 Had it very good plot or partial regression plot is one of the more useful diagnostics plots after regression analysis. This plot also demonstrates some features of regression analysis. So let's take a look at what partial regression plot actually does and we need some data and a regression model to do the plot. So we have the data here, the prestige data and we run a regression of prestige on income, education and share of women. And then we do the partial regression plots or added variable plots and they're shown here. So these plots we usually do them for every independent variable but these are basically three plots, three independent plots and we'll just be looking at the first one now because the other ones are done the exact same way. So this is the first added variable plot or partial regression plot. Why it's a partial regression plot will become clear in a few moments. But the idea here is that we have a line that goes with the data. So this is a scatter plot of data and then there's a regression line. So what are these data about? So these are not our observations. It is our education, conditional others, prestige, conditional on others. So understanding what these observations are, what these points here signify and what the line signifies. It's useful to understand how this is actually calculated and it is very simple to calculate. So this is an R code for my own added variable plot. The idea of added variable plot is that you first regress one of the independent variables, prestige on other independent variables, income and women here. Then we regress the dependent variable on the other independent variables except prestige. And then we take the residuals. So we take residual of this regression analysis here and then residual for this other regression analysis here. Those of you who don't understand R, the education here is the dependent variable, then income and women are the independent variables. So it's pretty simple to understand. Just a slightly different way of writing education equals beta one times income plus beta two times women plus beta zero. Then we run a regression analysis where we simply have the prestige residual as the dependent variable, the residual of education as the dependent variable and we do a scatter plot and we draw the regression line. Then the result is our parser regression plot. So this plot here is our built-in plot and this is what the R command does. So what the AD plot command does in R and this is using the built-in plot command. So we can do the exact same plot. The diagnostic plot for regression analysis just adds a grid here and it adds nicer labels for the plot and the plot axis. So this is exactly the same otherwise. So the plot here explains or tells us what is the relationship between our education and prestige when we eliminate all other variables from the model. So it tells us what is the bivariate relationship after we control for other variables. We can also view this or consider this from the Venn diagram perspective. So the Venn diagram perspective on regression analysis is that we have the dependent variable here. We have that's the prestige, then we have the independent variables. This is the education and this is the other variables. So the prestige and education are correlated. This area here is the correlation and we want to know what part of this overall correlation is unique to education and prestige and not accounted for by these other variables. So we can see that there's some overlap between all the variables and there are some unique relationships between all of these variables. And this signifies two different variables here. What we do here is that we regress prestige on these other variables. We regress education on these other variables and then we take the residual. So the residual is the part here if this is a multivariate regression model. The residual is the part that the other variables don't explain. So if we regress education on these other variables, prestige on these other variables and we take the residuals, what remains is this. So we have the residual of prestige, residual of education and now the added variable plot tells us graphically about these bivariate relationships. So importantly the correlation between these two variables is now the regression coefficient if we are using standardized estimates. So we get as the correlation tells us this regression coefficient and that's how. So here this income conditional on others is this area after we have eliminated all the variation that the other variables explain. Prestige here on the y-axis is this prestige residual after we eliminated the influence of all other variables from the data. Also another interesting feature is that the regression coefficient if we have regress prestige on education income on women and then we regress the residual from the added variable plot regression on the other residual. So we have the residual of prestige and residual of education. This regression coefficient here is exactly the same as the regression coefficient here. So you can calculate our regression coefficient this way as well. So you can take variation away one by one and then you get the final regression coefficient for the final variable would be the same as the regression coefficient if we entered all variables at the same time. So we can check that correlation is here same as here. The standard errors differ because here we assume that the effects of education income on women are known but here they are estimated so this is slightly different for that reason. So that regression coefficient is actually the slope of this line. So why is this useful? It is useful because it allows you to graphically present how one variable influences the dependent variable. And when you have a line then the slope tells you everything that you need to know about the line but when you have more complicated relationships like when you fit a log-transport dependent variable or log-transport independent variable or you fit the u-shape where you have a square of variable then you can use the same kind of plotting for those when you don't have a line but you have a curve and then you can check how that curve explains the data controlling for all other variables. So this is useful not only for diagnostics but also for interpretation and I have myself used this kind of plots in one paper that I've written for interpretation purposes. Also the idea that this regression coefficient is the same as regressing one residual and another allows you to understand what this paper by Agunis and Wandeberg are saying. So they are saying that if we have lots of controls in the model then we are basically just analyzing residuals from a model where the dependent variables is first regressed on those controls and the independent variable is regressed on those controls as well. So we are analyzing the relationship between two residuals. Whether that is problematic or not is something that I will not go into in this video. But it's technically correct to say that this is just a residual.