I have used a full SEM- (containing latent variables) for analysis. My Questions (See my simple example to better understand the questions.):
Example: Consider three LV's A (with indicators a1, a2, and a3), B (with indicators b1, b2) and C (with indicators c1, c2). A has an arrow to B and C, and B also has an arrow to C.
1- What does the weight of each LV to its indicators mean and what is its application in model interpretation? In fact, what is the benefit of knowing indicators of a LV if we could not quantitatively interpret them (i.e., when we can’t say anything about the effect of a1, a2, or a3 on C in the following example)? If so, why we don't use the factor analysis method (EFA) in a path model?
2- How can we find a value for a LV from its indicators while LV's are explaining their indicators!?
3- Is it possible to say how a LV would change by a change in its indicators?
I don't use spss or Mplus, but hope the general discussion of my answers will be helpful to you. I added how to do what you ask in R using the lavaan package (which is free)
1a) indicators of a LV are interpreted as effects of the unobserved LV plus some error (a1=lamba_a1*A + error_a1, a2=lambda_a2*A+error_a2, etc), thus the causal hypothesis is that all correlation among the indicators is due to common dependence on the LV. To make sure the model is identified, we usually set one of the lambdas to 1. This sets the scale of the latent variable (i.e. sets what units we want to measure it. For example, if we set lambda_a1=1, the we consider the LV is measured in units of a1 (i.e. [a1]) then the other other lambdas have units of [a2]/[a1] and [a3]/[a1], which allows you to interpret them.
1b) knowing the indicators of an LV allows you to estimate the effects of the LVs on one another, which according to the model you describe, is the scientific question of interest. The indicators of A have no effect of C, they are correlated with C through the common dependence on A (this is what the model you describe asserts). An EFA assumes all the indicators are influenced by each of the LVs, which have no distinct causal relationship among themselves. It estimates the LVs the tries to rotate the loadings to try to minimize the number of indicators for each LV. Using the method suggests that you have no theory about which indicators are affected by which LVs and that you are trying to discover it...An entirely different exercise than the SEM suggested above.
2) Given the fitted model, it's easy to estimate the LV value for each row of data. Most software will be able to return these to you. for example if using the lavaan package in R, if F is the fitted model object, typing "predict(F)" will return the estimates for each LV.
3) The causal language in this question seems backward to me. the model supposes that the LV is the cause of the indicators, but it is possible estimate what the LV value would be for an observed change in the indicators. (i.e. the indicators don't change the LV value, the reflect a changed LV value). again, returning to lavaan , 'predict(F, newdata=new.dat)' will give you these values where 'new.dat' is a data.frame the with the same column headers as were in the data.frame used to fit the model, but in which the values of the indicators have been changed to reflect the situation for which you want the new estimate of the LV.
I hope you find this helpful.
Dear Donald
Thanks a lot for you answers which I found very useful. However, I have two more questions according to your answers:
2) I have used the Amos Graphics for modeling. Is it possible to estimate the LV value for each row of data in this software? If possible, please let me know how.
Is it possible to calculate the LV manually? If so, what should I put instead of A in the abovementioned example?
3) I didn’t actually get what you mean. You mean I can say how much the LV would change by for example a unit change in its indicators (i.e., something like marginal effect or elasticity in regression modes)? How?
Thanks for your help.
I don't use AMOS, but this is from the amos support (http://www-01.ibm.com/support/docview.wss?uid=swg21600240):"To impute factor scores to a new file, Choose 'Data Imputation' from the bottom of the Analyze menu in AMOS. In the 'AMOS Data Imputation' dialog that opens, there are 3 imputation methods available. The default method is regression imputation, which will return the same factor scores that would be computed by applying the factor score weights in SPSS compute commands, as described below. Values will also be imputed for missing data on the observed variables. The alternate imputation methods are stochastic regression and Bayesian imputation."
Alternatively, you can compute them by hand with maximum likelihood or least squares.For each row in your data (i), you have N equations for each factor score Ai, where N is the number of indicators of the latent variable. taking derivatives and rearranging gives, for the ith row, Ai=Sumn(Lambdan*xn,i)/Sumn(Lambdan^2), where Sumn means take a sum over n (i.e. each indicator of the latent var), Lambda are the estimated coefs for each indicator from the Amos fit, and xn,i is the ith values of the nth indicator of A (i.e. your observed data). To get the A values for each row, you can do a bit of linear algebra: X*Lambda/(Lambda^T*Lambda), where X is the i*n matrix for the observed indicators of the latent variable A. and Lambda is the n*1 vector of the estimated values from Amos.
You can use this to estimate the counterfactual on A (i.e. what is the best estimate of what A would have been had I observed X=X' instead of the X I actually observed), in that case, just plug X' into the linear equation above.
1. The weight indicates how much that particular LV reflected on its indicator
2.LV is latent or unobservable. Thus, you observe it nor can you estimate it exactly
3. The indicators cannot explain its latent construct. Only the other way round can
Dear Donald
Thanks again for your answer.
1- I have tried to better show my question in a file attached. Please find it.
Attached (an excel file) I have tried to apply your formula for the latent variable “A” with three indicators and for ten records as an example. Please see if I am right or not and also how can I calculate the LV for all records?
Moreover, you have introduced two formulas which are different with each other in denominator! Is it correct? Can you also introduce any reference for the formulas above so that I could refer to them in my work?
2- I did not understand what you meant by:
"You can use this to estimate the counterfactual on A (i.e. what is the best estimate of what A would have been had I observed X=X' instead of the X I actually observed), in that case, just plug X' into the linear equation above."
3- Would you also please give me some hints about my third question in the previous message (i.e., ... I can say how much the LV would change by for example a unit change in its indicators (i.e., something like marginal effect or elasticity in regression modes)? How?)
Thank you.
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