Analyzing Effects with Non-Linear Structural Equation ...€¦ · Analyzing Eﬀects with...

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


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


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


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


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


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 )


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) ,


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)


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.


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



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.



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



2

3

4

5

1 2 3 4 5Self−Efficacy (Xi)

E(e

ta|X

=0,

Xi)

Control Group

2

3

4

5


E(e

ta|X

=1,

Xi)

Treatment Group



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) .



−1

0

1

2


Con

ditio

nal T

reat

men

t−E

ffect

Plot of the Effect Function


Results


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


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


Discussion

Thank you for your attention!


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