In regression, mean response (or expected response) and predicted response, also known as mean outcome (or expected outcome) and predicted outcome, are values of the dependent variable calculated from the regression parameters and a given value of the independent variable. The values of these two responses are the same, but their calculated variances are different.The concept is a generalization of the distinction between the standard error of the mean and the sample standard deviation.
In simple linear regression (i.e., straight line fitting with errors only in the y-coordinate), the model is
yi=\alpha+\betaxi+\varepsiloni
yi
xi
\alpha
\beta
\hat{y}d=\hat\alpha+\hat\betaxd,
yd=\alpha+\betaxd+\varepsilond
Expressions for the values and variances of
\hat\alpha
\hat\beta
Since the data in this context is defined to be (x, y) pairs for every observation, the mean response at a given value of x, say xd, is an estimate of the mean of the y values in the population at the x value of xd, that is
\hat{E}(y\midxd)\equiv\hat{y}d
\operatorname{Var}\left(\hat{\alpha}+\hat{\beta}xd\right)=\operatorname{Var}\left(\hat{\alpha}\right)+\left(\operatorname{Var}
2 | |
\hat{\beta}\right)x | |
d |
+2xd\operatorname{Cov}\left(\hat{\alpha},\hat{\beta}\right).
This expression can be simplified to
\operatorname{Var}\left(\hat{\alpha}+\hat{\beta}xd\right)
| ||||
=\sigma |
+
\left(xd-\bar{x | |
\right) |
2}{\sum(xi-\bar{x})2}\right),
where m is the number of data points.
To demonstrate this simplification, one can make use of the identity
\sum(xi-\bar{x})2=\sum
2 | |
x | |
i |
-
1 | |
m |
\left(\sum
2 | |
x | |
i\right) |
.
The predicted response distribution is the predicted distribution of the residuals at the given point xd. So the variance is given by
\begin{align} \operatorname{Var}\left(yd-\left[\hat{\alpha}+\hat{\beta}xd\right]\right)&=\operatorname{Var}(yd)+\operatorname{Var}\left(\hat{\alpha}+\hat{\beta}xd\right)-2\operatorname{Cov}\left(yd,\left[\hat{\alpha}+\hat{\beta}xd\right]\right)\\ &=\operatorname{Var}(yd)+\operatorname{Var}\left(\hat{\alpha}+\hat{\beta}xd\right). \end{align}
The second line follows from the fact that
\operatorname{Cov}\left(yd,\left[\hat{\alpha}+\hat{\beta}xd\right]\right)
\operatorname{Var}\left(\hat{\alpha}+\hat{\beta}xd\right)
Since
2 | |
\operatorname{Var}(y | |
d)=\sigma |
\begin{align} \operatorname{Var}\left(yd-\left[\hat{\alpha}+\hat{\beta}xd\right]\right)&=\sigma2+
| ||||
\sigma |
+
\left(xd-\bar{x | |
\right) |
2}{\sum(xi-\bar{x})2}\right)\\[4pt] &=\sigma2\left(1+
1 | |
m |
+
(xd-\bar{x | |
) |
2}{\sum(xi-\bar{x})2}\right). \end{align}
See main article: article and Confidence interval. The
100(1-\alpha)\%
yd\pm
t | |||||
|
\sqrt{\operatorname{Var}}
y
y
\hat\alpha
\hat\beta
\alpha+\betaxd
This is analogous to the difference between the variance of a population and the variance of the sample mean of a population: the variance of a population is a parameter and does not change, but the variance of the sample mean decreases with increased sample size.
The general case of linear regression can be written as
yi=\sum
n | |
j=1 |
Xij\betaj+\varepsiloni
Therefore, since
yd=\sum
n | |
j=1 |
Xdj\hat\betaj
n | |
\operatorname{Var}\left(\sum | |
j=1 |
Xdj\hat\betaj\right)=
n | |
\sum | |
i=1 |
n | |
\sum | |
j=1 |
XdiSijXdj,
where S is the covariance matrix of the parameters, given by
| ||
S=\sigma |
-1.