I am trying to calculate a latent variable interaction using the unconstrained product indicator approach (Marsh 2004) with double centering (Lin et al. 2010).

My data: two latent predictor variables A,B (both are 5 point likert scales). The issue now is, that A consists of 4 items but B is only one item. I am not sure how to handle this, i have several ideas for the product indicators:

  1. Using only one item from A, which leads to only one product indicator (a1*b1)and obviously loss of information
  2. Reusing indicator b1 for all possible product indicators (a1*b1,a2*b1...). The problem I see here is: all product indicators have residual covariance, which causes errors in lavaan when I try to include the covariance in my SEM model.
  3. Taking the mean of the four indicators of A, resulting in only one product indicator (a_m*b1). I am not sure if this is a valid approach?
#model for approach 1
interaction.model_1 <- 'Y =~ y1 + y2 + y3 + y4
                        A =~ a1c + a2c + a3c + a4c
                        B =~ b1c
                        INT =~ prod1
                        Y ~ A + B + INT'
#model for approach 2
interaction.model_2 <- 'Y =~ y1 + y2 + y3 + y4
                        A =~ a1c + a2c + a3c + a4c
                        B =~ b1c
                        INT =~ prod1 + prod2 + prod3 + prod4
                        Y ~ A + B + INT
                        prod1 ~~ prod2 + prod3 + prod4
                        prod2 ~~ prod3 + prod 4
                        prod3 ~~ prod4'

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