Summary Brown

Summary Brown (2006)

Summary Brown (2006): Chapter 1, introduction: Uses of confirmatory factor analysis: Confirmatory factor analysis (CFA) is a type of structural equation modeling (SEM) that deals specifically with measurement models; the relationship between observed measures or indicators and latent variables or factors. CFA is hypothesis-driven CFA should be conducted prior to the specification of an SEM model. EFA: exploratory factor analysis. no theoretical framework. Psychometric evaluation of test instruments: CFA is almost always used to examine the latent structure of an instrument. CFA also assists in the determination of how a test should be scored. (e.g. more or one scale). CFA can also be used other aspects of psychometric evaluation, such as scale reliability. Construct validation: a construct is a theoretical concept. CFA is an analytic tool for construct validation. The results can provide evidence of the convergent and discriminant validity of theoretical constructs.

- Convergent validity: different indicators of theoretically similar constract are strongly interrelated.

- Discriminant validity: indicators of theoretically dinstinct constructs are not highly intercorrelated.

CFA can be used in a multitrait-multimethod matrices sort of way. Method effects: often, some of the covariation of observed measures is due to sources other than the substantive latent factor. For example because of shared method variance. This is called a method effect. EFA is incapable of estimating method effects. CFA can specify method effects as part of the error theory of the measurement model. Measurement invariance evaluation: another key strength of CFA is the ability to determine how well measurement models generalize across groups of individuals across time. measurement invariance evaluation. Test biases can be addressed in CFA by multiple-groups solutions and MIMIC (multiple indicators, multiple causes) models. Why a book on CFA: In applied SEM research, most of the work deals with measurement models (CFA). However most books on SEM do not provide advanced applications of CFA. This book is written to provide an in-depth treatment of the concepts, procedures, pitfalls and extensions of this methodology. n.b. the paragraph about the content of the book is not in this summary. Chapter 2, the common factor model & exploratory factor analysis: Factor analysis has become one of the most widely used multivariate statistical procedures in applied research endeavors across a multitude of domains. A factor is an unobservable variable that influences more than one observed measure and that accounts for the correlations among these observed measures. Common factor model: postulates that each indicator in a set of observed measures is a linear function of one or more common factors and one unique factor. Thus factor analysis partitions the variance of each indicator in two parts: common variance and unique variance.


Factor analysis can be exploratory or confirmatory (EFA or CFA). EFA is a data-driven approach, CFA s a theory-driven approach. Accordingly, EFA is typically used earlier in the process of scale development and constructu validation, whereas CFA is used in later phases after the underlying structure has been established on prior empirical and theoretical grounds. EFA can be done is SPSS and SAS, by embedding the sample correlation matrix in the body of the syntax. Of particular interest in interpreting the results is the factor matrix. This contains the factor loadings: completely standardized estimates of the regression slopes for predicting the indicators from the latent factor. Squaring factor loadings provides the estimate of the amount of variance in the indicator accounted for by the latent variable; this is often called communality. A path diagram of an example one-factor measurement model is provided:

The latent factor is presented with a circle (letter eta). The indicators are represented by rectangles. The unidirectional arrows represent the factor loadings (lambda). The epsilon is used to relate unique variances to the indicators. A fundamental equation of the common factor model is:

This can also be presented as:

Or:

Or:

I need to read page 18 to 20 again, because I do not understand what they say. after lecture. Procedures of EFA: The overriding objective of EFA is to evaluate the dimensionality of a set of multiple indicators by uncovering the smallest number of interpretable factors needed to explain the correlations among them. There are no a priori restrictions on the pattern of relationships between observed measures and latent variables. After determining EFA is the best approach, one needs to determine the indicators to include, the size and nature of the sample. Moreover, a specific method to estimate the factor model needs to be selected, as well as an appropriate number of factors, rotation technique shall also be selecting. Factor extraction: there are many methods that can be used such as: maximum likelihood, principal factors, weighted least squares, unweighted least squares, generalized least squares, imaging analysis, minimum residual analysis, alpha factoring. With continuous indicators the most used factor extraction method are ML and PF(principal factors). ML allows for statistical


evalution of how well the factor solution is able to reproduce the relationships among the indicators in the input data. However, ML estimation requires the assumption of multivariate normal distribution of these variables. PF is free of distributional assumptions and is less likely than ML prone to improper solutions. However, PF does not provide goodness-of-fit indices. PCA is frequently miscategorized as an estimation method of common factor analysis. However, PCA relies on a different set of quantitive methods than EFA. PCA aims to account for the variance in the observed measures rather than explain the correlations among them. Some critics suggest PCA can be better than EFA. However Fabrigar et al. point out this is not the case. If the overriding rationale and empirical objectives of an analysis are in accord with the common facor model, then it is conceptually and mathematically inconsistent to conduct PCA; EFA is more appropriate. Factor selection: the results of the initial analysis are used to determine the appropriate number of factors to be extracted in subsequent analysis. Here problems of underfactoring or overfactoring come into play. Research suggest overfactoring is less severe, but can also provide serious problems with interpreting the results. The decision for the appropriate amount of factors should receive careful consideration by certain statistical guidelines. First, factors in the solution should be well defined. Secondly, the solution should also be evaluated with regard to whether trivial factors exist in the data. (e.g. error measures). It is also important to note that the number of factors (m) that can be extracted by EFA is limited by the number of observed measures (p). for example, using PF the maximum number of factors that can be extracted is p 1. In ML, the number of parameters that are estimated in the factor solution(a) must be equal to or less than the number of elements in the input correlation or covariance matrix(b) . these things can be calculated:

