You have two questions. The simpler one is does a non-significant Chi-sq mean the set of coefficients for males and females are not "significantly" different? Assuming the coding is all correct and the models all estimated accurately then yes.

The harder question is what to do with this information. First, basing a decision only a single number is often not good, particularly if the number is a p-value given all the problems with the meaning of p discussed in the literature (e.g., Cohen's The Earth is Round, p < .05). Information criteria are often also reported (e.g., BIC, AIC) and cross-validation may also be useful. Second, it will depend on the purpose of model choice. For example, if you plan to add further variables and their interactions with gender you might wish to keep them in. Another reason to keep them in is if you strongly believe they should be in. Third, some people prefer simpler models to more complex (and often more accurate models), see for example http://statweb.stanford.edu/~tibs/ftp/tibs-copss.pdf; for more details see http://web.stanford.edu/~hastie/StatLearnSparsity/. This will depend on what you are using the model for. And there are other arguments about this which I will leave to others to bring up.  This said, if you were looking for a simple answer to this hard question, many people (and this is the usual practice) drop variables from a model when they are not significant.

You have two questions. The simpler one is does a non-significant Chi-sq mean the set of coefficients for males and females are not "significantly" different? Assuming the coding is all correct and the models all estimated accurately then yes.

The harder question is what to do with this information. First, basing a decision only a single number is often not good, particularly if the number is a p-value given all the problems with the meaning of p discussed in the literature (e.g., Cohen's The Earth is Round, p < .05). Information criteria are often also reported (e.g., BIC, AIC) and cross-validation may also be useful. Second, it will depend on the purpose of model choice. For example, if you plan to add further variables and their interactions with gender you might wish to keep them in. Another reason to keep them in is if you strongly believe they should be in. Third, some people prefer simpler models to more complex (and often more accurate models), see for example http://statweb.stanford.edu/~tibs/ftp/tibs-copss.pdf; for more details see http://web.stanford.edu/~hastie/StatLearnSparsity/. This will depend on what you are using the model for. And there are other arguments about this which I will leave to others to bring up.  This said, if you were looking for a simple answer to this hard question, many people (and this is the usual practice) drop variables from a model when they are not significant.

Dear Daniel,

Does this mean that if the chi-square's increase from model 1 (no constraints) to model 2 (with constraints) was significant (in both cases p value would be significant), it would mean that the differences between regression coefficients would be statistically significant? Thank you for clarifying. I know much more than just months ago but I see what a long way ahead... Thank you. 

The lack of a significant difference means that there is no improvement in fit gained by including the additional parameters associated with adding two sets of coefficients, one for males and one for females, instead of using just one set to represent everyone. In other words, there is no interaction. 

Note you could also test this using just regression equations by using a dummy variable to represent gender, and then introducing an interaction effect that multiplies that dummy times the independent variable(s) of interest. You can find numerous descriptions of this approach if you search the net for a combination of: regression dummy variables interaction.

Hi Witold,

Sometimes you will get a significant difference between two models with none of the coefficients being significant, and sometimes the opposite. They are testing different things (the biggest difference being one may have many degrees of freedom, and the others often one each). There is also the issue of controlling for type 1 error with multiple p values. But, the way you describe it is how it usually works out for big enough samples. For some estimation methods the significance tests for individual coeffficients are less trusted than comparing the models, so often researcher create two models where the only difference is the coefficient of interest is either estimated or not.

Hi Daniel,

as far as I know, and I refer to the book longitudinal data analysis from Todd Little, you should have a close look at the CFI. IF the CFI increases more than 0,01 (1%), this means that there is a substatntial increase in model fit.

Bests

Have you included in your model an independent variable which could mask the effect of the sex variable? Have you forgotten a potential independent variable which could artificially increase the error? You could use the SWEEP operator to look at the many possible partial correlations between variables.

Dear Till,

I agree, CFI is another measure often used, and there are others. I worry equating quantifying verbal phrases like "substantial" with numeric ones on ResearchGate because people from different disciplines read these comments and "substantial" in terms of model fit will mean different things to a stock market trader trying to get any advantage, a psychologist using rating scales, an astronomer trying to detect exoplanets, etc. The relationship seems good for Todd's and my field (psychology and educational large data sets) though.

Dan

Measures such as CFI were created to deal with situations where the chi-square value was significant, but the overall fit appeared to be adequate. Most of those indices deal with the problem that chi-square is highly sensitive to the number of degrees of freedom, so that data from large samples often produces significant chi-square values.

In other words, many analysts feel direct chi-square assessments of models are too "conservative" and thus use things like CFI to provide a more "realistic" assessment of fit. In the present case, the actual chi-square indicates no difference, so there is little reason to delve into alternative fit indices (especially when they require subjective judgments to assess how much difference is enough to merit further attention).