I'm currently running an bifactor model (R - lavaan) with one general factor and three domain specific factors (X1, X2, X3). The dataset contains 18 items, of which it is assumed in theory that six questions each load on one specific domain factor.

As the variables are measured with a Six-Point Likert scale, I used the MLR estimator.

I used the following code:

model_bifactor <- "
X1 =~ item_1 + item_2 + item_3 + item_4 + item_5 + item_6
X2 =~ item_7 + item_8 + item_9 + item_10 + item_11 + item_12
X3 =~ item_13 + item_14 + item_15 + item_16 + item_17 + item_18

General =~ item_1 + item_2 + item_3 + item_4 + item_5 + item_6 + item_7 + item_8 + item_9 + item_10 + item_11 + item_12 + item_13 + item_14 + item_15 + item_16 + item_17 + item_18
"

bifactor_fit <- cfa(model = model_bifactor,
data = data,
estimator = "MLR",
missing = "ML",
orthogonal = TRUE)

summary(bifactor_fit, standardized = TRUE, fit.measures = TRUE)


After running the code, all six factor loadings on X3 where non-significant (p > 0.05), although the factor loadings would be relatively high (i.e. item_15)

Latent Variables:
Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
X3 =~
item_13                 1.000                               0.026    0.025
item_14                 0.367    1.422    0.258    0.796    0.010    0.011
item_15                35.466   72.098    0.492    0.623    0.931    0.732
item_16                25.382   51.014    0.498    0.619    0.666    0.552
item_17                31.542   63.858    0.494    0.621    0.828    0.638
item_18                17.973   36.203    0.496    0.620    0.472    0.345


Moreover, in contrast to the other two domain specific factors and the general factor, the variance of X3 is non-significant ((p > 0.05)

Variances:
Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
X1           0.397    0.068    5.806    0.000    1.000    1.000
X2           0.340    0.049    6.902    0.000    1.000    1.000
X3           0.001    0.003    0.244    0.807    1.000    1.000
General      0.223    0.047    4.741    0.000    1.000    1.000



Regarding these results I got two questions:

1. How can we explain, that the factor loadings of item 13-18 are non-significant, altough they are relatively high?
2. How can I interpret the non-significance of the factor loadings and variance of X3? Does this indicate, that this domain specific factor X3 does not exist (at least in my data)?

Alternatively, it might be problematic to use reference indicators for identification. Perhaps item_13's correlation with other indicators is mostly accounted for by the general factor, making its standardized loading nearly 0 on X3. I typically see researchers use factor standardization for identification instead in bifactor model applications. You can set this automatically in lavaan using std.lv=TRUE. Perhaps that would let the other X3 indicators have more of a voice regarding whether X3 contributes substantial variance (and specifically to which indicators).