Predictions

Once we have built a model, $\hat{y}=X\hat{\beta }$, we can calculate predicted y, ${\hat{y}}_0$ values for a new set of predictors, $x_0$.

Predictions

Once we have built a model, $\hat{y}=X\hat{\beta }$, we can calculate predicted y, ${\hat{y}}_0$ values for a new set of predictors, $x_0$.

$${\hat{y}}_0=x_0^T\hat{\beta }$$

Predictions

Once we have built a model, $\hat{y}=X\hat{\beta }$, we can calculate predicted y, ${\hat{y}}_0$ values for a new set of predictors, $x_0$.

$${\hat{y}}_0=x_0^T\hat{\beta }$$

For example, if we fit a model $\hat{y}=1.2+2.5x_1+3x_2$

And would like to know the predicted value for someone with $x_1=3$ and $x_2=2$, we would calculate

Predictions

Once we have built a model, $\hat{y}=X\hat{\beta }$, we can calculate predicted y, ${\hat{y}}_0$ values for a new set of predictors, $x_0$.

$${\hat{y}}_0=x_0^T\hat{\beta }$$

For example, if we fit a model $\hat{y}=1.2+2.5x_1+3x_2$

And would like to know the predicted value for someone with $x_1=3$ and $x_2=2$, we would calculate

$${\hat{y}}_0=\left[\begin{array}{ccc}1& 3& 2\end{array}\right]\left[\begin{array}{c}1.2\\ 2.5\\ 3\end{array}\right]$$

Predictions

There are ✌️ kinds of predictions that can be made from regression models

Predictions

There are ✌️ kinds of predictions that can be made from regression models

• A predicted mean response

Predictions

There are ✌️ kinds of predictions that can be made from regression models

• A predicted mean response
• A prediction of a future observation

Example

• What would a chicken who eats sunflowers weigh on average?

Example

• What would a chicken who eats sunflowers weigh on average?
• Suppose you want to feed your chicken sunflowers, what will your chicken's predicted weight be?

Example

• What would a chicken who eats sunflowers weigh on average?
• Suppose you want to feed your chicken sunflowers, what will your chicken's predicted weight be?

Prediction of the mean response

The prediction is $x_0^T\beta$, estimated by $x_0^T\hat{\beta }$.

Prediction Intervals

• There is an important conceptual difference here

Prediction Intervals

• There is an important conceptual difference here
• parameters (like ${\beta }_1$, ${\beta }_2$, etc) are considered fixed but unknown (they are not random) which is why we interpret confidence intervals like we do

Prediction Intervals

• There is an important conceptual difference here
• parameters (like ${\beta }_1$, ${\beta }_2$, etc) are considered fixed but unknown (they are not random) which is why we interpret confidence intervals like we do
• A future observation is a random variable. Therefore, we are saying there is a 95% chance that the future value falls within this interval

Prediction Intervals

Which is larger?

$$\left(x_0^T\right(X^{T}X{)}^{-1}{x}_0){\sigma }^2$$

or

$$(1+x_0^T(X^{T}X{)}^{-1}{x}_0){\sigma }^2$$