Can these residuals be regarded heteroscadastic?
I have tested the residuals are normally distributed.
The number of data points in your residual plot is a bit small to know for sure, but I see nothing there that strongly suggests heteroscedasticity. The warning flag for a violation of homoscedasticity would typically be a "funnel-shaped" scatter of points in the plot -- short/narrow vertical bands of data points on one side of the plot (either the left or right side), getting taller/wider as you move toward the other side. Your plot seems to show a fairly random scatter, with no pronounced funnel pattern, at least not in the few data points you have. BTW -- the non-significant departure from normality you apparently got is in part due to the small sample size. But, based on the residual plot above, I tend to agree that non-normality is probably not a concern in these data. I would worry more about the sample size. If your regression includes more than 2 predictor variables, the results from such a small sample would be suspect.
The number of data points in your residual plot is a bit small to know for sure, but I see nothing there that strongly suggests heteroscedasticity. The warning flag for a violation of homoscedasticity would typically be a "funnel-shaped" scatter of points in the plot -- short/narrow vertical bands of data points on one side of the plot (either the left or right side), getting taller/wider as you move toward the other side. Your plot seems to show a fairly random scatter, with no pronounced funnel pattern, at least not in the few data points you have. BTW -- the non-significant departure from normality you apparently got is in part due to the small sample size. But, based on the residual plot above, I tend to agree that non-normality is probably not a concern in these data. I would worry more about the sample size. If your regression includes more than 2 predictor variables, the results from such a small sample would be suspect.
While I agree with what Burke said I would notice that the spread changes in different areas of the plot. Hence l might run this data through a Box-Cox routine just to see if a transformation was recommended. Better safe than sorry I guess. David Booth
Zhanchao -
I am curious to know what these data represent, with predicted values both negative and positive. Would you please tell us where these data originate?
If you look at the negative predicted numbers, since there are more of them, and change the sign, you could use the following to assess heteroscedasticity, though with such a small sample size you may have to use a default coefficient of heteroscedasticity based on subject matter experience.
https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx
I had not run across negative predictions. Perhaps you could assume symmetry, and then besides changing the negative predictions to positive, also include the positive predictions. The procedure at the URL above - with references given there - calls for using the absolute value of the estimated residuals, and, assuming symmetry, you could also use the absolute values of the predicted y values.
If you apply the above "tool," it would be interesting to see your results. As Burke indicated, [much] more data would be helpful.
Cheers - Jim
PS -
Are your estimated residuals in the same units as the predicted y values? The estimated residuals seem very large. I am curious as to the subject matter.
PSS -
Here are suggestions for using weighted least squares regression:
https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression
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