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Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
SEM 1: Confirmatory Factor AnalysisWeek 3 - Measurement invariance and latent growth models
Sacha Epskamp
16-04-2019
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
What’s missing from thesepictures?
Yerkes argued that histests measured nativeintellectual ability, inother words, innateintelligence which wasunaffected by cultureand educationalopportunities.
http://www.holah.karoo.
net/gouldstudy.htm
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
What’s missing from thesepictures?
Yerkes argued that histests measured nativeintellectual ability, inother words, innateintelligence which wasunaffected by cultureand educationalopportunities.
http://www.holah.karoo.
net/gouldstudy.htm
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
In response to research studying IQ differences between dutchmajority and minority groups:
Again, the subtest with the largest cultural component(Learning Names) shows the largest between-groupdifference. That is, the Learning Names subtest containsseveral Dutch names from various fairy tales, which maybe unfamiliar to children of Moroccan or Turkish descent.
Wicherts, J. M., & Dolan, C. V. (2010). Measurement invariance in
confirmatory factor analysis: An illustration using IQ test performance of
minorities. Educational Measurement: Issues and Practice, 29(3), 39-47.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Measurement invariance
• Plenty of research investigating group differences insum-scores, without taking measurement into account
• Sometimes even followed up by e.g., evolutionary theories onwhy groups differ
• Many pressing research questions can be answered withmeasurement invariance testing
• E.g., should students with dyslexia get more time on exams?
• CFA allow for fine-grained and fair group comparison, withoutrequiring estimated latent variable scores (e.g., sum-scores)!
• Before we can do this, we need to extend CFA withmeanstructure and multi-group analysis.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Including the mean structure:
yyy i = τττ + ΛΛΛηηηi + εεεi
yyy ∼ N(µµµ,ΣΣΣ)
ηηη ∼ N(ααα,ΨΨΨ)
εεε ∼ N(000,ΘΘΘ),
In which τττ (sometimes ννν) is an intercept vector and ααα a vector oflatent means. The model-implied variance–covariance matrix andmeans vectors become:
ΣΣΣ = ΛΛΛΨΨΨΛΛΛ> + ΘΘΘ
µµµ= τττ + ΛΛΛααα
µµµ should resemble sample means yyy as closely possible. Updated fitfunction:
FML = trace(SSSΣΣΣ−1
)+ (yyy −µµµ)>ΣΣΣ−1(yyy −µµµ)− ln |SSSΣΣΣ−1| − p,
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
ΣΣΣ = ΛΛΛΨΨΨΛΛΛ> + ΘΘΘ
µµµ = τττ + ΛΛΛααα
• Number of observations: p(p + 1)/2 variances andcovariances and p means!
• Total number of observations: p(p + 3)/2!
• Number of added parameters:• p intercepts in τττ• Latent means in ααα
• τττ can cancel ααα out, hence we typically need to identify ααα = 000• Or fix enough intercepts in τττ to obtain an identified model
• p more observations, and p more parameters. This is why wenormally ignore means!
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
1 θ11ψ11
λ21
θ22λ31
θ33
η1 y1 ε1
y2 ε2
y3 ε3
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
1 θ11ψ11
λ21
θ22λ31
θ33
0 τ1
τ2
τ3
η1 y1 ε1
y2 ε2
y3 ε31
Typically identified with ααα = 000.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Latent Growth Model
Score of person i on time t:
yti = inti + slopei × t + εti
Since we know t (e.g., 1, 2, 3 . . .), we can treat the intercept andslope as latent variables and plug t into factor loadings!
