Michael C Neale Shaunna Clark NIDA Workshop VIPBG/VCU ...
Michael C Neale Shaunna Clark NIDA Workshop VIPBG/VCU ...
Michael C Neale Shaunna Clark NIDA Workshop VIPBG/VCU ...
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Tuesday, October 23, 12<br />
Measurement<br />
Invariance<br />
<strong>Michael</strong> C <strong>Neale</strong><br />
<strong>Shaunna</strong> <strong>Clark</strong><br />
<strong>NIDA</strong> <strong>Workshop</strong> <strong>VIPBG</strong>/<strong>VCU</strong><br />
October 23 2012
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•<br />
•<br />
Tuesday, October 23, 12<br />
Measurement Invariance<br />
What is it, and why should I care?<br />
How does one detect it?<br />
Is it possible to correct for it?<br />
•<br />
If so, how?<br />
Specify set of structural equation models
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Tuesday, October 23, 12<br />
Measurement Invariance<br />
Want to measure same thing in different<br />
populations<br />
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•<br />
•<br />
•<br />
•<br />
Males & females<br />
Young & old<br />
Those with genotypes AA, Aa and aa<br />
Cases & Controls<br />
Occult heterogeneity
•<br />
Tuesday, October 23, 12<br />
Are Sum Scores Sufficient?<br />
If and only if the following conditions hold:<br />
1. The items are unidimensional<br />
- only one latent trait underlies the scores<br />
on the set of items (or symptoms), and<br />
conditional on this latent trait, the items<br />
are statistically independent
Tuesday, October 23, 12<br />
Sum Score Sufficiency 2<br />
2. Expected values of the item responses have<br />
identical functional relations to the latent<br />
trait<br />
- Implies equal factor loadings in linear<br />
latent factor models for continuous items<br />
or equal discrimination parameters in<br />
item-response theory models for<br />
dichotomous items
Sum Score Sufficiency 3<br />
3. Variance not explained by the latent trait<br />
- Can be difficult to assess with binary items<br />
Tuesday, October 23, 12<br />
(residual variance) is equal for all items.
- Paper given by Roger Millsap SMEP 2012<br />
- Monotonic as long as residuals of observed<br />
- Monotonicity is not sufficient for sufficiency<br />
Tuesday, October 23, 12<br />
Sum Scores Monotonic with<br />
Factor Scores?<br />
measures are conditionally independent of<br />
the latent factors
Table 1<br />
Testing for MI Measurement invariance 29<br />
Equality constraints imposed across groups in steps towards strict factorial invariance<br />
No. Description factor loadings residual variances intercepts factor means<br />
1 Configural invariance free free free fixed at 0<br />
2 Metric/weak invariance invariant free free fixed at 0<br />
3 Equal residual variances invariant invariant free fixed at 0<br />
4 Strict factorial invariance invariant invariant invariant free 1<br />
Note: Each step is nested under the previous one; Underlined restrictions are tested in each<br />
Dolan, C. V., Oort, F. J., Stoel, R. D., and Wicherts, J. M. (2009). Testing<br />
Measurement Invariance in the Target Rotated Multigroup<br />
Exploratory Factor Model. Structural Equation Modeling, 16(2):295–<br />
314.<br />
Wicherts J & Dolan CV (In Press) Educational Measurement: Issues<br />
step; free: freely estimated within each group; invariant: parameters estimated equally across<br />
groups; Factor (co)variances are freely estimated throughout. 1 Modeled as between-group<br />
differences in factor means by restricting factor means in one arbitrary group to equal zero.<br />
and Practice<br />
Tuesday, October 23, 12
Tuesday, October 23, 12<br />
Simple Single Factor Model<br />
ψ 1<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
