The fitted value is the predicted y or , which is the mean response value for the given predictor values using the estimated regression equation.
The standard error of the fitted values in the mixed model are the square roots of the diagonal elements of the matrix that follows:
where
Notation
Term
Description
Zi
n x mi martix of known codings for the ith random effect in the model
Z'
transpose of Z
y
response data
X
design matrix, including the constant
variance component of the ith random factor
variance component for error
In
identity matrix with n rows and columns
Prediction interval
The range in which the predicted response for a new observation is expected to fall. The calculation of the prediction interval depends on whether you compute the interval for the marginal fit or for the conditional fit.
Marginal fit
where
The degrees of freedom for the t-statistic are given by this formula:
where
Conditional fit
where
The degrees of freedom for the t-statistic are:
where
Notation
Term
Description
1–α/2 quantile from the t distribution with the given degrees of freedom
vector of the new values of the random predictors
variance component for error
vector of new values of the fixed predictors
variance component of the ith random factor
Im
identity matrix with m rows and columns
m
number of columns in the design matrix to represent the ith random term in the model
c
number of random effects in the model
Zi
n x midesign matrix for the ith random effect in the model
Z'i
transpose of Zi
Confidence interval
The range in which the estimated mean response for a given set of predictor values is expected to fall.
The standard error of the fitted values in the mixed model are the square roots of the diagonal elements of this matrix:
where
The degrees of freedom use this formula when batch is a random factor:
where
Notation
Term
Description
t1-α/2, df
1–α/2 quantile from the t distribution with the given degrees of freedom
standard error of the fitted value
X
design matrix, including the constant
X'
transpose of X
variance component for error
variance component of the ith random factor
Zi
n x mi matrix of known codings for the ith random effect in the model
Zi'
transpose of Zi
In
identity matrix with n rows and columns
xi
predictor values for the fit or prediction
W
asymptotic variance-covariance matrix of the variance component for error
, which is the mean response value for the given predictor values using the estimated regression equation.
