# Regression when the value of one independent variable is dependent on another independent variable

I need to predict the number of retweeets a tweet receives as a function of (1) whether there is a hyperlink within the tweet's text and (2) the position of the hyperlink within the tweet. If a tweet doesn't contain a hyperlink then the position of the hyperlink within the tweet is a null value. Example data for this situation would be:

numberOfRetweets     hyperlinkInTweet?     positionOfHyperlink
(dependent var)    (binary, 0=no, 1=yes)   (integer, 1 to 135)
5                      0                    null
10                      1                     50
35                      1                     12
0                      0                    null
0                      1                      5
:                      :                      :


If a tweet doesn't contain a hyperlink (hyperlinkInTweet = 0) then it cannot have a position within the tweet text where the hyperlink begins (positions range from 1 to 135 inclusive). I've toyed with the idea of using zero in place of null, but I have reservations about doing this. Exploratory analysis of the data indicates that about half of the tweets contain a hyperlink.

I'm wondering if this is a nested regression problem, but would be grateful for a second opinion on the best course of action for this problem. Many thanks.

• What type of model do you use? A Poisson-GLM? Your question is solved by using the hyperlink dummy and the product of the dummy with the position as regressors (like an interaction). – Michael M Jun 19 '14 at 9:38

I think you can substitute zero for your null values, in this context. Let us see what this means. The poisson regression model is $$\DeclareMathOperator{\P}{Pr} \P(Y_i=y_i) = e^{-\lambda_i} \frac{\lambda_i^y}{y_i!}$$ with $\lambda_i >0$ the parameter. For each observation $i$, we have the model $\lambda_i = \exp(x_i^T \beta) = \exp(\eta_i)$ where $\eta_i$ is called the linear predictor. Here we have $\beta=(\beta_0, \beta_1, \beta_2)$ and $x=(1, x_1, x_2)$ where \begin{align} x_1 &= \text{1 if no hyperlink, 0 if hyperlink} \\ x_2 &= \text{0 if no hyperlink, else length of hyperlink (or some non-linear function of length)} \end{align} Then, for your example data (line 1,2,3) we get \begin{align} \eta_1 &= \beta_0 +\beta_1 \\ \eta_2 &= \beta_0 + 50\beta_2 \\ \eta_3 &= \beta_0 + 12\beta_2 \\ \end{align} (Probably you will need some non-linear function of length, such as log. Some plot may tell you!). This looks to me as an eminently sensible and interpretable model. The parameter $\beta_1$ can be seen as a kind of "equivalent length of hyperlink" for the cases without a hyperlink.