Introduction to Confirmatory Factor Analysis

Introduction to Confirmatory Factor Analysis In Exploratory FA, the analyst has no strong prior notion of the structure of the factor solution the goal is to infer factor structure from the patterns of correlation in the data Exploratory FA suggests hypotheses, but does not justify knowledge. Confirmatory FA Confirmatory FA is a theory-testing model as opposed to a theory-generating method In confirmatory FA, the analyst tests if the prior notion regarding the factor solution is consistent with the patterns in the data. The model, or hypothesis, that is tested specifies which variables will be correlated with which factors and which factors are correlated. The hypothesis is usually based on a strong theoretical and/or empirical foundation. With confirmatory FA, it is possible to measure the goodness-of-fit of a particular model and to statistically test the adequacy of the model fit. Confirmatory FA also offers the analyst a more viable method for evaluating construct validity. A construct is said to be reliably measured (have construct validity) when multiple measurements of the same construct using different methods are highly correlated. Ex. Menezes and Elbert (1979), as cited in Lattin, Carroll and Green (2003) used multiple measurement methods to examine multiple characteristics of the stores in a major local grocery chain. They selected two traits for their study: store appearance (A) and product assortment (P). Each trait was measured using three different measurement methods: the Likert scale, the semantic differential scale, and the Stapel scale: 1. Likert scale Selection is wide. SA GA MA MD GD SD Where SA = strongly agree, GA =generally agree, MA = moderately agree and SD, GD, MD correspond to disagree. 2. Semantic differential scale Extremely Quite Slight Slight Quite Extremely Wide Selection Limited Selection 36

3. Stapel scale +3 +2 +1 Wide Selection -1-2 -3 Menezes and Elbert tested whether any one scale is any more reliable and concluded that that there were no marked differences across the 3 methods. Confirmatory FA can also be used to test for discriminant validity, which in the context of the Menezres and Elbert example, holds when the two traits store appearance and product assortment can be separated into distinct factors. Confirmatory FA Procedure 1. The analyst begins with a correlation matrix or a variance/covariance matrix. 2. The analyst then proposes competing models, based on theory or existing data, that are hypothesized to fit the data. The models may specify the degree of correlation, if any, between each pair of common factors; the degree of correlation between individual variables and one or more factors, which particular pairs of unique factors are correlated. The different models are determined by "fixing" or "freeing" specific parameters such as: the factor loadings or coefficients, the factor correlation coefficients, and the variance/covariance of the error of measurement. These parameters are set according to the theoretical expectation of the analyst. One fixes a parameter by setting the parameter at a specific value based on one's expectations. One frees a parameter by allowing the parameter to be estimated during the analysis. 3. The competing models or hypotheses about the structure of the data are then tested against one another with the use of goodness-of-fit statistics. The fit statistics determine how well the competing models fit the data, or explain the covariation among the variables. Examples of these statistics include the chi square/degrees of freedom ratio, the Bentler comparative fit index (CFI), the parsimony ratio, and the Goodness-of-fit Index (GFI). 37

Example. Holzinger and Swineford s (1939) study on psychological testing of 7 th and 8 th grade children: X 1 = paragraph comprehension X 2 = sentence completion X 3 = word meaning X 4 = addition X 5 = counting dots The correlation matrix based on a sample of 145 children is given by 1.000.722.714.203.095 1.000.685.246.181 R = 1.000.170.113 1.000.585 1.000 An exploratory FA with two factors in a sense hypothesizes that all five variables load on both factors: X 1 = γ 11 F 1 + γ 12 F 2 + ε 1 X 2 = γ 21 F 1 + γ 22 F 2 + ε 2 X 3 = γ 31 F 1 + γ 32 F 2 + ε 3 X 4 = γ 41 F 1 + γ 42 F 2 + ε 4 X 5 = γ 51 F 1 + γ 52 F 2 + ε 5 But we may believe that two factors underlie test performance: verbal ability and quantitative ability. Moreover, we may believe that the two factors are distinct, but are positively correlated. The following FA model may thus be hypothesized: X 1 = γ 11 F 1 + ε 1 X 2 = γ 21 F 1 + ε 2 X 3 = γ 31 F 1 + ε 3 X 4 γ 42 F 2 + ε 4 X 5 = γ 52 F 2 + ε 5 Corr(F 1,F 2 ) = ϕ 12. We thus fixed γ 12 = γ 22 = γ 32 = γ 41 = γ 51 = 0. We can further specify that var(f 1 )=var(f 2 )=1, while allowing Corr(F 1,F 2 ) = ϕ 12 to be freely estimated. Note that with this model, we have 15 pieces of information (the five 1 s on the main diagonal and the 10 off-diagonal elements of R) with which to estimate 11 model parameters: the five γ loadings, ϕ 12 and the five specific variances (variances of the ε s). 38

