In This Topic
Confidence interval
The range in which the estimated mean response for a given set of predictor values is expected to fall.
Formula
Notation
| Term | Description |
|---|---|
 |  |
 | fitted response value for a given set of predictor values |
| α | type I error rate |
| n | number of observations |
| p | number of model parameters |
| S 2(b) | variance-covariance matrix of the coefficients |
| s 2 | mean square error |
| X | design matrix |
| X0 | vector of given predictor values with 1 column and p rows |
| X'0 | transpose of the new vector of predictor values with 1 row and p columns |
Fit
Notation
| Term | Description |
|---|---|
 | fitted value |
| xk | kth term. Each term can be a single predictor, a polynomial term, or an interaction term. |
| bk | estimate of kth regression coefficient |
Standard error of fitted value (SE Fit)
The standard error of the fitted value in a regression model with one predictor is:
The standard error of the fitted value in a regression model with more than one predictor is:
For weighted regression, include the weight matrix in the equation:
When the data have a test data set or K-fold cross validation, the formulas are the same. The value of s2 is from the training data. The design matrix and the weight matrix are also from the training data.
Notation
| Term | Description |
|---|---|
| s2 | mean square error |
| n | number of observations |
| x0 | new value of the predictor |
 | mean of the predictor |
| xi | ith predictor value |
| x0 | vector of values that produce the fitted values, one for each column in the design matrix, beginning with a 1 for the constant term |
| x'0 | transpose of the new vector of predictor values |
| X | design matrix |
| W | weight matrix |
Regression equation
For a model with multiple predictors, the equation is:
y = β0 + β1x1 + … + βkxk + ε
The fitted equation is:
In simple linear regression, which includes only one predictor, the model is:
y=ß0+ ß1x1+ε
Using regression estimates b0 for ß0, and b1 for ß1, the fitted equation is:
Notation
| Term | Description |
|---|---|
| y | response |
| xk | kth term. Each term can be a single predictor, a polynomial term, or an interaction term. |
| ßk | kth population regression coefficient |
| ε | error term that follows a normal distribution with a mean of 0 |
| bk | estimate of kth population regression coefficient |
 | fitted response |
Prediction interval
The prediction interval is the range in which the fitted response for a new observation is expected to fall.
Formula
Notation
| Term | Description |
|---|---|
| s(Pred) |  |
 | fitted response value for a given set of predictor values |
| α | level of significance |
| n | number of observations |
| p | number of model parameters |
| s 2 | mean square error |
| X | predictor matrix |
| X0 | vector of given predictor values with 1 column and p rows |
| X'0 | transpose of the new vector of predictor values with 1 row and p columns |
