#Calculate residual plus

So if we go back to the graph, that's what we've drawn here for this graph: here u_7 is the average distance of the pupils in school 7 from the overall regression line, and u_3 is the average distance of the pupils in school 3 from the overall regression line. And r_j, the raw residual, is just the mean of r_ij for group j.

#Calculate residual plus

So r_ij is our estimate of the combined level one and level two residual- it's our estimated u_j plus our estimated e_ij. So that's just the same as for a single level model, just like a single level residual, so far.

And to do that, we define this r_ij- notice that's slightly different, not r_j the raw residual but r_ij, and r_ij is the difference between the observation and the predicted value- it's the distance between the data point and the overall regression line. So how do we actually calculate these multilevel residuals? Well, first of all we have to calculate something called the 'raw residual', r_j. And that's the same for all the other schools. And similarly in the green school, they all share the level two residual, u_3, but they have their own individual level one residuals, e_7,3, e_4,3, e_5,3 and so on. So, going back to the graph, we have in pinky purple e_1,7, that's the residual for pupil 1 in school 7, the level one residual, and you can see that's the distance between the data point and the line for school 7, and then in green, we've got e_5,3, that's the level one residual for pupil 5 in school 3, and again you can see that's the distance between the observation for pupil 5 in school 3, and the line for school 3.Īnd now if we add in the other residuals for the pupils in those two schools, you can see that in school 7, the pinky-purple school, all of the pupils share the same level two residual, u_7, but they each have their own individual level one residual: e_3,7, e_4,7, e_6,7 and so on. So then the other residual that we have is the estimate for e_ij, the level one residual, and that's the distance from the line for the group to the data point. So if we have a look at the graph again, we can imagine that these are exam results, so along the x-axis we have the score at age 11 and on the y-axis we have the score at age 16, and the colours show which school each pupil belongs to, so two points that are the same colour are two pupils from the same school, and we can add in now the lines for each school, and now the level two residuals, so in pinky purple we have u_7, that's the level two residual for school 7, you can see it's the distance between the line for school 7 and the overall regression line, and then in green we have u_3, that's the level two residual for school 3, and again you can see that's the distance between the overall regression line and the line for school 3. So the level 2 residual, the estimate for u_j, is just the distance from the overall regression line to the line for the group. So now, going to the multilevel case, we have the same basic idea, but it's a bit more complicated now, because we have two error terms, so that means we're going to have two residuals: an estimate for u_j and an estimate for e_ij. So here we have a graph with lots of data points and a regression line, and if we now add on a couple of residuals, here we have in pinky-purple e_43, that's the residual for observation 43, and you can see that's just the distance between the data point and the regression line, and then over on the left, in green, we have e_20, that's the residual for the 20th observation, and again you can see it's the distance between the data point and the regression line. If we want a visual way to think of this, then the residual is simply the distance between the data point and the regression line. So, we can write it like this in symbols- y_i hat is the predicted value of y and y_i is the observed value of yĪnd that means we can calculate the residuals like this, taking the predicted value from the observed value.

So in other words it's an estimate for the error term e_i.

So in this case, the residual for each observation is just the difference between the value of y predicted by the equation and the actual value of y that we observe. Let's begin by revising residuals for a single level model. To watch the presentation go to Residuals - listen to voice-over with slides and subtitles (If you experience problems accessing any videos, please email See also - Residuals FAQs.Residuals A transcript of Residuals presentation, by Rebecca Pillinger