Because there are more observations (15) than unknown parameters (11), this model is overdetermined. In general, with an overdetermined model, it will not be possible to fit the observed values exactly. Having more observations than model parameters is only a necessary, but not sufficient, condition for each model parameter to be identified. However, if the model exhibits a maximally simple structure (i.e., each variable loads on only one factor), error terms are independent, and the number of factors is small relative to the number of measures, then the model is identified. Estimation Methods and Fitting Functions There are various estimation methods available for CFA. These methods all obtain parameter estimates collected in a vector θhat by iteratively minimizing a certain function F[S, Σ(θhat)] so that S and Σ(θhat) are close or, equivalently, the elements in the residual matrix S - Σ(θhat) are small. F is called the fitting function and its form depends on the estimation method used. Five estimation methods are available under PROC CALIS of SAS. These are Unweighted least squares (ULS) Generalized Least Squares (GLS) Normal theory Maximum Likelihood (ML) Weighted Least Squares (WLS, ADF) Diagonally Weighted Least Squares (DWLS). The default is GLS. Let S denote the sample correlation or covariance matrix for a sample size N and let C be the predicted moment matrix [i.e., C =Σ(θhat) ]. The fitting functions are as follows F ULS = 0.5 Tr(S C) 2 F GLS = 0.5 Tr(S -1 (S C)) 2 F ML = Tr(SC -1 )- n +log(det C) log(det S), where n = number of manifest variables F WLS = Vec(s ij - c ij )' W -1 Vec(s ij - c ij ), where Vec(s ij - c ij ) denotes the vector of the n(n+1)/2 elements of the lower triangle of the symmetric matrix S - C, and W = (w ij,kl ) is a positive definite symmetric matrix with n(n+1)/2 rows and columns If the moment matrix S is a correlation (rather than a covariance) matrix, the default setting of W = (w ij,kl ) is the estimators of the asymptotic covariances correlations S = (s ij ). of the 39

F DWLS = Vec(s ij - c ij )' diag(w -1 )Vec(s ij - c ij ) Using only the diagonal elements of W -1 eases computational burden. But statistical properties of DWLS estimates are still not known. The Fit Statistics Chi square statistic and Chi square/degrees of freedom ratio The chi square test tests the hypothesis that the model is consistent with the pattern of covariation among the observed variables. Specifically, it tests the null hypothesis that there is no statistically significant difference in the observed covariance matrix and the theoretical covariance structure matrix implied by the model. Under Ho: Σ = Σ(θ) and certain distributional assumptions (e.g., multivariate normality if the ML method is used), χ 2 = (n -1) F[S, Σ(θhat)] is asymptotically chi-square with degrees of freedom equal to the difference (c p) where c is the number of nonredundant variances and covariances of the observed variables and p the number of parameters to be estimated. Here n is the sample size. In the case of the chi-square statistic, smaller rather than larger values indicate a good fit. Suppose an SEM model with 5 degrees of freedom yields a discrepancy function of χ 2 = 16.37 with a p-value of 0.01. This implies that the model with 5 degrees of freedom is significantly worse than if it had estimated 5 more parameters and had 0 degrees of freedom. The chi-square statistic is very sensitive to sample size. When the sample size is large, even the smallest discrepancies between the fitted model matrix and the observed data matrix will be judged significant. In such situations, it is unclear whether the statistical significance of the chi square statistic is due to poor fit of the model or to the size of the sample. Thus, alternative fit statistics were developed. Because of its limitations, some researchers advocate that the chi-square not be used as a formal test statistic. They instead suggest that divide the Chi-square value be divided by its numbers of degrees of freedom to obtain the Chi square ratio. rule of thumb well-fitted: ratio < 2 acceptable: ratio < 3 not acceptable: ratio > 5 40

