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Measurement models Subjective Well-being in Later Life
Dr. Bram Vanhoutte
CCSR, University of Manchester
5 Dec 2013 - methods@manchester
Latent variables in health inequalities research
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Overview
• Intro – Why do we need measurement models?
• Theory – How do measurement models work?
• Practice – Which programs?
• Example – Subjective wellbeing in later life
• Some conclusions
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A. Intro Why do we need measurement models?
A) Because (social) scientists often work with ‘invisible’, (unobservable, latent) concepts, measured through multiple observed indicators
– Example Depression
• Straightforward question might not “work”
• Refer to symptoms such lack of sleep, feeling down, low energy, feeeling as if everything is an effort, feeling sad…
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A. Intro Why do we need measurement models?
B) Because the way we measure a concept matters, introduces “error” (which can be modelled!)
– Example:
• Acquiescence: tendency to say yes
• Counter with negatively worded items…
• But then meaning of item changes!
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A. Intro Why do we need measurement models?
C) Sets the stage for more advanced questions:
– What’s the structure of the latent concept
• Number of dimensions, hierarchical structure?
– Do different groups answer in the same way
• Equivalence of measures across gender, agegroups, countries
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B. Theory Notation
Latent variables, factors, constructs
Observed variables, measures, indicators, manifest variables
Direction of influence, relationship from one variable to another
Association not explained within the model
η & ξ
x & y
λ & β
ζ Unexplained Error in Model
δ & ε Measurement Errors
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• Two forms of factor analysis, both aimed at reducing (observed) data (into latent constructs)
• EFA is seen as data driven, and CFA as theory driven (BUT)
• EFA useful to determine number of factors and explore which item belongs where.
• This presentation focuses on CFA
B. Theory Exploratory VS Confirmatory?
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A Confirmatory Factor Model
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An Exploratory Factor Model
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1. Define a model
• Map items onto latent concept
• Model error?
2. Test how good a model fits the data
3. Evaluate model
• Substantively
• Statistically
4. Adapt?
B. Theory CFA / Measurement analysis:
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Simple Example: Trust in Individuals
δξΛx x
Trust in Individuals
people aren’t
helpful
(x1)
people can
not be trusted
(x2)
people are
Fair
(x3)
1
ξ1
δ1 δ2 δ3
21212 x
313 x
)( 111 VAR
)(00
)(0
)(
3
2
1
VAR
VAR
VAR
λ11 λ21
11111 x
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B. Theory Simple CFA
• Latent concept(s) and observed measures mapped beforehand
• Usually indicators only load on 1 latent construct (no crossloadings)
• 1 parameter already defined (“fixed”) • Indicator is “caused” by latent concept and
error • Error terms uncorrelated
• Model based on variance covariance matrix
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B. Theory Model Identification
• To estimate the parameters of a model, it needs to be at least just identified.
• This means there are as least as much unknowns (parameters to be estimated) as there are knowns (var and covar)
• Knowns = p(p+1)/2 , where p = number of observed vars
• To estimate model fit, we need over-identification, for ex. 3 indictors for one latent factor
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1. Define a model
• Map items onto latent concept
• Model error?
2. Test how good a model fits the data
3. Evaluate model
• Substantively
• Statistically
4. Adapt?
B. Theory CFA / Measurement analysis:
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• Originally: chi square test + degrees of freedom – With large samples trivial differences become significant
• Absolute measures of fit – How good does model reproduce the data?
– Root Means Square Error of Approximation (RMSEA)
• Good fit =<.08 , Excellent fit =<.06
• Incremental fit indexes – Where is model situated between best model and baseline
model
– Comparative Fit Index (CFI) , good fit >.90 , Excellent >.95
B. Theory Model fit indexes
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1. Define a model
• Map items onto latent concept
• Model error?
2. Test how good a model fits the data
3. Evaluate model
• Substantively
• Statistically
4. Adapt?
B. Theory CFA / Measurement analysis:
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• Model doesn’t seem to fit !
– Double check everything (sample used for estimation/model specified/item coding/…)
– What does theory say?
• Other models?
• Possible measurement effects ?
– What do the stats say?
• Do parameters make sense?
• Is your fit way off, or near the boundaries of acceptability
• Modification indexes flag parameters “under pressure”
– Adapt model ?
B. Theory Model evaluation
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1. Define a model
• Map items onto latent concept
• Model error?
