Setting "std.lv = TRUE" in higher-order CFA model

[Jeremy Eberle]

Hello,

For the higher-order CFA model below, my understanding is that std.lv fixes the variances of the (exogenous) second-order factors cost1 and cost2 to 1 and fixes the residual variances of the (endogenous) first-order factors TE, OE, LV, and EM to 1.

mod <- "TE    =~ TE1 + TE2 + TE3 + TE4 + TE5
        OE    =~ OE1 + OE2 + OE3 + OE4
        LV    =~ LVA1 + LVA2 + LVA3 + LVA4
        EM    =~ EM1 + EM2 + EM3 + EM4 + EM5 + EM6
        cost1 =~ TE + OE
        cost2 =~ LV + EM"

fit <- cfa(mod, std.lv = TRUE, estimator = "MLR", missing = "ml")

summary(fit, standardized = TRUE)

I have a couple questions about this:

1. I can see in the "Estimate" column of the summary that the residual variances in addition to the variances are fixed to 1 (in yellow). However, in the "Std.lv" and "Std.all" columns, the residual variances of the first-order factors are no longer 1 (in orange), but rather equal to 1 - R^2, where R is the value in green from the "Latent Variables" section (e.g., for TE, 1 - .919^2 = .155). How can the residual variances change if they were fixed to 1?

2. Given that I set std.lv to TRUE, I also expected the factor loadings in the "Estimate" column above to be the same as those in the "Std.lv" column, but they are different. I would be grateful for any thoughts about what might be going on or what I'm missing.

Thank you!

Jeremy

[Terrence Jorgensen]

How can the residual variances change if they were fixed to 1?

They didn't "change".  The argument std.lv=TRUE only fixes the latent residual variances to 1.  The standardized solution is a transformation, to results as though all variables had a TOTAL (not residual) variance = 1.

Terrence D. Jorgensen    (he, him, his)
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam
http://www.uva.nl/profile/t.d.jorgensen