# Methods and formulas for coefficients for Stability Study for random batches

Select the method or formula of your choice.

## Coefficients in mixed models

In matrix terms, the vector of coefficients is:

### Notation

TermDescription
Xthe design matrix, including the constant
X'the transpose of X
Ythe response data
the inverse of
the identity matrix with n rows and columns
the variance ratio for the ith random effect in the model
the n x mi matrix of known codings for the ith random effect in the model
mithe number of levels for the ith random effect
c

the number of random effects in the model

c = 2 for the model with time, batch, and the time*batch interaction

c = 1 for the model with time and batch

## Standard errors of the coefficients in mixed models

The standard errors of the coefficients depend on the test method for the fixed effects. For further details on the notation, go to the Methods section and the Tests of Fixed Effects section.

### Kenward-Roger's approximation

The standard errors of the estimated coefficients are the square roots of the dialgonal elements of the matrix .

where

### Satterthwaite Approximation

The standard errors are the square roots of the diagonal elements of the matrix, .

where

## Degrees of freedom for coefficients

The following hypotheses are for the tests of the coefficients:
The following equations give the degrees of freedom for the coefficient:
where

For further details on the notation, go to the Methods section and the Tests of Fixed Effects section.

## Confidence intervals for coefficients

The (1 − α)% confidence limits for the coefficient has the following equation:

### Notation

TermDescription
the estimated coefficient
the 1 − α/2 percentile from the t distribution with df degrees of freedom
the standard error of the estimated coefficient

## T-value

### Notation

TermDescription
test statistic for the coefficient
estimated coefficient
standard error of the estimated coefficient

## p-value (P)

The following equation gives the two-sided p-value for the null hypothesis that a coefficient equals 0:

### Notation

TermDescription
The probability that under the null hypothesis T is less than the absolute value of the calculated . Here, T follows a t-distribution with df degrees of freedom.
The t value for the coefficient.
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