How do I interpret this fitted vs residuals plot?

enter image description here

I don’t really understand heteroscedasticity. I would like to know whether my model is appropriate or not according to this plot.


As @IrishStat commented you need to check your observed values against your errors to see if there are issues with variability. I’ll come back to this towards the end.

Just so you get an idea of what we mean by heteroskedasticity: when you fit a linear model on a variable y you are essentially saying that you make the assumption that your yN(Xβ,σ2) or in layman’s terms that your y is expected to equate Xβ plus some errors that have variance σ2. This practically your linear model y=Xβ+ϵ, where the errors ϵN(0,σ2).
OK, cool so far let’s see that in code:

set.seed(1);            #set the seed for reproducability
N = 100;                #Sample size
x = runif(N)            #Independant variable
beta = 4;               #Regression coefficient
epsilon = rnorm(N);     #Error with variance 1 and mean 0
y = x * beta + epsilon  #Your generative model
lin_mod <- lm(y ~x)  #Your linear model

so right, how do my model behaves:

x11(); par(mfrow=c(1,3));   #Make a new 1-by-3 plot
title("Simple Residual Plot - OK model")
acf(residuals(lin_mod), main = ""); 
title("Residual Autocorrelation Plot - OK model");
plot(fitted(lin_mod), residuals(lin_mod)); 
title("Residual vs Fit. value - OK model");

which should give you something like this:
enter image description here
which means that your residuals do not seem to have an obvious trend based on your arbitrary index (1st plot – least informative really), seem to have no real correlation between them (2nd plot – quite important and probably more important than homoskedasticity) and that fitted values do not have an obvious trend of failure, ie. your fitted values vs your residuals appear quite random. Based on this we would say that we have no problems of heteroskedasticity as our residuals appear to have the same variance everywhere.

OK, you want heteroskedasticity though. Given the same assumptions of linearity and additivity let’s define another generative model with “obvious” heteroskedasticity problems. Namely after some values our observation will be much more noisy.

epsilon_HS = epsilon;               
epsilon_HS[ x>.55  ] = epsilon_HS[x>.55 ] * 9       #Heteroskedastic errors

y2 = x * beta + epsilon_HS      #Your generative model
lin_mod2 <- lm(y2 ~x)            #Your unfortunate LM

where the simple diagnostic plots of the model:

 par(mfrow=c(1,3));   #Make a new 1-by-3 plot
 title("Simple Residual Plot - Fishy model")
 acf(residuals(lin_mod2), main = ""); 
 title("Residual Autocorrelation Plot - Fishy model");
 plot(fitted(lin_mod2), residuals(lin_mod2)); 
 title("Residual vs Fit. value - Fishy model");

should give something like:
enter image description here
Here the first plot seems a bit “odd”; it looks like we have a few residuals that cluster in small magnitudes but that is not always a problem… The second plot is OK, means we have not correlation between your residuals in different lags so we might breathe for a moment. And the third plot spills the beans: it is dead clear that as we got to higher values our residuals explode. We definitely have heteroskedasticity in this model’s residuals and we need to do something about (eg. IRLS, Theil–Sen regression, etc.)

Here the problem was really obvious but in other cases we might have missed; to reduce our chances of missing it another insightful plot was the one mentioned by IrishStat: Residuals versus Observed values, or in for our toy problem at hand:

 plot(y, residuals(lin_mod) ); 
 title( "Residual vs Obs. value - OK model")
 plot(y2, residuals(lin_mod2) ); 
 title( "Residual vs Obs. value - Fishy model")

which should give something like:

enter image description here
here the first plot seems “relatively OK” with only a somewhat hazy upward trend in the residuals of the model (as Scortchi mentioned see here as to why we are not worried). The second plot though exhibits this problem fully. It is crystal clear we have errors that are strongly dependent on the values of our observed values. This manifesting in issues with the coefficient of determination R2 of our models at hand; eg. the “OK” model having an adjusted R2 of 0.5989 while the “fishy” one of 0.03919. Thus we have reasons to believe model misspecification might be an issue. (Thanks to Scortchi for pointing out the misleading statement in my original answer.)

In fairness of your situation, your residuals vs. fitted values plot seems relative OK. Checking your residuals vs. your observed values would probably be helpful to make sure you are on the safe side. (I did not mention QQ-plots or anything like that as not to perplex things more but you may want to briefly check those too.) I hope this helps with your understanding of heteroskedasticity and what you should look out for.

Source : Link , Question Author : kanbhold , Answer Author : usεr11852

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