Response variables in the human sciences are often categorical and not continuous. Traditionally, such categorical data are often treated as continuous because linear models are simple to implement and interpret. In recent years, nonlinear models have been developed to account for the categorical nature in item responses, which can be classified as two major formulations. In the IRT literature, the conditional-probability formulation is often used; whereas in the statistical literature, the latent-response formulation is common. The use of link functions (Equation 9) can integrate these two formulations.
Parameters in NSEM can be estimated with WinBUGS and Mplus. In a series of simulations, it can be concluded that: (a) The parameters of NSEM can be recovered very well by WinBUGS and Mplus; (b) WinBUGS is slightly more efficient than Mplus when data information is weak (e.g., 10 dichotomous items per latent trait) because WinBUGS considers prior information but Mplus does not, and prior information is more important when data information is weak; and (c) Table 9. Parameter estimates and standard errors (in parentheses) the structural parameters with different estimation approaches in Example 2
Note. γ
1is regression coefficient of θ
3on θ
1; γ
2is regression coefficient of θ
3on θ
2;
iscorrelation between θ
1and θ
2; Var1 is the variance of θ
1; Var2 is the variance of θ
2; Var3 is the residual variance of θ
3.
Parameter MLE PV Mplus WinBUGS
γ
10.23 (0.03) 0.32 (0.06) 0.29 (0.07) 0.30 (0.06) γ
20.39 (0.04) 0.48 (0.08) 0.50 (0.09) 0.51 (0.10)
0.44 (0.04) 0.54 (0.14) 0.53 (0.16) 0.55 (0.14)
Var1 0.28 (0.02) 7.43 (0.75) 8.14 (0.77) 7.38 (0.86)
Var2 0.19 (0.02) 3.31 (0.28) 3.69 (0.31) 3.35 (0.32)
Var3 0.11 (0.01) 1.48 (0.31) 1.59 (0.31) 1.31 (0.29)
Mplus needs less computer time than WinBUGS.
When original item responses are not accessible, one may adopt the PV approach to estimate the structural parameters in SEM. The simulation results reveal that (a) the PV approach can recover the structural parameters as satisfactorily as Mplus and WinBUGS when original item responses are accessible; (b) if measurement error in person measures is ignored by adopting the EAP or MLE approach, the estimation for the structural parameters was seriously biased (the regression parameters and the correlation parameters were underestimated); the shorter the test (the larger the measurement error), the worse the estimation.
The two empirical examples demonstrate how measurement models and NSEM can be fit to self-report item responses. It was found that the proposed models have a good fit in terms of the posterior predictive p-values of the Bayesian chi-squares. The PV approach yields estimates for the structural parameters that are very similar to Mplus and WinBUGS, whereas the MLE approach yields estimates that are much smaller than those from the PV approach, Mplus and WinBUGS. Moreover, the standard errors produced by the MLE approach were much smaller than those produced by the other three approaches, which is mainly because measurement error is ignored in the MLE approach.
The main idea of the PV approach is to consider measurement error properly.
Actually, it is possible to apply this idea to the MLE or EAP approach. In practice, many IRT computer programs proved point estimates (e.g., MLE or EAP), together with their standard errors. In the EAP and MLE approaches used in this
study, we purposely ignored the standard errors in order to mimic the common practice in which the point estimates are treated as true values. To consider measurement error when point estimates and their standard error are provided, one may draw a random sample of 5 (or more) from a normal distribution with the mean equal to the point estimate (e.g., MLE or EAP) and the standard deviation equal to the standard error, and then adopt Equations 18 and 19 to yield a final result, as in the PV approach. This revised approach may not perform as appropriately as the PV approach because the MLE is only asymptotically normal and the posterior distribution may not be normally distributed. Even so, the consideration of measurement error will be helpful and necessary. More studies are needed to assess the performance of the revised MLE or EAP approach.
WinBUGS is more flexible than Mplus in allowing users to specify customized models and various item response functions. Many practitioners may find it very difficult to convert Mplus results into IRT parameters. The popular 3-parameter logistic model as well as many other complicated item response functions such as the linear logistic test model (Fischer, 1983), linear partial credit model (Fischer & Ponocny, 1994), faceted model (Linacre, 1989), ordered partition model (Wilson, 1992), unfolding IRT model (Andrich, 1996; Luo, 1998;
Roberts, Donoghue, & Laughlin, 2000), testlet response models (Wainer, Bradlow,
& Wang, 2007), and multidimensional models (Reckase, 2009) can be fit by WinBUGS, but not by Mplus. Moreover, WinBUGS considers information from all individual persons, whereas Mplus considers only summarized information in the variance-covariance matrix. Thus, Mplus is not statistically optimal. However,
the flexibility of WinBUGS comes at a price. Users have to learn its syntax and be familiar with Bayesian statistics.
The structural part of NSEM in this study consists of only three latent traits.
In practice, the number of latent traits is often much lager and their relationships are often more complicated. The nonlinear relationship in the NSEM used in this study occurs only between item responses and latent traits. NSEM can be extended to accommodate nonlinear relationships among latent traits, multilevel data structures and mixture distributions (Lee, 2007; Lee & Zhu, 2002; Mooijaart
& Bentler, 2010). Future studies can be conducted to evaluate how WinBUGS will perform under complicated NSEM and develop computer programs that are more efficient and user-friendly than WinBUGS.
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#NSEM_ERP(GRM)
Appendix: WinBUGS commands for Example 2
exp(a[k]*(theta3[j]-step[k,2]))/(1+exp(a[k]*(theta3[j]-step[k,2])))- theta3 [j] <-lam1*theta [j,1]+lam2*theta [j,2]+e[j]
e [j] ~ dnorm(0,tau3) }
# Priors
tau3 ~ dgamma(0.001,0.001); theta3var <- 1/tau3 for( k in 1:T ){
step[k,1] ~ dnorm(0, 0.25 )I(, step[k,2])
step[k,2] ~ dnorm(0, 0.25 )I(step[k,1], step[k,3]) step[k,3] ~ dnorm(0, 0.25 )I(step[k,2], step[k,4]) step[k,4] ~ dnorm(0, 0.25 )I(step[k,3],)
ph[1:2,1:2]~ dwish( R[1:2,1:2],4) ; phi[1:2,1:2]<- inverse(ph[1:2,1:2]) 1,2,-2,-1,1,2,-2,-1,1,2), .Dim = c(9, 4)), tau3=1,lam1=0.8,lam2=0.7,a = c(NA,1,1,NA,1,1,NA,1,1),
ph=structure( .Data=c(1,-0.2,-0.2,1), .Dim=c(2,2)))