Polynomial factor models: non-iterative estimation via method-of ...

Polynomial factor models: non-iterative estimation via method-of-moments Florian Schuberth12

Rebecca B¨ uchner3 Karin Schermelleh-Engel3 Theo K. Dijkstra4 1 University

2 University

of Twente, The Netherlands

3 University 4 Unversity

of W¨ urzburg, Germany

of Frankfurt, Germany

of Groningen, The Netherlands

March 17, 2017 SEM Meeting, Ghent MoMpoly

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Overview

1

Polynomial factor model

2

Non-iterative method-of-moments

3

Monte Carlo simulation

4

MoMpoly package

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Structural model

Example of a polynomial factor model: η4 =γ1 η1 + γ2 η2 + γ3 η3 + γ11 (η12 − 1) + γ22 (η22 − 1)+ γ12 (η1 η2 − E(η1 η2 )) + γ112 (η12 η2 − E(η12 η2 )) γ123 (η1 η2 η3 − E(η1 η2 η3 )) + ζ1 .

(1)

All latent variables (LVs) are standardized, and the structural error term ζ1 is independent of the LVs on the right-hand side of the equation, i.e., ζ1 is independent of η1 , η2 , and η3 .

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Measurement model Each LV is connected to at least two indicators and each block of indicators is connected to one LV only:

yi = λi ηi + i .

(2)

The indicators are standardized and the measurement errors are mutually independent and independent of the η’s. The correlation matrix of the indicators of one block can be calculated as E(yi yi0 ) = Σii = λi λ0i + Θi ,

(3)

where the covariance matrix of the measurement errors Θi is diagonal. The correlation matrix of the indicators of different blocks (i 6= j): E(yi yj0 ) = Σij = ρij λi λ0j .

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Proxies and weights To estimate the model parameters, we build a proxy for each latent variable as weighted linear combination of its indicators. The weights used to build the proxy for latent variable i are obtained as X wˆi ∝ eij Sij wj (one-step weights), (5) j6=i

where wj is an arbitrary vector of the same length as yj and ˆ i are scaled: eij = sign(wi0 Sij wj ). Both weight vectors wj and w 0 0 wj Sjj wj = 1 and wˆi Sii wˆi = 1. ˆ i is The probability limit of w q ˆi ) = w ¯ i = λi / λ0i Σii λi = λi /ci . plim(w

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Estimation of the loadings Calculate the correction factor cˆi such that the squared difference between the off-diagonal elements of

Sii and cˆi wˆi cˆi wˆi 0

(7)

is minimized. As a result, we obtain s wˆi0 (Sii − diag(Sii ))wˆi , cˆi = wˆi0 (wˆi wˆi0 − diag(wˆi wˆi0 ))wˆi

(8)

where diag(Sii ) isp the diagonal matrix of Sii . Since plim(ˆ ci ) = λ0i Σii λi , the factor loadings can be consistently ˆi . ˆi = λ estimated by cˆi w

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Relationship between latent variables and proxies We define a population proxy: ¯ i0 yi = (w ¯ i0 λi )ηi + w ¯ i0 i = Qi ηi + δi , η¯i = w

(9)

where Qi (quality) is the correlation between the proxy and its latent variable. The δ’s (as the ’s) have zero means and are mutually independent and independent of the η’s. ¯ i in Qi , we obtain Replacing λi by ci w ¯i . ¯ i0 w ¯ i0 λi = ci w Qi = w

(10)

We can estimate the quality by ˆ i0 w ˆi . Qˆi = cˆi w

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Relationship between latent variables and proxies Relationship between the correlation of the proxies and the correlation between the latent variables: E(¯ ηi η¯j ) = Qi Qj E(ηi ηj ),

(12)

where E(¯ ηi η¯j ) is estimated by the sample covariance between the proxies i and j. Using PLS weights, this is already known and published: for linear models by [Dijkstra, T.K. & Henseler, J., 2015], and for polynomial models by [Dijkstra, T. K. & Schermelleh-Engel, K., 2014]. New: the use of one-step weights implementation in the MoMpoly package in R [Schuberth, F. & Dijkstra, T. K.& Schamberger, T., 2017] MoMpoly

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Non-iterative method-of-moments Starting point is a (regression) equation with a two-way interaction term: η3 = γ1 η1 + γ2 η2 + γ12 (η1 η2 − E(η1 η2 )) + ζ1 .

