Multiple Indicator Multiple Cause (Structural Equation Modeling)
Introduction of MIMIC model
“ Conventional wisdom” in psychological wisdom is that a latent variable is the cause of the measured variable
MIMIC model was first described by Swedish statistician Dr. Joreskog and A.S Goldberger in 1975.
This Modeling technique belongs to Structural Equation Modeling family.
MIMIC modeling is all about study of latent variables.
Measuring latent variables that are whether unobservable or not properly measured.
Estimation of casual link based on theoretical hypothesis.
In statistics latent variables are variables that are not directly observed but rather inferred from other variables and directly measured.
Latent variables can be identified by Confirmatory Factor Analysis, Exploratory Factor Analysis etc.
X1 Y1 L Latent variable X2 Y2 Casual link
MIMIC model consist of two parts
Structural part : The structural part shows how the latent variables are estimated through the observed variable
Measurement Part: The measurement part displays the casual link among the latent variables and observed causes.
L = α 1x1 + α 2x2 + α 3x3 ………+ α kxk + u
(where L= latent variable, α = set of structural coefficient, x = set of obs. Variables, u = set of error)
y1 = β 1L + v1
y2 = β 2L + v2
yp = β pL + vp (where L= latent variable, β = set of measurement coefficient, y = set of obs. causes, v = set of error)
If we combine the two equations:
Y = β ( α x + u) + v
= π x + e
Say π = β α , e = β u + v ,
And has covariance matrix : Ω = E(ee’)= E[( β u + v ) [( β u + v )]
= σ ²( β β ’) + Θ ²
Y = π x + e
As this model characterizes a casual relationship between latent variable and set of variables we can also identify the model with graphical form of representation using path analysis.
A path analysis is a pictorial representation of a system of simultaneous equation.
Casual links among variables are represented by unidirectional arrows. Their direction implies relationship from independent to dependent variable. The strength of these links are shown by the regression coefficients.
Simple associations among variables are represented by curve lines, the strength of these links are defined by correlation coefficients.
x1 ..... ..... xk y1 yp L α 1 α k ...... β 1 β p ...... Casual link Simple association
Example : Social status and social participation in a sample of 530 women was studied. It was assume that income , occupation and education explain social participation. Social participation was measured by church attendance , membership and friends seen .
Say spart = social participation
cattn = church attendance
memb = membership
friends = friends seen
Correlation matrix : income occup edu cattn memb friends
Path Analysis income occupation education church attendance member friends social participation 0.11 0.045 0.16 1.58 0.86 1.00 0.304 0.344 0.305
Thus one percent increase in income implies an estimated increase in social participation by about 0.11 percent. Similarly 1.58 percent increase in membership is measured by one percent increase in social participation.