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Regarding the multiple linear regression: I read that the magnitude of the residuals should not increase with the increase of the predicted value; the residual plot should not show a ‘funnel shape’, otherwise heteroscedasticity is present. In contrast, if the magnitude of the residuals stays constant, homoscedasticity is present.

In this residual plot I see that the magnitude of the residuals change with the increase of the predicted value, so does that mean that heteroscedasticity is present? I do not see this typical funnel shape. Can homoscedasticity or heteroscedasticity be derived from this residual plot?

enter image description here

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On residual plots, you can diagnose the residual variance by looking at the dispersion around the average. It's hard to tell because of the density of points on your plot, but the dispersion does not look dramatically heterogeneous.

However, the average of the residuals is not constant across predicted values (the cloud is "tilted"), indicating some strong non-linearity. This is to me the biggest issue revealed by the plot. Another issue is the neatly delimited aspect on the top right side of the cloud, which usually suggests that the dependent variable is (semi-)bounded with a high concentration of values at the boundary. In that case, you may want to transform your data or use a different type of model, such as a generalized linear model.

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    $\begingroup$ The distribution of residuals is so odd that I suspect some binning of data was done. We need to see a high-resolution histogram of $Y$. If $Y$ is partially discrete, then ordinal regression (with no further binning) is called for. $\endgroup$ – Frank Harrell Jun 22 '19 at 11:15
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Your graph shows a clear violation of model assumptions assumed in linear regression. Your data do indeed appear somewhat heteroscedastic. Funnel shapes are not the only shapes on these plots that are indicators of heteroscedasticity.

Recall that in ordinary linear regression, the model assumes that the errors of the model are assumed normally distributed with mean zero and a constant variance of $\sigma^2$ (i.e. $\epsilon_i \sim N(0, \sigma^2)$). If you see anything other than an essentially random pattern around of predicted values vs. residuals (i.e. $\hat{\epsilon}$ around the zero line), you likely have non-linearity of the response function and some heteroscedasticity implying the model assumptions for OLS are violated.

Putting aside the issue of non-linearity and other potential model assumption violations, you could always check for non-constant error variance with a formal statistical test, depending on how many points you actually have there (for example, the r function ncv.test will perform the Breusch-Pagan test which is a statistical test of the null hypothesis of constant error variance against the alternative that the error variance changes with the level of the response.)

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    $\begingroup$ We can't be 100% sure because the cloud is so much dense on the pic. Cloud outlines, outliers - they don't necessarily discard homoscedasticity overall. Of course, if one does not insist the distribution of errors must be, in practice, but normal. $\endgroup$ – ttnphns Jun 22 '19 at 7:55
  • $\begingroup$ You're right -- I toned down and revised my comments a bit. $\endgroup$ – StatsStudent Jun 22 '19 at 8:18

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