# Variables in a regression which could be zero or intentionally missing

I have a very specific regression problem involving a data set with a general pattern of missing data. The data set has several variables and corresponding binary variables determining whether those variables are expected to have a value or not.

Basically if the binary variable has a value of 1, there would be a value (could potentially be zero, positive or negative) for the respective variable. If the binary variable has a value of 0, the corresponding variable would have no value. i.e. NULL or .

The structure of the data set corresponds to the real world scenario where some companies offer certain services while others do not.

An example dataset

Dependent variable - Company Value                  (y1)
Independent variable 1 - Total sales of product A   (x1)
Binary variable 1 - Does the company sell Product A (b1)
Independent variable 2 - Total sales of product B   (x2)
Binary variable 2 - Does the company sell Product B (b2)

y1  x1  b1  x2  b2

30  46  1   .   0
50  0   1   82  1
20  18  1   .   0
40  .   0   0   1


My actual data set consists of c.1000 records and there are 5 independent variables with their associated binary variables

Is it appropriate to incorporate all of the variables and binary variables into a single regression model and if so, do I add them in as separate terms or as interaction terms? There are potentially 32 combinations of product/service offerings in my data set so I don't see how I could easily split this into multiple models.

You should not include both the binary variables and the sales values in your regression model as there is a direct relationship between your binary variables and the sales of the respective products. This violates the independence assumption of the model.

You need to first understand the reason for missing values in the variables x1 and x2. If they are missing because they was no sales or if they are missing because of some data issues.

If they are missing because there was no sales recorded, that means you can impute the missing values by 0. But if they are missing because of some data handling issues, consider building your model on x1,x2 and treat the missing values with some missing imputation techniques before training your model.

No, it will not work. In linear regression cases with missing are omitted. So all your cases with x1 and x2 omitted will be dropped, resulting in b1 and b2 containing only 1s. Independent variables that are a constant will not stay in the regression. At the end it is just the same as if you only put x1 and x2 in the model.

I think the technique was perhaps misunderstood. Usually, some people will just model b1 and b2 first to see of presence of data or not predicts the dependent variable. Then, another model with x1 and x2 will be done and its interpretation carried out with the first regression's results in mind.

Yet, this is generally considered a biased approach as well. One of such related work is Indicator and Stratification Methods for Missing Explanatory Variables in Multiple Linear Regression by Michael P. Jones. Other work such as directed acyclic graph has also shown that this is a biasing technique.

Perhaps multivariate missing imputation would be useful?