Specifying a 3 Factor Parallel CFA Model based on tau-equivalent model

[Stephanie Kliethermes]

Hi, 

I'm still learning the ropes of CFA and am currently  trying to (minimally) show partial tau-equivalency for a 3 -factor CFA model.  I'm conducting goodness of fit tests for 4 measurement models (congeneric, partial tau-equivalency, tau-equivalency and parallel).  

My question is whether or not I can actually specify a parallel model when an item is cross loaded on 2 factors (see motivation 1 below).  Below is how I am currently specifying the parallel model but I'm not sure this is entirely correct given the different loadings on motivation1 in the Factor 2 and Factor 3. 

Does anyone have insight as to whether nor not this is an appropriate specification? 

Thanks,

Steph

modeloverall <- '
  # Factor indicators
  Factor1 =~ a*social1 + a*social2 + a*social3 +a*social4
  Factor2 =~ b*motivation1  + b*motivation3  + b*motivation10+ b*motivation11+ b*motivation12
  Factor3 =~ c*physical1 + c*physical4+c*physical5 + c*motivation1  
 
  # Factor covariances
  Factor1 ~~ Factor2
  Factor1 ~~ Factor3
  Factor2 ~~ Factor3

  motivation1 ~~ k*motivation1
  motivation3 ~~ k*motivation3
  motivation10 ~~ k*motivation10
  motivation11 ~~ k*motivation11
  motivation12 ~~ k*motivation12


 physical1 ~~ k*physical1
  physical4 ~~ k*physical4
  physical5 ~~ k*physical5
 
  social1 ~~ f*social1
  social2 ~~ f*social2
  social3 ~~ f*social3
  social4 ~~ f*social4