# residuals plots from four linear models

I've made 4 linear models. For each of these models, I've plotted the residuals against the fitted values.

• First plot: generalised linear model with quasibinomial link function
• Second plot:generalised linear model with quasibinomial link function
• Third plot:linear regression
• Fourth plot: linear regression

I'm aware that, to satisfy the assumptions of linear models, residuals should not should any patterns, should be normal distributed around zero etc.

Which of these residual plots appear to satisfy assumptions of linear models and which do not?    • All of these are at least somewhat suspect. They seem to have clear bounds, which isn't really consistent w/ the standard OLS regression model (but may not pose too much of a problem in practice). Also, there may be some non-linearity--ie, mis-specified functional form. The wiggly-ness of the fitted line on the plots is driven by a smoothing parameter, you don't necessarily know how literally to take it. You can get another perspective on that same information by getting the acf & pcf of those residuals. What are these data / the models? – gung Mar 13 '14 at 15:54
• Sorry first two are from a generalised linear model, last two are from linear regression. Post edited. Last plot is using the diamonds dataset from ggplot2, other three are my own data. – luciano Mar 13 '14 at 16:18
• How can a link function be quasibinomial? – Scortchi Mar 13 '14 at 16:19
• Sorry modelled using the quasibinomial distribution – luciano Mar 13 '14 at 16:25
• So dichotomous data? – Scortchi Mar 13 '14 at 16:32

## 1 Answer

Numbers 1, 2 and 4 are clearly wrong. If number 3 is just from a smaller sample, it is wrong in the same way, in ways that aren't apparent because you don't have enough data. But in practice these assumptions are always broken. The key is to understand the consequences of brokenness and figure out how much they matter for the problem at hand. At very least you've got the linear approximation to the conditional expectation function, which is often useful in itself, even absent trustworthy confidence intervals.

• Sorry see edited post - only last two are from linear regression. Also, why are they wrong? How do i figure out consequences of broken assumptions – luciano Mar 13 '14 at 16:19
• If you use the library mgcv, the gam.check function plots residuals vs linear predictors, for the case of GLMs. I find that really helpful. Regarding the consequences of broken assumptions: in most cases it's a problem for inference (i.e.: confidence intervals). In some cases, such as heteroskedasticity, the expectation of your estimate is fine, but the variance will be off and you'll get an imprecise estimate (and the confidence intervals will also be wrong. This is all intro econometrics/stats. It is a little bit dangerous to jump into GLMs without really knowing OLS. – generic_user Mar 13 '14 at 18:30