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Structural Equation Modelling (SEM) dengan LISREL Konsep & Tutorial book . Read 3 reviews from the world's largest community for readers. Ada dua pe. LISREL is a registered trademark of Scientific Software International, Inc. . This document is intended as a tutorial to familiarize new users of LISREL for. International Standard Book Number: (Hardback) (Paperback) . LISREL–PRELIS Missing Data Example. Author: JASMINE BLATTLER Language: English, Spanish, Hindi Country: East Timor Genre: Religion Pages: 542 Published (Last): 26.02.2016 ISBN: 600-6-27142-299-5 ePub File Size: 15.57 MB PDF File Size: 12.20 MB Distribution: Free* [*Free Regsitration Required] Downloads: 44447 Uploaded by: RAISA

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A beginner's guide to structural equation modeling Home A beginner's guide to structural equation modeling. Randall E. Cover Design: Textbook Production Manager: Full-Service Compositor: Text and Cover Printer:

Similarly, a ratio variable, for example, weight, has the property of scale that implies equal intervals and a true zero point weightlessness. Therefore, ratio variables also permit mathematical operations of computing a mean and a standard deviation. Our use of different variables requires us to be aware of their properties of scale and what mathematical operations are possible and meaningful, especially in SEM, where variance-covariance correlation matrices are used with means and standard deviations of variables.

Different correlations among variables are therefore possible depending upon the level of measurement, but create unique problems in SEM see chap. Restriction of Range Data values at the interval or ratio level of measurement can be further defined as being discrete or continuous. For example, the number of continuing education credits could be reported in whole numbers discrete.

In contrast, a continuous variable is reported using decimal places; for example, a students' grade point average would be reported as 3. Joreskog and Sorbom provided a criterion in the PRELIS program based on research that defines whether a variable is ordinal or interval based on the presence of 15 distinct scale points.

Other factors that affect the Pearson correlation coefficient are presented in this chapter and discussed further in chapter 3. Missing Data The statistical analysis of data is affected by missing data values in variables. It is common practice in statistical packages to have default values for handling missing values.

The researcher has the options of deleting subjects who have missing values, replacing the missing data values, and using robust statistical procedures that accommodate for the presence of missing data.

SEM software programs handle missing data differently and have different options for replacing missing data values. Table 2. These options can dramatically affect the number of subjects available for analysis and the magnitude and the direction of the correlation coefficient, and can create problems if means, standard deviations, and correlations are computed based on different sample sizes. Listwise deletion of cases and pairwise deletion of cases are not always recommended due to the possibility of TABLE 2.

Mean substitution works best when only a small number of missing values is present in the data, whereas regression imputation provides a useful approach with a moderate amount of missing data. Amos uses full information maximum likelihood estimation in the presence of missing data, so it does not impute or replace values for missing data.

The value to be substituted for the missing value of a single case is obtained from another case that has a similar response pattern over a set of matching variables. In multivariable data sets, where missing values occur on more than one variable, one can use multiple imputation of missing values with mean substitution, delete cases, or leave the variables with defined missing values as options in the dialog box.

In addition, the multiple imputation procedure implemented in LISREL uses either the expected maximization EM algorithm or Monte Carlo Markov chain MCMC; generating random draws from probability distributions via Markov chains approaches to replacing missing values across multiple variables. We assume the data to be missing at random MAR with an underlying multivariate normal distribution. We must know the number of variables in the raw data file.

We must also select Data, then Define Variables, and then select We should examine our data both before Table 2. This provides us with valuable information about the nature of the missing data. We also highly recommend comparing SEM analyses before and after the replacement of missing data values to fully understand the impact missing data values have on the parameter estimates and standard errors.

A comparison of EM and MCMC is also warranted in multiple imputations to determine the effect of using a different algorithm for the replacement of missing values. We have also noticed that selecting matching variables with a higher correlation to the variable TABLE 2.

