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 residual is the difference between an observed value and the corresponding fitted value. This part of the observation is not explained by the model. The residual of an observation is:

Term | Description |
---|---|

y_{i} | i^{th} observed response value |

i^{th} fitted value for the 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 part of the observation due to the whole variation (after accounting for model terms) in a split-plot design.

Term | Description |
---|---|

fitted value for the full model (including the whole plot error term as well as fixed terms) | |

fitted value using only the fixed effects terms, not the whole plot error term |

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:

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 |

The standard errors of the coefficients are the square roots of the diagonal elements of the covariance matrix:

The standard error of the fitted value at a given point (used for confidence intervals) is:

The standard error that is used in the prediction intervals is:

Term | Description |
---|---|

subplot variance component, calculated as MSE(SP) | |

X | n × p design matrix for effects of factors, covariates, blocks, and the whole plot error term |

the whole plot variance component, which in a balanced design has this formula: | |

m | the number of subplots within a whole plot |

Z | n × w matrix of whole plot indicators (all 1's and 0's) |

n | number of rows of data |

p | number of coefficients |

w | number of whole plots |

x | row vector of predictor levels |

covariance matrix of β | |

β | vector of coefficients |

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 |

X'_{0} | transpose of the new vector of predictor values |

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} | matrix of given predictor values |

X'_{0} | transpose of the new vector of predictor values |