2. Test how good a model fits the data
3. Evaluate model
• Substantively
• Statistically
4. Adapt?
B. Theory CFA / Measurement analysis:
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Simple Example: Trust in Individuals
δξΛx x
Trust in Individuals
people aren’t
helpful
(x1)
people can
not be trusted
(x2)
people are
Fair
(x3)
1
ξ1
δ1 δ2 δ3
21212 x
313 x
)( 111 VAR
)(00
)(0
)(
3
2
1
VAR
VAR
VAR
λ11 λ21
11111 x
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C. Practice
• Different programs can be used for CFA
– Mplus, AMOS, Stata (a bit), LISREL, EQS, R
-->Structural equation modeling software
– Mplus most advanced and many possibilities, but requires some learning and gives extensive output.
– Stata add-on “runmplus”
– Possible to do path analysis, growth curve analysis, multilevel analysis as well
•
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D. Example
• Measuring subjective wellbeing in later life
– Investigate concept of later life wellbeing
– using common measures of well-being
– in a second order factor analysis
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Epicurus/Aristippus Aristotle
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Hedonic well-being
• Philosophical roots in Aristippus of Cyrene, Epicurus, Bentham, Mill
– Well-being is maximalisation of pleasure, minimalisation of suffering
• Affective and cognitive aspect (Diener 1984)
– Both + and – affect, based on moods and emotions
– Individual assessment of quality of life, based on internal criteria (Life satisfaction)
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Hedonic
Well-being
Positive
Affect
Affective Cognitive
+ -
Negative
Affect
CES-D SWLS
Domain
specific Holistic
Hedonic Well-being
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Eudemonic well-being
• Different operationalisations, with similar subdimensions: – Psychological Well-being (Ryff & Singer, 1998) – Self-determination Theory (Ryan & Deci, 2000) – In later life: CASP (Hyde, Wiggins, Higgs & Blane, 2003)
• Philosophical roots in Aristotle: • Well-being is about developing one-self and realising one’s potential (Maslow 1968; Erikson 1959)
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Eudemonic Well-being
Eudemonic
Well-being
Autonomy & Self-
realisation Control Pleasure
CASP CASP15
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Data + methods
• Data:
– English Longitudinal Study of Ageing, age 50+
– Wave 3 Self-completion questionnaire (n=8244)
• Second order cfa
– First establish first order constructs
– Then investigate second order constructs (=factors that determine first order factors)
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Examine first order factors: CES-D • CES-D :
– Theory : Depression in later life is commonly more somatic and less severe, symptoms might also be due to stresses of later life rather than depression
• =>1 or 2 dimensions?
• =>Measurement effects negative items?
CES-D
x1 x2 x6
δ1 δ2 δ6
x7
δ7
… x1 x2 x3
δ1 δ2 δ3
Mood
x4 x5 x6
δ4 δ5 δ6
x7
δ7
OR Somatic
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Outcome CES-D first order analysis
RMSEA CFI
1 factor (Depressive Symptoms) .077 .971
With error correlations .065 .981
2 factors (Somatic / Mood Symptoms) .053 .987
With error correlations .035 .995
•Test different models in a CFA framework •Examine outcomes and fit statistics
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Possible Second order Models
Model 1 Model 2 Model 3 Model 4
GHQ Anxiety
Subjective Well-
being
Hedonic Well-
being
Affective Well-
being
Hedonic Affective
Well-being
GHQ Social
dysfunction
GHQ Loss of
confidence
CES-D Somatic
CES-D Mood
SWLS Present
Cognitive Well-
being
Hedonic Cognitive
Well-being SWLS Past
CASP Control
&Autonomy
Eudemonic Well-
being
Eudemonic Well-
being CASP Self-
Realisation
CASP Pleasure
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Second order Results
RMSEA CFI
Model 1 – 1 dimension of SWB 0.080 0.902
Model 2 – 2 dimensions: hedonic/eudemonic 0.075 0.913
Model 3 – 2 dimensions: affective/cognitive 0.062 0.940
Model 4 – 3 dimensions:
affective/cognitive/eudemonic 0.057 0.951
Although all models show acceptable fit, best fit is three dimensional model
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3 Dimensional Second Order Model
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Conclusions example
• 3 dimensional model of wellbeing in later life, distinguishing emotional, cognitive and eudemonic aspects of wellbeing
• Strong relation between cognitive and eudemonic measures, slightly weaker relationship of both concepts with affective wellbeing
– > satisfaction and autonomy do not necessarily mean the same thing as good mental health
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Conclusions in general Measurement models
• Transforms multiple categorical indicators in
an (interval) latent variable
• Inform us about the structure of the concepts we use and test substantial theory
• Make it possible to model measurement effects and reduce measurement error
• Starting point more than endpoint ?
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Want to learn more ?
• Mplus online tutorials are quite good to learn the ropes
– More info on statmodel.com
• There is a one day short course on latent factor analyses I will be giving on februar 7th
– More info on ccsr.ac.uk/courses