Linear regression with low variations in the explanatory variables Below is a scatter plot of some data with the best fit OLS line. From $R^2$ we can see it is a quite a poor fit. But is there a way to predict before running the regression that the results won't be reliable? My gut-feeling is that because we've got the cluster of data points on 1.5, 2 and 2.5, the linear regression is not a good model. But I am looking for a more formal explanation. 
 A: You're asking several different questions. 


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*Why the predictor takes on particular distinct values (multiples of 0.25, it seems) could be an important detail. Ideally you would have information on why that happens from the data source or from a client or boss, say that the numbers arise from division by 4 and there is a good reason for that. At worst, even with no detail on that point in this case the number of distinct values still seems large enough to detect an overall pattern. 

*Calling $R^2 = 0.5306$ quite poor may or may not be echoed by people who know about the data in that field and what to expect. There are fields where anything less than say $0.9$ could mean nothing but disaster. There are  fields where anything higher than that means a bogus or even tautologous relationship (just two different versions of the same underlying variable) or fraud, because correlations just aren't that strong otherwise. 

*To my eye, the scatter suggests curvature, so I would try a quadratic next. I would also want substantive detail on whether a relationship with a turning point makes sense and also on whether the response can ever be negative. 

*One way to tell how well a regression will work when there is just one predictor, or you can plot in two dimensions as here, is to look at the scatter plot. Otherwise I don't know what kind of prior analysis you're reaching for. 

*Reliable in many statistical contexts means reproducible rather than strong. A reliable relationship will show up in similar form in a similar sample. 
I don't think that there is much scope to find good models without context on the data and how they arose. 
