Methods and formulas for variance ratios in confidence intervals in Crossed Gage R&R Study

With operator

When the operator and interaction term are included, there are two possible calculation methods. Minitab first calculates the bounds using the modified large-sample (MLS) method. If certain conditions are not met during the calculations, then Minitab uses the Satterthwaite approximation. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

MLS method

The two conditions for the existence of the lower and the upper bounds using the MLS method are:

If the two conditions are not satisfied, Minitab cannot use this method to construct the lower and upper bounds. Minitab will use the Satterthwaite approximation to calculate the lower and upper bounds.

Satterthwaite approximation

The formulas for the lower and upper bounds still hold, except L and U are defined as follows:

Without operator term

Without interaction term

When the operator and interaction term are included, there are two possible calculation methods. Minitab first calculates the bounds using the modified large-sample (MLS) method. If certain conditions are not met during the calculations, then Minitab uses the Satterthwaite approximation. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

MLS method

The two conditions for the existence of the lower and the upper bounds using the MLS method are:

If the two conditions are not satisfied, Minitab cannot use this method to construct the lower and upper bounds. Minitab will use the Satterthwaite approximation to calculate the lower and upper bounds.

Satterthwaite approximation

The formulas for the lower and upper bounds still hold except L and U are defined as follows:

Notation

Term	Description
	the α *100 percentile of the chi-square distribution with nq degrees of freedom
J	the number of operators
I	the number of parts
K	the number of replicates

With interaction tern

There are two possible calculation methods. First, Minitab calculates the bounds using the modified large-sample (MLS) method. If certain conditions are not met during the calculations, then Minitab uses an alternate approximation. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

MLS method

The lower and upper bounds for an approximate (1- α) * 100% confidence interval are calculated by solving quadratic equations.

If B²– 4AC < 0, there is no solution to the quadratic equation. In this case, Minitab uses the second method to estimate the confidence intervals.

Second method

The lower and upper bounds for an approximate (1 – α) * 100% confidence interval are calculated as follows:

Without interaction term

There are two possible calculation methods. First, Minitab calculates the bounds using the modified large-sample (MLS) method. If certain conditions are not met during the calculations, then Minitab uses an alternate approximation. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

MLS method

The lower and upper bounds for an approximate (1 – α) * 100% confidence interval are calculated by solving quadratic equations. Lower bound, L, equals J times the smaller solution to the following equation.

Upper bound, U, equals J times the larger solution to the following equation.

If B²– 4AC < 0, there is no solution to the quadratic equation. In this case, Minitab uses the second method to estimate the confidence intervals.

Second method

The lower and upper bounds for an approximate (1 – α) * 100% confidence interval are calculated as follows:

Notation

Term	Description
	the α *100 percentile of the chi-square distribution with nq degrees of freedom
J	the number of operators
I	the number of parts
K	the number of replicates
a	I
b	J
c	(IJ – I –J)
d	IJ(K-1)
e	I – 1

Notation

Term	Description
	the α *100 percentile of the chi-square distribution with nq degrees of freedom
J	the number of operators
I	the number of parts
K	the number of replicates

Notation

Term	Description
	the α *100 percentile of the chi-square distribution with nq degrees of freedom
J	the number of operators
I	the number of parts
K	the number of replicates

With operator and interaction term

There are two possible calculation methods. First, Minitab calculates the bounds using the modified large-sample (MLS) method. If certain conditions are not met during the calculations, then Minitab uses an alternate approximation. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Without operator term

Lower bound = 1 – (the lower bound for the ratio of the repeatability variance and the total variance)

Upper bound = 1 – (the upper bound for the ratio of the repeatability variance and the total variance)

Without interaction term

Notation

Term	Description
	the α *100 percentile of the chi-square distribution with nq degrees of freedom
J	the number of operators
I	the number of parts
K	the number of replicates