# Does this graph imply a violation of homoscedasticity?

I assume that this graph doesn't support the assumption of homoscedasticity. Am I right? Does it make sense to carry out another test to be sure?

y-axis: Regression Standardized Residual, x-axis: Regression Standardized Predicted Value, dependent variable: Score of a questionnaire measuring rage attacks, n=156 • Welcome to the site. For those of us who don't speak German (I think it's German) could you translate the axis label? I think I understand most of it, but not "geschatzer Wert" and "think I understand" is not really good. – Peter Flom Dec 16 '19 at 13:56
• Sure, it's in my text: y-axis: Regression Standardized Residual, x-axis: Regression Standardized Predicted Value. – AppleSeed Dec 16 '19 at 14:05
• You should have a look at this – MGP Dec 16 '19 at 14:13
• Does this answer your question? Chart indicates homoscedasticity but Breusch-Pagan test p<.001 – Nick Cox Dec 16 '19 at 14:27
• There is a presumably a constraint on the response, say no score is lower than zero. Hence, residuals must plot above some line that in spirit is residual = minimum observed $-$ fitted. (Standardization affects details only.) This alone inhibits or prohibits homoscedasticity. If there is an upper bound only, all the more reason to consider a generalized linear model with appropriate link rather than plain regression. For more, see the thread suggested just above as duplicate. – Nick Cox Dec 16 '19 at 14:32

## 1 Answer

Two essential assumptions of regression are being unbiased and variance independence with observations. Mathematically, $$\varepsilon \in \mathbb{R}^{n}$$ is:

$$\varepsilon \sim \mathcal{N} (0, \sigma^{2} I_{n}).$$

In your case, it seems that the estimated variance increase with observations. Two possibilities : the first one is a problem of heteroscedasticity and maybe a problem of sampling.

I hope it helps.