# Linear regression: Intercept isn't significant

I built a model of multiple linear regression to predict the revenues at the box office of a movie: the variable $Y$ is revenue while the variables $X$ are two: Activity (number of tweets per day) and sentiment (connotation of these tweets).

The problem is that the intercept is not significant. I thought of a possible explanation: the intercept in practice is the value of the variable $Y$ when the variables $X$ assume values of 0; but because of the very strong development of social networks, it is virtually impossible not to have even a tweet (and then a wave of activity) about a movie that just came out, I thought that the significance or not the intercept could be neglected because the assumed value has no meaning in reality (a tweet of 0 would never be satisfied, given the market today). In this way, I consider the intercept a mere constant for the calculation of the line, without worrying about its significance (because in some cases would take negative values, which is impossible for revenues).

The interpretation may be correct?

It is not necessarily a problem that an intercept is not significant(ly different from zero) and indeed that may be scientifically or practically what you expect. But much more can be said.

The estimate of the intercept for any linear regression that includes one will be determined by all the data, especially including those furthest away from it! The estimation thus pays no attention to whatever is substantively meaningful or interpretable (physically, biologically, economically, and so forth). In many projects what happens at or near the origin to the position of the fit may be at least partly a side-effect of the leverage exerted by very large positive values, either of the response or of the predictors. (In principle, exactly the same leverage effect could be observed for large negative values, but that is less common in practice.) Intuition can be gained by looking at scatter plots in 2 or 3 dimensions and imagining how the fitted line or plane needs to move to satisfy a least squares criterion (unless, naturally, some other fitting method is used).

Nevertheless it is often a good idea to think about whether the intercept estimate is consistent with known or expected behaviour of the response or outcome variable as it changes with predictors. This is especially important whenever there are values close to zero for the predictors, but less important when no value is close to their origin, when the intercept is in effect just an extrapolation of the fitted line or plane away from the mass of the data. When doing this, drawing graphs never hurts and can be invaluable for understanding what is happening. (It is still extraordinarily common in my experience to encounter researchers who have fitted a regression and not plotted their data. This is most common, it seems, in certain social sciences marked by a primitive belief that testing every hypothesis in sight is not only necessary but also sufficient to elucidate what has been done.)

The example here of box office revenues is typical of many response variables which cannot be negative and in practice will usually be positive. Concern that a negative intercept might be produced can be linked to a deeper concern. A linear regression $y = Xb$ is often not a good idea for such responses any way as in principle it will predict negative values for some values of the predictors. Here using a generalised linear model with logarithmic link is often much better, and may capture curvature and heteroscedasticity in the data in a way that is consistent with the underlying science.

With a nod to completeness, note also that many people prefer to omit the intercept from a linear regression and force the fitted surface through the origin and/or to use other functional forms (power laws or power functions) that do something equivalent. There are costs as well as benefits to either strategy.

I think you understand correctly :)

The intercept isn't significant because there isn't sufficient statistical evidence that it's different from zero. And as you say, it seems reasonable that a Movie not mentioned by anyone would make no money.

• Now imagine instead of revenue, we are looking at the response variable Y = revenue - \$10m. Then is the intercept still not significant? I don't think so. The problem is that 0 has a special role here, and the closer the true intercept is to 0, the more "likely" one sees insignificance. Also I disagree with OP's interpretation of statistical insignificance of the intercept. Even without data around X=0, one should be able to get very confident estimate of the intercept, under the linearity assumption, provided more data is used. – John Jiang Sep 24 '15 at 22:59