Goodness-of-fit index (GFI) and adjusted goodness-of-fit index (AGFI) The good of fit index "is a measure of the relative amount of variances and covariances jointly accounted for by the model". It can be thought of as being roughly analogous to the multiple R squared in multiple regression. The closer the GFI is to 1.00, the better is the fit of the model to the data. GFI = 1 - F[S, Σ(θhat)] F[S, Σ(θ)] Here the numerator is the minimum value of the fitting function F for the hypothesized model, while the denominator is the minimum value of F when there is no hypothesized model. Thus GFI measures how much better the model fits compared to no model at all. Rules of thumb: good fit: GFI > 0.95 acceptable fit: GFI > 0.90. The adjusted goodness of fit statistic (AGFI) is based on a correction for the number of degrees of freedom in a less restricted model obtained by freeing more parameters. It penalizes more complex models (i.e., those with more parameters to be estimated) with a downward adjustment in the fit index. c AGFI = 1 - df h F[S, Σ(θhat)] F[S, Σ(θ)] Here c is as before and df h = c p, the df for the hypothesized model. Rules of thumb: good fit: AGFI > 0.90 acceptable fit: AGFI > 0.80. Both the GFI and the AGFI are less sensitive to sample size than the chi square statistic. The Normed and Nonnormed Fit Index and Their Extensions These measures fit by comparing a hypothesized model to some reasonable but more restrictive (fewer parameters to be estimated) baseline model. The independence model is usually taken to be the baseline model. Under the independence model (1) no latent variables are hypothesized to underlie the observed variables; (2) observed variables are measured without error; and the observed variables are independent. Let F h denote the minimum value of the fitting function F for the hypothesized model with associated degrees of freedom df h. Similarly, let F i denote the minimum value of F for the hypothesized model with associated degrees of freedom df i. 41

χ 2 2 i - χ h F h NFI = = 1 -. 2 χ i F i The Normed Fit Index (NFI) is normed to a 0 to 1 range with large values ( >.90) indicating a better fit. However, the NFI is affected by sample size and might not equal 1 even if the correct model is hypothesized. Thus the NFI model often underestimates model fit especially for small samples. The Nonnormed Fit Index (NNFI) penalizes more complex (or less restrictive) models by a downward adjustment and rewards more parsimonious (more restrictive models) by an upward adjustment. As its name suggests, the NNFI is not normed and may take values outside the 0 to 1 range. χ 2 2 i df i. - χ h df h NNFI =. χ 2 i df i. - 1 The normed Comparative Fit Index (CFI) was introduced to address the deficiency of the NFI, while the nonnormed Fit Index (FI) was introduced to address the deficiency of the NNFI. Let l h = [(n 1 ) F h - df h ] and l i = [(n 1 ) F i df i ]. Then CFI = 1 - l 1 l 2 where l 1 = max ( l h, 0) and l 2 = max ( l h, l i, 0). l h FI = 1 -. l i CFI is restricted to a 0 to 1 range while FI is not. Comparisons of CFI and FI with NFI and NNFI show that underestimation of fit happens less with CFI than with NFI and that FI behaves better than NNFI at the endpoints of the [0,1] interval in that it is not as frequently negative as NNFI and when it exceeds 1, it does so by a smaller amount. CFI is also less affected by sample size than NFI. Values above 0.90 are considered to be indicative of good overall fit for NFI, NNFI CFI and FI. 42