Total sample size 28 Number of missing values 0 1 Number of cases 19 9 Effective sample sizes [univariate in diagonal and pairwise bivariate off diagonal ]: Frequency Percent 19 9 VAR1 Outliers Outliers or influential data points can be defined as data values that are extreme or atypical on either the independent X variables or 32 CHAPTER 2 dependent Y variables variables or both.

Outliers can occur as a result of observation errors, data entry errors, instrument errors based on layout or instructions, or actual extreme values from self-report data. Because outliers affect the mean, the standard deviation, and correlation coefficient values, they must be explained, deleted, or accommodated by using robust statistics e. Sometimes, additional data will need to be collected to fill in the gap along either the Y or the X axis.

EQS has an interesting feature: In addition, EQS permits three-dimensional 3D rotation of factor analysis axes to visualize the pattern of coordinate points. Linearity A standard practice is to visualize the coordinate pairs of data points of continuous variables by plotting the data in a scatterplot. These bivariate plots depict whether the data are linearly increasing or decreasing. The presence of curvilinear data reduces the magnitude of the Pearson correlation coefficient, even resulting in the presence of zero correlation.

Recall that the Pearson correlation value indicates the magnitude and the direction of the linear relationships between pairs of data. Figure 2.

Nonnormality Inferential statistics often rely on the assumption that the data are normally distributed. Data that are skewed lack of symmetry or more frequently occurring along one part of the measurement scale will affect the variance-covariance among variables. In addition, kurtosis flatness in data will impact statistics. Leptokurtic data values are more peaked than the symmetric normal distribution, whereas platykurtic data values are flatter and more dispersed along the X axis, but have a consistent low frequency on the Y axis, that is, the frequency distribution of the data appears rectangular in shape.

Nonnormal data can occur because of the scaling of variables ordinal rather than interval or the limited sampling of subjects. Possible solutions for skewness are to resample more participants or perform a permissible linear transformation, for example, square root, reciprocal, logit, or probit.

Our experience is that a probit data transformation works best in correcting skewness. Kurtosis in data is more difficult to resolve; however, leptokurtic data can be analyzed using elliptical estimation techniques in EQS.

Platykurtic data are the most problematic and require additional sampling of subjects or bootstrap methods available in the SEM software programs. The factor loading matrices must be full rank and have no rows with zeros in them to be able to compute structure coefficients, that is the variance-covariance matrices must be positive definite Wothke, Researchers should use the built-in menu options to examine, graph, and test for any of these problems in the data prior to conducting any SEM model analysis.

Basically, researchers should know their data characteristics. Data screening is a very important first step in structural equation modeling. The next chapter illustrates in more detail issues related to the use of correlation and variance-covariance in SEM models. We provide specific examples to illustrate the importance of topics covered in this chapter.

Amos uses which command to import data sets? File, then Data Files b. File, then Open c. File, then New 2. EQS uses which command to import data sets? File, then New 3. File, then New 4. Define the following levels of measurement. Nominal b. Ordinal c. Interval d. Ratio 5. Mark each of the following statements true T or false F. Amos can compute descriptive statistics. EQS can compute descriptive statistics. Explain how each of the following affects statistics: Restriction of range b.

Missing data c. Outliers d. Nonlinearity e. Scales and statistics: Parametric and non-parametric. Psychological Bulletin, 58, User's reference guide.

Stevens, S. On the theory of scales of measurement.

Science, , Wothke, W. Nonpositive definite matrices in structural equation modeling. Long Eds. Newbury Park, CA: File, then Data Files 2. File, then Open 3. Restriction of range: A set of scores that are restricted in range implies reduced variability. Variance and covariance are important in statistics, especially correlation. Missing data: A set of scores with missing data can affect the estimate of the mean and standard deviation. It is important to determine whether the missing data are due to data entry error, are missing at random, or are missing systematically due to some other variable e.

