Measurement and StructureModelsCarlo Magno, PhDCounseling and Educational Psychology DepartmentDe La Salle University-Manila
Bivariate linear regression Y = a + BXMultiple regression Y = a + B1X1 + B2X2 + B2X3Path Model Y = B1X1 + B2X2 + e1Structural Equations Model F2 = B1F1 + e1
Example 1 2.63* Organization and Delta1 Planning 2.201* 4.10* Delta2 Student Interaction 2.83* 2.11* Delta3 Evaluation 1.99* Teacher 5.79* 3.80* Performance in Instructional NSTP Delta4 Methods 3.03* 2.93* Delta5 Course Outcome 2.95* 2.45* Delta6 Learner- 2.00* centeredness Delta7 2.74* Communication X2=3.42, GFI=.97, RMSEA=.01What do you call this model?What analysis is used in this model?What are the three things that you interpret in thismodel?
Example 2 DELTA2 DELTA2 25.12* 17.69* KC RC 7.22* 7.86* Metacognition 2.12*X2=4.51, GFI=.99, RMSEA=.00 5.00* Critical ZETA Thinking 1 1.00 0.67* 0.86* 0.74* 0.40* Inference Recognition Deduction Interpretation Evaluation of of Arguments Assumption 7.37* 2.02* 6.16* 5.02* 3.57* EPSILON1 EPSILON2 EPSILON3 EPSILON4 EPSILON5 1. How many measurement models are shown? 2. How are the two measurement models linked in the figure? 3. Where are the errors located in the figure? 4. What the things interpreted in the figure?
Structural Equations ModelingGoes beyond regression models: Test several variablesand latent constructs with their underlying manifestvariables.Provide a way to test the specified set of relationshipsamong observed and latent variables as a whole andallow theory testing for causality even whenindependent variables are not experimentallymanipulatedTheentire model is tested if the data fits the specifiedmodel.Takes into account errors of measurement.
Variables in a Structural Model:3. Manifest variables: Directly observed or measured. Boxes are used to denote manifest variables.4. Latent variables: Not directly observed; we learn about them through the manifest variables that are supposed to “represent” them. Ovals are used to denote latent variables.
Symbols used in Structural Models Manifest Variable Latent Variable Direction of an effect/Parameter Estimate Relationship/Covariance
Some Levels of Analyzing Structural Models:2. Confirmatory factor analysis3. Causal modeling or path analysis • Independent/exogenous variables • Dependent/endogenous variables
Confirmatory Factor Analysis Example 3 2.63* Organization and Delta1 Planning 2.201* 2.11* Delta2 Learner- Centeredness 2.95* 2.43* Set 1 Delta3 Evaluation 1.99* 2.74* 2.00* 1.00 Delta4 Communication 4.10* Delta5 Student Interaction 2.83* 2.93* Set 2 Delta6 Course Outcome 3.03* 5.79* 3.80* Delta7 Teaching method Common Factor Model/Multifactor Model1. What critical estimate is determined in a common factor model?2. When do we say that set 1 and set 2 are common factors?
Confirmatory Factor AnalysisExample 4 2.11* Evaluation 1.996* Delta1 2.74* 2.00* Set 1 Delta2 Communication 2.33* 1.00 Delta3 Learner- 2.20* Centeredness Set 2 2.43* 2.95* 1.00 Delta4 Organization & Plan 1.00 4.10* Delta5 Course Outcome 2.83* 5.79* Delta6 Student Interaction 3.80* Set 3 2.93* 3.03* Delta7 Teaching method Common Factor Model/ Multifactor Model
Path Model (Example 5) Self-efficacy .48* E 1 1.0 .28* Deep Approach Metacognition .20* Surface Approach χ2=10.03, df=3, χ2/df=3.34, GFI (.98), adjusted GFI (.92), RMSEA= .08 1. What type of variables are studies in a path model? 2. What is the difference between a path and a structural model? 3. What is the similarity between a path and structural model? 4. Where is the error located?
