Methods and formulas for coefficients for Stability Study for random batches

Select the method or formula of your choice.

In This Topic

Coefficients in mixed models
Standard errors of the coefficients in mixed models
Degrees of freedom for coefficients
Confidence intervals for coefficients
T-value
p-value (P)

Coefficients in mixed models

In matrix terms, the vector of coefficients is:

Notation

Term	Description
X	the design matrix, including the constant
X'	the transpose of X
Y	the response data
	the inverse of

	the identity matrix with n rows and columns
	the variance ratio for the i^th random effect in the model
	the n x m_i matrix of known codings for the i^th random effect in the model
m_i	the number of levels for the i^th 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

Term	Description
	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

Term	Description
	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

Term	Description
	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.