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
1 1 1 1 1 1 1 2 3 4 5 6
ψ11 ψ22ψ21
θ11 θ22 θ33 θ44 θ55 θ66
α1 α2int slope
yt=1 yt=2 yt=3 yt=4 yt=5 yt=6
ε1 ε2 ε3 ε4 ε5 ε6
1
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Multi-group CFA
Suppose we measure two groups (e.g., control and experimental ormajority and minority population). We can then form two seperatemodels, first for group 1:
ΣΣΣ1 = ΛΛΛ1ΨΨΨ1ΛΛΛ>1 + ΘΘΘ1
µµµ1 = τττ1 + ΛΛΛ1ααα1
With observed variance–covariance matrix SSS1 and observed meansyyy1. For group 2:
ΣΣΣ2 = ΛΛΛ2ΨΨΨ2ΛΛΛ>2 + ΘΘΘ2
µµµ2 = τττ2 + ΛΛΛ2ααα2
With observed variance–covariance matrix SSS2 and observed meansyyy2. Twice as many observations and parameters.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
For m groups with sample size n1, n2, . . . from a total samplen = n1 + n2 + . . ., the fit function follows as:
FML =m∑
g=1
ngnF(g)ML,
in which F(g)ML indicates the fit function in a particular group.
• Number of observations (including mean structure): means,variances and covariances per group!
• m × p × (p + 3)/2
• Number of parameters: Total number of parameters over allgroups
• Parameters constrained equal over groups are only countedonce
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ111
θ111
ψ111
λ211
θ221
λ311
θ331
α11
τ11 τ21 τ31
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ112
θ112
ψ112
λ212
θ222
λ312
θ332
α12
τ12 τ22 τ32
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
Model still needs to be scaled:
• Scaling of variance: λ111 = λ112 = 1 or ψ111 = ψ112 = 1
• Scaling of mean: α11 = α12 = 0
By imposing equality constrains, differences in means and variancesof latent variables may become identified!
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ111
θ111
ψ111
λ211
θ221
λ311
θ331
α11
τ11 τ21 τ31
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ112
θ112
ψ112
λ212
θ222
λ312
θ332
α12
τ12 τ22 τ32
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
Model still needs to be scaled:
• Scaling of variance: λ111 = λ112 = 1 or ψ111 = ψ112 = 1
• Scaling of mean: α11 = α12 = 0
By imposing equality constrains, differences in means and variancesof latent variables may become identified!
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ111
θ111
1
λ211
θ221
λ311
θ331
0
τ11 τ21 τ31
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ112
θ112
1
λ212
θ222
λ312
θ332
0
τ12 τ22 τ32
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
3 means, 3 variances and 3 covariances per group = 18observations. 3 residual variances; 3 factor loadings and 3 intereptsper group = 18 parameters; DF = 0.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
The goal of multi-group CFA is to measure differences in meansand (co)variances (networks) between groups:
Group 1Group 2
ηThat is, we wish to perform tests for homogeneity:
• ααα1 = ααα2 = ααα3 = . . . = ααα
• ΨΨΨ1 = ΨΨΨ2 = ΨΨΨ3 = . . . = ΨΨΨ
Without measurement error, that would be similar MANOVA,Box’s test, Levene’s test, etcetera.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
But...
• The latents means are not identified
• The latent variances are arbitrary due to scaling
• We don’t know if the test measures the same constructs inboth group.
It turns out these problems can both be solved by imposingsequential equality constrains and testing for increasing levels ofmeasurement invariance:
Name Additional constrains Allows to test
Configural invariance Same zeroes in ΛΛΛ1, ΛΛΛ2, . . .Weak invariance ΛΛΛ1 = ΛΛΛ2 = . . . = ΛΛΛ ΨΨΨ1 = ΨΨΨ2 = . . . = ΨΨΨStrong invariance τττ 1 = τττ 2 = . . . = τττ ααα1 = ααα2 = . . . = 000Strict invariance ΘΘΘ1 = ΘΘΘ2 = . . . = ΘΘΘ Full homogeneity
Note: ααα2,ααα3, . . . are identified if strong invariance holds.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
But...
• The latents means are not identified
• The latent variances are arbitrary due to scaling
• We don’t know if the test measures the same constructs inboth group.