(p*m) indicates the number of factor loading. [(m*(m+1)]/2) indicates the number of factor variances and covariances. P corresponds to the number of residual variances. M2 reflects the number of restrictions that are required to identity the EFA model. Factor selection is often guided by the eigenvalues generated from either the unreduced correlation matrix or the reduced. Eigenvalues summarize the variance in the indicators explained by the successive factors. See table 2.2. for an example. Eigenvalues in SPSS get also presented for R, listed under the initial statistics heading. Thus, eigenvalues guide the factor selection process by conveying whether a given factor explains a considerable portion of the total variance of the observed measures. Three analyses are based on eigenvalues: Kaiser-Guttman rule, scree test, parallel analysis. Towards the Kaiser-Guttman rule there is a lot of criticism: leads to over/underfactoring. Scree test has also some problems regarding objectivity. However when sample size is large and factors are well-defined, this is less the case. Also the parallel analysis has some arbitrary outcomes. Moreover, when ML analysis is used, one can look at the goodness-of-fit test provided. The goal of this approach is to identify the solution that reproduces the observed correlations considerably better than more parsimonious models, but is able to reproduce these observed relationships equally or nearly as well as more complex solutions. N.B. it should be nted EFA is largely an exploratory procedure, thus the results of an initial EFA should be considered cautiously. Factor rotation: The extracted factors are rotated, to foster their interpretability. This is done on the principle of simple structure: the most readily interpretable solutions in which each factor is defined by a


subset of indicators, and each indicator has high loading on only one factor. There is no explicit guideline of what counts as a salient factor loading. There are two types of rotation: orthogonal and oblique. Orthogonal implies factors are to be uncorrelated, oblique rotation allows of intercorrelation of factors. Othrogonal rotation is often used, however it has some drawbacks. When factors are correlated orthogonal rotation provided misleading solutions. Oblique rotation is preferred because it provides a more realistic representation of how factors are interrelated. NOTE: SPSS provides pattern matrixes and structure matrixes. Loadings in the structure matrix will typically be larger than those in the pattern matrix because they are inflated by the overlap in the factors. The pattern matrix is most often interpreted. It should be noted that communality of an indicator does not change with rotation. It only makes the solution more interpretable. Factor scores: Afte an appropriate factor solution has been established, the researcher may wish to calculate factor scores using the factor loadings and factor correlations. A factor score is the score that would have been observed for a person if it had been possible to measure the latent factor directly. Coarse factor scores: unweighted composites of the raw scores of indicators. Alternatively, factos scores can be estimated by multivariate methods. (least squares regression approach) there is however indeterminacy in the common factor model; an infinitie number of sets of factor scores can be computed. There is no way of discerning which set of scores is most accurate. However there are some things that influence indeterminacy: ratio items-factors, size of item communality. Grice has specified three criteria for evaluating the quality of factor scores: 1. Validity coefficients. 2. Univocality the extent to which the factor scores are excessively correlated with other factors in the same analysis. 3. Correlational accuracy, how closely the correlations among factor scores correspond to the correlations among the factors. Chapter 3, introduction to CFA: The purpose of CFA is to identify latent factors that account for the variation and covariation among a set of indicators. In CFA the researcher must prespecify all aspects of the factor model. Standardized and unstandardized solutions: In EFA the tradition is to completely standardize all variables. CFA also produces a completely standardized solution, but much of the analysis does not standardize the latent or observed variables. CFA typically analyses the variance-covariance matrix, so CFA also provides an unstandardized solution. (Note: a covariance can be calculated by multiplying the correlation of two indicators by their SDs. ) Understandardized means of indicators can also be included in CFA as data. This allows for estimation of the means of the laten factors and the intercepts of the indicators. The intercept is the predicted value of the indicator when the latent factor is zero. Indicator cross-loadings/model parsimony: EFA and CFA differ markedly in the manner by which indicator cross-loadings are handled in solutions entailing multiple factors. 1. Factor rotation does not apply to CFA. 2. Indicator crossloadings are fixed to zero. indicators load on only one factor. See table 3.1 for differences in outcome between EFA and CFA. Another consequence of fixing cross-loadings to zero is that factor correlation estimates in CFA tend to be of higher magnitude than in EFA solutions.