The Parsimonious Fit Indices and the Parsimony Ratio The parsimony fit indices (Parsimonious Goodness of Fit Index or PGFI and Parsimony Normed Fit Index or PNFI) and the parsimony ratio take into consideration the number of parameters estimated in the model. The fewer number of parameters necessary to specify the model, the more parsimonious is the model. By multiplying the parsimony ratio by a fit statistic, an index of both the overall efficacy of the model explaining the covariance among the variables and the parsimony of the proposed model is obtained. df h PGFI = GFI df n PNFI = df h df i NFI where df h is the degrees of freedom associated with the hypothesized model, df n = c, the degrees of freedom when no model is hypothesized (the number of nonredundant variances and covariances of the observed variables) and df i is the degrees of freedom for the independence baseline model. Both the AGFI and PGFI adjust the GFI downward for model complexity, but the PGFI provides a harsher penalty than the AGFI for hypothesizing a less restrictive (therefore less parsimonious) model. Muelller (1996) provides an example where both the GFI and NFI are very high (0.999 and 0.997, respectively) but PGFI and PNFI are 0.2 and 0.332, respectively. Root Mean Square Error of Approximation (RMS or RMSEA) This is a standardized summary of the average covariance residuals. Covariance residuals are the differences between the observed and model-implied covariances. Perfect model fit: RMSEA=0 Close fit : RMSEA < 0.05 Akaike's Information Criterion (AIC) Bozdogan's Consistent version of the AIC (CAIC) Browne-Cudeck Criterion (BCC) Bayes Information Criterion (BIC) AIC and CAIC address the issue of parsimony in the assessment of model fit are used to compare two or more models, with smaller values representing a better fit of the hypothesized model AIC : a modification of the standard goodness of fit Chi-square that includes a penalty for complexity provides a quantitative method for model selection, whether or not the models are hierarchical 43

is defined as: AIC= log e (L) + 2q where log e (L) = log-likelihood function q = number of parameters in the model rationale: Adding additional parameters (and increasing the complexity of a model) will always improve the fit of a model; however, improvement in fit may not be enough to justify the added complexity To use: AIC is computed for each candidate model Choose model with lowest AIC. The AIC has the best literature supporting its use is based on strong underlying statistical reasoning has been subjected to numerous Monte Carlo simulations and is found to be well behaved reflects the extent to which parameter estimates from the original sample will cross-validate in future samples CAIC: a modification of the AIC that takes into account the sample size assigns greater penalty to model complexity than either the AIC or BCC, but not as great as the BIC The AIC, CAIC and BCC assume that no true model exists, but try to find the best one among those being considered. BCC: was developed specifically for the analysis of structural eqns. There is some evidence that the BCC may be superior to the other measures. It imposes a slightly greater penalty for model complexity than does the AIC. BIC: assigns more penalty for complexity than the AIC, BCC and CAIC and hence has a greater tendency to pick parsimonious models. has been found in Monte Carlo simulations to perform comparably to the AIC assumes that the true model is in the set of candidate models and that the goal of model selection is to find the true model. requires that the sample size to be very large. In most outputs, these measures are applied to three models: (1) the "default" model, aka the model that you have specified (2) the independence model (one in which it is assumed that there are no correlations between any of the variables), and (3) the saturated model, one in which there are as many parameters as there are observations. 44

In a sense, one is looking to see how bad your model is compared with a "worst-case" scenario. The closer the AIC, CAID, BCC or BIC are to the saturated value and the further from the independence model, the better the fit. The more common use is when one develops two reasonably fit models; these measures can be used to help decide which model to use. Hoelter's critical N. This fit statistic focuses on the adequacy of sample size, rather than on model fit. Its purpose is to estimate a sample size that would be sufficient to yield an adequate model fit for a χ 2 test. A model is considered adequately fitting if the Hoelter's N is greater than 200. Guiding Principles in Assessing Model Fit 1. Use a strong underlying theory as the primary guide to assessing model adequacy since the objective of structural equation modeling in general, and CFA in particular, is to understand a substantive area. 2. If possible, formulate several alternative models prior to analysis rather than considering a single model. 3. Whenever possible, compare fit results to that obtained in previous studies for the same or similar models. 4. In addition assessing the fit of the whole model, consider assessing the fit of components of the model. 5. Use different measures of fit from different families of measures. 6. Use overall fit indices that take the degrees of freedom of the model into account and that depend as little as possible on the sample size. Interpreting Confirmatory Factor Analyses It is important to remember when interpreting the findings from a confirmatory factor analysis that more than one model can be determined that will adequately fit the data. Therefore, finding a model with good fit does not mean that the model is the only, or optimal model for that data. 45