ψ 2<br />
μ 3<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3
ψ 1<br />
Tuesday, October 23, 12<br />
μ 1<br />
V 1<br />
1<br />
Strict Factorial Invariance<br />
λ 11<br />
μ 2<br />
Males Females<br />
ψ 2<br />
μ 3<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3<br />
ψ 1<br />
μ 1<br />
1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
μ F<br />
ψ 2<br />
μ 3<br />
λ 21<br />
V F<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3
Failure of Configural Invariance<br />
ψ 1<br />
Tuesday, October 23, 12<br />
λ 12<br />
μ 1<br />
1<br />
F2<br />
V 1<br />
1<br />
Males Females<br />
λ<br />
22<br />
λ<br />
31 λ21<br />
μ 2<br />
ψ 2<br />
μ 3<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3<br />
ψ 1<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
ψ 2<br />
μ 3<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3
ψ 1<br />
Tuesday, October 23, 12<br />
μ 1<br />
Failure of Metric Invariance<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
ψ 2<br />
μ 3<br />
Males Females<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3<br />
ψ 1<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
ψ 2<br />
μ 3<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3
ψ 1<br />
Tuesday, October 23, 12<br />
Failure of Residual Invariance<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
ψ 2<br />
μ 3<br />
Males Females<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3<br />
ψ 1<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
ψ 2<br />
μ 3<br />
λ 21<br />
1<br />
F<br />
V 2<br />
λ 31<br />
ψ 3<br />
V 3
Tuesday, October 23, 12<br />
Continuous Variable (Age) Invariance<br />
Single Factor Model Moderated Means<br />
ψ 1<br />
1<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
μ 2<br />
δ F<br />
μ F<br />
ψ 2<br />
μ 3<br />
λ 21<br />
D F<br />
F<br />
V 2<br />
Age i<br />
Age i<br />
δ 1<br />
β F<br />
λ 31<br />
ψ 3<br />
δ 2<br />
1<br />
L<br />
V 3<br />
δ 3<br />
D S
Tuesday, October 23, 12<br />
Continuous Variable (Age) Invariance<br />
Single Factor Model Moderated Means and Variances<br />
ψ 1<br />
1<br />
μ 1<br />
V 1<br />
1<br />
λ 11<br />
δ F<br />
μ F<br />
W<br />
δ 3<br />
1.0<br />
β 3<br />
D F<br />
F<br />
Age i<br />
D S<br />
μ 3<br />
β F<br />
λ 31<br />
Age i<br />
1<br />
L<br />
V 3<br />
ψ 3
Three methods of scoring<br />
• Sum score<br />
• Simple & Practical<br />
• Widely Used<br />
• Maximum likelihood factor score<br />
• More complex (need computer)<br />
• Less widely used<br />
• Can test assumptions<br />
• Neither - use SEM framework for testing<br />
Tuesday, October 23, 12
Tuesday, October 23, 12<br />
Non-Invariance Effects<br />
Sum Scores vs. ML Factor Scores 26
3. Revise scale<br />
Tuesday, October 23, 12<br />
Sequence of MNI testing<br />
1. Model effects of covariates<br />
on factor mean & variance<br />
2. Model effects of covariates<br />
on factor loadings & thresholds<br />
1 beats<br />
2?<br />
No<br />
2. Identify which loadings &<br />
thresholds are non-invariant<br />
Yes<br />
Measurement<br />
invariance: Sum<br />
or MLE z-scores<br />
MNI: Compute<br />
ML factor scores<br />
using covariates<br />
18
Tuesday, October 23, 12<br />
Estimates of (a) Factor Loadings and (b)<br />
Thresholds of Nicotine Dependence Items<br />
Plotted by Gender and Measurement Instrument<br />
(FTQ or FTND Scale)
Tuesday, October 23, 12<br />
Estimated Nicotine Dependence Item<br />
Characteristic Curves for 20-Year-Old<br />
Females
Psychometric Factors Model<br />
Tuesday, October 23, 12<br />
F<br />
Twins' factors correlate; so do their residuals<br />
a 1<br />
A C E A C E<br />
M1 M2 M3 M4 M5 M6<br />
M1 M2 M3 M4 M5 M6<br />
A C E A C E A C E A C E A C E A C E A C E A C E A C E A C E A C E A C E<br />
