Question

Hi,

Could someone please give detail explanation of the following graphs. I try to understand what assumptions are being violate here ie (constant variance, independent residuals, linearity) sample size is n=270 observation. I think the variance in top left graph is not constant; there is huge gap in middle of data, but i do not know how to explain formally,

I really looking for someone who has a strong understanding of statistics to provide justification. I want to understand if assumptions being violated. nomral qq looks fine becasue no major deivation away from normality.  

Can you please exaplin how in this example, resdiduals vs fitted, cooks distance, and leverage plot all relate.

Please give justification with answer ,, I want to better understand, many thanks.

Residuals vs Fitted Normal Q-Q 1140 50 60 70 80 Fitted values Theoretical Quantiles Leverage plot of hat values Cook's distance 50 100 150 200 250 50 100 150 Obs. number

Answer 1

The Residual vs Fitted plot checks if the relation between variables is linear and whether there is homoscedasticity. A good plot is one where the points are distributed around the 0-line randomly and in an even spread(for homoscadasticity). In any regression you cannot predict the error of an upcoming point but studying previous data you can say if it is random. This plot checks the consistency of observed residuals with random error. If there is a pattern, then it suggests that error is not entirely random, there are physical and assignable causes that are affecting the value of the response variable.

This assignable error can be due to leverage points or outlier observations in the data. Leverage Points are (if simply put) extreme observations in the predictor values and Outliers are extreme response variable values. Both these values can unduly affect the regression line that hampers future prediction.

The Leverage Plot of Hatvalues is a measure of distance between each X and the mean of X. It indicates potential predictor value outliers(Leverage Points) in its plot. Cooks distance shows how much our model will change if a particular observation would be deleted. So a high Cooks Distance would indicate a potential outlier in Y.

The graphs are related in a sense that the first graph checks for the error distribution. And the next two plots are then used to detect the data points that may be responsible.

In the given graphs, the residual vs fitted plot clearly indicates a pattern and an uneven spread which means it violates assumptions. Then moving on to the Leverage Plot, we do not find any point at any extremity so we can assume there are no Leverage values, and from the Cooks Distance plot, we see 3 observations (58,114,268) that are possible outliers(Y) and the most influential among them is Observation 114.