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I applied a linear regression (continuous response, two continuous predictors and one categorical variable). The plot of residuals and fitted values is something like two different clusters. Can I say that the assumptions are satisfied in this example?

enter image description here

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    $\begingroup$ What are your thoughts? $\endgroup$ – information_interchange Jul 20 '18 at 15:57
  • $\begingroup$ Assumptions to do what exactly? $\endgroup$ – Michael M Jul 20 '18 at 17:13
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    $\begingroup$ There's no assumption about the pattern in the x's. It's the conditional distribution of the y's (or the residuals on the y-axis in the case of the plot) that you need to look at. $\endgroup$ – Glen_b Jul 20 '18 at 17:16
  • $\begingroup$ @MichaelM. for example having non-linear patterns? We can look at residual plots from a ‘good’ model and a ‘bad’ model. The good model data are simulated in a way that meets the regression assumptions very well, while the bad model data are not. Please correct me if I am wrong. $\endgroup$ – stat Jul 20 '18 at 19:05
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If you see any pattern in a residuals vs fitted plot different from "the good one" (which is the case when values spread randomly in an horizontal band around 0), this means that "something remains to be explained" and that the assumptions of the linear model are not fully satisfied (cf. https://data.library.virginia.edu/diagnostic-plots/ or https://onlinecourses.science.psu.edu/stat501/node/277 for instance). In your particular case, I think you miss some supplementary binary variable in your model.

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    $\begingroup$ This does not sound right. The two clusters are probably just levels of a strong binary predictor present in the model. $\endgroup$ – Michael M Jul 20 '18 at 16:15
  • $\begingroup$ @ Michael M , thanks so much for the comment. It means that there is no violation of assumptions (any kind of pattern) in this example and we can report the estimated coefficients and P-values? $\endgroup$ – stat Jul 20 '18 at 17:07
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    $\begingroup$ @MichaelM - why not expand that comment into an answer? It looks right to me! $\endgroup$ – jbowman Jul 20 '18 at 18:08

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