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Let's say we would work with this simplified path model.

lavaan -> '
  # measurement model
  A =~ ItemA + ItemB + ItemC
  B =~ ItemD + ItemE + ItemF
  C =~ ItemG + ItemH + ItemI

  # path model
  A ~ B + C'

How would I then test that B and C do not have a quadratic relationship with A.

Would the following be correct? I square the items of B and C and store them under new variable names ("sqr") in my dataset. And include them in my model as follows.

lavaan ->'
  # measurement model
  A =~ ItemA + ItemB + ItemC
  B =~ ItemD + ItemE + ItemF
  C =~ ItemG + ItemH + ItemI

  B_sqr =~ ItemD_sqr + ItemE_sqr + ItemF_sqr
  C_sqr =~ ItemG_sqr + ItemH_sqr + ItemI_sqr

  # path model
  A ~ B + B_sqr + C + C_sqr'

Then I look to see if R-squared gets higher by adding the quadratic terms and see if the quadratic parameters become significant.

Or would the following option also work:

lavaan ->'
  # measurement model
  A =~ ItemA + ItemB + ItemC
  B =~ ItemD + ItemE + ItemF
  C =~ ItemG + ItemH + ItemI

  # path model
  A ~ B + I(B^2) + C + I(C^2)'

Or is the I operator only applicable in lm objects?

What do you think about all this?

HHKK
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  • Or would the following option also work: lavaan -> " # Measurement model A =~ ItemA + ItemB + ItemC B =~ ItemD + ItemE + ItemF C =~ ItemG + ItemH + ItemI # Path model A ~ B + I(B^2) + C + I(C^2) Or is the I operator only applicable in lm objects? – HHKK Apr 04 '23 at 08:27
  • The `I()` operator only works in `formula` objects, and operates on observed variables. You could use the product-indicator method for estimating latent interaction effects (a quadratic effect is a variable moderating its own effect, so mean-center your indicators and square them to create indicators of a quadratic factor). The `semTools::indProd()` function is helpful for this, and you can read the references on its help page. – Terrence Apr 05 '23 at 07:49
  • Ah okay, so the following would be correct to get the indicators: dat <- indProd(df[ , -1], var1 = 14:18, var2 = 14:18). And I then put these indicators in here in my measurement model. lavaan ->' # Measurement model A =~ ItemA + ItemB + ItemC B =~ ItemD + ItemE + ItemF C =~ ItemG + ItemH + ItemI B_sqr =~ mc_ItemD_sqr + mc_ItemE_sqr + mc_ItemF_sqr C_sqr =~ mc_ItemG_sqr + mc_ItemH_sqr + mc_ItemI_sqr # path model A ~ B + B_sqr + C + C_sqr' (*mc = mean-centred) – HHKK Apr 05 '23 at 11:39
  • Dependent variable ~ latent variable + latent variable_sqr + manifest variable A + manifest variable B .... 1. If I have found out that the square latent variable has a significant effect on the dependent variable, but the linear latent variable does not. Do I then simply throw out the linear latent variable and replace it with the square latent variable or do I keep both in the path model? 2. Can I parcel a square latent variable in the same way as a linear one or does it lose its quadratic effect? – HHKK Apr 06 '23 at 10:43
  • Syntax looks fine, but you should keep your lower-order terms even if not significant, to ensure the quadratic effect is unbiased. The linear slope is the (simple) slope of the tangent line touching the curve when that predictor = 0. Forcing it to be 0 just makes your curve bend suboptimally. Of course, a quadratic effect is just a simple tool to explore the possibility of nonlinearity; it is not necessarily the "correct" functional form of the predictor's effect. – Terrence Apr 06 '23 at 12:09
  • ah okay! My model described here is only a simplification of my actual model. The latent variable with conventional, non-squared items is divided into parcels. Should I also parcel out the counterpart, i.e. the latent variable based on quadratic items (or choose the raw items) before I include it in the path model, or would that have an effect on the quadratic effect? Actually, it should have no effect on it. – HHKK Apr 07 '23 at 13:25
  • If you are parceling, I would just make one parcel to simplify everything. – Terrence Apr 08 '23 at 11:57
  • So then as follows: mod_parcel <- ' # Measurement models A =~ par1 + par2 + par3 + par4 A_sqr =~ par5 B =~ par6 + par7 + par8 C =~ Item1 + Item2 + Item3 + Item4 + Item5 # Path model C ~ A + A_sqr + B – HHKK Apr 11 '23 at 07:06
  • By "one parcel", I mean just calculate a scale mean for each construct. Then there is no parcel allocation variability. – Terrence Apr 11 '23 at 08:10

0 Answers0