Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan
Analyzing Effects with Non-Linear StructuralEquation Mixture Modeling in blavaan
Introduction
Mayer, A., Umbach, N., Flunger, B., & Kelava, A. (2017). Effect Analysis Using Non-Linear Structural Equation Mixture Modeling. StructuralEquation Modeling.
The EffectLiteR-NSEMM Approach:• Definitions of average and conditional effects of a treatment/intervention• Computations of average and conditional effects (based on model parameters)• Latent continous and discrete (mixture) variables• Higher-order interactions, quadratic or more general non-linear relationships• Based on a Bayesian model• Open Source Software• Greater insight into the effectiveness of the intervention (e.g., for situations where the
treatment is non-effective for extreme values of a latent covariate but is effective for mediumvalues)
2 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Introduction
Example• Latent outcome variable η measured by multiple indicators• Latent covariate ξ measured by multiple indicators• Binary treatment variable X with values 0 and 1
For example, we consider regressions such as the followingE (η | X , ξ) = γ00 + γ01 ξ + γ02 ξ
2 + γ10 X + γ11 X ξ + γ12 X ξ2 ,
⇒ Other forms of non-linearity are possible as well
3 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Introduction
X = 0
ξ
Y11
Y21
Y31
Y41
ε11
ε21
ε31
ε41
η
Y12
Y22
ξ2
ε12
ε22
1
λ1
λ2
λ3
1
1
α01
α02 ζ0
X = 1
ξ
Y11
Y21
Y31
Y41
ε11
ε21
ε31
ε41
η
Y12
Y22
ξ2
ε12
ε22
1
λ1
λ2
λ3
1
1
α11
α12 ζ0
4 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Average Effects in a Manifest Moderated Regression Model
Taking one step back, consider a manifest moderated regression model:
E (Y | X , Z ) = γ00 + γ01 Z + γ10 X + γ11 X Z
• The average effect AE10 of treatment condition X = 1 vs. X = 0 is defined asAE10 = E
[E (Y | X = 1,Z )− E (Y | X = 0,Z )
]• It can be computed based on regression coefficients and the unconditional expectation of Z :
AE10 = γ10 + γ11 E (Z )• A more common way to obtain the average effect is to mean center the continuous covariate
Z• In linear models, both strategies give the exact same average effect but potentially different
standard errors• A conditional treatment effect is defined as CE10(z) = γ10 + γ11 z
⇒ It is not as straightforward to extend it to models with conditional non-linearitiesand latent variables
5 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Quadratic and Other Non-Linear Terms
Conditional linearity assumption:• The conditional linearity assumption in the classic moderated regression model may be wrong• In such cases, we may erroneously conclude based on the conditional linearity model that
there are no differential treatment effects
For conditional quadratic relationships, a better model is:E (Y | X , Z ) = γ00 + γ01 Z + γ02 Z 2 + γ10 X + γ11 X Z + γ12 X Z 2
• The general definition of average and conditional effects remain the same• AE10 = γ10 + γ11 E (Z ) + γ12 E (Z 2).• It is no longer as easy to use mean centering to obtain average effects• The squared centered covariate no longer has mean zero• Double mean centering can be used but does not generalize to more complex forms of
non-linearity and latent variables
6 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Latent Variables and Multigroup SEM
Latent Variables:• We oftentimes have latent variables in the social sciences• The manifest models are not easy to generalize to latent variable models• Complex interactions involving latent variables require specialized modeling techniques
We distinguish four scenarios for products of different types of variables with differentconsequences for modeling:(a) Product of a continuous latent variable and a continuous (latent) variable: Distribution
analytic approaches, product indicator approaches, Bayesian approaches, factor scores(b) Product of a continuous latent variable and a categorical manifest variable: Multigroup
structural equation model(c) Product of a categorical latent variable and a continuous (latent) variable: Latent class
model (direct and indirect approaches)(d) Product of a categorical latent variable and a categorical manifest variable: Combination of
the multigroup approach and the latent class model
7 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
The EffectLiteR Approach
Mayer, A., Dietzfelbinger, L., Rosseel, Y., & Steyer, R. (2016). The EffectLiteR Approach for Analyzing Average and Conditional Effects.Multivariate Behavioral Research.
In short, we define a conditional effect function whose values are the conditionaleffects of intervention X = t, t = 1, 2, . . . , p, compared to the control group X = 0:
CEt0(K ,Z ) = E (η | X = t,K ,Z )− E (η | X = 0,K ,Z )
Average and conditional effects:E [CEt0(K ,Z )] Average effect (AVE )CEt0(k, z) Conditional effect given a value z of Z and a value k of KE [CEt0(K ,Z ) | X = t∗] Effect given X = t∗ (Effect on the treated)E [CEt0(K ,Z ) | X = 0] Effect on the non-treatedE [CEt0(K ,Z ) |K = k] Effect given a value K = kE [CEt0(K ,Z ) | Z = z ] Effect given a value Z = zE [CEt0(K ,Z ) | V = v ] Effect given a value V = v , where V = f (X ,K ,Z )
8 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
The EffectLiteR Approach
Group-invariant measurement modely = ν + Λη + ε
Structural model for group (X = x ,K = k)η = αxk + Bxk η + ζ
=
αxk0µxk1µxk2...µxkq
+
0 αxk1 αxk2 . . . αxkq0 0 0 . . . 00 0 0 . . . 0... ... ... . . . ...0 0 0 . . . 0
ηξ1ξ2...ξq
+
ζ0ζ1ζ2...ζq
Group sizes for group (X = x ,K = k)
P(X = x ,K = k) = f (κxk) ,
9 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
The NSEMM Approach
The NSEMM model consists of a measurement model and a structural model. Bothof them can potentially be latent class-specific, as in traditional structural equationmixture modeling (SEMM). In addition to classic SEMM, the NSEMM approach addsnon-linear terms for the latent predictors ξ in the structural model, so that theresulting model equations are
Measurement model for latent class cy = νc + Λc η + ε
Structural model for latent class cη = αc + Bc η + Γc h(ξ) + ζ ,
⇒ h(·) is a function that maps the vector ξ to a vector of product terms of ξ (e.g.,interactions or quadratic terms)
10 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
The Combined Model
In order to combine the EffectLiteR and the NSEMM approaches, we need acombination of latent classes and observed groups. The variable C∗ can be considereda more general version of K (from ELR) also including latent classes.