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Model Modification When a confirmatory analysis fails to fit the observed factor structure with the theoretical structure, the analyst can 1. Re-specify another a priori model, based on theory and your knowledge; or 2. Modify the original model to improve the fit. Improving the fit can be done by: (a) (b) examining the fitted model for regression weights or covariances that are not significant and pruning back these covariances or paths. Rerunning the model and checking if the fit is now adequate. However, post-hoc model modifications should make substantive sense and should be theoretically justifiable. Two types of model specification errors 1. External omission of important variables from the model Can be identified based on knowledge of the substantive area 2. Internal omission of important relations within the model Indicators of possible internal specification errors: 1. Modification indices (MI) The MI is an estimate of the decrease in the χ 2 statistic (which is actually a badness of fit measure) if a fixed parameter is freed. 2. Expected parameter change statistics (EPC) The EPC is an estimate of the value of the freed parameter. Suggestions: Large MI, large EPC : free the parameter Large MI, small EPC : keep parameter fixed Small MI, small EPC : keep parameter fixed Other suggestion: free those with large MI indices Consequences of post hoc model modification: 1. Model modification based on these measures might lead to several competing models that should be compared on substantive as well as statistical merits. 2. Fit statistics of a modified structure that was applied to the same data set that yielded the MIs and EPCs will be biased upward. These do not necessarily indicate that a structure that better reflects the true structure in the population is obtained; they 47

could simply indicate that a model that is particularly suited to the sampled data at hand has been obtained. Solution to overfitting to the sample data: cross validate the modified structure with a new and independent sample. Split the sample randomly into calibration and validation subsamples and compute the cross validation index (CVI) due to Cudeck and Brown. The CVI measures the distance between the unrestricted variance-covariance matrix from the validation sample S v and the model-implied variance-covariance matrix from the calibration sample Σ(θ c hat). That is CVI = F[S v, Σ(θ c hat)]. The smaller the value of CVI, the better the estimated predictive validity of the model. One can compare the competing models based on CVI, or compare the CVI obtained for the restructured model to that obtained from the saturated model (an unrestricted model with df=0) or the independence model. If the sample data cannot be split, use the expected value of the cross validation index (ECVI) given by ECVI = F[S, Σ(θ c hat)] + ( 2p / (n 1)) where F[S, Σ(θ c hat)] is the minimum value of the fitting function for the hypothesized structure, p is the number of parameters to be estimated based on the model and n is the sample size. Comparing the ECVI value for the hypothesized model to the values obtained for the independence and saturated models shows that ECVI s < ECVI h < ECVI i implying that technically the saturated model has better predictive validity than the hypothesized model. But if the differences between the more parsimonious hypothesized model and the saturated model is small, the former may be taken to be an adequate approximation to reality. Example 1 from SAS manual: Second-Order Confirmatory Factor Analysis A second-order confirmatory factor analysis model is applied to a correlation matrix of Thurstone reported by McDonald (1985). Using the LINEQS statement, the three-term secondorder factor analysis model is specified in equations notation. The first-order loadings for the three factors, F1, F2, and F3, each refer to three variables, X1-X3, X4-X6, and X7-X9. One second-order factor, F4, reflects the correlations among the three first-order factors. The second-order factor correlation matrix P is defined as a 1 1 identity matrix. Choosing the second-order uniqueness matrix U2 as a diagonal matrix with parameters U21-U23 gives an unidentified model. To compute identified maximum likelihood estimates, the matrix U2 is defined as a 3 3 identity matrix. The following code generates results that are partially displayed in the output that follows. data Thurst(TYPE=CORR); Title "Example of THURSTONE resp. McDONALD (1985, p.57, p.105)"; _TYPE_ = 'CORR'; Input _NAME_ $ Obs1-Obs9; Label Obs1='Sentences' Obs2='Vocabulary' Obs3='Sentence Completion' Obs4='First Letters' Obs5='Four-letter Words' Obs6='Suffices' Obs7='Letter series' Obs8='Pedigrees' Obs9='Letter Grouping'; 48

datalines; Obs1 1......... Obs2.828 1........ Obs3.776.779 1....... Obs4.439.493.460 1...... Obs5.432.464.425.674 1..... Obs6.447.489.443.590.541 1.... Obs7.447.432.401.381.402.288 1... Obs8.541.537.534.350.367.320.555 1.. Obs9.380.358.359.424.446.325.598.452 1. ; proc calis data=thurst method=max edf=212 pestim se; Title2 "Identified Second Order Confirmatory Factor Analysis"; Title3 "C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide"; Lineqs Obs1 = X1 F1 + E1, Obs2 = X2 F1 + E2, Obs3 = X3 F1 + E3, Obs4 = X4 F2 + E4, Obs5 = X5 F2 + E5, Obs6 = X6 F2 + E6, Obs7 = X7 F3 + E7, Obs8 = X8 F3 + E8, Obs9 = X9 F3 + E9, F1 = X10 F4 + E10, F2 = X11 F4 + E11, F3 = X12 F4 + E12; Std F4 = 1., E1-E9 = U11-U19, E10-E12 = 3 * 1.; Bounds 0. <= U11-U19; run; Output: Second-Order Confirmatory Factor Analysis Example of THURSTONE resp. McDONALD (1985, p.57, p.105) Identified Second Order Confirmatory Factor Analysis C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation Parameter Estimates 21 49