Structural Equations Model (Example 6) DELTA DELTA DELTA DELTA DELTA DELTA 2 3 4 5 6 7 100.43 71.46 57.11 34.94 71.92 88.10 * Conditional * Procedural * Planni * Monitori * Information * Debugging Knowledge Knowledge Manageme Strategy ng ng nt 6.88* 7.07* 9.25* 7.91* 7.24* Declarative 9.03* Evaluati Knowledge 6.27* on Metacogniti 25.12 82.57 on * 78.39 * DELTA * DELTA 1 8 2.10* 5.19* Critical ZETA Thinking 1 0.67* 0.86* 0.74* 0.40* Inferen Recognition Deductio Interpretati Evaluation ce of n on of Assumption Argument 7.27* 2.06* 6.15* 5.03* s 3.57* EPSILON EPSILON EPSILON EPSILON EPSILON 1 2 3 4 51. What variable is exogenous?2. What variable is endogenous?
Path Model (Example 7 & 8) E 1 1.0 Self-efficacyExample 7 .17* .51* E 2 1.0 .30* School Ability Metacognition E Self-efficacy 4 .51* 1.0 MetacognitionExample 8 School Ability .30* Self-efficacy X School Ability 1. What is being demonstrated by self-efficacy in the example 7? 2. What is the difference between example 7 and example 8?
Procedures (Summary)2. Specify the measurement models for the exogenous latent variables (i.e., which manifest variables represent which latent variables?)3. Likewise, specify the measurement models for the endogenous latent variables.4. Specify the paths from the exogenous to the endogenous latent variables.
DELTA: residuals of exogenous manifest variablesEPSILON: residuals of endogenous manifest variables
Testing for Goodness of Fit If the entire model approximates the population. The degree to which the solution fit the data would provide evidence for or against the prior hypothesis. A solution which fit well would lend support for the hypothesis and provide evidence for construct validity of the attributes and the hypothesized factorial structure of the domain as represented by the battery of attributes.
Goodness of Fit Noncentrality Interval Estimation Single Sample Goodness of fit Index
Goodness of Estimate Goodness of fit index Estimatefit index required requiredRMSEA .08 and below Joreskog GFI .90 and above Bentler-Bonett (1980) Normed Fit Index James-Mulaik-Brett Parsimonious Fit IndexRMS .08 and below Akaike Information Compare nested Criterion modelsMcDonald’s .90 and above Schwarzs Bayesian Lowest value isIndex of Criterion the best fittingNoncentrality model Bollens RhoPopulation .90 and above Independence Model Close to 0, notGamma Index Chi-square and df significant Browne-Cudeck Cross Requires two Validation Index samples
Noncentrality Interval Estimation Represents a change of emphasis in assessing model fit. Instead of testing the hypothesis that the fit is perfect, we ask the questions (a) "How bad is the fit of our model to our statistical population?" and (b) "How accurately have we determined population badness-of-fit from our sample data."
Noncentrality Indices Steiger-Lind RMSEA -compensates for model parsimony by dividing the estimate of the population noncentrality parameter by the degrees of freedom. This ratio, in a sense, represents a "mean square badness-of-fit." Values of the RMSEA index below .05 indicate good fit, and values below .01 indicate outstanding fit
Noncentrality Indices McDonalds Index of Noncentrality-The index represents one approach to transforming the population noncentrality index F* into the range from 0 to 1. Good fit is indicated by values above .95.
Noncentrality Indices The Population Gamma Index- an estimate of the "population GFI," the value of the GFI that would be obtained if we could analyze the population covariance matrix Σ. For this index, good fit is indicated by values above .95.
Noncentrality Indices Adjusted Population Gamma Index (Joreskog AGFI) - estimate of the population GFI corrected for model parsimony. Good fit is indicated by values above .95.
Single Sample Goodness of fit Index Joreskog GFI. Values above .95 indicate good fit. This index is a negatively biased estimate of the population GFI, so it tends to produce a slightly pessimistic view of the quality of population fit. Joreskog AGFI. Values above .95 indicate good fit. This index is, like the GFI, a negatively biased estimate of its population equivalent.