Unique variances: The CFA framework offers the researcher the ability to specify the nature of the relationships among the measurement errors of the indicators. CFA differentiates among relationships of unique variances, unlike EFA. CFA makes it possible to estimate such relationships when this specification is substantively justified and other identification requirements are met. Thus the error measurement can be random, but can also be specified as an correlated error between two indicators. This may be justified on the basis of method effects, social desirability and sorts. Error variances can be modeled in various ways: correlated uniqueness approach, correlated methods. Note EFA is not able to specify correlated errors and is thus limited. Model comparison: CFA allows for the comparison of above reviewed possible restrictions. Nested model: subset of the free parameters of another model parent model. thus, they differ in their number of freely estimated versus constrained parameters. Freely estimation allows the analysis to find the values for the parameters in the CFA solution that optimally reproduce the variances and covariances of the input matrix. Fixed parameters, means that a researcher assigns specific values. Constrained parameters mean that parameters are not exactly specified, but other restrictions on the magnitude of these values are placed. Accordingly the fit of such a model can e statistically compared to the fit of an model with free parameters and other nested models. Purposes and advantages of CFA: As is made clear in the past sections, every aspect of the CFA model is specified in advance. These modeling flexibility and capabilities of CFA afford sophisticated analyses of construct validity. In addition, CFA offers a very strong analytic framework for evaluating the equivalence of measurement model across distinct groups. Frequently, CFA is used as a precursor to SEM models that specify structural relationships among the latent variables. SEM can be broken down into two major components: the measurement model (i.e. CFA), and the structural model. (see figure 3.2). thus, whereas relationships among the latent variables are allowed to freely intercorrelate in the CFA model, the exact nature of the relationships is specified in the structural model. often the structural model is more parsimonious than the measurement model because it attempts to reproduce the relationship among latent variables with fewer freely estimated parameters. Poor fit of the SEM can be due to poor fit of the CFA, as well as of the structural model. This is the reason why often CFA is often done first, to create a good measurement model. Parameters of an CFA model: All CFA models contain factor loadings, unique variances, and factor variances (note this can also be specified to zero). We already know what all of these thing mean, so Im not going to explain this again. A CFA may also include error covariances (also called correlated uniquenesses, correlated residuals, correlated errors). This implies that two indicators covary for reasons other than the shared influence of the latent factor. The CFA model can be expanded to include analysis of mean structures; the parameters also strive to reproduce to observed sample means of the indicators. These models also include parameter estimates of the indicator intercepts and the latent factor means. Latent variables in CFA can be either exogenous(not caused by other variables) or endogenous (caused by other variables). See also figures 3.3. and 3.4 for LISREL notation for latent X and Y specifications. Even if a perfect measurement model is the case, some researcher choose to specify the analysis as a latent Y solution. This creates greater simplicity, and corresponds with the way


statistical papers use terms. Note: lowercase Greek symbols correspond to specific parameters, whereas capital Greek letters reflect an entire matrix.

Read page 56 -59 for detailed explanation of the several symbols used. Above are the symbols provided for measurement models. Structural components of a model also have their own notation. Gamma: regressions between latent X and Y variables. Beta: directional effects among endogenous variables. Fundamental equations of a CFA model: CFA aims to reproduce the sample variance-covariance matrix by the parameter estimates of the measurement solution. Example based on figure 3.5, inserted below. There are indicators (Xn) and latent constructs (Xi). Because X4 until X6 load on one single factor they are said to be congeneric. thus, to calculate the variance of an congeneric indicator the following formula applies: VAR(X2) = 22 = x212 11 + 2 This is in this example: 802(1) + .36 Because this is an completely standardized solutions, the variance is 1. The squared factor loading is the communality the variance explained by the factor. (also depicted as 2)


Moreover, the predicted covariance between two indicators can also be calculated, with the following formula: COV(X2, X3) = 3,2 = x21 11 x31 . It should be noted that this does not include the correlated error. When this parameter is estimated it should be summed up by the past equation. Thus for COV(X5, X6) = 6,5 = x52 22 x62 + 65. CFA model identification: In order to estimate the parameters in CFA, the measurement model must be identified. This is the case, when on the basis of known information it is possible to obtain a unique set of parameter estimates. Scaling the latent variable: Every latent variable must have its scale identified. In CFA this is done in two ways. First, the researcher may fix the metric of the latent variable the same as one of the indicators, the marker/reference indicator. In the second method, the variance of latent variables is fixed to a specific value, usually 1.00. consequently standardized solutions are produced. Statistical identification: The parameters of a CFA model can be estimated only if it does not exceed the number of pieces of information in the input variance-covariance matrix. See for the following part , figure 3.6, page 64. Underidentified: when the number of unknown parameters exceed the number of pieces of known information. (x +y = 7). Thus there are an infinite number of values that are possible. Generally what is known are the variances and covariances. When the amount of unknown parameters is the same as number of pieces of known information, the model is called just-identified. Note that fixed parameters do not count as unknown information or known information. However, restrictions of parameters should be reasonable on basis of evidence or theory. A just-identified model can be described with the following formula: x + y = 7 & 3x-y = 1. It should be noted that this kind of solutions always have a perfect fit. A goodness-of-fit model does not apply. It should be noted that this will only be a good model if the errors of the indicators are not correlated. A model is overidentified when the number of known exceeds the number of freely estimated model parameters. The difference in this constitutes the models degrees of freedom. To count the number of elements of the input matrix the following formula is very handy: b = p (p+1)/2, where p is the number of indicators included in the input matrix. It is however easier to count the parameters estimated in the model. In overidentified models the goodness-of-fit evaluation applies. Overidentified models rarely fit the data perfectly, although the available known information indicates that there is one best value for each freely estimated parameter. An solution can also be empirically underidentified. This means the solution is just- or overidentified, but aspects of the input matrix prevent the analysis from obtaining a unique and valid set of parameter estimates. (for example covariance of zero). This will lead to a fail in the computer software, or an improper solution, for example an Heywood case. Guideline for model identification: On the basis of the preceding discussion, some basic guidelines can be summarized:

1. Regardless of the complexity of the model latent variables must be scaled by specifying marker indicators or fixing the variance of the factor, usually to 1.

