Hi!

I want to construct a path analysis model that can account for measurement
error in totally aggregated parcels, which refer to parcels where all of
the items in a scale are summed or averaged. If I am not mistaken, Bollen
(1989) advocates the following formula for computing the error variance of
each parcel:

(1−α(parcel))×variance(parcel),

such that α refers to Cronbach's alpha, which is a measure of reliability.

What follows is the following path analysis model in theory (i.e., in
practice the formulas are replaced with actual numbers):

path.inf.final <- specifyModel()
pRU -> sRU, test1
pRU -> rRU, test2
sRU -> rRU, test3
sRU -> power_alt, gam1
pRU -> power_alt, gam2
rRU -> power_alt, gam3
sRU -> ms_alt, gam7
pRU -> ms_alt, gam8
rRU -> ms_alt, gam9
sRU <-> sRU,  NA, (1 - alpha(sRU))*(variance(sRU))
pRU <-> pRU, NA, (1 - alpha(pRU))*(variance(pRU))
rRU <-> rRU, NA, (1 - alpha(rRU))*(variance(rRU))
power_alt -> em, iota1
power_alt <-> power_alt, NA, (1 - alpha(power_alt))*(variance(power_alt))
ms_alt -> em, zeta1
ms_alt <-> ms_alt, NA, (1 - alpha(ms_alt))*(variance(ms_alt))
em -> relM1, eta1
em <-> em, NA, (1 - alpha(em))*(variance(em))
relM1 <-> relM1, lam1

After testing this model, the model fits were unnaceptable and package sem
could not compute modification instances. In contrast, when I do not fix
error variances, after a few modifications to the model, I am able to reach
a theoretical cogent model that has acceptable model fits.

I have several questions on this regard:

   1. Am I understanding Bollen (1989) correctly and using the formula
   correctly in my path analysis model?
   2. I also heard that rather than Bollen's formula, I could use alpha or
   alpha squared. Is this correct? If so, would it follow that I just input it
   into the model in much the same manner like I used Bollen's formula. I also
   tried this approach and reached unsatisfactory results.
   3. The dependent variable in the path analysis model is binary and I
   could argue that there is no measurement error associated with it. How
   could I fix the error variance for this variable, i.e., relM1, such that
   the aforementioned is accounted for.

Any help would be much appreciated.

References:

   - Bollen, K. A. (1989). Structural equations with latent variables. New
   York: Wiley.

Thank you!,
~Kino

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