CHAPTER CONCEPTS Second-order factor model Model specification
Model identification Model estimation Model testing Model interpretation
In this chapter we expand our understanding of measurement models.
Measurement models which define the latent variables used in structural mod-els will require most of your time, knowledge, and skill. Model specification is the first important step in establishing a measurement model. If the indicator variables for the latent variables are not properly selected and specified based on prior theory and research, then the measurement model will not adequately reflect the variable relations in the covariance matrix or yield a valid latent vari-able construct.
When the measurement model is not specified correctly, which often happens when too many parameter estimates are indicated for the number of variables, model identification fails. Model identification requires that degrees of freedom be equal to or greater than 1 (df ≥ 1). SEM software programs show the degrees of freedom, and generally report when it is zero (just-identified model) or negative (under-identified model).
Model estimation is also problematic in that continuous variables, ordinal vari-ables, or a mixture of both are used in the measurement model. When this occurs a different sample matrix (asymptotic covariance matrix) and estimation method
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(diagonally weighted least squares) are usually required to achieve robust esti-mates of parameter estiesti-mates. LISREL9, using raw data, will automatically perform robust estimation of standard errors and chi-square goodness-of-fit measures under non-normality, and routinely provide the asymptotic matrix and robust chi-square statistic. In the presence of missing data, the FIML (full infor-mation maximum likelihood) estiinfor-mation method will be used. All of this works for both continuous and ordinal variables. Mplus also provides correct matrices, automated start values, and different estimation methods for categorical (ordinal) and continuous variables.
Model testing involves reporting the chi-square, degrees of freedom, and p value. The chi-square statistic is sensitive to sample size, with large sample sizes over-reporting significance and small sample sizes under-reporting significance.
Consequently, other fit indices (although not statistically based) are used to sub-stantiate model fit, model comparison, or model parsimony. These other fit indi-ces generally use the chi-square value, degrees of freedom, and sample size in their calculations. We have also suggested that parameter estimates be examined for statistical significance. It is possible to have a global chi-square value that is non-significant, thus indicating a good data to model fit, yet one of the param-eter estimates may not be statistically significant. Remember that the chi-square statistic is a test of the overall model fit.
Please be aware that our discussion will only scratch the surface of the many excit-ing new developments in structural equation modelexcit-ing related to measurement models. Some of these new applications have been included in chapters of books (Marcoulides & Schumacker, 1996, 2001; Schumacker & Marcoulides, 1998) and journal articles. In addition, the newest versions of SEM software, for example, LISREL9, have included these capabilities with software examples and further explanations. Please check your SEM software for any new features. Our intention is to provide a basic understanding of these topics to further your interest in the structural equation modeling approach.
MODEL SPECIFICATION
A second-order factor model is specified where the three first-order factors explain a higher-order factor structure. Theory plays an important role in justifying a higher-order factor. Visual, verbal, and speed are three psychological factors that most likely indicate a second-order factor, namely Ability. An ability second-order factor model is therefore hypothesized and diagrammed in Figure 9.1 using data from Jöreskog and Sörbom (1993).
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MODEL IDENTIFICATION
Recall from Chapter 7 that the number of distinct values in the S matrix (sample covariance matrix) is p(p + 1)/2. For the 9 indicator variables in the second-order factor model: p(p + 1)/2 = 9(10)/2 = 45 distinct values (variances and covariances).
The number of free parameters (saturated model – all paths specified) is p(p + 3)/2. For the saturated model: p(p + 3)/2 = 9(12)/2 = 54 total free parameters in the model that could be estimated. Consequently, if we estimated all parameters in the model, we would have a negative degrees of freedom (df = 45 – 54 = –9), and the model would not be identified. The number of parameters to be estimated in the model must be less than 45 to have a positive degrees of freedom (df ≥ 1).
visual
visperc
cubes
lozenges
wordmean paragrap
sentence
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err_c
err_l
err_p
err_s
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s-c caps addition
countdot
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Figure 9.1: ABILITY SECOND-ORDER FACTOR MODEL (AMOS)
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The second-order factor model has degrees of freedom = 23. The parameters to be estimated in the model are 10 factor loadings, 3 second-order factor loadings, and 9 error variances (22 parameters). The degrees of freedom for the model is computed as 45 – 22 = 23. The model is over-identified, that is, fewer param-eter estimates than distinct values in the matrix, which is what we desire in SEM analyses. A just-identified model would have the number of parameters estimated equal to the number of distinct values, thus df = 45 – 45 = 0. An under-identified model would have more parameters estimated than distinct values, thus a negative degrees of freedom, df = 45 – 54 = –9.