Single Sample Goodness of fit Index Akaike Information Criterion. This criterion is useful primarily for deciding which of several nested models provides the best approximation to the data. When trying to decide between several nested models, choose the one with the smallest Akaike criterion. Schwarzs Bayesian Criterion. This criterion, like the Akaike, is used for deciding among several models in a nested sequence. When deciding among several nested models, choose the one with the smallest Schwarz criterion value.
Single Sample Goodness of fit Index Browne-Cudeck Cross Validation Index. Browne and Cudeck (1989) proposed a single sample cross-validation index as a follow-up to their earlier (Cudeck & Browne,1983). It requires two samples, i.e., the calibration sample for fitting the models, and the cross- validation sample. Independence Model Chi-square and df. These are the Chi-square goodness-of-fit statistic, and associated degrees of freedom, for the hypothesis that the population covariances are all zero.
Single Sample Goodness of fit Index Bentler-Bonett (1980) Normed Fit Index. measures the relative decrease in the discrepancy function caused by switching from a "Null Model" or baseline model, to a more complex model. This index approaches 1 in value as fit becomes perfect. However, it does not compensate for model parsimony. Bentler-Bonett Non-Normed Fit Index. This comparative index takes into account model parsimony. Bentler Comparative Fit Index. This comparative index estimates the relative decrease in population noncentrality obtained by changing from the "Null Model" to the kth model.
Single Sample Goodness of fit Index James-Mulaik-Brett Parsimonious Fit Index. Compensate for model parsimony. Basically, it operates by rescaling the Bentler-Bonnet Normed fit index to compensate for model parsimony. Bollens Rho. This comparative fit index computes the relative reduction in the discrepancy function per degree of freedom when moving from the "Null Model" to the kth model. Bollens Delta. This index is similar in form to the Bentler-Bonnet index, but rewards simpler models (those with higher degrees of freedom).
Noncentrality Estimates(SIR-NSTP Models) Model 1 Model 2 Model 3 Point Lower Point Point Lower Estim Upper 90 Estim Upper Lower Estim UpperNoncentrality 90% at 90% % at 90% 90% at 90%Fit indices CI e CI CI e CI CI e CIPopulationNoncentralityParameter 0.156 0.222 0.303 0.158 0.224 0.305 0.162 0.228 0.31Steiger-LindRMSEA Index 0.106 0.126 0.147 0.11 0.131 0.153 0.121 0.144 0.168McDonaldNoncentrality Index 0.859 0.895 0.925 0.859 0.894 0.924 0.857 0.892 0.922PopulationGamma Index 0.92 0.94 0.957 0.92 0.94 0.957 0.919 0.939 0.956Adjusted Populatio nGamma Index 0.841 0.881 0.914 0.827 0.87 0.907 0.793 0.844 0.887
Example(metacognition and critical thinking)Sing Sample Fit Indices Model 1 Model 2Joreskog GFI 0.926 0.915Joreskog AGFI 0.841 0.879Akaike Information Criterion 0.394 0.817Schwarzs Bayesian Criterion 0.612 1.210Browne-Cudeck Cross 0.398 0.831Validation IndexIndependence Model Chi- 786.533 1382.03Square 4Independence Model df 21.000 78.000
Single Sample Fit Indicers Model 1 Model 2Bentler-Bonett Normed Fit 0.919 0.898IndexBentler-Bonett Non-Normed Fit 0.892 0.928IndexBentler Comparative Fit Index 0.933 0.941James-Mulaik-Brett 0.569 0.737Parsimonious Fit IndexBollens Rho 0.868 0.875Bollens Delta 0.934 0.941
Reference for the Goodness of fitfor CFAStatSoft, Inc. (2005). STATISTICA electronic manual. Tulsa OK: Author.Arbuckle, J. L. (2005). Amos: 6.0 User’s guide. USA: Amos Development Corp.