Boletín 3130 Efecto en la auditoría por incumplimiento de una entidad con leyes y

We are now going to summarize different strategies to test hypotheses (for instance, about average total effects) within the framework of structural equation modeling. This short survey is included here to show the similarity between the Wald–test used in Steyer and Partchev (2008) and the test statistic applied by Flory (2008). The basic statistical model for the structural equation modeling approach was already introduced in section 3.1.3 (see Bollen, 1989, for details). For the discussion of different test statistics for hypotheses about the average total effect, we assume that the parameters of the covariate-treatment regression are estimated with a maximum likelihood (ML) based estimation method by minimizing, e. g., Equation (3.39), or by the more general full information maximum likelihood equivalent (Arbuckle, 1996).

Within the framework of structural equation modeling different strategies for testing hypotheses about constraints have been developed and adapted from general statistical theory. The following different ap-proaches to test hypotheses about the average total effect are described in the next subsection: The

likeli-3.3 Structural Equation Modeling 77

hood ratio test, the Wald–test, the Lagrange Multiplier test, and the test statistic based on the standard error of the estimated average total effect (z–test).

Nonlinear Constraints Statistical inference about the average total effect based on a covariate-treatment regression with covariate-treatment interaction can be drawn with the help of nonlinear constraints within the framework of structural equation modeling. Providing an appropriate model specification for the covariate-treatment regression, we can express the hypothesis H0: ATE10= 0 as a nonlinear constraint, i. e., as a function of the estimated model parameters:

H0: c(bθ) = 0. (3.41)

Technically, different constraints can be specified for a constrained estimation of structural equation models, i. e., for the minimization of Equation (3.39) with additional restrictions (see for a summary of the different constraints, e. g., Kline, 2005). To differentiate the general linear hypothesis, which is assumed to be linear in its parameters (see, subsection 3.2.3), we use the term nonlinear constraint to emphasize that c(bθ) might be a nonlinear function of (estimated) model parameters. However, as explained in the following paragraph, we do not generally claim that a constrained estimation is performed with respect to the nonlinear straint in Equation (3.41). Instead, when referring to the average total effect we use the term nonlinear con-straint as a synonym for the nonlinear function of estimated model parameters of the covariate-treatment regression (which yields an estimator of the average total effect).

Likelihood Ratio Test Most computer packages for structural equation modeling are capable of estimat-ing constrained structural equation models (for exampleLISREL, Jöreskog & Sörbom, 1996 - 2001,EQS, Bentler, 1995, andMplus, L. K. Muthén & Muthén, 1998 - 2007). With this option, the parameters of the structural equation model are estimated by simultaneously minimizing Equation (3.39) and satisfying Equation (3.41) [see, for example, Tang & Bentler, 1998]. Hence, a test statistic based on the value of the likelihood function for the restricted estimation of bθ (LR) and the value of the likelihood function for the estimation of bθ without the restriction (LU) can be constructed. Comparing the two values of the likelihood function allows one to test hypotheses about the average total effect. This strategy is known as the likeli-hood ratio test (sometimes referred to as the χ²-difference test). Given that the null hypothesis is true, the following test statistic

LR = −2ln µLR

LU

¶

(3.42)

is χ²–distributed for sufficiently large samples, with degrees of freedom equal to the number of restrictions imposed by c(bθ) [see, for the general properties of the likelihood ratio test, Greene, 2007, and Bollen, 1989,

3.3 Structural Equation Modeling 78

for the likelihood ratio test for latent variable models]. The underlying idea is that if the restrictions im-posed by the nonlinear constraint are valid, they should not lead to a large reduction in the value of the log-likelihood function. The ratio ^L_L^R

U must be between zero and one because both likelihoods are positive and the unrestricted optimum is always superior to the restricted one.

Wald Test A practical shortcoming of the likelihood ratio test described in the last paragraph is that the estimation of a restricted model is required to obtain LRin addition to the estimation of the unrestricted model for LU. The constrained model is misspecified provided that the null hypothesis is true. For that reason, the risk of non-convergence is high and likelihood ratio tests are inconvenient to some degree.

Hypotheses about the average total effect can be tested alternatively based on the unconstrained model only. The underlying idea is generally known as the Wald–test (Wald, 1943). By translating the hy-pothesis in Equation (3.41) in to the form c(bθ) = q, that is with q = 0, the following test statistic can be derived

Wald =£

c( ˆθ) − q¤£

acov¡

c( ˆθ) − q¢¤−1£

c( ˆθ) − q¤

, (3.43)

where acov¡

c(bθ) − q¢

, is the asymptotic variance-covariance matrix of the constraint obtained from the es-timation of LU. Under the null hypothesis and for large samples, the test statistic in Equation (3.43) is χ²–distributed with degrees of freedom equal to the number of restrictions imposed by c(bθ) = q. The Wald–

test is asymptotically equivalent to the likelihood ratio test (DasGupta, 2008).

Lagrange Multiplier Test In contrast to the Wald–test (which is based on the unrestricted LU), the re-stricted LRis utilized for the Lagrange multiplier test. In textbooks about structural equation modeling, the Lagrange multipliers are well known as modification indices (see, for example, Kaplan, 2000). As described in detail by Greene (2007), two terms are necessary to compute the Lagrange multiplier test:

LM =

Ã∂ ln L( ˆθR)

∂ ˆθR

!T

£I( ˆθR)¤₋₁Ã

∂ ln L( ˆθR)

∂ ˆθR

!