A set of scores with an outlier extreme score can affect the estimate of the mean and standard deviation. It is important to determine whether the outlier is an incorrect data value due to data entry error, represents another group of persons, or potentially requires the researcher to gather more data to fill in between the range of data. Researchers have generally analyzed relationships in data assuming linearity. Linearity is a requirement for the Pearson correlation coefficient.

Consequently, a lack of linearity that is not included in the statistical model would yield misleading results. Skewness, or lack of symmetry in the frequency distribution, and kurtosis, the departure from a normal distribution, affect inferential statistics, especially the mean, the standard deviation, and correlation coefficient estimates.

Data transformations, especially a probit transformation, can help to yield a more normally distributed set of scores. Unstandardized 3.

## Wijanto lisrel

Shortly thereafter, Charles Spearman used the correlation procedure to develop a factor analysis technique. The correlation, regression, and factor analysis techniques have for many decades formed the basis for generating tests and defining constructs.

Today, researchers are expanding their understanding of the roles that correlation, regression, and factor analysis play in theory and construct definition to include latent variable, covariance structure, and confirmatory factor measurement models. The relationships and contributions of Galton, Pearson, and Spearman to the field of statistics, especially correlation, regression, and factor analysis, are quite interesting Tankard, In fact, the basis of association between two variables, that is, correlation or covariance, has played a major role in statistics.

The Pearson correlation coefficient provides the basis for point estimation test of significance , explanation variance accounted for in a dependent variable by an independent variable , prediction of a dependent variable from an independent variable through linear regression , reliability estimates test-retest, equivalence , and validity factorial, predictive, concurrent. The partial and part correlations further permit the identification of specific bivariate relationships between variables that allow for the specification of unique variance shared between two variables while controlling for the influence of other variables.

Partial and part correlations can be tested for significance, similar to the Pearson correlation coefficient, by simply using the degrees of freedom, n — 2, in the standard correlation table of significance values or an F test in multiple regression that tests the difference in R2 values between full and restricted models see Tables A. Although the Pearson correlation coefficient has had a major impact in the field of statistics, other correlation coefficients have emerged depending upon the level of variable measurement.

The types of correlation coefficients developed for these various levels of measurement are categorized in Table 3. Therefore, you may need to check a popular statistics book or look around for a computer program that will compute the type of correlation coefficient you need, for example, the phi or the point-biserial coefficient.

In SEM analyses, the Pearson coefficient, tetrachoric or polychoric for several ordinal variable pairs coefficient, and biserial or polyserial for several continuous and ordinal variable pairs coefficient are typically used see PRELIS for the use of Kendall's tau-c or tau-b, and canonical correlation.

The SEM software programs permit mixture models, which use variables with ordinal and interval-ratio levels of measurement see chap. Although SEM software programs are now demonstrating how mixture models can be analyzed, the use of variables with different levels of measurement has traditionally been a problem in the field of statistics e.

In this chapter we describe the important role that correlation covariance plays in structural equation modeling. We also include a discussion of factors that affect correlation coefficients and the assumptions and limitations of correlation methods in structural equation modeling. The key factors are the level of measurement, restriction of range in data values variability, skewness, kurtosis , missing data, nonlinearity, outliers, correction for attenuation, and issues related to sampling variation, confidence interval, effect size, significance, and power addressed in bootstrap estimates.

Level of Measurement and Range of Values Four types or levels of measurement typically define whether the characteristic or scale interpretation of a variable is nominal, ordinal, interval, or ratio Stevens, In structural equation modeling, each of these types of scaled variables can be used.

However, it is not recommended that they be included together or mixed in a correlation covariance matrix. Until recently, SEM required variables measured at the interval or ratio level of measurement, so the Pearson product-moment correlation coefficient was used in regression, path, factor, and structural equation modeling.

The interval or ratio scaled variable values should also have a sufficient range of score values to introduce variance. If the range of scores is restricted, the magnitude of the correlation value is decreased.