It turns out these problems can both be solved by imposingsequential equality constrains and testing for increasing levels ofmeasurement invariance:
Name Additional constrains Allows to test
Configural invariance Same zeroes in ΛΛΛ1, ΛΛΛ2, . . .Weak invariance ΛΛΛ1 = ΛΛΛ2 = . . . = ΛΛΛ ΨΨΨ1 = ΨΨΨ2 = . . . = ΨΨΨStrong invariance τττ 1 = τττ 2 = . . . = τττ ααα1 = ααα2 = . . . = 000Strict invariance ΘΘΘ1 = ΘΘΘ2 = . . . = ΘΘΘ Full homogeneity
Note: ααα2,ααα3, . . . are identified if strong invariance holds.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Measurement Invariance
• Two compare two groups on their mean or variance, it isimperative that the test measures the same construct in bothgroups, and is not biased.
• Group membership may influence the latent trait, but shouldnot influence individual items.
• If measurement invariance does not hold, then test scores(e.g., sum-scores) can not be used to compare groups in ameaningful way
• If partial invariance holds (most parameters are equal acrossgroups), multi-group CFA can be used to test for homogeneityin means and variances
• But interpretation gets harder the more measurementinvariance is violated!
Canonical reference: Mellenbergh, G. J. (1989). Item bias anditem response theory. International journal of educationalresearch, 13(2), 127-143.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Wicherts, J. M., & Dolan, C. V. (2010). Measurement invariance in
confirmatory factor analysis: An illustration using IQ test performance of
minorities. Educational Measurement: Issues and Practice, 29(3), 39-47.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Wicherts, J. M., & Dolan, C. V. (2010). Measurement invariance in
confirmatory factor analysis: An illustration using IQ test performance of
minorities. Educational Measurement: Issues and Practice, 29(3), 39-47.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ111
θ111
1
λ211
θ221
λ311
θ331
0
τ11 τ21 τ31
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ112
θ112
1
λ212
θ222
λ312
θ332
0
τ12 τ22 τ32
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
Configural invariance: Does the same model fit in both groups?
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
# Load the package:
library("lavaan")
# Load data:
data("HolzingerSwineford1939")
Data <- HolzingerSwineford1939
# Model:
Model <- '
visual =~ x1 + x2 + x3 + x9
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
x3 ~~ x5
'
# Fit configural:
conf <- cfa(Model, Data, group = "school")
fitMeasures(conf,
c("rmsea","cfi","tli","rni","rfi","ifi","srmr","gfi"))
## rmsea cfi tli rni rfi ifi srmr gfi
## 0.069 0.964 0.942 0.964 0.871 0.965 0.049 0.997
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ11
θ111
1
λ21
θ221
λ31
θ331
0
τ11 τ21 τ31
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ11
θ112
ψ112
λ21
θ222
λ31
θ332
0
τ12 τ22 τ32
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
Weak Invariance: Are factor loadings the same? — Differences invariance (and covariances) of latent variable now interpretable!
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
weak <- cfa(Model, Data, group = "school",
group.equal = "loadings")
anova(conf, weak)
## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## conf 44 7452.2 7689.4 75.646
## weak 51 7444.6 7655.9 82.094 6.4487 7 0.4884
(7 DF difference: −7 free factor loadings)
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ11
θ111
1
λ21
θ221
λ31
θ331
0
τ1 τ2 τ3
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ11
θ112
ψ112
λ21
θ222
λ31
θ332
α12
τ1 τ2 τ3
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
Strong Invariance: Are the intercepts the same? — Latent meandifference now identified!
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
strong <- cfa(Model, Data, group = "school",
group.equal = c("loadings","intercepts"))
anova(conf, weak, strong)
## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## conf 44 7452.2 7689.4 75.646
## weak 51 7444.6 7655.9 82.094 6.449 7 0.4884
## strong 57 7473.2 7662.2 122.638 40.544 6 3.561e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(6 DF difference: −9 intercepts, but +3 latent variable means)
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Where is the misfit?