Functions (Observations) 45 Lower Bounds 9 Upper Bounds 0 GCONV convergence criterion satisfied. Fit Function 0.1802 Goodness of Fit Index (GFI) 0.9596 GFI Adjusted for Degrees of Freedom (AGFI) 0.9242 Root Mean Square Residual (RMR) 0.0436 Parsimonious GFI (Mulaik, 1989) 0.6397 Chi-Square 38.1963 Chi-Square DF 24 Pr > Chi-Square 0.0331 Independence Model Chi-Square 1101.9 Independence Model Chi-Square DF 36 RMSEA Estimate 0.0528 RMSEA 90% Lower Confidence Limit 0.0153 RMSEA 90% Upper Confidence Limit 0.0831 ECVI Estimate 0.3881 ECVI 90% Lower Confidence Limit. ECVI 90% Upper Confidence Limit 0.4888 Probability of Close Fit 0.4088 Bentler's Comparative Fit Index 0.9867 Normal Theory Reweighted LS Chi-Square 40.1947 Akaike's Information Criterion -9.8037 Bozdogan's (1987) CAIC -114.4747 Schwarz's Bayesian Criterion -90.4747 McDonald's (1989) Centrality 0.9672 Bentler & Bonett's (1980) Non-normed Index 0.9800 Bentler & Bonett's (1980) NFI 0.9653 James, Mulaik, & Brett (1982) Parsimonious NFI 0.6436 Z-Test of Wilson & Hilferty (1931) 1.8373 Bollen (1986) Normed Index Rho1 0.9480 Bollen (1988) Non-normed Index Delta2 0.9868 Hoelter's (1983) Critical N 204 Manifest Variable Equations Obs1 = 0.5151* F1 + 1.0000 E1 Std Err 0.0629 X1 t Value 8.1868 Obs2 = 0.5203* F1 + 1.0000 E2 Std Err 0.0634 X2 50

t Value 8.2090 Obs3 = 0.4874* F1 + 1.0000 E3 Std Err 0.0608 X3 t Value 8.0151 Obs4 = 0.5211* F2 + 1.0000 E4 Std Err 0.0611 X4 t Value 8.5342 Obs5 = 0.4971* F2 + 1.0000 E5 Std Err 0.0590 X5 t Value 8.4213 Obs6 = 0.4381* F2 + 1.0000 E6 Std Err 0.0560 X6 t Value 7.8283 Obs7 = 0.4524* F3 + 1.0000 E7 Std Err 0.0660 X7 t Value 6.8584 Obs8 = 0.4173* F3 + 1.0000 E8 Std Err 0.0622 X8 t Value 6.7135 Obs9 = 0.4076* F3 + 1.0000 E9 Std Err 0.0613 X9 t Value 6.6484 Latent Variable Equations F1 = 1.4438* F4 + 1.0000 E10 Std Err 0.2565 X10 t Value 5.6282 F2 = 1.2538* F4 + 1.0000 E11 Std Err 0.2114 X11 t Value 5.9320 F3 = 1.4065* F4 + 1.0000 E12 Std Err 0.2689 X12 t Value 5.2307 The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation Variances of Exogenous Variables Variable Paramete r Estimat e Standar d Error t Value F4 1.00000 E1 U11 0.18150 0.02848 6.37 E2 U12 0.16493 0.02777 5.94 E3 U13 0.26713 0.03336 8.01 51

