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

For all variance components, lower and upper bounds for variance components must not be negative values. If the bounds calculated using the formulas are negative, then they are set to zero.

For all ratios between 0 and 1, lower and upper bounds should also be between 0 and 1. If the bounds are outside the range, they are set to 0 or 1 accordingly.

Notation

Term	Description
	the α *100 percentile of the chi-square distribution with nq degrees of freedom
Fα(nq, nγ)	the α *100 percentile of the F distribution with nq and nγ degrees of freedom
I	the number of parts
J	the number of operators
K	the number of replicates

For degrees of freedom:

Parts: n₁=I–1

Operators: n₂=J–1

Parts*Operators: n₃=(I–1)(J–1)

Replicates: n₄=IJ(K–1)

MSPart = S₁²

MSOperator = S₂²

MSPart*Operator = S₃²

MSReplicates = S₄²

Minitab calculates the lower and upper bounds for an exact (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

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

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

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

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1–α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Formulas

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

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Formulas

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

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

With operator

Without operator

Without interaction

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

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

With operator

Without operator

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