Basically, as a group of subjects becomes more homogeneous, score variance decreases, reducing the correlation value between the variables. This points out an interesting concern, namely, that there must be enough variation in scores to allow a correlation relationship to manifest itself between variables. The meaningfulness of a correlation relationship will depend on the variables employed; hence, your theoretical perspective is very important. You may recall from your basic statistics course that a spurious correlation is possible when two sets of scores correlate significantly but are not meaningful or substantive in nature. The probit transformation appears to be most effective in handling univariate skewed data. Consequently, the type of scale used and the range of values for the measured variables can have profound affects on your statistical analysis in particular, on the mean, variance, and correlation. The scale and range of a variable's numerical values affect statistical methods, and this is no different in structural equation modeling.

Nonlinearity The Pearson correlation coefficient indicates the degree of linear relationship between two variables. It is possible that two variables can indicate no correlation if they have a curvilinear relationship. Thus, the extent to which the variables deviate from the assumption of a linear relationship will affect the size of the correlation coefficient. It is therefore important to check for linearity of the scores; the common method is to graph the coordinate data points.

The linearity assumption should not be confused with recent advances in testing interaction in structural equation models discussed in chapter You should also be familiar with the eta coefficient as an index of nonlinear relationship between two variables and with the testing of linear, quadratic, and cubic effects.

Consult an intermediate statistics text e. The heuristic data set in Table 3. In the first data set, the Y values increase from 1 to 10 and the X values increase from 1 to 5, then decrease from 5 to 1 nonlinear. The restriction of range in values can be demonstrated using the fourth heuristic data set in Table 3.

The fifth data set indicates how limited sampling can affect the Pearson coefficient. Missing Data A complete data set is also given in Table 3. The Pearson correlation coefficient changes from statistically significant to not statistically significant. More importantly, in a correlation matrix with several variables, the various correlation coefficients could be computed on different sample sizes.

If we used listwise deletion of cases, then any variable in the data set with a missing value would cause a subject to be deleted, possibly causing a substantial reduction in our sample size, whereas pairwise deletion of cases would result in different sample sizes for our correlation coefficients in the correlation matrix. Researchers have examined various aspects of how to handle or treat missing data beyond our introductory example using a small heuristic data set.

One basic approach is to eliminate any observations where some of the data are missing, listwise deletion. Pairwise deletion excludes data only when they are missing on the variables selected for analysis. However, this could lead to different sample sizes for the correlations and related statistical estimates. Missing completely at random MCAR implies that data are missing unrelated statistically to the values that would have been observed.

Missing at random MAR implies that data values are missing conditional on other variables or a stratifying variable. A third situation, nonignorable data, implies probabilistic information about the values that would have been observed. For MCAR data, mean substitution yields biased variance and covariance estimates, whereas FIML and listwise and pairwise deletion methods yield consistent solutions. For MAR data, FIML yields estimates that are consistent and efficient, whereas mean substitution and listwise and pairwise deletion methods produce biased results.

When missing data are nonignorable, all approaches yield biased results; however, FIML estimates tend to be less biased. It would be prudent for the researcher to investigate how parameter estimates are affected by the use or nonuse of a data imputation method.

Basically, FIML is the recommended parameter estimation method when data are missing in structural equation model analyses. Outliers The Pearson correlation coefficient is drastically effected by a single outlier on X or Y. For example, the two data sets in Table 3. In EQS, open the data set manul7. After clicking OK, one sees the following scatterplot: Notice the outlier data point in the upper right-hand corner at the top of the scatterplot. If we brush this data point and then drop it in the black hole in the upper left corner, our regression calculations are automatically updated without this outlier data point.

To brush the outlier data point, use the left mouse button and drag from the upper left to the lower right as if forming a rectangle. The outlier data point should turn red once you release the left mouse button, as indicated in the first diagram.