MODEL ESTIMATION
The sample data are nine psychological variables that identified three common factors (Visual, Verbal, and Speed). The second-order factor model hypothesized that these three common factors indicate a higher-order second factor, Ability. We will be using a correlation matrix, rather than a variance–covariance matrix, so the variables have mean = 0 and standard deviation = 1. Essentially, this places the variables on the same scale of measurement, but we run the risk of the standard errors being inflated. If the standard errors are inflated, then a parameter estimate may not be statistically significant or biased (recall, t or z = parameter estimate divided by standard error). The maximum likelihood estimation method was used (default value) with the assumption of multivariate normality.
MODEL TESTING
The second-order factor model is now ready to be tested for model fit. The second-order factor model is based on an example in LISREL9, SPLEX folder (EX5A.spl). The second-order factor model includes the Ability latent variable and sets the variance of this higher-order second factor to 1.0. We have changed this model to include “S-C CAPS” loading on two latent variables, Visual and Speed (Figure 9.1). The selected model fit indices indicated that the hypothesized second-order factor model has an acceptable fit (χ2 = 28.744, p = .189, df = 23;
RMSEA = .04; GFI = .958).
MODEL INTERPRETATION
The measurement equations with unstandardized factor loadings and stand-ard errors indicated statistically significant parameter estimates. There are eight measurement equations for indicators of a single factor, and one equation with
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an indicator (“SC-CAPS”) for the two latent variables. The computer program output is shown as follows:
Measurement Equations (unstandardized)
VIS PERC = 0.708*Visual, Errorvar.= 0.498 , R² = 0.502
Standerr (0.0899)
Z-values 5.546
P-values 0.000
CUBES = 0.483*Visual, Errorvar.= 0.766, R² = 0.234 Standerr (0.102) (0.100)
Z-values 4.726 7.644 P-values 0.000 0.000
LOZENGES = 0.650*Visual, Errorvar.= 0.578 , R² = 0.422 Standerr (0.110) (0.0908)
Z-values 5.905 6.361 P-values 0.000 0.000
PAR COMP = 0.868*Verbal, Errorvar.= 0.247 , R² = 0.753
Standerr (0.0511)
Z-values 4.825
P-values 0.000
SEN COMP = 0.830*Verbal, Errorvar.= 0.311 , R² = 0.689 Standerr (0.0723) (0.0535)
Z-values 11.481 5.820 P-values 0.000 0.000
WORDMEAN = 0.825*Verbal, Errorvar.= 0.319 , R² = 0.681 Standerr (0.0723) (0.0539)
Z-values 11.412 5.929 P-values 0.000 0.000
ADDITION = 0.681*Speed, Errorvar.= 0.536 , R² = 0.464
Standerr (0.0928)
Z-values 5.778
P-values 0.000
COUNTDOT = 0.859*Speed, Errorvar.= 0.262, R² = 0.738 Standerr (0.147) (0.113)
Z-values 5.848 2.320 P-values 0.000 0.020
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S-C CAPS = 0.457*Visual + 0.419*Speed, Errorvar.= 0.467 , R² = 0.533 Standerr (0.101) (0.0931) (0.0724)
Z-values 4.519 4.499 6.446
P-values 0.000 0.000 0.000
The measurement model results can also be indicated as standardized output in SEM programs. This permits a comparison of the factor loadings. Table 9.1 displays the standardized factor loadings for the three latent variables. The fac-tor loadings can be considered validity coefficients, that is, they measure how much the indicator variable defines the latent variable. Ideally, the factor load-ings should be .60 (60%) or higher, so the CUBES factor loading is lower than desired. The “SC-CAPS” indicator variable shares two latent variables, so we would combine these for our communality estimate, thus these values being lower than .60 is not necessarily undesirable. The factor loadings of the three indicator variables (PAR COMP; SEN COMP; WORDMEAN) for the Verbal latent variable are desirable. When we average the standardized factor load-ings, it indicates how much factor variance is explained. For example: (.868 + .830 + .825)/3 = .841, so 84% of the Verbal factor variance is explained (16% is unexplained).
