# Regression results have unexpected upper bound

I try to predict a balance score and tried several different regression methods. One thing I noticed is that the predicted values seem to have some kind of upper bound. That is, the actual balance is in $[0.0, 1.0)$, but my predictions top at about $0.8$. The following plot shows the actual vs. the predicted balance (predicted with linear regression):

And here are two distribution plots of the same data:

Since my predictors are very skewed (user data with power law distribution), I applied a Box-Cox transformation, which changes the results to the following:

Although it changes the distribution of the predictions, there is still that upper bound. So my questions are:

• What are possible reasons for such upper bounds in prediction results?
• How can I fix the predictions to correspond to the distribution of the actual values?

Bonus: Since the distribution after the Box-Cox transformation seems to follow the distributions of the transformed predictors, is it possible that this is directly linked? If so, is there a transformation I can apply, to fit the distribution to the actual values?

Edit: I used a simple linear regression with 5 predictors.

• I'm really interested to see where this goes. This is just a linear regression model? How many predictors? Commented Mar 23, 2015 at 12:49
• As a side note: As your outcome variable is bounded by 0 and 1, a simple linear regression model will likely predict values outside of those bounds which is of course invalid. There are other options to consider in this case. Commented Mar 23, 2015 at 13:49
• Bounded input implies bounded output for a linear model. What are the bounds on the (transformed) predictors? Can you show us a summary table of the model fit? Commented Mar 23, 2015 at 13:58
• Mennny: All you really need (to start with) are the coefficient values and the bounds on the predictors. By matching signs one-by-one, you can quickly determine the minimum and maximum prediction (assuming the predictors will always satisfy the bounds, either implicitly or explicitly). Commented Mar 23, 2015 at 16:43
• @cardinal: I checked the bounds of the predictors and was able to confirm your assumption. With the given (untransformed) predictors the maximum prediction is ~0.79. Can you please "copy/paste" your comment as an answer so that I can accept it? How can I proceed? I guess this shows that there is no linear relationship between my predictors and the outcome? Commented Mar 24, 2015 at 9:00