Solved – Residual Plots and Transformations in Linear Regression

heteroscedasticityregressionresiduals

What does it typically mean when the plot of residuals vs. fitted values in a linear regression forms a parabola symmetric about the y-axis (for both convex and concave parabolas)? How can one infer from the shape of the residual plot the transformation that will need to be made (if one exists) to satisfy homoscedasticity?

Best Answer

If you're referring to a shape like this:

enter image description here

Then that doesn't indicate a problem with heteroskedasticity, but lack of fit (perhaps suggesting the need for a quadratic term in the model, for example).

If you see a shape like this:

enter image description here

that does indicate a problem with heteroskedasticity.

If your plot doesn't look like either, I think you're probably going to have to show us.

Related Solutions

Solved – How to assess multilevel model assumptions using residual plots

A multilevel model is defined as $y = Xβ + Zη + ǫ$

Thus there are 3 different kinds of residuals:

Marginal residuals: $y − Xβ\ (= Zη + ǫ)$
Conditional residuals: $y − Xβ − Zη\ (= ǫ)$
Random effects: $y − Xβ − ǫ\ (= Zη)$

Marginal residuals:

Should be mean 0, but may show grouping structure
May not be homoskedastic!
Good for checking fixed effects, just like linear regression.

Conditional residuals:

Should be mean zero with no grouping structure
Should be homoskedastic!
Good for checking normality of ǫ, outliers

Random effects:

Should be mean zero with no grouping structure
May not be be homoskedastic!
Good for checking normality of , outliers

In R (if results is an mer object), the command residuals(results) gives you the conditional residuals.

# checking the normality of conditional residuals:
qqnorm(resid(results), main="Q-Q plot for conditional residuals")

# checking the normality of the random effects (here random intercept):
qqnorm(ranef(resuls)$Name_of_group_variable$`(Intercept)`, 
       main="Q-Q plot for the random intercept")

The answer is partly copied from the following PowerPoint slide deck pdf.

Solved – heteroscedasticity, residual vs. independent X variables in a multiple regression

You have count data - use a model appropriate for this: Based on your description of the data, and your residual plot, I suggest that your response variable is a proportion value based on a fixed denominator, which means that it is based on an underlying set of count data (i.e., positive integers up to a fixed known maximum value). That is why you get lines of values in your residual plot when you use OLS estimation. In such cases, the error term in them model is not normally distributed, and you will probably get a better fit from a model designed for count data (e.g., a binomial GLM).

Best Answer

Related Solutions

Solved – How to assess multilevel model assumptions using residual plots

Solved – heteroscedasticity, residual vs. independent X variables in a multiple regression

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