Mean_and_predicted_response

Variance of the mean and predicted responses

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In regression, mean response (or expected response) and predicted response, also known as mean outcome (or expected outcome) and predicted outcome, 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. The concept is a generalization of the distinction between the standard error of the mean and the sample standard deviation.

Background: simple linear regression

In simple linear regression (i.e., straight line fitting with errors only in the y-coordinate), the model is

y_{i}=\alpha +\beta x_{i}+\varepsilon _{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 mean, and 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}+\varepsilon _{d}\,

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

Variance

Variance of the 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 ${\hat {E}}(y\mid x_{d})\equiv {\hat {y}}_{d}\!$ . The variance of the mean response is given by

\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}+2x_{d}\operatorname {Cov} \left({\hat {\alpha }},{\hat {\beta }}\right).

This expression can be simplified to

\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

\sum (x_{i}-{\bar {x}})^{2}=\sum x_{i}^{2}-{\frac {1}{m}}\left(\sum x_{i}\right)^{2}.

Variance of the 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

{\begin{aligned}\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{aligned}}

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

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

{\begin{aligned}\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{aligned}}

Confidence intervals

The $100(1-\alpha )\%$ confidence intervals are computed as $y_{d}\pm t_{{\frac {\alpha }{2}},m-n-1}{\sqrt {\operatorname {Var} }}$ . 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 $y$ 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 sample size.

General case

The general case of linear regression can be written as

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

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