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.
where
is the response variable, 
is the explanatory variable, εi is the random error, and 
and 
are parameters. The predicted response value for a given explanatory value, xd, 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 xd, 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
The predicted response distribution is the predicted distribution of the residuals at the given point xd. So the variance is given by
The second part of this expression was already calculated for the mean response. Since
(a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by
confidence interval
s 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 
values does not shrink when one samples from it, because the random variable εi does not decrease, but the variance mean of the 
does shrink with increased sampling, because the variance in 
and 
decrease, so the mean response (predicted response value) becomes closer to 
.
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.
Therefore since
the general expression for the variance of the mean response is
where M is the covariance matrix
of the parameters, given by
.
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple 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 iswhere
is the response variable, is the explanatory variable, εi is the random error, and and are parameters. The predicted response value for a given explanatory value, xd, is given bywhile the actual response would be
Expressions for the values and variances of
and are given in linear regressionLinear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
.
Mean response is an estimate of the mean of the y population associated with xd, that is
. The variance of the mean response is given byThis expression can be simplified to
To demonstrate this simplification, one can make use of the identity
The predicted response distribution is the predicted distribution of the residuals at the given point xd. So the variance is given by
The second part of this expression was already calculated for the mean response. Since
(a fixed but unknown parameter that can be estimated), the variance of the predicted response is given byConfidence intervals
The confidence intervalConfidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...
s 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 values does not shrink when one samples from it, because the random variable εi does not decrease, but the variance mean of the does shrink with increased sampling, because the variance in and decrease, so the mean response (predicted response value) becomes closer to .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 asTherefore since
the general expression for the variance of the mean response iswhere M is the covariance matrix
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
of the parameters, given by
.