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.

Contents 1 Straight line regression 2 Confidence intervals 3 General linear regression 4 References

[edit] Straight line regression

In straight line fitting the model is

where y_i is a dependent variable, x_i is an independent variable and α and β are parameters. The predicted response value for a given value, x_d, of the independent variable is given by

while the actual response would be

Expressions for the values and variances of and are given in linear regression.

Mean response is an estimate of the mean of the y population associated with x_d, that is . The variance of the mean response is given by

This expression can be simplified to

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

Predicted response is the expected range of values of y at some confidence level. The predicted response distribution is the predicted distribution of the residuals at the given point x_d. So the variance is given by

The second part of this expression was already calculated for the mean response. Since , the variance of the predicted response is given by

[edit] Confidence intervals

The 100(1 − α)% confidence intervals are computed as . Thus, the confidence interval for predicted response is wider than the interval for mean response. This is expected intuitively – the variance population of y values does not shrink when one samples from it, but the variance mean of the y does shrink with increased sampling. This is analogous to the difference between the variance of a population and the variance of the mean of a population.

[edit] General linear regression

The general linear model can be written as

Therefore since the general expression for the variance of the mean response is

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

.

[edit] References

Draper, N.R., Smith, H. (1998) Applied Regression Analysis. Wiley. ISBN 0-471-17082-8

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