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I am running a SEM using lavaan package in R. The idea is to check different mediation effects of social class of origin, cognitive abilities, education and personality on the log-income of the offsprings. Basically, this is the path-scheme I designed:

path_male <- '
#latent variable 
ability =~ c_cgwri_dv + c_cgwrd_dv + c_cgs7ca_dv + c_cgvfc_dv + c_cgna_dv

#Paths 
ability ~ origin_class
education ~ ability + agree + conscientiousness + extrov + neurotic + openness + origin_class 
log_income ~ education + origin_class + ability + agree + conscientiousness + extrov + neurotic + openness + job_type 

#correlations
c_cgwri_dv ~~ c_cgwrd_dv
c_cgs7ca_dv ~~   c_cgna_dv
ability  ~~ origin_class
conscientiousness~~origin_class
agree~~origin_class
openness~~ origin_class
neurotic~~ origin_class
extrov~~ origin_class
agree~~conscientiousness
agree~~openness
agree~~neurotic
agree~~extrov
conscientiousness~~openness
conscientiousness ~~neurotic
conscientiousness ~~extrov
openness~~neurotic
openness~~extrov
neurotic~~extrov
'

The resulting estimates are controversial. Indeed, all the measures to check the fit are quite good: CFI= 0.978; TLI=0.96; RMSEA=0.028; SRMR=0.020;
but the p-value of the chi-squared is 0.000 suggesting that the model is a poor-fit.

I also used the modification indices to check what additional paths/correlations are necessary to improve the fit, but those with high values (around 20/30) are not theoretically meaningful. What could be the reason of this contradiction between chi-squared and the other measures of fit?

Thanks

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2 Answers 2

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No contradiction. Those fit measures are all OK to good.

Your sample size is high. The measures are telling you that there is a statistically significantly discrepancy between your model and data, but that the discrepancy might not be large enough to be concerned about.

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I just go through the Saris-Satorra (2009) approach focusing on the MI and EPC. Therefore, with not relevant misspecification detected with the command:

 modificationindices(fit, sort. = TRUE, power = TRUE, delta = .4, op = "~~") 

(and repeat it for loadings), I can accept with confidence the model even if the p-value on the chi-squared is less than 0.05.

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