Create a new categorized variable from a continuous variable, then use an interaction of both variables in Logistic Regression

Question

I'm a stat newbie and not good at English, but I will try my best to explain my problems.

• First, I have a dependent variable (y) and 5 independent variables (x1, x2, x3, x4, x5)
• x1 and x2 are categorical. x3, x4, x5 are continuous (integer).
• Focus on x5, its histogram looks like this:

• Then, I categorize x5 using 5 as a cutpoint and call this new categorical variable x5_cat
• Next, I use x1, x2, x3, x4, x5 and an interaction between x5 and x5_cat as independent variables to predict y by logistic regression
• Code in R looks like this: glm(y ~ x1 + x2 + x3 + x4 + x5 + x5:x5_cat, data = train, family = 'binomial')

Because of my limited knowledge of statistics, I can't explain why my solution is wrong. Can anyone help me ?

---

Edited :

The cutpoint cuts x5 into 2 categories that change from a non linear distribution into 2 nearly linear distributions, one is positive and another is negative slope. I think the coefficient of the interaction term can tell me whether there is a different effect of x5 on y when x5 is below the cutpoint and when x5 is above the cutpoint.

Answer 1

Without knowing why you want to create an interaction it's hard to give you a full answer, so I will give you a general one.

Creating dummy variables out of interval data usually needs some theoretical (or technical) reasoning behind it. So if $x_5$ is delay in minutes of trains, you might want to create a dummy in which: $x=\begin{cases} 1, &\text{if x > 5}. \\ 0, &\text{else}. \end{cases}$

If a delay of 5 minutes or more is penalized by the train authority. In this example, any delay between 0 and 5 minutes have no real difference. The cut-point is what we care about.

An interaction is the multiplication of two variables, which means that we have a reason to believe that the effect of $x_1$ on $y$ is varies according to the values of $x_2$. In such a case, we may say that (using a common example) that the effect of adding sugar to water ($x_1$) on sweetness ($y$) varies depending of stirring ($x_2$).

Using a base variable ($x_5$) and interacting it with a dummy version of itself ($x_{5cat}$) doesn't make very much sense. Assume we are regressing a continuous, a dummy and interaction variables: $\hat{y}=a+\beta_1x_1 + \beta_2x_2 + \beta_{12}x_1x_2$

the meaning of $\beta_1$ is that this will be the addition (or raise in log odds or whatever) to $\hat{y}$ with every unit increase in $x_1$, when $x_2=0$ (because of the interaction term). $\beta_2$ is the average difference between category $1$ and category $0$ when $x_1=0$.

So here is the problem. If $x_2$ if a dummy of $x_1$, than $\beta_2$ is not defined because category $1$ does not exist when $x_1=0$