Lisrel Matrix Notation

Newsom 1 USP 655 SEM Winter 2012 LISREL Matrix Notation LISREL (which stands for “linear structural relations”) involves eight matrices that organize the causal paths, loadings, correlations, and error terms in any model. Although this makes a cumbersome syntax, it is used in the mathematical description of SEM in the vast majority of statistical articles. A shortened, simpler four-matrix version (shaded rows below) can be used instead for specifying models in LISREL and is presented in some articles (see page 2). The figure below contains an example that illustrates all of the Greek letters.

ζ2

ψ ζ η2 23 ξ1 1 β21 γ11 ζ3

φ12 η1 β31 ξ2 γ12 η3

λx42 λx62 λy43 λy63 λx52 λy53 x4 x5 x6 y4 y5 y6

θδ4 θδ5 θδ6 θε4 θε5 θε6

(x1 through x3 and y1 through y3 are omitted from the diagram)

Parameter symbol English Matrix symbol Description (lowercase Greek Spelling (capital Greek letter) Letter)

λx lambda-x Λx loadings for exogenous variables

λy lambda-y Λy loadings for endogenous variables

φ phi Φ variance & covariances of exogenous latent variables ψ psi Ψ endogenous disturbance, covariances among endogenous disturbances γ gamma Γ causal path from exogenous to endogenous β beta Β causal path

θδ theta delta Θδ measurement errors for exogenous variables

θε theta epsilon Θε measurement errors for endogenous variables

ξ xi (ksi) not used as matrix, only in exogenous latent variables naming factors (see Φ matrix) η eta not used as matrix, only in endogenous latent variables naming factors (see Ψ matrix) ζ zeta not used as matrix, only in disturbances for endogenous variables naming disturbance (see Ψ matrix)

Note: Shaded symbols are used in four-matrix notation and syntax. This approach is much simpler and is equivalent to the eight matrix. Psi represents the variances and covariances for exogenous variables. Newsom 2 USP 655 SEM Winter 2012 LISREL “All Y” Matrices

The “all-y” Lisrel notation is a much simpler system. Many researchers who use Lisrel, use the all- y format for syntax programs. There are also a number of statistical papers that use the all-y matrices to present information. The all-y notation does not distinguish between exogenous and endogenous. Variances and covariances among exogenous variables are now represented in the Psi matrix. To distinguish between disturbances for endogenous variables (or their covariances) and variances for exogenous variables (and the covariances), one simply determines if the element in the psi matrix is associated with an exogenous or endogenous variable in the model.

ζ4

ψ ζ η4 45 η1 3 β43 β31 ζ5

ψ12 η3 β53 η2 β32 η5

λy42 λy52 λy62 λy10,5 λy11,5 λy12,5

y4 y5 y6 y10 y11 y12

θε4 θε5 θε6 θε10 θε11 θε12

(y1 through y3 and y7 through y10 are omitted from the diagram)

Parameter symbol English Matrix symbol Description (lowercase Greek Spelling (capital Greek letter) Letter) lambda-y loadings for all measured variables λy Λy

ψ psi Ψ variance & covariances of exogenous latent variables, endogenous disturbance, covariances among endogenous disturbances β beta Β causal path theta epsilon measurement errors for all measured variables θε Θε

η eta not used as matrix, only in exogenous and endogenous latent variables naming factors (see Ψ matrix) ζ zeta not used as matrix, only in disturbancesfor endogenous variables naming disturbance (see Ψ matrix)

Note: Zetas and etas are simply used for naming. For example, the disturbance, zeta, is presented as an element in the psi matrix. Eta is used only for naming variables, the variance for an exogenous variable is a diagonal element in the psi matrix.