The structural equations in the output indicate the strength of relationship between the first-order factors and the second-order factor, Ability. Visual (.987) is indicated as a stronger measure of Ability, followed by Verbal (.565) and Speed (.395); with all three being statistically significant (z > 1.96). Therefore, Ability is predominantly a function of visual perception of geometric configurations with complementary verbal skills and speed in completing numerical tasks. The com-puter output is shown as follows:
Table 9.1: Second-order Factor Model (Standardized Solution)
Indicator Variables Visual Verbal Speed
VIS PERC .708
CUBES .483
LOZENGES .650
PAR COMP .868
SEN COMP .830
WORDMEAN .825
ADDITION .681
COUNTDOT .859
S-C CAPS .457 .419
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Structural Equations
Visual = 0.987*Ability, Errorvar.= 0.0257, R² = 0.974 Standerr (0.228) (0.400)
Z-values 4.324 0.0643
P-values 0.000 0.949
Verbal = 0.565*Ability, Errorvar.= 0.681, R² = 0.319 Standerr (0.140) (0.170)
Z-values 4.029 4.010
P-values 0.000 0.000
Speed = 0.395*Ability, Errorvar.= 0.844, R² = 0.156 Standerr (0.131) (0.226)
Z-values 3.010 3.729
P-values 0.003 0.000
SUMMARY
The second-order factor model permits other factors to represent a higher-order construct. We first examined the indicators for the first-order factors, which had statistically significant factor loadings. We then examined the second-order Ability factor, which was defined by Visual, Verbal, and Speed first-order factors. We examined the factor loadings for these three first-order factors, and found them to be statistically significant indicators of Ability. The second-order factor model had a good data to model fit.
We hope that our discussion of this SEM application has provided you with a basic overview and introduction to second-order factor methods. We encourage you to read the references provided at the end of the chapter and run the exercise provided in the chapter. We further hope that the basic introduction in this chapter will permit you to read the research literature and better understand second-order factor models, which should support various theoretical perspectives. Attempting a few basic models will help you better understand the approach; afterwards, you may wish to conduct this SEM application in your own research.
EXERCISE
The psychological research literature tends to suggest that drug use and depres-sion are leading indicators of suicide among teenagers. (Note: Set variance of Suicide = 1 for model identification purposes.) Given the following data set infor-mation, create and run an SEM program to conduct a second-order factor ana-lysis. The basic program setup is shown as follows without the model equations:
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Suicide second order factor analysis
Observed Variables: drug1 drug2 drug3 drug4 depress1 depress2 depress3 depress4
Sample Size 200 Correlation Matrix 1.000
0.628 1.000
0.623 0.646 1.000 0.542 0.656 0.626 1.000
0.496 0.557 0.579 0.640 1.000 0.374 0.392 0.425 0.451 0.590 1.000
0.406 0.439 0.446 0.444 0.668 .488 1.000
0.489 0.510 0.522 0.467 0.643 .591 .612 1.000
Means 1.879 1.696 1.797 2.198 2.043 1.029 1.947 2.024
Standard Deviations 1.379 1.314 1.288 1.388 1.405 1.269 1.435 1.423
Latent Variables: drugs depress Suicide
The suicide second-order factor model is displayed in Figure 9.2. This provides the basis for writing the model equations in the second-order factor analysis program.
drug1
drug2
drug3
drug4
depress1
depress2
depress3
depress4 drugs
depress Suicide
d1
d2
Figure 9.2: SUICIDE SECOND-ORDER FACTOR MODEL (AMOS)
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SUGGESTED READINGS
chan, d. W. (2006, Fall). Perceived multiple intelligences among male and female chinese gifted students in Hong Kong: the structure of the student multiple intelligences profile. The Gifted Child Quarterly, 50(4), 325–338.
cheung, d. (2000). evidence of a single second-order factor in student ratings of teaching effectiveness. Structural Equation Modeling: A Multidisciplinary Journal, 7, 442–460.
Marsh, H. W., & Hocevar, d. (1985). Application of confirmatory factor analysis to the study of self-concept: First- and higher order factor models and their invariance across groups. Psychological Bulletin, 97(3), 562–582.
rand, d., conger, r. d., Patterson, g. r., & ge, X. (1995). it takes two to rep-licate: A mediational model for the impact of parents’ stress on adolescent adjustment. Child Development, 66(1), 80–97.
REFERENCES
Jöreskog, K. g., & Sörbom, d. (1993). LISREL 8: Structural equation modeling with the SIMPLIS command language. chicago: Scientific Software international.
Marcoulides, g., & Schumacker, r. e. (eds.). (1996). Advanced structural equation modeling: Issues and techniques. Mahwah, nJ: lawrence erlbaum Associates.
Marcoulides, g., & Schumacker, r. e. (eds.). (2001). New developments and techniques in structural equation modeling. Mahwah, nJ: lawrence erlbaum Associates.
Schumacker, r. e., & Marcoulides, g. A. (1998). Interaction and nonlinear effects in structural equation modeling. Mahwah, nJ: lawrence erlbaum.