F
Tuesday, October 23, 12<br />
Advantages<br />
• Multiple groups<br />
• Test for equality of loadings<br />
• Test for equality of thresholds<br />
• Test for equal factor means<br />
• Test for equal factor variances<br />
• Can handle ordinal items<br />
• Can deal with missing data ‘CCC’ model<br />
• Structured clinical interviews
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Tuesday, October 23, 12<br />
Disadvantages<br />
Gets cumbersome with multiple latent<br />
factors<br />
Gets slow with lots of latent factors<br />
Gets slow if blocks of non-independent<br />
items get large (e.g., large pedigrees)
Tuesday, October 23, 12<br />
Sex limitation<br />
• Common BG questions<br />
• Are genetic/environmental variance components<br />
equal for males & females<br />
• Do same genetic/common environmental factors<br />
influence males & females<br />
• Common psychometric question:<br />
• Do items perform equivalently in males and females<br />
• Measurement invariance
•<br />
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Tuesday, October 23, 12<br />
Psychometric Sex Limitation<br />
What are the implications of failure of MI for<br />
tests of scalar sex-limitation?<br />
What are the implications of MI failure for<br />
non-scalar sex-limitation?<br />
How should we resolve any implications<br />
encountered?
Tuesday, October 23, 12<br />
Standard practice in BG<br />
• Multiple item questionnaire<br />
• Compute factor score<br />
• Compute sum score<br />
• Clinical interview<br />
• Ask stem items<br />
• Ask probe items if stems met<br />
• Use DSM or other criteria to diagnose<br />
disorder<br />
• Use endophenotype measures on calibrated<br />
quantitative scale
The Gory Details<br />
Implications of absence of measurement invariance for detecting sex<br />
limitation and genotype by environment interaction<br />
Tuesday, October 23, 12<br />
Gitta H. Lubke 3 Conor V. Dolan 2 <strong>Michael</strong> C. <strong>Neale</strong> 1<br />
1 Virginia Commonwealth University 2 University of Amsterdam<br />
3 University of Notre Dame<br />
Twin Research 2004 (June)
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Tuesday, October 23, 12<br />
Definition of Measurement<br />
Mellenbergh 2002<br />
Invariance<br />
Conditional on factor scores (eta), observed<br />
scores Y have identical distribution
Tuesday, October 23, 12<br />
Item scores<br />
Sum score S f
Tuesday, October 23, 12<br />
Predicted correlation for sum<br />
scores
Predicted Covariances<br />
Same effect on covariances of difference in km vs kf<br />
Tuesday, October 23, 12<br />
as for difference am vs af
Opposite sex DZ twin pair<br />
Male<br />
Loadings<br />
Tuesday, October 23, 12<br />
M<br />
Twins' factors correlate; so do their residuals<br />
a 1<br />
A C E A C E<br />
M1 M2 M3 M4 M5 M6<br />
M1 M2 M3 M4 M5 M6<br />
A C E A C E A C E A C E A C E A C E A C E A C E A C E A C E A C E A C E<br />
F<br />
Female<br />
Loadings
Simulation<br />
MZ f DZ f MZ m DZ m DZ o<br />
sum score .18 (.07) .09 (.07) .42 (.06) .21 (.07) .14 (.05)<br />
unequal loadings .50 (.20) .25 (.19) .50 (.06) .25 (.08) .25 (.09)<br />
Factor loadings .3 .4 .3 .4 for males .6 .7 .8 .9 for<br />
Tuesday, October 23, 12<br />
Estimated Twin Correlations: Sum Score vs. Multivariate Analysis<br />
females
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Tuesday, October 23, 12<br />
<strong>Shaunna</strong> <strong>Clark</strong>...<br />
Example Script
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Tuesday, October 23, 12<br />
Obtain the MI script<br />
Practical<br />
Maybe run it; maybe not<br />
Interpret the Output<br />
Consider possible modifications
Change language