In Applied Linear Statistical Models (Kutner, Nachtsheim, Neter, Li) one reads the following on the coefficient of partial determination:
A coefficient of partial determination can be interpreted as a coefficient of simple determination. Consider a multiple regression model witht two $X$ variables. Suppose we regress $Y$ on $X_2$ and obtain the residuals: $$e_i(Y|X_2) = Y_i - \hat Y_i(X_2)$$ where $\hat Y_i(X_2)$ denotes the fitted values of $Y$ when $X_2$ is in the model. Suppose we further regress $X_1$ on $X_2$ and obtain the residuals: $$e_i(X_1|X_2) = X_{i1}-\hat X_{i1}(X_2)$$ where $\hat X_{i1}(X_2)$ denotes the fitted values of $X-1$ in the regression of $X_1$ on $X_2$. The coefficient of simple determination $R^2$ between these two sets or residuals equals the coefficient of partial determination $R^2_{Y 1| 2}$. Thus this coefficient measures the relation between $Y$ and $X_1$ when both of these variables have been adjusted for there linear relationships to $X_2$.
Where $$R^2_{Y1|2} = \frac{\text{SSR}(X_1|X_2)}{\text{SSE}(X_2)} = \frac{\text{SSE}(X_2) - \text{SSE}(X_1,X_2)}{\text{SSE}(X_2)}$$
But I don't understand what they mean. I interpret $R^2_{Y1|2}$ as the relative decrease in $\text{SSE}$ when $X_1$ is added to the model (already containing $X_2$).
I've tried to implement the statement in R:
set.seed(1)
n <- 100
e <- rnorm(n,0, 10)
beta0 <- 30
beta1 <- 3
beta2 <- 7
x1 <- rnorm(n,10, 4)
x2 <- rnorm(n,2, 5)
y <- rep(beta0,n) + beta1*x1+beta2*x2 +e
SSTO <- sum( (y-mean(y))^2)
fit.Full <- lm(y~x1+x2)
summary(fit.Full)
R.squared.Full <- summary(fit.Full)$r.squared
SSE.x1x2 <- SSTO * (1 - R.squared.Full)
fit.reduced <- lm(y~x2)
summary(fit.reduced)
res1 <- y - as.vector(predict.lm(fit.reduced))
SSE.x2 <- sum(res1^2)
R.squared.partial <- 1 - SSE.x1x2/SSE.x2
## = 0.6203542
## Fitting the predictors
fit.x <- lm(x1~x2)
res2 <- x1-as.vector(predict.lm(fit.x))
plot(res1~res2)
fit.res <- lm(res1~res2)
summary(fit.res)
## and indeed R^2 = 0.6204
Indeed, those values are equal!
But why? And how should I interpret this?
Like 62.04% of the variation in the residuals of the reduced model is explained by...'?