E4 U14 0.30150 0.05102 5.91 E5 U15 0.36450 0.05264 6.93 E6 U16 0.50642 0.05963 8.49 E7 U17 0.39032 0.05934 6.58 E8 U18 0.48138 0.06225 7.73 E9 U19 0.50509 0.06333 7.98 E10 1.00000 E11 1.00000 E12 1.00000 The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation Squared Multiple Correlations Variable Error Variance Total Variance R- Square 1 Obs1 0.18150 1.00000 0.8185 2 Obs2 0.16493 1.00000 0.8351 3 Obs3 0.26713 1.00000 0.7329 4 Obs4 0.30150 1.00000 0.6985 5 Obs5 0.36450 1.00000 0.6355 6 Obs6 0.50642 1.00000 0.4936 7 Obs7 0.39032 1.00000 0.6097 8 Obs8 0.48138 1.00000 0.5186 9 Obs9 0.50509 1.00000 0.4949 10 F1 1.00000 3.08452 0.6758 11 F2 1.00000 2.57213 0.6112 12 F3 1.00000 2.97832 0.6642 52

Example 2 : CFA on needs data SAS Program ods html file='e:\mvar\0708\calisout_html'; proc calis data=e.need tech=nr nobs=285 all; lineqs imp1 = lam1 F1 + E1, imp2 = lam2 F1 + E2, imp3 = F1 + E3, imp4 = lam4 F1 + E4, imp5 = lam5 F1 + E5, imp6 = lam6 F2 + E6, imp7 = lam7 F2 + E7, imp8 = F2 + E8, imp9 = lam9 F2 + E9, F1 = gam1 F3 + D1, F2 = gam2 F3 + D2; STD E1-E9= the1-the9, D1-D2=psi1-psi2, F3=1; COV D1 D2 = phi; run; ods html close; run; Note: the default estimation method is ML. tech=nr specifies that the Newton Raphson algorithm be used in optimization. The following output shows that the data do not follow a multivariate normal distribution. None of the marginal distributions has zero kurtosis. Mardia s multivariate kurtosis is positive, indicating leptokursis. Violating the multivariate normality assumption in Generalized Least Squares and Maximum Likelihood usually leads to the wrong approximate standard errors and incorrect fit statistics based on the χ 2 value. In general the parameter estimates are more stable against violation of the assumption of multivariate normality. 53

The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation Observations 285 Model Terms 1 Variables 9 Model Matrices 4 Informations 45 Parameters 21 Variable Mean Std Dev Skewness Kurtosis IMP1 4.63860 0.50272-0.99713 0.81035 IMP2 4.60702 0.50346-0.60675-1.23090 IMP3 4.63158 0.49758-0.72046-1.06096 IMP4 4.61053 0.53658-1.48328 5.44282 IMP5 4.63860 0.48851-0.67111-1.31892 IMP6 4.58596 0.50051-0.43565-1.60273 IMP7 4.61404 0.48768-0.47097-1.79080 IMP8 4.57193 0.50968-0.45265-1.41305 IMP9 4.55088 0.53901-0.74738 0.25909 Mardia's Multivariate Kurtosis 144.3764 Relative Multivariate Kurtosis 2.4583 Normalized Multivariate Kurtosis 86.6076 Mardia Based Kappa (Browne, 1982) 1.4583 Mean Scaled Univariate Kurtosis -0.0706 Adjusted Mean Scaled Univariate Kurtosis 0.1200 54

The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation Fit Function 0.5899 Goodness of Fit Index (GFI) 0.8774 GFI Adjusted for Degrees of Freedom (AGFI) 0.7701 Root Mean Square Residual (RMR) 0.0390 Parsimonious GFI (Mulaik, 1989) 0.5849 Chi-Square 167.5285 Chi-Square DF 24 Pr > Chi-Square <.0001 Independence Model Chi-Square 2517.5 Independence Model Chi-Square DF 36 Specifying that weighted least squares estimation, which is equivalent to Browne's asymptotically distribution-free estimation, yields much improved fit results. WLS is invoked by typing proc calis data=e.need method=wls nobs=285 all; The CALIS Procedure Covariance Structure Analysis: Weighted Least-Squares Estimation Fit Function 0.1997 Goodness of Fit Index (GFI) 0.9998 GFI Adjusted for Degrees of Freedom (AGFI) 0.9996 Root Mean Square Residual (RMR) 0.1411 Parsimonious GFI (Mulaik, 1989) 0.6665 Chi-Square 56.7052 Chi-Square DF 24 Pr > Chi-Square 0.0002 Independence Model Chi-Square 9113.5 Independence Model Chi-Square DF 36 55