Invariant measurement modely = ν + Λη + ε
Structural modelη = αxc∗ + Bxc∗ η + Γxc∗ h(ξ) + ζ
Model for group sizesP(X = x ,C∗ = c∗) = f (κxc∗) .
⇒ We assume a group- and class-invariant measurement model. For group sizesdifferent models are possible, including multinomial models, Poisson models, orBayesian approaches.
11 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Illustrative Example
Research Question:• We use an empirical example from a quasi-experimental field study in 16 ninth grade physics
classrooms with one experimental group and one control group• We want to examine how effective the treatment (manipulated autonomy support by the
teacher) is in reducing students’ state of boredom and whether that treatment effectdepends on students’ self-efficacy• We modeled the relationship between boredom and self-efficacy in each treatment group as
quadratic• We hypothesize a u-shaped effect function, where the treatment is most effective in reducing
boredom for medium values of self-efficacy and less effective for extreme values (high or low)of self-efficacy
12 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Illustrative Example
Measures:• Boredom: We used the latent variable boredom as the dependent variable η. It was
measured by a scale with two indicators. The response format was a five-point Likert scalewhich ranged from 1 (not at all true) to 5 (completely true).• Treatment Variable: The autonomy-supportive intervention X encompassed both a training
for the physics teacher and a standardized teaching unit for the whole class of students. Thephysics teachers in the control condition did not receive training and the teaching unit forthe control condition did not entail autonomy-supportive strategies.• Self-efficacy: The scale ‘self-efficacy’ consisted of four items. We modeled a latent predictorξ with these four indicators.
13 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Illustrative Example
X = 0
ξ
Y11
Y21
Y31
Y41
ε11
ε21
ε31
ε41
η
Y12
Y22
ξ2
ε12
ε22
1
λ1
λ2
λ3
1
1
α01
α02 ζ0
X = 1
ξ
Y11
Y21
Y31
Y41
ε11
ε21
ε31
ε41
η
Y12
Y22
ξ2
ε12
ε22
1
λ1
λ2
λ3
1
1
α11
α12 ζ0
14 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Illustrative Example
2
3
4
5
1 2 3 4 5Self−Efficacy (Xi)
E(e
ta|X
=0,
Xi)
Control Group
2
3
4
5
1 2 3 4 5Self−Efficacy (Xi)
E(e
ta|X
=1,
Xi)
Treatment Group
15 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Illustrative Example
The effect model is based on the regression:E (η | X , ξ) = γ00 + γ01 ξ + γ02 ξ
2 + γ10 X + γ11 X ξ + γ12 X ξ2 ,
Because the chosen functional form of both treatment-group specific regressions isquadratic, the effect function, which is their difference, is also quadratic:
CE10(ξ) = E (η | X = 1, ξ)− E (η | X = 0, ξ)= γ10 + γ11 ξ + γ12 ξ
2
Unconditional expectations of the covariate (needed for average effects) can becomputed:
E (ξ) = α00 · P(C = 0) + α10 · P(C = 1) ,E (ξ2) = E (ξ2 | C = 0) · P(C = 0) + E (ξ2 | C = 1) · P(C = 1) .
16 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Illustrative Example
−1
0
1
2
1 2 3 4 5Self−Efficacy (Xi)
Con
ditio
nal T
reat
men
t−E
ffect
Plot of the Effect Function
17 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Results
18 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Results
19 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Results
Results:• By visual inspection, we see a quadratic relationship in the treatment group, but not in the
control group• Notice that due to the non-linear relationship, the average effect is not identical to the effect
at the mean of the covariate: Compare the average effect (Table) with the value of theeffect function at the mean of ξ, which is E (ξ) = 2.84, in the effect plot• Another interesting aspect of our empirical example is that when we use the standard
specification for a moderated regression model without the quadratic term, the coefficient ofthe interaction term is very close to zero and its credibility interval includes zero(γ11 = 0.01,CI = [−0.24, 0.24]). So, based on the misspecified model, we woulderroneously conclude that the autonomy-supportive intervention is equally effective foreverybody and we would miss an important aspect of the differential effectiveness
20 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Discussion
Some points:• We presented an approach for comprehensive analysis of the effectiveness of interventions
based on non-linear structural equation mixture modeling• Non-linear effects can be helpful in examining treatment effects• Choosing the appropriate type of non-linearity (for example, u-shaped or logistic curves) is
an issue which needs to be addressed separately• Direct approach for mixtures could be helpful for testing for unobserved heterogeneity in
treatment effects and for detecting sub-groups• We applyied a mixture distribution to approximate the non-normality of the latent variables.
This strategy relaxes the strong distributional assumptions of non-linear models
21 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017
Discussion
Thank you for your attention!
22 of 22 Analyzing Effects with Non-Linear Structural Equation Mixture Modeling in blavaan —Axel Mayer [email protected] — RWTH Aachen University — WorkingGroup Structural Equation Modeling — 16.03.2017