# Fixing error variance in a path analysis to model measurement error in scales using sem package

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:

$$\big(1 - \alpha(\text{parcel})\big)\times \text{Var}(\text{parcel}),$$

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

What follows is the following path analysis model in theory:

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.
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are you calculating cronbach's alpha outside and prior to the code snippet you've provided? if not, this is a problem because in order to compute internal consistency using alpha(X), X needs to be a data matrix containing multiple variables. essentially, you're re-scaling error variances to reflect internal consistency. which begs the question - why bother parceling? –  allinonemove Mar 22 '12 at 14:19
@allinonemove Of course! Otherwise it would not run. –  Joaquin Aguilar Mar 23 '12 at 2:50
Where in Bollen (1989) did you find the formula for computing the error variance of each parcel? –  histelheim Aug 2 '12 at 18:30