structural equation modeling covariance calculate

#1
Bildschirmfoto 2021-08-10 um 16.00.01.png
Hey all,

i don't know how to calculate the covariance between Xi and ETA(n) with the 6 factor loadings and gamma in the middle of the picture. The Variance of the latent variables is 1, so it is the fully standardised solution.. Furthermore i want to calculate the covariance between x2 and y2. Also the explained variances of the covariance would be interesting for me. Does anyone know how I do the calculations?

Best regards,
Bernd
 
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spunky

Can't make spagetti
#2
View attachment 3538
Hey all,

i don't know how to calculate the covariance between Xi and ETA(n) with the 6 factor loadings and gamma in the middle of the picture. The Variance of the latent variables is 1, so it is the fully standardised solution.. Furthermore i want to calculate the covariance between x2 and y2. Also the explained variances of the covariance would be interesting for me. Does anyone know how I do the calculations?

Best regards,
Bernd
What do you know about:

(i) Covariance algebra?
(ii) Wright's rules of path analysis?
 

spunky

Can't make spagetti
#4
Very well. Let's start with the basics and build up from it.

Can you show me how to express Xi (you don't mention which X the i is indexing so I'm leaving it like that) in terms of the parameters of the model? Like, what would be the equation for Xi (as in Xi = blah + blah + blah).
 
#5
Xi does not mean x1, x2 or x3. XI is the 14th letter of the Greek alphabet and in the picture above its the "ξ". In Addition to this in the model The residual variance of Eta(η) is chosen so that the total variance of Eta is 1. The l1, l2, l 3 and so on you can see the regression weights.
 

spunky

Can't make spagetti
#6
Oh, I see. I guess I knew that one as "ksi" in English.

Perfect, so if you can construct the equation and you're familiar with the rules of covariance algebra, then you have everything that you need to answer your question.
 

spunky

Can't make spagetti
#8
Well, now that you clarified that "Xi" is actually \(\xi\), this may actually be easier to figure out. Here's a hint.

In the simple regression model:

\(Y = \beta_0 + \beta_1X + \epsilon\)

It is not difficult to show that the regression coefficient \(\beta_1\) can be expressed purely in terms of variances and covariances of model parameters as:

\(\beta_1= \frac{cov(X,Y)}{\sigma^2_{X}}\)

Using your path diagram, can you see what goes in place of \(\beta_{1}, \sigma^2_{X}, cov(X,Y)\) ?
 
#9
I'm not so sure.
σ
2
X = should be the variance and the variance is 1

β
1= is a regressions weight and it could be one of them: l1, l2, l3, l4, l5, l6

and the cov(X,Y) is the covariance between the two latent variables ?
 

spunky

Can't make spagetti
#10
I'm not so sure.
σ
2
X = should be the variance and the variance is 1
Correct

β
1= is a regressions weight and it could be one of them: l1, l2, l3, l4, l5, l6
Are you sure about that? Which variables are exogenous and which are endogenous under the path analysis model you're showing?

and the cov(X,Y) is the covariance between the two latent variables ?
Correct. This is the what you want to solve for using the information you have.
 
#11
The exogenous variables are the independent variables (x1, x2 ..) and the endogenous variables are the two latent variables. Maybe the g1 is our β1 because we want the covariance between the two latent variables?
 
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spunky

Can't make spagetti
#13
The exogenous variables are the independent variables (x1, x2 ..) and the endogenous variables are the two latent variables. Maybe the g1 is our β1 because we want the covariance between the two latent variables?
So this is correct (indeed \(g_{1}\) is \(\beta_{1}\) in my example) but it is clear now that some concepts need further clarification.

You've mentioned the x's are the independent/exogenous variables and \(\xi\) and \(\eta\) are endogenous.

So let's go back to the basic rules of path analysis. When you have a one-headed arrow say like this \(X \rightarrow Y\) which one in this small example is exogenous/independent and which one is endogenous/dependent?
 

spunky

Can't make spagetti
#15
Great! Now let's look back at your path analysis. Using what you just learned, are \(\xi, \eta\) exogenous or endogenous? I'm sure once you realize this, you can use the formula I gave you above to find the covariance.
 
#16
The ξ is exogenous and the η is endogenous ? But i'm still not sure how i can relate this to the formula. Is the covariance just 0,36 in this example, because ß1 is 0,36 and the variance is 1 ?
 
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spunky

Can't make spagetti
#18
Maybe one Information is missing : The sample size was very small: n = 20. I don't know if we need the sample size.
Wait. This path analysis came from an *actual* dataset? Do you have access to this data?

This is probably easier then than doing all of this! I thought all you had access to was the path diagram.