2. Regardless of the complexity of the model, the number of pieces in the input matrix, must be equal or exceed the number of freely estimated model parameters.


3. In the case of a one-factor model, a minimum of three indicators is required, the it is just-identified. With four of more indicators, the model can be overidentified, and the goodness of fit can be used.

4. With two or more factors and at least two indicators per laten construct, the solution will be overidentified. However a minimum of three indicators is recommended, given susceptiblility to empirical underidentification.

Estimation of CFA model Paramaters: The objective for CFA is to obtain estimates for each parameter of the measurement model that produce a predicted variance-covaraince matrix that resembles the sample variance=covariance matrix as closely as possible. In overidentified models perfect fit will rarely be achieved. This process entails a fitting function; a mathematical operation to minimize the difference between and S. Most likely used is Maximum Likelihood(ML). Determinant: a single number that reflects a generalized measure of variance for the entire set of variables contained in the matrix Trace: sum of values on the diagonal. ML tries to minimize the differences between these matrix summaries for S and . This leads to the following formula: FML = ln|S| ln|| + trace[(S)(-1)] p. This procedure is iterative: the program begins with an initial set of parameter estimates (initial estimates) and repeatedly refines these estimates in an effort to reduce the value of Fml. (iterations). The program stops when a set of parameter estimates cannot be improved upon to further reduce Fml. (convergence). When the program does not converge this could mean the model is misspecified. ML is widely used because it possesses desirable statistical properties (e.g. provides SEs and goodness-of-fit indices). It should however be noted that ML is more prone to Heywood cases and distorted solutions if misspecification of the model has been the case. ML is based on certain assumptions: 1. Large sample size, 2. Continuity of measures, 3. Multivariate normal distribution. Non-normality can result in biases standard errors and poorly behaved chi-square test, and even incorrect parameter estimates. In this case MLM may be better. If measures are not continuous or non-normal WLS, WLSMV and ULS may be more appropriate. Illustration: example of previous text. You can read it from page 76. Descriptive goodness-of-fit indices: classically used is 2= Fml(N-1). However in practice it is rarely used as a sole index of model fit. This is because: 1. The underlying distribution is often not 2 distributed. 2. It is inflated by sample size. 3. Is it based on the very stringent hypothesis that S = Many other indices work with reasonable fit. Fit indices can be broadly characterized as three categories: absoluate fit, parsimony fit, relative fit (also called comparative/incremental fit) Absolute fit: they evaluate the reasonability of the hypothesis that S = , without taking into account other aspects such as fit in relation to more restricted solutions. Chi-square is the obvious example. Another is standardized root mean square residual (SRMR): the average discrepancy between the correlations observed in input matrix and the correlation depicted by the model. Root mean square residual (RMR), reflects the average discrepancy between observed and predicted covariances. This can however be difficult to interpret. Parsimony correction: these indices incorporate a penalty function for poor model parsimony. Most often used in root mean square error of approximation (RMSEA). It is an index that relies on the noncentral chi-square distribution, this includes a noncentrality parameter (NCP), which expresses the degree of model misspecification. It can be estimated as 2 df. Thus when the


model is perfect NCP is zero. RMSEA is an error of approximation index. Note that RMSEA is sensitive to the number of model parameters, but relatively insensitive to sample size. Note that there has been developed a statistical test of closeness of model fit. close fit is operationalized as RMSEA values less than or equal to .05. this is congruent with p>.05. although some methodologists argue for stricter guidelines. Comparative fit: evaluate the fit of a user-specified solution in relation to a more restricted, nested baseline model. the baseline model is a null, or independence model. these indices provide favorable outcomes. For example the comparative fit index (CFI), with values closer to 1.0 implying good model fit. It is also based on the noncentrality parameter. Another index is called the Tucker-Lewis index (TLI), which can compensate for the model complexity. Guidelines for interpreting goodness-of-fit indices: It should be noted that all indices have their strengths and weaknesses. TLI and RMSEA tend to falsely reject models when N is small, SRMR does not appear to perform well in CFA models based on categorical indicators. However overall they have satisfactory performances. Also, although the first step is looking at these indices there is more to a good model, such as examining potential areas of localized strain and interpretability. Based on all these things a reasonable good fit is obtained in instances where 1. SRMR values are close to .08 or below. RMSEA values are close to .06 or below and CFI and TLI values are close to .95 or greater. However when one of these values does not point to a good model, the model should not be rejected instantly. Maybe you should look to the appendixes provided at the end of chapter 3. I dont know how important they are. Chapter 4, specification and interpretation of CFA models: The concepts introduced in the previous chapters are now illustrated and extended in the context of a full example of a CFA measurement model. very important for this, is figure 4.1, page 104. (also placed here) Although the model is basic, numerous predictions underlie this model specification. 1. All measurement error is presumed to be unsystematic. 2.


