Formative models
Message/Author
 MBH posted on Sunday, July 06, 2014 - 3:41 pm
I have two questions regarding formative model. I feel as though formative models have not been covered well anywhere.

Formative models needs supplemented with additional variables for the model to be identified. In case when using a formative model in SEM, does the factor scale of the latent variable still have a mean of 0 and variance of 1 if the syntax are following:

n1 by;
n1 on y1* y2 y3;
n1@1;

X1 on n1 X2 X3 X4;

a) Here does the latent variable have a mean of 0 and variance defined by metric of either X1 X2 X3 or X4 or by variance of 1?

- And on the following

n1 by;
n1 on y1@1 y2 y3;
n1@0;

X1 on n1 X2 X3 X4

b) here does the latent variable have a mean of 0 and variance defined by metric of either X1 X2 X3 or X4 or by variance of y1?
 Linda K. Muthen posted on Sunday, July 06, 2014 - 4:01 pm
You refer to n1;. This is a residual variance. Formative factors have residual variances of zero.
 MBH posted on Sunday, July 06, 2014 - 4:37 pm
Thanks Linda,

So I am correct when I say the interpretation of the latent variable is following:

it is a continuous score that has a mean of 0 and residual variance is 1?
 MBH posted on Sunday, July 06, 2014 - 4:38 pm
Sorry I meant mean of zero and residual variance of 0?
 Linda K. Muthen posted on Sunday, July 06, 2014 - 4:59 pm
The mean is the sum of the regression coefficient times the indicator for all formative indicators. A formative factor is basically a weighted sum.
 Linda K. Muthen posted on Sunday, July 06, 2014 - 4:59 pm
The residual variance is zero.
 MBH posted on Sunday, July 06, 2014 - 5:05 pm
I see, so is the regression thereafter interpreted then?

does it still mean that when the formative factor is regressed on a variable say

X1 X2 on formative-model;

are they still interpreted as one unit increase in formative-model would increase X1 by for example 0.5 and one unit increase in formative-model would increase X2 by say 04?
 Linda K. Muthen posted on Monday, July 07, 2014 - 8:39 am
You say the formative factor regressed on. This means it is a dependent variable. You show it as an independent variable. Which do you mean? If you mean an independent variable, what is the scale of x1 and x2.
 MBH posted on Monday, July 07, 2014 - 9:02 am
Formative model would need dependent 2 variables to be identified.

a)
For illustration purposes lets say:

Formative model -----> X1 (continuous variable)
Formative model -----> X2 (categorical variable)

Would the scale of X1 and X2 need to be identified and additionally determined to interpret the coefficient the model has on X1 and X2?

b)
For the following formative model:

Y1 -----> formative model
Y2 -----> formative model
Y3 -----> formative model

How would each factor loading (Y1 Y2 Y3) be interpreted if the factor is @0? Are they still 1 unit increase?
 Bengt O. Muthen posted on Monday, July 07, 2014 - 9:40 am
A formative model needs one DV to be identified, not two.

It is a little confusing when you (seemingly) use X for DVs and Y for covariates - Mplus notation is the other way around.

a) I assume that you view X1 and X2 as DVs here. I don't know what you mean by "scale of X1 and X2", but the model is ok with X1 and X2 being categorical or cont's.

b) I don't know what you mean by "the factor is @0". Perhaps you are referring to the residual variance of f being fixed at zero.
 MBH posted on Monday, July 07, 2014 - 9:48 am
Is there a good source which discusses how formative models results are interpreted?
 Linda K. Muthen posted on Monday, July 07, 2014 - 11:19 am
You should look at articles by Ken Bollen in the SEM journal.