## Structural Equation Modelling (SEM) dengan LISREL Konsep & Tutorial by Setyo Hari Wijanto

To drag the outlier data point to the black hole, place the mouse pointer on the data point, depress the left mouse button, and drag the outlier data point to the black hole. Once you release the left mouse button, the outlier data point should drop into the black hole, as indicated in the second diagram.

The regression equation is automatically updated with an R2 value of. This indicates the impact of a single outlier data point. Other outlier data points can be brushed and dragged to the black hole to see cumulative effects of other outlier data points. Correction for Attenuation A basic assumption in psychometric theory is that observed data contain measurement error. A test score observed data is a function of a true score and measurement error. A Pearson correlation coefficient has different values depending on whether it is computed with observed scores or the true scores where measurement error has been removed.

The Pearson correlation coefficient can be corrected for attenuation or unreliable measurement error in scores, thus yielding a true score correlation; however, the corrected correlation coefficient can become greater than 1.

When this happens, either a condition code or a non-positive definite error message occurs stopping the structural equation software program. Non-Positive Definite Matrices Correlation coefficients greater than 1. In other words, the solution is not admissible, indicating that parameter estimates cannot be computed. Correction for attenuation is not the only situation that causes non-positive definite matrices to occur Wothke, Sometimes the ratio of covariance to the product of variable variance yields correlations greater than 1.

The following variance-covariance matrix is nonpositive definite because it contains a correlation coefficient greater than 1. This can be caused by correlations greater than 1. A Heywood case also occurs when the communality estimate is greater than 1. Possible solutions to resolve this error are to reduce communality or fix communality to less than 1.

Regression, path, factor, and structural equation models mathematically solve a set of simultaneous equations typically using ordinary least squares OLS estimates as initial estimates of coefficients in the model. However, these initial estimates or coefficients are sometimes distorted or too different from the final admissible solution.

When this happens, more reasonable start values need to be chosen. It is easy to see from the basic regression coefficient formula that the correlation coefficient value and the standard deviation values of the two variables affect the initial OLS estimates: Sample Size A common formula used to determine sample size when estimating means of variables was given by McCall For example, given a random sample of ACT scores from a defined population with a standard deviation of , a desired confidence level of 1.

In structural equation modeling, however, the researcher often requires a much larger sample size to maintain power and obtain stable parameter estimates and standard errors. The need for larger sample sizes is also due in part to the program requirements and the multiple observed indicator variables used to define latent variables.

Hoelter proposed the critical N statistic, which indicates the sample size that would make the obtained chi-square from a structural equation model significant at the stated level of significance. This sample size provides a reasonable indication of whether a researcher's sample size is sufficient to estimate parameters and determine model fit given the researcher's specific theoretical relationships among the latent variables.

SEM software programs estimate coefficients based on the user-specified theoretical model, or implied model, but also must work with the saturated and independence models. A saturated model is the model with all parameters indicated, whereas the independence model is the null model or model with no parameters estimated. If the sample size is small, then there is not information to estimate parameters in the saturated model for a large number of variables.

Consequently, the chi-square fit statistic and derived statistics such as Akaike's information criterion AIC and the root-mean-square error of approximation RMSEA cannot be computed.

In addition, the fit of the independence model is required to calculate other fit indices such as the comparative fit index CFI and the normal fit index NFI. Ding, Velicer, and Harlow found numerous studies e. Boomsma , recommended , and Hu, Bentler, and Kano indicated that in some cases 5, is insufficient!

### See a Problem?

Many of us may recall rules of thumb in our statistics texts, for example, 10 subjects per variable or 20 subjects per variable.

In our examination of the published research, we found that many articles used from to subjects, although the greater the sample size, the more 50 CHAPTER 3 likely it is that one can validate the model using cross-validation see chap. For example, Bentler and Chou suggested that a ratio as low as 5 subjects per variable would be sufficient for normal and elliptical distributions when the latent variables have multiple indicators and that a ratio of at least 10 subjects per variable would be sufficient for other distributions.