Neuroticism and extraversion are presumed to be correlated, but no directionality of such relationships. This is often done in CFA. Using a simple formula provided in chapter 3 it can be readily determined that the input matrix contains 36 pieces of information. The measurement model contain 17 freely estimated parameters. Thus 19 dfs are in the model, and goodness-of-fit indices will apply. Model specification, substantive justification: CFA requires specification of a measurement model that is well grounded by prior empirical evidence and theory. This is because CFA entails more constraints than other approaches. Defining the metric of latent variables: Model specification also entails defining the metric of latent factors. In applied research often the market indicator approach is used. Although in practice this is often done without consideration, one should consider carefully which indicator to use. Data screening and selection of the fitting function: The vast majority of CFA and SEM analyses in the applied research literature are conducted using ML estimation. As discussed ML assumptions entail: sufficient sample size, indicators that approximate interval-level scales, multivariate normality. See also chapter 9 and 10. Data screening should be conducted on the raw sample data. If the data are deemed suitable, the researcher has the option for using the raw data, a correlation matrix, variance-covariance matrix. If the correlation matrix is used, the indicator SDs must also be provided for converting into variances and covariances. If ML is not being used, a simple correlation or covariance matrix cannot be used. See chapter 9, for missing data. Running the CFA analysis: The CFA model can be fitted to the data after the previous steps have been settled. Table 4.1 provides syntax for programs for the example model. the method of setting the marker indicator varies across software programs. For us, using lisrel, it is important to note Lisrel uses the value (VA) command to fix the unstandardized loading to 1.0. Although Mplus needs the least amount of syntax, novice users are advised to become fully aware of the system defaults, to ensure that their models are specified as intended. The default of each of the listed programs is to automatically generate initial estimates to begin the iterations to minimize Fml. Model evaluation: One of the most important aspects of model evaluation occurs prior the the actual statistical analysis. The model has to be meaningful and useful on the basis of prior research and theory. Then can be looked at overall goodness of fit, the presence of localized areas of strain in the solution, interpretability, size, and statistical significance of the models parameter estimates. Overall goodness of fit: goodness-of-fit indices are examined to begin evaluating the acceptability of the model. if indices point to good fit, the following step of model evaluation can take place. However, if indices point to poor fit, subsequent aspects of fit evalution would be focused on diagnosing the sources of model misspecification. Occasionally, fit indices will provide inconsistent information about the fit. In these instances, greater caution is needed in determining the acceptability of the solution. Localized areas of strain: the previous indices provide only a global, descriptive indication of the goodness-of-fit. However, the goodness of fit indices can be acceptable, when in fact some


relationships among indicators have not been reproduced. Two statistics are frequently used to identify focal areas of misfit in CFA: residuals & modification indices. Residuals: these can be found in the residual variance-covariance matrix, which reflects the difference between the sample and the model-implied matrices. The residual matrix provides specific information about how well each variance and covariance was reproduced by the models parameter estimates. However fitted residuals can be difficult to interpret because they are affected by raw metric and dispersion of the observed measures. This is addressed by standardized residuals: fitted residuals divided by estimated standard errors. Because these standardized residuals can be interpreted as z-scores, the same practical cutoff points can be used to examine if there is a significant difference between the observed and the presumed variances. In practice, researcher may scan for standardized residuals larger than 1.96 ( = p


In CFA models where there are no cross-loading indicators, the completely standardized factor loading can be interpreted as the correlation between the indicator and the latent factor: squaring the completely standardized factor loading provides the proportion of variance of the indicator that is explained by the latent factors : communality. These squared factor loadings can be considered as estimates of the indicators reliability. Small factor covariances are usually not considered problematic, however covariances approaching 1.0 could lead to questioning the notion that the latent factors represent distinct constructs. Factor correlations exceeding .80 or .85 are often used as criterion to define poor discriminant validity. Interpretation and calculation of CFA model parameter Estimates: This section reviews on how the various parameter estimates in the example model are calculated and interpreted, see figure 4.2, page 132. The marker indicator method is used (N1, E1 are set to 1.000). in unstandardized solutions, factor loadings can be interpreted as unstandardized regression coefficients. Thus, in standardized solutions, the factor loadings are standardized regression coefficients. Squaring the standardized factor loading provides the proportion of variance in the indicator that is explained by the latent factor. There is also an easy method for transforming solutions to either unstandardized or standardized. The variance of the latent factor is calculated by squaring the completely standardized factor loading of the marker indicator and multiplying this result by the observed variance of the marker indicator. The standard deviations of the latent factor are calculated by simply taking the square root of the factor variances. The unstandardized error variances of the indicators can be calculated by multiplying the completely standardized residuals by the observed variance of the indicators.(2 = 2 * 22 , whereby the second d2 is standardized) They can also be calculated by squaring the completely standardized factor loadings, minus the observed variances of indicators. (2 = 22 22(212)) Factor covariances can be calculated by using the following formula: 21 = r21(SD1)(SD2). Unstandardized regression coefficient can be computed by: b = (ryxSDy) / (SDx), whereby ryx is the completely standardized factor loading and SDy the SD of the indicators and SDx the SD of the latent factor. Solutions can also be completely standardized. The standardized indicator error can be calculated by dividing the model-estimated error variance, by the observed variance. Factor intercorrelation can be calculated by dividing a factor covariance by the product of the SDs of the factors. Squaring a factor correlation provides the proportion of overlapping variance between two factors. A standardized regression coefficient can also be computed by the following formula: b* = (bSDx) /