. (3.44)

The matrix I¡bθ_R¢

denotes the information matrix, which is available for the maximum likelihood esti-mation within the framework of structural equation modeling [see Equation (3.47)]. The term

∂ ln L( ˆθR)

∂ ˆθR

= −bC^Tˆλ is zero if the constraint is valid, where ˆλ is the vector of estimated Lagrange multipliers (modification indices as computed by conventional program packages for the analysis of structural equa-tion models), and bC is the matrix of partial derivatives of the constraint with respect to the model parameters [see Equation (3.49)].

3.3 Structural Equation Modeling 79

The available program packages for the estimation of structural equation models do not give esti-mates of the Lagrange multipliers for every model developed in section 3.3 at this time. Furthermore, as mentioned above, the estimation of the restricted model is prone to non-convergence problems.

Test based on the Standard Error A fourth option to test the hypothesis of an average total effect different from zero is the so-called z–test based on the (estimated) standard error of the nonlinear constraint. For this test, the estimated average total effect, as a function of model parameters, is divided by its estimated stan-dard error. The ratio of a parameter estimate and its stanstan-dard error is asymptotically normally distributed (see, e. g., Wasserman, 2004). This property can be used to establish significance tests, or to construct a confidence interval for model parameters of structural equation models (see, e. g., Bollen, 1989):

z =| dATE10− ATE10|

S.E.( dATE10) . (3.45)

For the simple two group case of generalized analysis of covariance considered in this thesis, the nonlin-ear constraint yields a single restriction for the maximum likelihood estimation and hence the z–test is equivalent to the Wald–test described in the last paragraph. This equivalence can easily be verified because c(θ) − q = 0 can be substituted by dATE10− ATE10for the hypothesis ATE10= 0:

Wald =£

c( ˆθ) − q¤£

avar¡

c( ˆθ) − q¢¤₋₁£

c( ˆθ) − q¤

=£

ATEd10− ATE10¤£

acov¡

ATEd10− ATE10¢¤₋₁£

( dATE10− ATE10)¤

=

¡ATEd10− ATE10¢2

acov£

ATEd10¤ =

¡ATEd10ATE10¢2

S.E.( dATE10)² = z².

(3.46)

As given by Greene (2007), the test statistic of the Wald–test follows a χ²–distribution with one degree of freedom, which is the square of the standard normal distribution of z in Equation (3.45).

For some of the structural equation models developed in this thesis, the standard error of the nonlinear constraint, i. e., the standard error of the ATE–estimator, is not easily available without additional assump-tions. Accordingly, for quasi-experimental designs where the treatment variable X is a stochastic regressor, the derivation of an approximated standard error for the estimated average total effect was suggested by Nagengast (2006). In order to discuss the underlying assumption of this approach, we shall provide a re-view of the (multivariate) δ-method in the following paragraph.

(Multivariate) δ-method The δ-method is a very useful tool to derive the variance of a function of a ran-dom variable (Rao, 1973, p. 388, see also Oehlert, 1992, and Raykov & Marcoulides, 2004). This statistical

3.3 Structural Equation Modeling 80

tool can be applied generally for a random variable whose distribution depends on a real-valued parameter and for any function of the random variable which can be differentiated with respect to the parameter.

In its multivariate extension, the method involves two parts: the (asymptotic) variances and covari-ances of the incorporated random variables and the partial derivatives of the functions of the random variables with respect to the parameters (see, e. g., D. P. MacKinnon, 2008, p. 91 ff., for a non-technical description and Wasserman, 2004, for a comprehensive discussion). The variance of smooth functions of model parameters is approximated with the multivariate δ-method based on a first-order Taylor expansion (Bishop, Fienberg, & Holland, 1975, p. 487).

The asymptotic variance-covariance matrix of the estimated (unconstrained) parameters, acov( ˆθ), is the starting point for calculating a standard error for the estimated average total effect. Bollen (1989, p. 109 and appendix 4B therein) provided the formula for the asymptotic variance-covariance matrix for structural equation models estimated by the maximum likelihood fitting function [see Equation (3.39)] as

acov( ˆθ) = µ 2

N − 1

¶½ E

·∂²FML

∂θ∂θ^T

¸¾−1

. (3.47)

For the smooth function f ( ˆθ) = c(ˆθ) = dATE10 of model parameters, the asymptotic variance-covariance matrix is estimated from acov¡θb¢

as

 acov¡

c(bθ)¢

= bCacov¡θb¢Cb^T (3.48)

by pre- and post-multiplying with

C =b ∂c(bθ)

∂bθ^T

(3.49)

as the J × K matrix of partial derivatives of the constraint with respect to the K elements of the parameter vector θ (called the Jacobian), where K is the total number of parameters in the structural equation model and J is the number of groups (see, e. g., Raykov & Marcoulides, 2004, p. 628).

The standard error for the function of parameters (in our application the standard error of the ATE–

estimator) is obtained as the square root of the asymptotic variance

S.E.( dd ATE10) =q

 acov£

ATEd10¤

= q

 acov£

c( ˆθ)¤

. (3.50)

The δ-method reviewed here can be applied if an estimate of the variance-covariance matrix of the param-eter estimates is available for all paramparam-eters involved in the constraint.

3.3 Structural Equation Modeling 81

Summary The Lagrange multiplier test and the likelihood ratio test are based either on the restricted, or the restricted and the unrestricted model. As argued, this might be disadvantageous as the restricted model should by theory be misspecified under the null hypothesis. The Wald–test can be applied to test multiple or combined hypotheses based on the estimated parameters and their asymptotic variance-covariance matrix.

Therefore, we will focus on the Wald–test as the most flexible tool, which is equivalent to the test based on the standard error for all structural equation models studied in this thesis. This means that the statistics considered for the average total effect are at their core based on the asymptotic variance-covariance matrix of the parameter estimates, acov(bθ), and the value of the function c(bθ).