Cohen and Cohen , in describing correlation research, further presented the correlation between two variables controlling for the influence of a third. These correlations are referred to as part and partial correlations, depending upon how variables are controlled or partialed out.

Some of the various ways in which three variables can be depicted are illustrated in Fig. The diagrams illustrate different situations among variables where a all the variables are uncorrelated Case 1 , b only one pair of variables is correlated Cases 2 and 3 , c two pairs of variables are correlated Cases 4 and 5 , and d all of the variables are correlated Case 6.

It is obvious that with more than three variables the possibilities become overwhelming. A theoretical perspective is essential in specifying a model and forms the basis for testing a structural equation model. The partial correlation coefficient measures the association between two variables while controlling for a third, for example, the association between age and comprehension, controlling for reading level.

Controlling for reading level in the correlation between age and comprehension partials out the correlation of reading level with age and the correlation of reading level with comprehension. Part correlation, in contrast, is the correlation between age and comprehension level with reading level controlled for, where only the correlation between comprehension level and reading level is removed before age is correlated with comprehension level.

Whether a part or partial correlation is used depends on the specific model or research question. Possible three-variable relationships. For example, using the correlations in Table 3. Age 2. Comprehension 3. Reading level 1. A suppressor variable correlates near zero with a dependent variable but correlates significantly with other predictor variables. This correlation situation serves to control for variance shared with predictor variables and not the dependent variable. The partial correlation coefficient increases once this effect is removed from the correlation between two predictor variables with a criterion. Partial correlations will be greater in magnitude than part correlations, except when independent variables are zero correlated with the dependent variable; then, part correlations are equal to partial correlations.

The part correlation coefficient r1 2. In our example, the zero-order relationships among the three variables can be diagramed as in Fig. However, the partial correlation of age with comprehension level controlling for reading level is r A part correlation of age with comprehension level while controlling for the correlation between reading level and comprehension level is r1 2.

Bivariate correlations. Partial correlation area. These examples consider only controlling for one variable when correlating two other variables partial , or controlling for the impact of one variable on another before correlating with a third variable part.

Other higher order part correlations and partial correlations are possible e. Readers should refer to the references at the end of the chapter for a more detailed discussion of part and partial correlation. Part correlation area. A variance-covariance matrix is made up of variance terms on the diagonal and covariance terms on the off-diagonal.

If a correlation matrix is used as the input data matrix, most of the computer programs convert it to a variance-covariance matrix using the standard deviations of the variables, unless specified otherwise. The researcher has the option to input raw data, a correlation matrix, or a variance-covariance matrix. The SEM software defaults to using a variance-covariance matrix.

The correlation matrix provides the option of using standardized or unstandardized variables for analysis purposes. If a correlation matrix is input with a row of variable means and a row of standard deviations, then a variance-covariance matrix is used with unstandardized output. If only a correlation matrix is input, the means and standard deviations, by default, are set at 0 and 1, respectively, and standardized output is printed.

When raw data are input, a variance-covariance matrix is computed. Dividing the covariance between two variables covariance is the offdiagonal values in the matrix by the square root of the product of the two variable variances variances of variables are on the diagonal of the matrix yields the correlations among the three variables: Structural equation software uses the variance-covariance matrix rather than the correlation matrix because Boomsma found that the analysis of correlation matrices led to imprecise parameter estimates and standard errors of the parameter estimates in a structural equation model.

In SEM, incorrect estimation of the standard errors for the parameter estimates could lead to statistically significant parameter estimates and an incorrect interpretation of the model, that is, the parameter divided by the standard error indicates a critical ratio statistic ort-value see Table A. Browne , Jennrich and Thayer , and Lawley and Maxwell suggested corrections for the standard errors when correlations or standardized coefficients are used in SEM.

However, only one structural equation modeling program, SEPATH in the program Statistica, permits correlation matrix input with the analysis computing the correct standard errors. In general, a variance-covariance matrix should be used in structural equation modeling, although some SEM models require variable means e. The standardized coefficients are thought to be sample specific and not stable across different samples because of changes in the variance of the variables.

