# 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. Dec 16, 2019 at 13:56
• Sure, it's in my text: y-axis: Regression Standardized Residual, x-axis: Regression Standardized Predicted Value.
– user268744
Dec 16, 2019 at 14:05
• You should have a look at this
– MGP
Dec 16, 2019 at 14:13
• Does this answer your question? Chart indicates homoscedasticity but Breusch-Pagan test p<.001 Dec 16, 2019 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. Dec 16, 2019 at 14:32

#### You have bigger problems than heteroskedasticity here

Based on the straight diagonal line forming the lower bound of the residuals on the lower-left side of the plot, it appears that you are using a linear regression to deal with a non-negative response variable (possibly a count variable?). If this is the case then it would be usual to either use a count regression model or at least use a logarithmic transformation on your response variable in the model. This would usually give you a more appropriate model for this kind of data and would typically deal with the heteroskedasticity at issue.

Before worrying about heteroskedasticity, I recommend that you reconsider whether you are using the correct model. If your response is a count variable then I would recommend starting with a negative-binomial count regression. If your response is not a count variable, but is some other non-negative variable, I would recommend that you consider a log-linear regression (with an adjustment for response values of zero). You can also consider variations on these models, but that is where I would start.

• One thing I would like to add: the original poster DID look at the residuals plot, and decided that this is not how it should look. That is a good thing; I have seen many people not looking at it or ignoring it.
– W_vH
Oct 28, 2022 at 5:52

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.

• $\varepsilon \sim \mathcal N(0,sigma^2 I_n)$ is much stronger than being "unbiased" (this is not the usual terminology though, I'm guessing what you mean is that the errors are mean 0 conditional on regressors) and "variance independence with observations" (the usual terminology here should be "homoskedastic". Also, the answer is vague: what does it mean for those two assumptions to be "essential". For example, if the only issue is heteroskedasticity, OLS is still consistent (although conventional standard errors may be misleading). Oct 28, 2022 at 1:35