The predicted Y or is the mean response value for the given predictor values using the estimated regression equation.
Cross-validated fitted values
Cross-validated fitted values indicate how well your model predicts data. These values are similar to ordinary fitted values, which indicate how well your model fits the data. To obtain cross-validated fitted value for an observation, it must be removed from the data used to calculate the model and then the fit is calculated with the coefficient vector that is independent from the observation. The formula for the cross-validated fitted values is as follows:
Indicates that i observation was left out of the model calculation
the intercept for the model that does not include observation i
the predictor values
the coefficients for the model that does not include observation i
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:
ith observed response value
ith fitted value for the response
Cross-validated residuals measure the model's predictive ability and are used to calculate the PRESS statistic. Cross-validated residuals in PLS and least squares regression are conceptually similar, but their calculations differ.
In PLS, the cross-validated residuals are the differences between the actual responses and the cross-validated fitted values.
The cross-validated residual value varies based on how many observations are omitted each time the model is recalculated during cross-validation.
In least squares regression, the cross-validated residuals are calculated directly from the ordinary residuals.
observation omitted from the model calculation
cross-validated fitted value
Standardized residual (Std Resid)
Standardized residuals are also called "internally Studentized residuals."
ith diagonal element of X(X'X)–1X'
mean square error
transpose of the design matrix
Standard error of fitted value (SE Fit)
The standard error of the fitted value in a regression model with one
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.
value of the predictor
mean of the predictor
ith predictor value
vector of values
that produce the fitted values, one for each column in the design matrix,
beginning with a 1 for the constant term
transpose of the new vector of predictor
The confidence interval is the range in which the estimated mean response for a given set of predictor values is expected to fall. The interval is defined by lower and upper limits, which Minitab calculates from the confidence level and the standard error of the fits.
number of observations
number of predictors
mean square error
variance-covariance matrix of the coefficients
The prediction interval is the range in which the fitted response for a new observation is expected to fall.
fitted response value for a given set of predictor values