When y is negative in linear regression?

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Discussion Overview

The discussion revolves around the issue of obtaining negative predictions for 'y' in a linear regression model where 'y' represents time in seconds. Participants explore the implications of negative coefficients and the appropriateness of the linear regression model given the nature of the data.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether it is possible for 'y' to be negative, given that 'y' represents time in seconds.
  • Another participant seeks clarification on the structure of the linear regression, specifically whether the 8 variables refer to independent variables or data points.
  • A participant explains that negative coefficients for the independent variables can lead to negative predictions for 'y', raising concerns about the suitability of the linear model.
  • There is a suggestion to check the statistical significance of the coefficients, indicating that insignificant variables might lead to unreliable results.
  • One participant proposes using a model that ensures 'y' remains non-negative, such as transforming 'y' using the natural logarithm, which could provide a more appropriate fit for the data.

Areas of Agreement / Disagreement

Participants express differing views on the implications of negative predictions and the appropriateness of the linear regression model. There is no consensus on whether negative values for 'y' are acceptable or how to best address the issue.

Contextual Notes

Participants mention the potential for noise in the data affecting the coefficients and the need for statistical significance in the model, but do not resolve these concerns.

Who May Find This Useful

Individuals interested in linear regression modeling, particularly in contexts where the dependent variable has specific constraints, such as being non-negative.

xeon123
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I am using linear regression to predict 'y' based on 8 variables.
With my example, most the Betas that I got are negative. So, y, the value to predict, is negative.
To my data, y is a time in seconds, so I think it shouldn't be negative.

I my example in python, and I want to know if y should be negative, even when y is seconds, or my code is not correct.

Is is possible that y can be negative?
 
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It's not clear what you mean by a linear regression with 8 variables. Does this mean you are using 8 data points?
 
I mean that I use 8 independent variables to get y.

y = Beta1*x1 + Beta2*x2 + Beta3*x3 + Beta4*x4 + Beta5*x5 + Beta6*x6 + Beta7*x7 + Beta8*x8

And when I calculate the Betas to get a predicted y, \hat{y}, some of them are negative, making \hat{y} negative.
 
Last edited:
Have you checked the statistical significance of those betas? If some of them are just noise then you would expect to get nonsense results.

Even then statistical modeling with a linear fit is never going to be a perfect, it is entirely possible that if x1 is larger that the timing of y will be shorter, causing a negative Beta1 to appear. At that point you might question whether a linear model is a good one to use for the x1 variable.
 
A lot of statistics packages have stepwise regression algorithms. They start with a constant and the most significant independent variable (say Xm): Y = Beta0 + Betam * Xm. Then, one by one, add in the next most significant term, then the next, etc., till there are no statistically significant terms to add. That will allow you to include only those terms that are statistically significant.

If you know that Y can never be negative, you might want to try a model that will never go negative, like Y = exp( Beta0 + Beta1 * X1). For that, do a stepwise linear regression using the natural log of the Y data. That will give an expression ln(Y) = Beta0 + Beta1 * X1. Many statistics packages have these types of regressions as options.
 

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