Term | Description |
---|---|
fitted value | |
x_{k} | k^{th} term. Each term can be a single predictor, a polynomial term, or an interaction term. |
b_{k} | estimate of k^{th} regression coefficient |
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 s^{2} is from the training data. The design matrix and the weight matrix are also from the training data.
Term | Description |
---|---|
s^{2} | mean square error |
n | number of observations |
x_{0} | new value of the predictor |
mean of the predictor | |
x_{i} | i^{th} predictor value |
x_{0} | 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 |
Term | Description |
---|---|
e_{i} | i ^{th} residual |
i ^{th} observed response value | |
i ^{th} fitted response |
Standardized residuals are also called "internally Studentized residuals."
Term | Description |
---|---|
e_{i} | i ^{th} residual |
h_{i} | i ^{th} diagonal element of X(X'X)^{–1}X' |
s^{2} | mean square error |
X | design matrix |
X' | transpose of the design matrix |
Also called the externally Studentized residuals. The formula is:
Another presentation of this formula is:
The model that estimates the i^{th} observation omits the i^{th} observation from the data set. Therefore, the i^{th} observation cannot influence the estimate. Each deleted residual has a student's t-distribution with degrees of freedom.
Term | Description |
---|---|
e_{i} | i^{th} residual |
s_{(i)}^{2} | mean square error calculated without the i^{th} observation |
h_{i} | i ^{th} diagonal element of X(X'X)^{–1}X' |
n | number of observations |
p | number of terms, including the constant |
SSE | sum of squares for error |
The range in which the estimated mean response for a given set of predictor values is expected to fall.
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 |
X_{0} | 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 |
The prediction interval is the range in which the fitted response for a new observation is expected to fall.
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 |
X_{0} | 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 |