(SDy). Standardized estimates are helpful to the interpretation of models where latent factors are regressed onto categorical background variables. When factor variances are fixed to a specific value, this is usually 1. Many aspects of CFA solution do not change, but the unstadndardized estimates of the factor loadings, factor variances and factor covariances will. In addition, the unstandardized factor loadings will take on the same values of the standardized estimates. Cfa models with single indicators: CFA makes it possible to include single indicator variables in the analysis. These variables should not be interpreted as factors. When CFA is a precursor for SEM it is very important to include these single indicators is the measurement model. otherwise, specification error may occur in SEM. In CFA the relationship between these single indicators and latent factors can be examined. For these single indicators an error theory can be invoked, by fixing the unstandardized error of the indicator to some predetermined value. when this is zero, it is assumed the indicator is perfectly reliable. This is however not always the right way. Fortunately, measurement error can be incorporated by fixing its unstandardized error to some non-zero value, calculated on the basis of the measures sample variance estimate: x = VAR(X)(1 - ). An example is provided in the text, page 140/141. in this example there are two single indicators that load on pseudofactors. When this loading is squared it produces the reliability coefficient. Reporting a CFA study: Researchers should be aware of what information should be presented when reporting the results of a CFA study. The recommended information to present is lsisted in table 4.6, page 145, 146. There should also be a few notes made. First, most models are too complex to present them in a path diagram. Often a tabular format is used, specifically a p-by-m matrix of factor loadings. It should be noted that there is no gold standard for how a path diagram should be prepared. Some constants exist, but other things are more variable. The reader should always be aware of such differences. Another consideration is whether to present an unstandardized solution or a completely standardized solution. The convention has been the latter one. Although the completely standardized solution can be informative, the unstandardized solution may be preferred in some instances. If possible, the sample input data used in the CFA should be published in the research report, or made available upon request. Finally, it should be emphasized that the suggestion provided in table 4.6 are most germane to a measurement model conducted in a single group and thus must be adapted on the basis of the nature of the particular CFA study. Chapter 5, CFA model Revision and Comparison: Often a CFA model will need to be revised. Mostly this is done to improve the fit of the model. this means: the model does not fit well on the whole, does not reproduce some indicator relationships well, or does not produce uniformly interpretable parameter estimates. in addition, respecification is often conducted to improve the parsimony and interpretability of the CFA model. this will almost never lead to a better fit. Three types of respecification to improve parsimony, are multiple-groups solutions and higher-order factor models, and collapsing the highly overlapping factors. Sources of poor-fitting CFA solutions:


In a CFA model the main sources of misspecification are the number of factors, the indicators and the error theory. When the initial model is grossly misspecified, specification searches are not nearly as likely to be successful. But with small misspecification it can be successful. number of factors: this is in practice rarely the case. When it does, it is often because the researcher has moved into the CFA framework to quickly. However, there are some instances where EFA has the potential to provide misleading information. For example, when the relationships among indicators are better accounted for by correlated errors than separate factors. Thus while EFA recommends more factors, the variance can be explained by method effects. A CFA solution with too few factors will fail to adequately reproduce the observed relationships among the several indicators. One needs to pay attention to this, because results can point to need for extra correlated measurement errors, when instead the true model has one more factor. A model with the same indicators, but less factors, is a nested version of one with more factors. the nested model is always more constrained than the parent model. These solutions can be compared by using the chi-square statistic. When the chi-square difference is more than 3.84 it is said to provide a significant better fit to the data. However, there are some critics that suggest a factor model with fewer factors is not a nested model. and chi-square statistics should be interpreted the way they are. Moreover, modification indices represent the predicted decrease in model chi-square if a fixed or constrained parameter was freely estimated, thus one df difference. However, nested models differ mostly more than one df. Thus, these indices cannot be used for these differences. As with all here described, the number of factors should only be changed if there is a theory or rationale for doing so. When to many factors have been specified, this can be visible in correlations between factor approaching +/- 1; there is poor discriminant validity. In applied research this point is often placed at 0.85. An alternative of combining two highly correlated factors is to drop one of the factors and its constituent indicators. This could be beneficial if that factor only had a few indicators loading on it. Indicators and factor loadings: another potential source of CFA model misspecification is an incorrect designation of the relationships between indicators and the latent factors. this can occur in the following manner: 1. Indicator should load on more than one factor; 2. Indicator should load on another factor 3. Indicator has no relationship with any of the factors. fit diagnostics for these forms of misspecifications are presented in figure 5.1, page 168, as well as table 5.2. It should be noted that when only minor misspecifications are the case, specification searchers are more likely to be successful. Secondly, these results demonstrate that the acceptability of a model should not only be based on indices of overall model it. Third, the modification indexes and standardized EPC values do rarely correspond exactly to the actual change in the model chi-square and parameter estimates. This is because modification indexes are approximations of model change. It should be noted that parameters estimates should not be interpreted when the model is poor fitting. In this case Heywood cases can arise, through the iterative process, the parameters may take on out-of-range values to minimize Fml. Moreover, an indicator-factor relationship may be misspecified when an indicator loads on the wrong factor. This does not have to appear in the overall fit. But it will surely by available to see in the large standardized residuals and modification indexes. When two models are not nested, the chi-square difference test cannot be used. Another strategy can be used to compare the two solutions. In this case there will be looked at the overall goodness of fit, focal areas of ill fit, and interpretability/strength of parameter estimates. In addition, two other procedures for using chi-square with non-nested models have been developed; Akaike Information Criterion (AIC) & Expected Cross-Validation Index (ECVI). The


