
Scatterplot of 2 classes
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Mean S1|c2
Mean S1|c1
Mean S2|c1
Mean S2|c2
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Scatterplot of 2 classes
Closer means
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Mean S1|c2
Mean S1|c1
Mean S2|c1 Mean S2|c2
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Scatterplot of 2 classes
Latent heterogeneity: Factors or classes?
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Latent Profile Model
Class
Membership
probability
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m1|c1
m1|c1
m2|c1 m3|c1
m3|c1
m2|c1
mp|c1
mp|c1
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Sp|c1
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e1|c2
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Class 1: p
Class 2: (1-p)
Factor Mixture Model
Class
Membership
probability
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Class 1: p
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Class 2: (1- p)
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NB means omitted
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
R>0
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STEM
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Series of bivariate integrals
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m/2
t1i
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Π (  (x , x ) dx dx )j
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0 1 2
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i-1 i-1
Can work with p-variate integration, best if p<m 
“Generalized MML” built into Mx
Dependence 1
Did your use of it cause you physical problems or make
you depressed or very nervous?
Consequence: physical & psychological
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cannabis
cocaine
stimulants
sedatives
opioids
hallucinogens
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Extensions to More Complex
Applications
• Endophenotypes
• Linkage Analysis
• Association Analysis
Basic Linkage (QTL) Model
 = p(IBD=2) + .5 p(IBD=1)
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Pihat
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Q: QTL Additive Genetic
F: Family Environment
E: Random Environment
3 estimated parameters: q, f and e
Every sibship may have different model
Measurement
Linkage (QTL) Model

 = p(IBD=2) + .5 p(IBD=1)
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Pihat
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q f
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Q: QTL Additive Genetic
F: Family Environment
E: Random Environment
3 estimated parameters: q, f and e
Every sibship may have different model
Fulker Association Model
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Multilevel model
for the means
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LDL1
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Measurement Fulker Association Model (SM)
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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