22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1  3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                  

22 22 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                  

11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                  

A1 A2 A3 AAAA AAAA B1 B2 B3 BBBB BBBB A4 B4 B5 B6 A1 A2 A3 A5 A6 A7 AAAA AAAA A4 A8 A9 A10 A1 A2 A3 AAAA AAAA B2 B3 B4 BBBB BBBB B1 C1 C2 C3 CCCC CCCC One-Factor Model Two-Factor Model Three-Factor Model

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1  3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1  3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1  3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3 X10   10,3 Some factors are correlated Some factors are not correlated Uniqueness or Error terms are not Independent (correlated)

Correlation matrix Covariance matrix

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1  3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3 X10   10,3

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1 = 0.52  3,1 = 0.71  3,2 = 0.47 X10   CR =.782 VE =.473 CR =.600 VE =.449 CR =.823 VE =

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3 11 11

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9                    2,1  3,1  3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

List constructs that will comprise the measurement model. Determine if existing scales/constructs are available or can be modified to test your measurement model. If existing scales/constructs are not available, then develop new scales. Stage 1: Defining Individual Constructs

A A B B C C D D E E F F IQ2             1 1 Hypothesized Measurement Model: Two-Factor model of IQ IQ ,1 1,1 3,1 4,2 5,2 6,2

Stage 2: Developing the Overall Measurement Model Key Issues... Unidimensionality – no cross loadings Congeneric measurement model – no covariance between or within construct error variances Items per construct – identification Reflective vs. formative measurement models

A A B B C C D D E E F F IQ2             1 1 CFA Model: Two-Factor model IQ1 1 1 Congeneric measurement model: no covariance (correlation) between or within construct error variances Unidimensionality: No cross-loading Reflective measurement models Congeneric measurement model: Each measured variable is related to exactly one construct

A A B B C C D D E E F F IQ2             1 1 IQ1 1 1 CFA Model: Two-Factor model with correlate factor Cross-loading covariance between construct error variances Covariance within construct error variances measurement model is Not Congeneric : Each measured variable is not related to exactly one construct /errors are not independent

1 2 Model Identifications: Underidentified, Just-identified & Over-identified 11 11 X1 X2 11 11 X1 X2 X3 11 11 X1 X2 X3       X4             Parameter estimated = 4Parameter estimated = 6 Parameter estimated = 8

Stage 2: Developing the Overall Measurement Model Developing the Overall Measurement Model … In standard CFA applications testing a measurement theory, within and between error covariance terms should be fixed at zero and not estimated. In standard CFA applications testing a measurement theory, all measured variables should be free to load only on one construct. Latent constructs should be indicated by at least three measured variables, preferably four or more. In other words, latent factors should be statistically identified.

Stage 3: Designing a Study to Produce Empirical Results The ‘scale’ of a latent construct can be set by either: Fixing one loading and setting its value to 1, or Fixing the construct variance and setting its value to 1. Congeneric, reflective measurement models in which all constructs have at least three item indicators are statistically identified in models with two or more constructs. The researcher should check for errors in the specification of the measurement model when identification problems are indicated. Models with large samples (more than 300) that adhere to the three indicator rule generally do not produce Heywood cases.

Stage 4: Assessing Measurement Model Validity Assessing fit – GOF indices and path estimates (significance and size) Construct validity Diagnosing problems Standardized residuals Modification indices (MI) Specification searches

Stage 4: Assessing Measurement Model Validity Loading estimates can be statistically significant but still be too low to qualify as a good item (standardized loadings below |.5|). In CFA, items with low loadings become candidates for deletion. Completely standardized loadings above +1.0 or below -1.0 are out of the feasible range and can be an important indicator of some problem with the data. Typically, standardized residuals less than |2.5| do not suggest a problem. Standardized residuals greater than |4.0| suggest a potentially unacceptable degree of error that may call for the deletion of an offending item. Standardized residuals between |2.5| and |4.0| deserve some attention, but may not suggest any changes to the model if no other problems are associated with those items.

Stage 4: Assessing Measurement Model Validity The researcher should use the modification indices only as a guideline for model improvements of those relationships that can theoretically be justified. CFA results suggesting more than minor modification should be re- evaluated with a new data set (e.g., if more than 20% of the measured variables are deleted, then the modifications can not be considered minor).

A A B B C C D D E E F F IQ2             1 1 Hypothesized Measurement Model: Two-Factor model of IQ IQ ,1 1,1 3,1 4,2 5,2 6,2  2,2  1,1  3,3  4,4  5,5  6,6  2,1

Measurement Model of IQ & LISREL Matrix (LX, PH, TD)

Measurement Model of IQ & Data (Input Matrix) Fitting Models & Data

Measurement Model of IQ & LISREL Syntax

Result of Analysis (LISREL Path Model)

Result of Analysis (Goodness-of-Fit Index)