I have two linear regressions. The linear coefficients of both of them are very close to $1$. The first plot seems reasonable linear regression, while the second one is tricky since if the bottom left part or the top right part of the data (they are quite random) are fitted, we can not get a close-to-1 coefficient.

The question is how to distinguish the two regressions after I get their very similar regression coefficient values? Instead of by looking at the plots, a statistic is preferred to describe the difference of the two regressions. Any ideas?

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In part, you've answered your own question. You distinguish the regressions by looking at plots like the ones you've posted. R (and other programs) offer a set of default plots for linear models that is quite good.

Creating data that looks somewhat like yours:

x <- c(rnorm(100, -5, 1), rnorm(100, 5, 1))
y <- 2*x + rnorm(200)
m1 <- lm(y~x)

we can then run


And at least two of the plots show clear violations of assumptions.

If your question is how to write about this, in model 2 you would just say "the assumptions of regression were violated.".

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    $\begingroup$ In addition, your notation suggests that your data are time series and you are plotting successive values. That alone suggests that regression here is not the main way forward; it is to think what drives the time series. $\endgroup$ – Nick Cox Nov 27 '13 at 11:57
  • $\begingroup$ @Peter Flom Thank you very much for your answer. But I mean I want a statistical value (perhaps a statistic or a hypothesis test) to distinguish the two regressions, instead of using eyes. Any idea on that? $\endgroup$ – yanfei kang Nov 27 '13 at 13:01
  • $\begingroup$ The key point is that the regressions are very similar. What differs are the marginal and joint distributions of your variables. SD and kurtosis will differ, for example, for each variable. Still true that to see what is going on you need a graph. $\endgroup$ – Nick Cox Nov 27 '13 at 13:07
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    $\begingroup$ Well, after looking at the graphs, you could then test two regressions, one for larger values of $x_{t-1}$ and one for smaller values, and see that $R^2$ changes hugely compared to the whole set of values. Another option would be to plot a loess line to the data and then see how much the lines diverged; again, eyes would be key, but you could probably test the difference in the lines somehow. $\endgroup$ – Peter Flom Nov 27 '13 at 14:24

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