Methods and formulas for Predict for Stability Study for random batches

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Fitted value and SE fit for random batch

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

TermDescription
Zin x mi martix of known codings for the ith random effect in the model
Z'transpose of Z
yresponse data
Xdesign matrix, including the constant
variance component of the ith random factor
variance component for error
Inidentity 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

TermDescription
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
mnumber of columns in the design matrix to represent the ith random term in the model
c number of random effects in the model
Zin x mi design matrix for the ith random effect in the model
Z'itranspose 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

TermDescription
t1-α/2, df1–α/2 quantile from the t distribution with the given degrees of freedom
standard error of the fitted value
Xdesign matrix, including the constant
X' transpose of X
variance component for error
variance component of the ith random factor
Zin x mi matrix of known codings for the ith random effect in the model
Zi' transpose of Zi
Inidentity 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
cnumber of random effects in the model