# Interaction and negative binomial regression analysis for criminological thesis

#### eneirync

##### New Member
Hello,

For my master's thesis I'm conducting a research on the influence of morality and self-control on self-reported delinquency. Morality is a metric scale, while self-control is divided into three subscales, being temper, impulsivity and risk-taking behaviour, which are all metric scales. Self-reported delinquency is a count variable.

Since the dependent variabele (i.e. delinquency) is a count variabele, I've been advised to use negative binomial regression analysis. However, I want to examine the interaction effect between self-control and morality, but I've heard that negative binomial is not the most appropriate method of evaluating interaction effects. So how do I examine the interaction effect then? Would it be okay to evaluate the main effects through negative binomial, and then use OLS linear regression to examine the interaction effects? This feels kind of counterintuitive, since it nullifies the purpose of using negbin in the first place.

If so, do I make three interaction variables (like the first one being temper*morality, second being impulsivity*morality, etc...), or is this a bad idea?
Sorry if this sounds like a simple problem, but I feel like I'm really in a dark room right now, and any help/advice would be appreciated. I guess I'm most confused about using negbin and OLS all scrambled together, and the fact that I have to include so many interaction variables.

Cheers,

Elias

#### CB

##### Super Moderator
I've heard that negative binomial is not the most appropriate method of evaluating interaction effects.
Hiya, could you post the sources that have told you this? I haven't heard this before, so it'd be interesting to see the reasoning behind this idea.

Out of interest, when you say "delinquency", do you mean a count of actual crimes? And who is your sample?

#### eneirync

##### New Member
Hiya, could you post the sources that have told you this? I haven't heard this before, so it'd be interesting to see the reasoning behind this idea.

Out of interest, when you say "delinquency", do you mean a count of actual crimes? And who is your sample?
I'm having a hard time myself finding a source for the claim in academic journals and the likes, but my professor of statistics told me this. I have no clue whatsoever what is the reason behind this.

Delinquency is based on a summated (sp.?) Likert scale where respondents were asked for a number of crimes how many times they commited that particular act (where 0= never, ranging through 4= a lot). The sample consists of high school students.

In fact, I am using secondary data from the SPAN-project (http://www.pads.ac.uk/Web_Pages/Research_Pages/Collaborative_Studies/The_Hague.html).

#### CB

##### Super Moderator
I'm having a hard time myself finding a source for the claim in academic journals and the likes, but my professor of statistics told me this. I have no clue whatsoever what is the reason behind this.
Maybe bug your prof to tell you his/her reasoning? Since this seems an important issue for your project, you'd probably want to have good sound reasoning and references to back this idea up, rather than just someone's informal opinion (even an expert's...)

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I have not heard this either. Can you use ordinal regression given your dependent variable?

#### JesperHP

##### TS Contributor
Hiya, could you post the sources that have told you this?
I remembered a line about something similar in my a statbook by Verbeek although in this case the model under consideration is a probitmodel. Verbeek says something like the following:

Considering interaction terms a subtle issue emerges. Consider the model (with a probit link though):

$$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i1} x_{i2}$$

In this model it holds:

$$\frac{\partial \Phi(x_i ' \beta)}{\partial x_{i1}}= \phi(x_i ' \beta) (\beta_1 + \beta_3 x_{i2})$$

Assuming both $$\beta_1 > 0$$ and $$\beta_3>0$$ this would seem to suggest that $$P(y_i=1 \lvert x_i)$$ increases with $$x_{i1}$$ the marginal effect being larger when $$x_{i2}$$ is larger.

And then he says what must be the important part: THIS LATTER CONCLUSION IS NOT NECESSARILY CORRECT. Because $$x_{i2}$$ is correlated with $$\phi(x_i ' \beta)$$ it is possible for the marginal effect to decrease if $$x_{i2}$$ gets larger. He then says that the same holds in the logit model but not in the linear model. And cites a reference: Powers E.A. (2005) Interpreting the logit model .... Journal of Corporate Finance 11, 504-522.

Anyway this could suggest similar problems in the count - I am thinking - model where:

$$\frac{\partial E[y_i \lvert x_i]}{\partial x_{i1}} = exp( \beta ' x_i) (\beta_1 + \beta_3 x_{i2})$$

same correlation"problem".

There's some discussion of similar issues here:

http://www.stata-journal.com/sjpdf.html?articlenum=st0194

Last edited:

#### eneirync

##### New Member
Maybe bug your prof to tell you his/her reasoning? Since this seems an important issue for your project, you'd probably want to have good sound reasoning and references to back this idea up, rather than just someone's informal opinion (even an expert's...)
At the moment he said that, I didn't really give it much thought (it was a rather casual conversation), but now he's not in his office until september.

@hlsmith: I'd prefer sticking with negbin since my promotor advised using this method. Also, I think 'lowering' the delinquency variable would result in a loss of information? I'm also kind of reluctant to arbitrarily selecting cutting points for different categories.

@JesperHP: And what would this mean exactly for my problem? I'm not very 'stat savvy', although I'm eager to learn more.

Thank you all for your replies!