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I have try to conduct a second order factor consisting of five first order factors. However, some of the first order factors are related to each other negatively, while others are related to each other positively. I checked some of the books but I could not find the answer. When I conduct the analyses I get higher goodness of fit statistics which are above the cut points. So, does it make sense to make analyses with those factors?

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Sure. Nothing prevents the second order factor from loading on the first order factors in such a way that some of the covariances are negative. However, a second order model makes very stringent assumptions. Your first order model allows for 10 correlations between the factors. The second order model claims that these can all be computed from 5 loadings (or 4 loadings and a variance). That's a tall order. And any discrepancies in the first order model (say, manifest 1 from Factor 1 has a residual correlation with manifest 3 of Factor 4), will kick in to further reduce the fit of your model. A measure has to be very well designed and reflective of genuine underlying factors to fit a second order model.

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