# Group 1:
lavInspect(strong, "mu")[[1]] -
lavInspect(strong, "sampstat")[[1]]$mean
## x1 x2 x3 x9 x4 x5 x6 x7 x8
## 0.053 0.159 -0.227 0.051 -0.035 0.066 0.009 -0.182 0.077
# Group 2:
lavInspect(strong, "mu")[[2]] -
lavInspect(strong, "sampstat")[[2]]$mean
## x1 x2 x3 x9 x4 x5 x6 x7 x8
## -0.057 -0.130 0.166 -0.041 0.028 -0.032 -0.015 0.132 -0.064
Some other options are possile (e.g., ‘lavTestScore‘ in lavaan).
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Free two intercepts:
strong2 <- cfa(Model, Data, group = "school",
group.equal = c("loadings","intercepts"),
group.partial = c("x3 ~ 1","x7 ~ 1"))
anova(conf, weak, strong, strong2)
## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## conf 44 7452.2 7689.4 75.646
## weak 51 7444.6 7655.9 82.094 6.449 7 0.4884
## strong2 55 7442.4 7638.9 87.846 5.752 4 0.2185
## strong 57 7473.2 7662.2 122.638 34.792 2 2.786e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
λ11
θ11
1
λ21
θ22
λ31
θ33
0
τ1 τ2 τ3
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 1
λ11
θ11
ψ112
λ21
θ22
λ31
θ33
α12
τ1 τ2 τ3
η1
y1
ε1
y2
ε2
y3
ε3
1
Group 2
Strict Invariance: Are the residual variances the same?
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
strict <- cfa(Model, Data, group = "school",
group.equal = c("loadings","intercepts","residuals",
"residual.covariances"),
group.partial = c("x3 ~ 1","x7 ~ 1"))
anova(strong2, strict)
## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## strong2 55 7442.4 7638.9 87.846
## strict 65 7441.9 7601.3 107.398 19.552 10 0.03379 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
10 less parameters (residual variances). Strict invariance holdsadequatly (although debatable).
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Do the variances differ?
eqvars <- cfa(Model, Data, group = "school",
group.equal = c("loadings","intercepts","residuals",
"residual.covariances","lv.variances"),
group.partial = c("x3 ~ 1","x7 ~ 1"))
anova(strict, eqvars)
## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## strict 65 7441.9 7601.3 107.40
## eqvars 68 7437.5 7585.8 108.95 1.5561 3 0.6694
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
And the means?
eqmeans <- cfa(Model, Data, group = "school",
group.equal = c("loadings","intercepts","residuals",
"residual.covariances","lv.variances",
"means"),
group.partial = c("x3 ~ 1","x7 ~ 1"))
anova(eqvars, eqmeans)
## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## eqvars 68 7437.5 7585.8 108.95
## eqmeans 71 7460.6 7597.8 138.10 29.141 3 2.092e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Does the model fit still?
fitMeasures(eqvars,
c("rmsea","cfi","tli","rni","rfi","ifi","srmr","gfi"))
## rmsea cfi tli rni rfi ifi srmr gfi
## 0.063 0.954 0.951 0.954 0.880 0.954 0.067 0.995
Adequately, we may now substantively interpret these groupdifferences between the schools:
lavInspect(eqvars, "est")[[2]]$alpha
## intrcp
## visual 0.035
## textual 0.574
## speed -0.082
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Conclusion
• Means can be added to the CFA model
• In a single group, this can be used e.g., to model growth
• In multiple-group CFA, a CFA model is fitted to several groups
• Measurement invariance can be assessed stepwise:
Name Additional constrains Allows to test
Configural invariance Same zeroes in ΛΛΛ1, ΛΛΛ2, . . .Weak invariance ΛΛΛ1 = ΛΛΛ2 = . . . = ΛΛΛ ΨΨΨ1 = ΨΨΨ2 = . . . = ΨΨΨStrong invariance τττ 1 = τττ 2 = . . . = τττ ααα1 = ααα2 = . . . = 000Strict invariance ΘΘΘ1 = ΘΘΘ2 = . . . = ΘΘΘ Full homogeneity
Note: ααα2,ααα3, . . . are identified if strong invariance holds.
Introduction Mean-structure Latent growth Multi-group CFA Measurement invariance Conclusion
Thank you for your attention!