RMSEA Estimate 0.0693 RMSEA 90% Lower Confidence Limit 0.0461 RMSEA 90% Upper Confidence Limit 0.0928 ECVI Estimate 0.3476 ECVI 90% Lower Confidence Limit 0.2833 ECVI 90% Upper Confidence Limit 0.4389 Probability of Close Fit 0.0824 Bentler's Comparative Fit Index 0.9964 Akaike's Information Criterion 8.7052 Bozdogan's (1987) CAIC -102.9545 Schwarz's Bayesian Criterion -78.9545 McDonald's (1989) Centrality 0.9442 Bentler & Bonett's (1980) Non-normed Index 0.9946 Bentler & Bonett's (1980) NFI 0.9938 James, Mulaik, & Brett (1982) Parsimonious NFI 0.6625 Z-Test of Wilson & Hilferty (1931) 3.5454 Bollen (1986) Normed Index Rho1 0.9907 Bollen (1988) Non-normed Index Delta2 0.9964 Hoelter's (1983) Critical N 184 56

Manifest Variable Equations with Estimates IMP1 = 0.9368 * F1 + 1.0000 E1 Std Err 0.0216 lam1 t Value 43.3864 IMP2 = 0.9515 * F1 + 1.0000 E2 Std Err 0.0169 lam2 t Value 56.4608 IMP3 = 1.0000 F1 + 1.0000 E3 IMP4 = 0.9631 * F1 + 1.0000 E4 Std Err 0.0299 lam4 t Value 32.1829 IMP5 = 0.9631 * F1 + 1.0000 E5 Std Err 0.0179 lam5 t Value 53.6875 IMP6 = 0.9720 * F2 + 1.0000 E6 Std Err 0.0290 lam6 t Value 33.4809 IMP7 = 1.0435 * F2 + 1.0000 E7 Std Err 0.0262 lam7 t Value 39.8476 IMP8 = 1.0000 F2 + 1.0000 E8 IMP9 = 0.9451 * F2 + 1.0000 E9 Std Err 0.0281 lam9 t Value 33.5879 57

The CALIS Procedure Covariance Structure Analysis: Weighted Least-Squares Estimation Latent Variable Equations with Estimates F1 = 0.9422 * F3 + 1.0000 D1 Std Err 0.00653 gam1 t Value 144.3 F2 = 0.8910 * F3 + 1.0000 D2 Std Err 0.0153 gam2 t Value 58.0883 Other Differences Between EFA and CFA 1. Factor rotation does not apply to CFA. Identification restrictions associated with CFA are achieved in part by fixing most or all indicator cross-loadings to zero. Thus rotation is not necessary in CFA because simple structure is obtained by specifying indicators to load on just one factor. 2. CFA models are typically more parsimonious than EFA solutions. Only primary loadings and factor correlations are typically freely estimated. 3. Unlike EFA, CFA allows for ability to specify the nature of measurement errors (unique variances) of the indicators. In EFA, factor models are specified under the assumption that measurement error is random. CFA allows for the specification of correlation between the error terms of indicator (manifest) variables. This specification allows for the possibility that some of the covariation between the indicators is due to the shared influence of the underlying latent factor and some may be due to sources other than the common factor, e.g., common assessment method like reversed wording of questions, may have resulted in additional indicator covariation. Example. Self-Esteem Scale Positively worded items, e.g., I feel good about myself. Negatively worded items, e.g., At times I think I am no good at all. Application of EFA results in a positive self-evaluation dimension and a negative self-evaluation dimension Allowing for correlated errors results in a global self-esteem solution. 58

R E F E R E N C E S http://v8doc.sas.com/sashtml/stat/chap19/sect49.htm Jackson, J. L. et. al. (2005). Introduction to Structural Equation Modeling (Path Analysis). (From www) Lattin, J., Carroll, J. D., and Green, P.(2003). Analyzing Multivariate Data. Ca: Brooks/ Cole Thomson Learning. Mueller, Ralph O. (1996). Basic Principles of Structural Equation Modeling. New York: Springer Verlag. Stapleton, Connie D. (1997). Basic Concepts and Procedures of Confirmatory Factor Analysis in http://searcheric.org/ericdb/ed407416.htm. 59