The SweSAT Vocabulary (word): understanding of words and concepts. Data Sufficiency (ds): numerical reasoning ability. Reading Comprehension (read): Swedish reading comprehension. Diagrams, Tables and Maps (dtm): problem-solving test with information presented in of tables, graphs, and/or maps. General Information (gi): knowledge and information from many different areas. English Reading Comprehension (erc). English reading comprehension.

Two questions Does it makes sense to form a single sum of subtest scores which are so different? In other words, do the six subtests measure one and the same dimension of ability? How should the subtests be combined to achieve optimal prediction of success in higher education?

Covariance matrix

Purposes of regression analysis 1.To predict as accurately as possible scores on a dependent (or criterion) variable for one or more individuals from one or more independent (or predictor) variables. 2.To determine the relative degree of importance of the independent (or explanatory) variables in predicting the dependent variable.

A path model for the regression of Y on X

Basic definition of structural equation modeling Structural equation modeling involves identification of a set of parameters representing relations, variances and covariances for variables in a postulated model. From the postulated model and its parameters, an estimated (or implied) covariance matrix may be computed and this estimated covariance should as closely as possible approximate the observed covariance matrix. If the estimated matrix differs from the observed matrix we must conclude that the postulated model does not fit the data, and the model must be rejected.

Types of parameters Fixed parameter, which implies that a fixed value is assigned. Free parameter, which implies that a value of the parameter is to be estimated from data. Constrained parameter, which implies that a value of the parameter is to be estimated from data under the restriction that the estimated parameter value shall be equal to the value of one or more other parameters.

Instructions for the regression of mrk on word sem (word -> mrk)

Output from the simple regression model

Standardized output from the simple regression model sem (word -> mrk),stand

Instructions for the regression of mrk on read and word sem (word read -> mrk)

Standardized output from the multiple regression model

Instructions for the regression of mrk on all six subtests sem (word read gi erc ds dtm -> mrk)

Standardized output from the regression model with all six subtests

Problems with the regression analysis Confounded independent variables Errors of measurement in the independent variables

Regression of mrk on the sum of subtest scores

A latent variable model for the SweSAT

Estimating the model Needed: estimates of 12 parameters (5 regression coefficients, 6 error variances, 1 variance of the latent variable). Available: 21 elements of the covariance matrix (15 covariances and 6 variances). Express the known entities in terms of the unknown parameters through application of path rules, e. g.: –Cov(word, ds) = 1 Var(Gen) b3 –Cov(word, erc) = 1 Var(Gen) b5 –Var(word) = 1 Var(Gen) Var(word&) 1 Solve the 21 equations for the 12 unknown parameters.

Instructions for the latent variable model sem (Gen -> word read gi erc ds dtm)

Output from the latent variable model

Regressing MRK on the latent variable sem (Gen -> mrk word read gi erc ds dtm)

Does the one-factor model fit the data? Reproduce the covariance matrix from the estimated parameters (the implied matrix) and compare it with the observed matrix, e.g.: Cov(word, ds) = 1.0 x x 0.56 = 8.25 (observed value = 6.97) Cov(read, erc) = 0.72 x x 0.86 = 9.12 (observed value = 9.27) A chi-square test and other measures of model fit may be computed: Chi-square = , df = 9, p <.00

Problems with the Chi-square Goodness of Fit test The test is  2 distributed only when data has a multivariate normal distribution. When the sample size is large even trivial deviations between model and data cause the  2 test to be significant. When the sample size is small even important deviations from the true model may be undetected. A model with many free parameters has a better  2 value than a model with few free parameters. However, models with few free parameters are generally to be preferred over models with many free parameters. The researcher often uses the same data to test several, successively modified, alternative models. Such a practice will, however, cause the actual level of significance to be different than the nominal level.

The Root Mean Square Error of Approximation (RMSEA) The RMSEA measures the amount of discrepancy between model and data in the population, taking model complexity (i. e., number of estimated parameters) into account. –Values less than 0.05 indicate good fit, and values up to 0.07 or 0.08 may be accpted. –The Test of Close Fit tests the hypothesis that RMSEA < –A 90 % confidence interval of RMSEA may be contructed. The lower limit of interval should be less than.05 and the upper limit of the interval should not be higher than 0.07.

Testing the fit of the one-factor model

Modification indices The modification index expresses the expected improvement of the  2 test with 1 df when a fixed or constrained parameter is turned into a free parameter. –Modifications must be theoretically sound and substantively interpretable. –Modifications which involve extensions of the latent variables cannot be suggested. However, modification indices which suggest introduction of a covariance between the residuals of two observed variables often indicate a need to introduce a new latent variable.

Computing modification indices

Modifying the model

An oblique two-factor model

Instructions for the two-factor model sem (Verb -> word read gi erc) /// (Reas -> ds dtm)

Parameter estimates for the two-factor model

Instructions for the two-factor regression model sem (Verb -> word read gi erc) /// (Reas -> ds dtm) /// (Verb Reas -> mrk)

Estimates for the latent regression model