The standardized coefficients are useful, however, in determining the relative importance of each variable to other variables for a given sample. Other reasons for using standardized variables are that variables are on the same scale of measurement, are more easily interpreted, and can easily be converted back to the raw scale metric.

The Amos and EQS programs routinely provide both unstandardized and standardized solutions. Even if the assumptions and limitations of using the Pearson correlation coefficient are met, a cause-and-effect relationship still has not been established.

The following conditions are necessary for cause and effect to be inferred between variables X and Y Tracz, These three conditions may not be present in the research design setting, and in such a case, only association rather than causation can be inferred.

In structural equation modeling, the amount of influence rather than a cause-and-effect relationship is assumed and interpreted by direct, indirect, and total effects among variables, which are explained later in a structural equation model example. Philosophical and theoretical differences exist between assuming causal versus inference relationships among variables, and the resolution of these issues requires a sound theoretical perspective. Bullock, Harlow, and Mulaik provided an in-depth discussion of causation issues related to structural equation modeling research; see their article for a more elaborate discussion.

We feel that structural equation models will evolve beyond model fit into the domain of model testing. Model testing involves the use of manipulative variables, which, when changed, affect the model outcome values, and whose effects can hence be assessed.

This approach, we believe, best depicts a causal assumption. This discussion included various types of bivariate correlation coefficients, part and partial correlation, variable metrics, and the assumptions and limitations of causation in models.

Structural equation modeling programs typically use a variance-covariance matrix and include features to output the type of matrices they use. The use of various correlation coefficients and subsequent conversion into a variancecovariance matrix will continue to play a major role in structural equation modeling, especially given mixture models see chap.

The chapter also presented numerous factors that affect the Pearson correlation coefficient, for example, restriction of range in the scores, outliers, skewness, and nonnormality.

SEM software also converts correlation matrices with means and standard deviations into a variancecovariance matrix, but if attenuated correlations are greater than 1. Non-positive definite error messages are all too common among beginners because they do not screen the data, thinking instead that structural equation modeling will be unaffected.

Another major concern is when OLS initial estimates lead to bad start values for the coefficients in a model; however, changing the number of default iterations sometimes solves this problem. Compare the variance explained in the bivariate, partial, and part correlations of Exercise 1.

## A beginner's guide to structural equation modeling

Explain causation and provide examples of when a cause-and-effect relationship could exist. Structural equation modeling in practice: A review and recommended two step approach. Psychological Bulletin, , Anderson, C. A comparison of five robust regression methods with ordinary least squares regression: Relative efficiency, bias, and test of the null hypothesis.

Understanding Statistics, 2, Arbuckle, J. Full information estimation in the presence of incomplete data. Schumacker Eds. Mahwah, NJ: Amos 4.

Beale, E. Missing values in multivariate analysis. Bentler, P. Practical issues in structural equation modeling. Sociological Methods and Research, 16, Boomsma, A. Wold Eds. Causality, structure, prediction Part I pp. Sociometric Research Foundation. Browne, M. Covariance structures. Hawkins Ed. Cambridge University Press. Bullock, H. Causation issues in structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 1, Cohen, J.

Hillsdale, NJ: Collins, L. Best methods for the analysis of change: Recent advances, unanswered questions, future directions.

Washington, DC: American Psychological Association. Crocker, L. Introduction to classical and modern test theory. Ding, L. Effects of estimation methods, number of indicators per factor, and improper solutions on structural equation modeling fit indices. A Multidisciplinary Journal, 2, Ferguson, G. Statistical analysis in psychology and education 6th ed. Hinkle, D. Applied statistics for the behavioral sciences 5th ed.

Houghton Mifflin. Outliers lie: An illustrative example of identifying outliers and applying robust methods. Multiple Linear Regression Viewpoints, 26 2 , Hoelter, J.