# Undesirable direction of beta coefficient in simple linear regression

The beta coefficient for independent variables surprisingly become negative, and with significant p-value.

e.g. smoking (independent) and risk of lung cancer (dependent) Regression coefficient for smoking is surprisingly negative, where it suppose to be positive (since we expect that an increase in smoking will increase the chance of getting lung cancer)

This is a SIMPLE linear regression analysis, no other variables included.

So, what are the reasons of getting unexpected or wrong sign of coefficient in simple linear regression, considering data cleaning and transformation have been done?

• This is much too vague to be answerable, but take a look at stat.columbia.edu/~gelman/stuff_for_blog/…. Dec 3 '14 at 6:01
• The are many things that could have gone wrong, but my first action would be to look very carefully at the smoking variable: what numerical value have smokers and what numerical value have non-smokers? If non-smokers are 1 and smokers 0 then you would expect a negative coefficient, i.e. not smoking decreases your risk for long-cancer. Dec 3 '14 at 10:43

If there was no errors during data collection and cleaning, and you did all the statistics stuff correctly then that is what your data "say". If you insist that your prior knowledge suggests that there is a high probability of the opposite result and you want to incorporate this prior knowledge in your statistical model, then we are talking Bayesian...

But seriously: I would use few Bayesian models with different informative and uninformative priors to check if the results hold. You could use a prior pointing in positive direction, an uninformative one, and/or others - if results under different priors would be roughly the same then it would suggest that this is your result. Check a paper by Spiegelhalter et al. (2004) (or book) for hints about using different priors for this kind of analysis.