Mean and predicted response

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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

Further information: Straight line fitting

In straight line fitting, the model is

y_i=\alpha+\beta x_i +\epsilon_i\,

where $y_{i}$ is the response variable, $x_{i}$ is the explanatory variable, ε_i is the random error, and $\alpha$ and $\beta$ are parameters. The predicted response value for a given explanatory value, x_d, is given by

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

while the actual response would be

y_d=\alpha+\beta x_d +\epsilon_d \,

Expressions for the values and variances of ${\hat {\alpha }}$ and $\hat\beta$ 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 x_d, is an estimate of the mean of the y values in the population at the x value of x_d, that is ${\hat {E}}(y|x_{d})\equiv {\hat {y}}_{d}\!$ . The variance of the mean response is given by

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

This expression can be simplified to

\text{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).

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

\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 x_d. So the variance is given by

\text{Var}\left(y_d - \left[\hat{\alpha} + \hat{\beta}x_d\right]\right) = \text{Var}\left(y_d\right) + \text{Var}\left(\hat{\alpha} + \hat{\beta}x_d\right) .

The second part of this expression was already calculated for the mean response. Since $\text{Var}\left(y_d\right)=\sigma^2$ (a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by

\text{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) = \sigma^2\left(1+\frac{1}{m} + \frac{\left(x_d - \bar{x}\right)^2}{\sum (x_i - \bar{x})^2}\right) .

Confidence intervals

Main article: Confidence interval

Further information: Prediction interval

The $100(1-\alpha)\%$ confidence intervals are computed as $y_d \pm t_{\frac{\alpha }{2},m - n - 1} \sqrt{\text {Var}}$ . 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 $y$ values does not shrink when one samples from it, because the random variable ε_i does not decrease, but the variance of the mean of the $y$ does shrink with increased sampling, because the variance in $\hat \alpha$ and ${\hat {\beta }}$ decrease, so the mean response (predicted response value) becomes closer to $\alpha + \beta x_d$ .

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

y_{i}=\sum _{j=1}^{n}X_{ij}\beta _{j}+\epsilon _{i}\,

Therefore, since $y_{d}=\sum _{j=1}^{n}X_{dj}{\hat {\beta }}_{j}$ the general expression for the variance of the mean response is

\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

\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.

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