(13)

The γ’s can be obtained by solving the following moment equations:      E(η1 η3 ) 1 E(η1 η2 ) E(η12 η2 ) γ1 2  E(η2 η3 )  =    1 E(η1 η2 ) γ2  , (14) 2 2 2 E(η1 η2 η3 ) E(η1 η2 ) − E(η1 η2 ) γ12 where the moments are given by: E(¯ ηi η¯j ) = Qi Qj E(ηi ηj ),

(15)

E(¯ ηi2 η¯j ) = Qi2 Qj E(ηi2 ηj ),

(16)

E(¯ ηi2 η¯j2 )

=

Qi2 Qj2 (E(ηi2 ηj2 )

− 1) + 1, and

(17)

E(¯ ηi η¯j η¯k ) = Qi Qj Qk E(ηi ηj ηk ). MoMpoly

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Non-iterative method-of-moments

So far, no distributional assumptions are necessary. We only assume that the moments exist, the measurement errors are mutually independent and independent of the η’s, and that the structural error term is independent of the η’s of the right-hand side of the equation.

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Non-iterative method-of-moments Adding quadratic terms to the equation with the two-way interaction term: η3 =γ1 η1 + γ2 η2 + γ11 (η12 − 1) + γ12 (η1 η2 − E(η1 η2 )) + γ22 (η22 − 1) + ζ1 .

(19)

Higher moments are required and therefore more assumptions are necessary, in particular, assumptions about the higher-order moments of the δ’s: E(η¯i 3 ) =Qi3 E(ηi3 ) + E(δi3 ), E(¯ ηi4 ) E(¯ ηi3 η¯j )

=Qi4 E(ηi4 ) + 6Qi2 (1 − Qi2 ) + E(δi4 ), =Qi3 Qj E(ηi3 ηj ) + 3 E(¯ ηi η¯j )(1 − Qi2 ).

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(20) and

(21) (22)

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Assumptions about the moments of the δ’s

In the following, we assume that δi , has the same higher-order moments as the normal distribution, i.e., E(δi3 ) =0, and

(23)

E(δi4 ) =3 [var(δi )]2 = 3(1 − Qi2 )2 .

(24)

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Joint normality of all exogenous variables Instead of using the general moments, we can assume joint normality for all exogenous variables ’s, ζ, and exogenous η’s . Considering again the equation of example above:      E(η1 η3 ) 1 E(η1 η2 ) E(η12 η2 ) γ1 2  E(η2 η3 )  =    1 E(η1 η2 ) γ2  . (25) 2 2 2 E(η1 η2 η3 ) E(η1 η2 ) − E(η1 η2 ) γ12 Assuming joint normality leads to:      E(η1 η3 ) 1 ρ12 0 γ1  E(η2 η3 )  =  1 0   γ2  , E(η1 η2 η3 ) 1 + ρ212 γ12

(26)

where ρ12 = E(¯ η1 η¯2 )/(Q1 Q2 ). Covariances between the LVs of the RHS and the dependent LV are calculated as above. In principle every recursive model can be estimated in this way. MoMpoly

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First simulation Setting: Structural model: η3 =0.3η1 + 0.4η2 + 0.12(η12 − 1) + 0.15(η1 η2 − 0.3) + 0.1(η22 − 1) + ζ1 .

(27)

Correlation between η1 and η2 is set to ρ12 = 0.3 Each latent variable is
connected with 3 indicators with λ0i = 0.9 0.85 0.8 Sample size of N = 400 and 500 runs Estimators: MoMpoly with one-step weights, and Latent Moderated Structural Equations (LMS) [Klein, A. & Moosbrugger, H., 2000]

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Results

Para. γ1 γ2 γ11 γ12 γ22 ρ12 λ11 λ12 λ13

true 0.30 0.40 0.12 0.15 0.10 0.30 0.90 0.85 0.80

mean MoMpoly1 0.298 0.403 0.118 0.147 0.104 0.300 0.898 0.854 0.795

mean LMS2 0.297 0.402 0.117 0.147 0.101 0.299 0.899 0.852 0.800

sd MoMpoly1 0.048 0.047 0.044 0.062 0.042 0.052 0.044 0.052 0.058

sd LMS2 0.047 0.045 0.041 0.059 0.039 0.052 0.017 0.018 0.022

1 Inadmissible 2 No

results are removed, therefore the results are based on 484 estimations. inadmissible results were produced.

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Second simulation Our approach can also deal with several recursive equations. For demonstration, we run a second simulation using the following model: η3 =0.5η1 − 0.4η2 + ζ1

(28)

η4 =0.35η1 + 0.2η3 + 0.1(η32 − 1) + ζ2

(29)

η5 =0.3η2 − 0.25η4 + 0.3(η3 η4 − 0.333) + ζ3

(30)

E(η1 η2 ) = 0.3 Each latent variable is
connected with 3 indicators with λ0i = 0.9 0.85 0.8 Sample size of N = 400 and 500 runs Estimator: MoMpoly with one-step weights

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Results Para. γ31 γ32 γ41 γ43 γ433 γ52 γ54 γ534 ρ12 λ51 λ51 λ51 1 Inadmissible

true 0.500 -0.400 0.350 0.200 -0.100 0.300 -0.250 0.300 0.300 0.900 0.850 0.800

mean1 0.502 -0.395 0.347 0.201 -0.099 0.301 -0.247 0.298 0.299 0.895 0.846 0.803

sd1 0.049 0.053 0.056 0.058 0.038 0.050 0.052 0.052 0.051 0.054 0.061 0.068

results are removed, therefore the results are based on 475 estimations.

