Interaction Term Significant, Main Terms Insignificant: Interpret Effects Negative Binomial Regression

[Antonia Fontaine]

Dear Statalists,

I'm trying to interpret the coefficient of a continuos-continuos interaction term, in a Negative Binomial Regression. The dependent variable is the number of car accidents, while the main terms are the assistance per capacity of the stadium in a football match and the expectation of winning that match (both take values from 0 to 100). I demeaned the main terms in the interaction, so that the main coefficients represent the effect on the dependent variable when the other main coefficient is at its mean. As I expected, the coefficient of the interaction is positive and significant. However, the main coefficients are both negative and not significant. I don't really now how to interpret these results.


(Std. Err. adjusted for 61 clusters in cl)
-----------------------------------------------------------------------------------------------------------
| Robust
accidents | Coef. Std. Err. z P>|z| [95% Conf. Interval]
----------------+------------------------------------------------------------------------------------------
P_100 | -.0049938 .0062891 -0.79 0.427 -.0173203 .0073326
I_100 | -.0030911 .0027701 -1.12 0.264 -.0085204 .0023382
K_100 | .0005024 .0002241 2.24 0.025 .0000632 .0009417
accidents_prev | .050766 .0383775 1.32 0.186 -.0244524 .1259844



P_100 is the expectation of winning the match, I_100 the relevance of the match (Assistance per capacity), and K_100 is the interaction terms(main terms demeaned). The mean of the expectation of winning is 46.9, while the mean of the relevance of the match is 49.6. I believe that the right way to interpret them is "the effect of a one percentage point increase in the expectation of winning on the number of accidents increase by 0.05024% for each one percentage point increase in the relevance of the match" and vice versa. Also, the effect of the expectation of winning turns positive for matches with a relevance greater than 59.5% (as exp(-0.00499+((59.5-49.6)\times 0.000502))=1) But, as the main coefficients are not significant, and don't now if I'm right. Moreover, I would like to now if the number of accidents increase with the expectation of winning and the relevance of the Match. Should I use margins?

Regards,

Antonia Fontaine.

[Clyde Schechter]

Your interpretations are correct. The lack of statistical significance of the "main" effects is irrelevant, just ignore it.

Remember that when you are using a model with a continuous by continuous interaction (say, X1 and X2), you are saying that the marginal effect of each of those variables is a linear function of the other variable. Because linear functions have the entire real line as both their domain and range, this implies that there will always be some value of variable X1 for which the marginal effect of X2 is zero and vice versa. Clearly for values of X1 that are close enough to that value, the marginal effect of X2 will be non-statistically significant. As it happens, in your situation, that value of X1 lies close to X1 (demeaned) = 0, and vice versa. In these models, you really just confuse yourself and others by even thinking about the statistical significance of these main effects. Unless your research goal is to specifically test hypotheses about the effect of X1 conditional on X2 = 0 and vice versa, these p-values should be ignored.

[Richard Williams]

Welcome to Statalist. If you read the FAQ, pt #12 will tell you how to ask Qs more effectively, e.g. use code tabs when posting commands and output.

Once you have interaction effects the significance of the main effects is usually not an issue. For an explanation, see

https://www3.nd.edu/~rwilliam/stats2/l53.pdf

[Antonia Fontaine]

Thank you both for your prompt responses. I'm still having some issues with my results... My hypothesis is that the higher the relevance of a lost football match and the higher the expectation of winning it (which is a proxy of frustration), the higher the number of car accidents (so the coefficient of the interaction is the expected). However, when also considering the main effects for illustrating the results, I do not obtain the expected effect, in terms of "counts"of the dependent variable. For example, if the expectation of winning and the relevance of a match is zero, the number of accidents would increase by a factor of 1.16(as exp((0-49.6)\times(0-46.9)\times 0.000502))=1.16), while if the expectation and relevance become 50% the number of accidents would decrease by a factor of 0.7(as exp((-0.00499\times50)+(-0.00309\times50)+(50-49.6)\times(50-46.9)\times 0.000502))=0.7). If both were 100%, they would increase by a factor of 1.7. Given the latter, and due to the fact that the data does not provide evidence to affirm that the main coefficients are negative, zero or positive, I'm not sure if my hypothesis is correct. Can I affirm that my hypothesis is correct for any value of the expectation of winning and the relevance of the match, just looking at the interaction effect?

Regards,

Antonia Fontaine

[Clyde Schechter]

So you need to think a bit more clearly about what your hypothesis is, and your use of language:

I'm still having some issues with my results... My hypothesis is that the higher the relevance of a lost football match and the higher the expectation of winning it (which is a proxy of frustration), the higher the number of car accidents (so the coefficient of the interaction is the expected)

contradicts itself. The hypothesis you set out there has nothing to do with an interaction model. These hypotheses would be tested in a non-interaction model.

An interaction model is designed to test a different bunch of hypotheses:
The higher the relevance of a lost football match, the larger the marginal effect of expectations of winning on traffic accidents. (Or, equivalently, the higher the expectations of winning, the larger the marginal effect of relevance of a lost football match on traffic accidents.)

Next, I think you are ill-advised to do all of these effect calculations by hand. It is easy to make mistakes. The -margins- command automates all of this for you. To use it, you will first have to re-do your regression using factor variable notation. If you read the excellent Richard Williams' https://www3.nd.edu/~rwilliam/stats/Margins01.pdf, you will learn about these and see worked examples that are relevant to your problem. I also think you will probably understand your model better if you use -marginsplot- after -margins- so you can literally see what your model does.