following formulas are needed: AIC = 2 2a, where a is the number of freely estimated parameters in the model. (note this how it is done in Lisrel, other programs use other formulas, see page 180). ECVI = (2/n) + 2(a/n), where n = N-1, and a the number of freely estimated parameters. The indices foster the comparison of the overall fit models, with adjusting for the complexity of each. Another possible problematic outcome is that an indicator does not load on any factor. This is also visible in the standardized residuals. In this case this indicator can be removed, but the overall fit of the model will probably not improve (because there is no influence of the indicator. Correlated errors: A CFA solution can also be misspecified with respect to the relationships among the indicator error variances. When no correlated error is assumed, the researcher is asserting that all the covariation among the indicators is due to the latent dimension, and all measurement error is random. Note that correlated error can occur and it is even possible to occur on items loading on different latent factors. Correlated error may arise from items that are very similarly worded, reverse-worded, or differentially prone to social desirability, and so forth. Unnecessary correlated errors can be detected be results indicating their statistical or nonsignificance. When correlated errors are not specified, standardized residuals and modification indices indicate that the relationship between these items has not adequately been reproduced. Because of the large sample sizes typically involved in CFA, the researcher will often encounter boredeline modification indices that suggest the model could be improved if correlated errors were added. However, the researcher should always take into account whether these modifications make theoretical sense or not. Improper solutions and nonpositive definite matrices: A measurement model should not be deemed acceptable if the solution contains one or more parameter estimates that have out-of-range values, also called Heywood cases or offending estimates. This can be for example a negative error variance or an standardized factor loading with a value greater than 1.0. A necessary condition for obtaining a proper CFA solution is that both the input variance-covariance matrix and the model-implied variance-covariance matrix are positive definite. See also appendix 3.3 (chapter 3). A determinant is a single number (scalar) that conveys the amount of nonredundant variance in a matrix. When this is zero, the matrix is said to be singular; one or more rows or columns in the matrix are linearly dependent on other rows and columns. Thus a singular matrix will not be positive definite. The condition of positive definiteness can be evaluated by submitting the variance-covariance matrix to PCA. If all the eigenvalues produced are greater than zero, the matrix is indefinite. Often a nonpositive definite input matrix is due to a minor data entry problem, or errors in reading the data into the analysis, such as a large amount of missing data and not the right approach towards them. Pairwise deletion of missing data can cause definiteness problems because the input matrix is computed on different subsets of the sample; listwise deletion can produce nonpositive definite matrix by decreasing the sample size. Another common cause for improper solutions is a misspecified model. In these situations is is often possible to revise the model using the fit diagnostic procedures described previously, or moving back to an EFA framework. Problems can also often arise when using small samples. Small samples are more prone to the influence of outliers. Moreover, the risk for negative variance estimates is highest in small samples when there are only two or three indicators per latent variable and when the communalities of


the indicators are low. Thus having more indicators per factor decreseas the likelihood of improper solutions. Note that the estimator approach should be appropriate for the data. Data that are non-normal or categorical data need other approaches, see also chapter 9. Moreover, the risk of nonconvergence and improper solutions is positively related to model complexity. Sometimes this can be rectified by removing some freely estimated parameters. See figure 5.2, for some examples of nonpositive definite matrices and improper solutions. (page 192,193) In practice, the problem of improper solutions is often circumvented by a quick fix method. When the LISREL program encounters an indefinite matrix, it invokes a ridge option; a smooting function to eliminate negative or zero eigenvalues. However this is not recommended, because they dismiss programs with the data or specification. EFA in the CFA Framework: a common sequence in scale development and construct validation is to conduct CFA as the next step after latent structure has been explored using EFA. however, sometimes the CFA leads to problems not encountered in EFA. there is also the option for the procedure of exploratory factor analysis within the CFA framework (E/CFA). This provides a more intermediate step. In this strategy, the CFA applies the same number of identifying restrictions used in EFA by fixing the factor variances to unity, freely estimating the factor covariances, and by selecting an anchor item for each factor whose cross-loadings are fixed to zero. This provides more information, such as statistical significance of cross-loadings and the potential presence of salient error covariances. An example is given, whereby table 5.6, and 5.7 present the results and syntax, page 194-198. There are two major differences with common CFA models: all factor loadings and cross-loadings are freely estimated & the metric of the latent actors is specified by fixing the factor variances to 1.0. see also table 5.8 for output. Although EFA may also furnish evidence of the presence of double-loading items, EFA does not provide any direct indications of the potential existence of salient correlated errors. Model identification revisited: Because latent variable software programs are capable of evaluating whether a given model is identified, it is often most practical to simply try to estimate the solution and let the computer determine the models identification status. It is however helpful to be aware of general model identification guidelines. Chapter 3, has made some comments regarding this issue. Moreover, the researcher should be mindful of the fact that specification of a large number of correlated errors may produce an underidentified model. as a general rule, for every indicator there should be at least one other indicator in the solution with which it does not share an error covariance. Equivalent CFA solutions: Another important consideration in model specification and evaluation is the issue of equivalent solutions. Equivalent solutions are solutions with identical goodness of fit and predicted covariance matrices. An example is given in the book:

