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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.


== Straight line regression ==


In straight line fitting, the model is
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:<math>y_i=\alpha+\beta x_i +\epsilon_i\,</math>
where <math>y_i</math> is the [[response variable]], <math>x_i</math> is the [[explanatory variable]], ''ε<sub>i</sub>'' is the random error, and <math>\alpha</math> and <math>\beta</math> are parameters. The predicted response value for a given explanatory value, ''x<sub>d</sub>'', is given by
 
:<math>\hat{y}_d=\hat\alpha+\hat\beta x_d ,</math>
while the actual response would be
:<math>y_d=\alpha+\beta x_d +\epsilon_d  \,</math>
 
Expressions for the values and variances of <math>\hat\alpha</math> and <math>\hat\beta </math> are given in [[linear regression]].
 
'''Mean response''' is an estimate of the mean of the ''y'' population associated with ''x<sub>d</sub>'', that is <math>E(y | x_d)=\hat{y}_d\!</math>. The variance of the mean response is given by
:<math>\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) .</math>
This expression can be simplified to
:<math>\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).</math>
 
To demonstrate this simplification, one can make use of the identity
:<math>\sum (x_i - \bar{x})^2 = \sum x_i^2 - \frac{1}{m}\left(\sum x_i\right)^2 .</math>
 
The predicted response distribution is the predicted distribution of the residuals at the given point ''x<sub>d</sub>''. So the variance is given by
:<math>\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) .</math>
 
The second part of this expression was already calculated for the mean response. Since <math>\text{Var}\left(y_d\right)=\sigma^2</math> (a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by
:<math>\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) .
</math>
 
== Confidence intervals ==
The <math>100(1-\alpha)% </math> [[confidence interval]]s are computed as <math> y_d  \pm t_{\frac{\alpha }{2},m - n - 1} \sqrt{\text {Var}} </math>. 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 <math>y</math> values does not shrink when one samples from it, because the random variable ''ε<sub>i</sub>'' does not decrease, but the variance of the mean of the <math>y</math> does shrink with increased sampling, because the variance in <math>\hat \alpha</math> and <math>\hat \beta</math> decrease, so the mean response (predicted response value) becomes closer to <math>\alpha + \beta x_d</math>.
 
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
:<math>y_i=\sum_{j=1}^{j=n}X_{ij}\beta_j + \epsilon_i\,</math>
 
Therefore since <math>y_d=\sum_{j=1}^{j=n} X_{dj}\hat\beta_j </math> the general expression for the variance of the mean response is
:<math>\text{Var}\left(\sum_{j=1}^{j=n} X_{dj}\hat\beta_j\right)= \sum_{i=1}^{i=n}\sum_{j=1}^{j=n}X_{di}M_{ij}X_{dj},</math>
where '''M''' is the [[covariance matrix]] of the parameters, given by
:<math>\mathbf{M}=\sigma^2\left(\mathbf{X^TX}\right)^{-1}</math>.
 
{{No footnotes|date=November 2010}}
 
== References ==
* {{cite book
|title = Applied Regression Analysis
|edition = 3rd
|last1= Draper |first1=N.R. |last2=Smith |first2=H.
|publisher = John Wiley
|year = 1998
|isbn = 0-471-17082-8}}
 
 
{{least squares and regression analysis}}
 
{{DEFAULTSORT:Mean And Predicted Response}}
[[Category:Regression analysis]]
[[Category:Estimation theory]]

Latest revision as of 19:43, 23 May 2014


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