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The MoMpoly package Model specification in lavaan syntax [Rosseel, Y., 2012], e.g., model=' ETA3~ETA1+ETA2+ETA1.ETA1+ETA1.ETA2+ETA2.ETA2 ETA1 =~ y1+y2+y3 ETA2 =~ y4+y5+y6 ETA3 =~ y7+y8+y9 ' The model can be estimated by the MoMpoly function from the MoMpoly package: MoMpoly(model, data)

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Capabilities of the MoMpoly package It is capable to deal with: single equations using the general moments (however, skewness and kurtosis of the δ’s from the normal distribution are used) the moments from the multivariate normal distribution (all exogenous variables: ’s, ζ, and exogenous η’s are jointly normally distributed)

correlated measurement error within a block several recursive equations using the moments from the multivariate normal distribution (all exogenous variables: ’s, and ζ’s, and exogenous η’s are jointly normally distributed) where the parameter estimates are retrieved from reduced form coefficient matrix are obtained from replacing equation by equation

several recursive equations using the general moments. The parameters are estimated equation by equation (limited to LVs up to higher-order 3) single-indicator latent variables MoMpoly

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Further options in the MoMpoly function and functions MoMpoly(model, data,criterionload=criterion.geometric, criterionpath=criterion.diff,normality=F, weightFun=weightFun.onestep, KindOfEstimation=approach.none,...) Further options in the MoMpoly function: different ways to obtain the correction factor cˆi different correction factor for loadings and path coefficients different weights: one-step weights, PLS weights, or prescribed weights Further functions: MoMpoly.boot(): combination of the MoMpoly function with the boot package MoMpoly.jack(): function to obtain Jackknife standard errors MoMpoly

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Further research

Elaboration of the measurement model via instrumental variable techniques Other higher-order moments for delta than those from the normal distribution) Robustness against model misspecifications

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Thank you! Questions/Comments?

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References Dijkstra, T. K. & Schermelleh-Engel, K., (2014) Consistent partial least squares for nonlinear structural equation models Psychometrika 79(4) 585 – 604. Dijkstra, T.K. & Henseler, J. (2015) Consistent and asymptotically normal PLS estimators for linear structural equations Computational Statistics & Data Analysis, 81, 10 – 23. Klein, A. & Moosbrugger, H. (2000) Maximum likelihood estimation of latent interaction effects with the LMS method Psychometrika, 65(4), 457 – 474. Rosseel, Y. (2012) lavaan: An R Package for Structural Equation Modeling Journal of Statistical Software, 48(2), 1 – 36. Schuberth, F., Schamberger, T. & Dijkstra, T. K. (2017) MoMpoly package (version 0.1.3.) available upon request. MoMpoly

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Results first simulation Para. γ1 γ11 γ12 γ2 γ22 ρ12 λ31 λ32 λ33 λ11 λ12 λ13 λ21 λ22 λ23

true 0.300 0.120 0.150 0.400 0.100 0.300 0.900 0.850 0.800 0.900 0.850 0.800 0.900 0.850 0.800

mean 0.298 0.118 0.147 0.403 0.104 0.300 0.896 0.850 0.797 0.898 0.854 0.795 0.898 0.850 0.800

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sd 0.048 0.044 0.062 0.047 0.042 0.052 0.034 0.039 0.041 0.044 0.052 0.058 0.037 0.044 0.049 March 17, 2017

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Results second simulation Para. γ31 γ32 γ41 γ43 γ433 γ52 γ534 γ54 ρ12 λ31 λ32 λ33

true 0.500 -0.400 0.350 0.200 -0.100 0.300 0.300 -0.250 0.300 0.900 0.850 0.800

mean 0.502 -0.395 0.347 0.201 -0.099 0.301 0.298 -0.247 0.299 0.904 0.847 0.797

sd 0.049 0.053 0.056 0.058 0.038 0.050 0.052 0.052 0.051 0.036 0.040 0.045

Para λ41 λ42 λ43 λ51 λ52 λ53 λ11 λ12 λ13 λ21 λ22 λ23

MoMpoly

true 0.900 0.850 0.800 0.900 0.850 0.800 0.900 0.850 0.800 0.900 0.850 0.800

mean 0.898 0.850 0.800 0.895 0.846 0.803 0.897 0.851 0.799 0.897 0.848 0.803

sd 0.040 0.043 0.048 0.054 0.061 0.068 0.035 0.040 0.042 0.041 0.046 0.049

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