Equivalent solutions cannot be compared based in the chi-square difference tests. Thus other options should be considered. For example based on logic or theory some solutions can be dismissed. SEM solutions can also be equivalent, see also figure 5.4, page 208.


It should be noted that in practice, researchers explicit recognition of equivalent models is almost never stated in articles. Chapter 10: In designing a CFA investigation, the researcher must address critical questions of how many cases should be collected to obtain an acceptable level of precision and statistical power of the models parameter estimates, as well as reliable indices of overall model fit. Some rules of thumb have been offered, but they were mostly arbitrary; such as minimal sample size (N >100), mimimum number of cases per each freed parameter (5 10), minimum number of cases per indicator. Requisite sample size depends on a variety of aspects such as study design, the size of the relationships among the indicators, the reliability of the indicators, the scaling and distribution of the indicators, estimator type, the amount and patterns of missing dat, the size of the model. The sample size affects the statistical power and the precision of models parameter estimates. Statistical power = 1 probability of Type II error. Mostly set at cut-off point of .80. Precision: the ability of the models parameter estimates to capture true population values. Satorra-saris method: This is the most widely used approach of conducting power analysis in SEM. It looks at the power of the chi-square difference test to detect specification errors associated with a single parameter. The researcher specifies two models: a model associated with a null hypothesis (H0) and a model associated with an alternative hypothesis (H1). H1 is set to reflect the true population values for each parameter. H0 is identical, except for the parameters to be tested. The covariance matrix of H1 is used as input for H0. A sample size of interest is used and the parameters of interest are misspecified, usually to 0. This will produce a nonzero model chi-square value, which represents a noncentrality parameter (NCP) of noncentral chi-square distribution. Using this NCP, the power of the test can be determined. An example is given on page 414 and 415. See also table 10.1 for LISREL syntax for creating a covariance matrix of H1. The covariance matrix used as input for H1 contains 1s on the diagonal and 0s on the off-diagonal. All parameters are fixed to population values using the value command (??), this will provide a fitted covariance matrix that will be used as the population matrix for the next stepts. The second step is an accuracy check. H1 is freely estimated using the fitted covariance matrix . this should provide a perfect fit of the model to the data. In the third step, two key alterations are made. The parameter of interest is misspecified by fixing it to zero. (H0 is specified) secondly, the sample size for which power is desired should be specified. If the value found in this analysis is not zero, this suggests that the misspecification is observed. In the next step this value is used as the NCP to calculate the power. There are several programs the calculate this and you will need the degrees of freedom (in this case the number of parameters focused on, for the example it is 1), the NCP and the critical chi-square value (for an of .05, it is 3.84) Some critics have suggest model-based approaches to power analysis require the researcher to make exact estimates of population values for each parameter in the model. This can be a constrain of the method, when there is no research to base these parameters on. A few misestimates of the parameter population values may undermine the power analysis. Monte Carlo approach: Recent developments in latent variable software packages permit researchers to use the Monte Carlo methodology to determine the power and precision of model parameters in context of a


given model, sample size and data set. This method also requires specification of an H1 model containing the population values of all parameters. On the basis of this, numerous samples are randomly generated. Some programs allow for specifying non-normality and missing data, as well as the capability of estimating in the context of noncontinuous indicators. For each sample the SEM model is estimated and all results are averaged together. These averages are used to determine the precision and power of the estimates. This approach is also illustrated with an example, with Mplus syntax. Because we dont use Mplus, I will not explain this. Muthen en Muthen have established the following criteria for determining adequate sample size: 1. Bias of the parameters and their standard errors do not exceed 10 % for any parameter in the model; 2. Parameters that are the specific focus of the power analysis, have a standard error that does not exceed 5%; 3. Coverage is between .91 and .98.; 4. Power of the salient model parameters is .80 or above. The percentage of parameter bias can be calculated by subtracting the population parameter value from the average parameter value, dividing this difference by the population value and then multiplying the result by 100. The bias in standard errors of parameters can also be calculated. By substracting the average of the estimated standard errors across replications from the standard deviation of the parameter estimate, dividing this difference by the standard deviation of the parameter estimate and then multiplying by 100. The rest is not of great importance in my opinion. Chapter 7, CFA with Equality Constraints, Multiple Groups, and Mean Structures: Overview of equality constrains: Parameters in a CFA solution can be freely estimated, fixed or constrained. A constrained parameter is unknown, but it cannot have any value. the most common form of constrained parameters are equality constraints, in which unstandardized parameters are restricted to be equal in value.

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