# Mean and predicted response

In linear regression, mean response and predicted response 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.

## Background

In straight line fitting, the model is

$\displaystyle{ y_i=\alpha+\beta x_i +\varepsilon_i\, }$

where $\displaystyle{ y_i }$ is the response variable, $\displaystyle{ x_i }$ is the explanatory variable, εi is the random error, and $\displaystyle{ \alpha }$ and $\displaystyle{ \beta }$ are parameters. The mean, and predicted, response value for a given explanatory value, xd, is given by

$\displaystyle{ \hat{y}_d=\hat\alpha+\hat\beta x_d , }$

while the actual response would be

$\displaystyle{ y_d=\alpha+\beta x_d +\varepsilon_d \, }$

Expressions for the values and variances of $\displaystyle{ \hat\alpha }$ and $\displaystyle{ \hat\beta }$ are given in linear regression.

## Mean response

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 $\displaystyle{ \hat{E}(y \mid x_d) \equiv\hat{y}_d\! }$. The variance of the mean response is given by

$\displaystyle{ \operatorname{Var}\left(\hat{\alpha} + \hat{\beta}x_d\right) = \operatorname{Var}\left(\hat{\alpha}\right) + \left(\operatorname{Var} \hat{\beta}\right)x_d^2 + 2 x_d \operatorname{Cov} \left(\hat{\alpha}, \hat{\beta} \right) . }$

This expression can be simplified to

$\displaystyle{ \operatorname{Var}\left(\hat{\alpha} + \hat{\beta}x_d\right) =\sigma^2\left(\frac{1}{m} + \frac{\left(x_d - \bar{x}\right)^2}{\sum (x_i - \bar{x})^2}\right), }$

where m is the number of data points.

To demonstrate this simplification, one can make use of the identity

$\displaystyle{ \sum (x_i - \bar{x})^2 = \sum x_i^2 - \frac 1 m \left(\sum x_i\right)^2 . }$

## Predicted response

The predicted response distribution is the predicted distribution of the residuals at the given point xd. So the variance is given by

\displaystyle{ \begin{align} \operatorname{Var}\left(y_d - \left[\hat{\alpha} + \hat{\beta} x_d \right] \right) &= \operatorname{Var} (y_d) + \operatorname{Var} \left(\hat{\alpha} + \hat{\beta}x_d\right) - 2\operatorname{Cov}\left(y_d,\left[\hat{\alpha} + \hat{\beta} x_d \right]\right)\\ &= \operatorname{Var} (y_d) + \operatorname{Var} \left(\hat{\alpha} + \hat{\beta}x_d\right). \end{align} }

The second line follows from the fact that $\displaystyle{ \operatorname{Cov}\left(y_d,\left[\hat{\alpha} + \hat{\beta} x_d \right]\right) }$ is zero because the new prediction point is independent of the data used to fit the model. Additionally, the term $\displaystyle{ \operatorname{Var} \left(\hat{\alpha} + \hat{\beta}x_d\right) }$ was calculated earlier for the mean response.

Since $\displaystyle{ \operatorname{Var}(y_d)=\sigma^2 }$ (a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by

\displaystyle{ \begin{align} \operatorname{Var}\left(y_d - \left[\hat{\alpha} + \hat{\beta} x_d \right] \right) & = \sigma^2 + \sigma^2\left(\frac 1 m + \frac{\left(x_d - \bar{x}\right)^2}{\sum (x_i - \bar{x})^2}\right)\\[4pt] & = \sigma^2\left(1 + \frac 1 m + \frac{(x_d - \bar{x})^2}{\sum (x_i - \bar{x})^2}\right). \end{align} }

## Confidence intervals

The $\displaystyle{ 100(1-\alpha)\% }$ confidence intervals are computed as $\displaystyle{ y_d \pm t_{\frac{\alpha }{2},m - n - 1} \sqrt{\operatorname{Var}} }$. Thus, the confidence interval for predicted response is wider than the interval for mean response. This is expected intuitively – the variance of the population of $\displaystyle{ y }$ values does not shrink when one samples from it, because the random variable εi does not decrease, but the variance of the mean of the $\displaystyle{ y }$ does shrink with increased sampling, because the variance in $\displaystyle{ \hat \alpha }$ and $\displaystyle{ \hat \beta }$ decrease, so the mean response (predicted response value) becomes closer to $\displaystyle{ \alpha + \beta x_d }$.

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 samples.

## General linear regression

The general linear model can be written as

$\displaystyle{ y_i=\sum_{j=1}^n X_{ij}\beta_j + \varepsilon_i\, }$

Therefore, since $\displaystyle{ y_d=\sum_{j=1}^n X_{dj}\hat\beta_j }$ the general expression for the variance of the mean response is

$\displaystyle{ \operatorname{Var}\left(\sum_{j=1}^n X_{dj}\hat\beta_j\right)= \sum_{i=1}^n \sum_{j=1}^n X_{di}S_{ij}X_{dj}, }$

where S is the covariance matrix of the parameters, given by

$\displaystyle{ \mathbf{S}=\sigma^2\left(\mathbf{X^{\mathsf{T}}X}\right)^{-1}. }$

## References

• Draper, N. R.; Smith, H. (1998). Applied Regression Analysis (3rd ed.). John Wiley. ISBN 0-471-17082-8.