# Appropriate regression-like model where the response is on half-integers

What is an appropriate model for the above scatter plot? I am not fully satisfied with a simple linear regression model. Any suggestions? Y in this problem is discrete in nature. It only increments by 0.5.

• It is very curious that the vast majority of responses are actually whole numbers and not half-integers. This suggests that the responses are favoring whole numbers. Could you say more about what these responses are and help us understand this very strange behavior? Also, are the values of the independent variable set by the experimenter or are they also observations (and therefore perhaps need to be modeled as random variables, too)?
– whuber
Mar 22, 2015 at 22:07
• My depend variable is change in price. Here prices are recorded up to 0.5 cents. Hence the behavior.
– deb
Mar 23, 2015 at 6:51
• deb, that useful information would be best mentioned in your question as well; people shouldn't need to read comments to know that. Mar 23, 2015 at 8:48
• Related question: stats.stackexchange.com/questions/21058/… Mar 23, 2015 at 10:14

You could modify your linear function with applying floor/ceiling on your dependent variable (see Wikipedia).

This way you would get a discrete step function that mimics your data (see image below).

Given your regression function $y = 0.009x - 0.002$, you have to transform it in the following way: $y = ⌊0.009(x+offset) - 0.002⌋$ and select the $offset$ in a way that maximizes your fit, probably somewhere around $50$.

A more general (and complicated) solution would be to define constant linear functions for every possible Y-value. Then, you have to select intervals on the X-scale to specify where you use each of your constant functions in order to maximize your fit. Obviously, not all constant functions will be used.

Given your plot, I would select something like this (as a rule of thumb):

$$f(x) = \left\{ \begin{array}{l l} -2 & \quad \text{x < -150} \\ -1 & \quad \text{-150 <= x < -25} \\ 0 & \quad \text{-25 <= x < 50} \\ 1 & \quad \text{50 <= x < 125} \\ 2 & \quad \text{x >= 125} \\ \end{array} \right.$$

• How exactly would you fit data to this model and why would it work at all well? What assumptions would you make about the residuals?
– whuber
Mar 22, 2015 at 22:05
• I am also not sure how the model you described would work. Can you please explain in little more details?
– deb
Mar 23, 2015 at 6:52
• OK, I have updated my answer with two possible procedures. Mar 23, 2015 at 7:55