I am estimating a multinomial model with two quantitative predictors, X1 and X2, and 3 responses. The responses are called neutral, positive and negative with neutral being the baseline. There are actually many models being estimated because I estimate the model over time and also for various parameter sets but that's not important. When I estimate a model, since neutral is the baseline and there is no interaction term, I get back coefficients

X1 negative
X2 negative

X1 positive
X2 positive

Usually the signs of the coefficients are what I would expect. Also, I've read about Anova so I think that I kind of understand what that is doing. But, what I'm confused about is the following: In some of the models, I can get back wald statistics for X1 say, where both the X1 negative Wald stat and the X1 positive Wald stat are not significant. Yet, the pvalue from the Anova for the X1 variable overall is significant ? Is this possible ? I think I'm not understanding the Anova output as well as I thought because, to me, that seems inconsistent ?

I understand that the Wald statistics for the particular variables are kind of analogous to the t-stats in a regular regression in that they are a function of the decrease in deviance conditional on all the rest of the variables being in the model. The pvalue in the Anova table I thought was kind of doing the same thing except not differentiating between the factors and just calculating the decrease in deviance due to X1 overall without regard to the particular factor response ?

If I'm right in my interpretation of the Anova output, then can that still happen ?

If I'm wrong about my interpretation, and it can happen, can someone tell me where to look for an explanation on why that can happen and possibly explain where my interpretation is wrong ? I just want to understand my output as best as I can.

If it can't happen, then it's puzzling because it is happening.

Thanks for any insights, comments or references. The output is not easily reproducible or else I would reproduce it here.

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