CFA of Second-Order Constructs
Message/Author
 Maik Hamann posted on Tuesday, August 26, 2014 - 5:47 am
Hello,

I would like to do a CFA for a second-order construct, which consists of
* three dimensions, which are first order constructs with at least 3 indicators each
* and two reflective indicators of the second-order construct

As a final test I would like to compare the model fit of a model based on the reflective second-order construct with the a model based on the formative second-order construct (with two reflective indicators because of model identification).

Additional information: Previous theoretical rationales and other empirical tests suggest
a formative second-order construct.

I would like to ask, if the two model specifications in MPLUS below are correct to do this?

Reflective second-order construct:
MODEL:
DIM1 BY ind1_1 ind1_2 ind1_3;
DIM2 BY ind2_1 ind2_2 ind2_3;
DIM3 BY ind3_1 ind3_2 ind3_3;
DIM3 BY ind4_1 ind4_2 ind4_3;

RSOC BY DIM1 DIM2 DIM3 DIM4 refl_ind1 refl_ind2;

Formative second-order construct:
MODEL:
DIM1 BY ind1_1 ind1_2 ind1_3;
DIM2 BY ind2_1 ind2_2 ind2_3;
DIM3 BY ind3_1 ind3_2 ind3_3;
DIM3 BY ind4_1 ind4_2 ind4_3;

RSOC BY refl_ind1 refl_ind2;
RSOC ON DIM1 DIM2 DIM3 DIM4;

Kind regards,

Maik
 Bengt O. Muthen posted on Tuesday, August 26, 2014 - 3:06 pm
There is nothing wrong with either setup. The second setup, however, could be done differently and perhaps more in line with formative thinking. The way you have it now, RSOC can be seen as measured by the 2 reflective indicators and simply regressed on the 4 dimensions. Instead, you could specify it more like formative modeling (see our Topic 1 handout, slide 244, Model 1) with 2 DVs:

MODEL:
DIM1 BY ind1_1 ind1_2 ind1_3;
DIM2 BY ind2_1 ind2_2 ind2_3;
DIM3 BY ind3_1 ind3_2 ind3_3;
DIM3 BY ind4_1 ind4_2 ind4_3;

RSOC ON DIM1@1 DIM2 DIM3 DIM4;
RSOC@0;

refl_ind1 refl_ind2 ON RSOC;

refl_ind1 WITH refl_ind2;

In this way you don't assume that refl_ind1 and refl_ind2 are conditionally independent given RSOC.