The VARCOMP Procedure

Gauge Repeatability and Reproducibility Analysis

In a typical gauge R&R experiment, each operator ($O_ j$) makes multiple observations on each of several similar parts ($P_ i$) from a monitored process. The statistical model used to describe the response variable is the balanced two-factor crossed random model with interaction

\[ y_{ijk} = \mu _ y + P_ i + O_ j + (PO)_{ij} + E_{ijk} \]

where $i = 1,\dots ,p,\; j = 1,\dots ,o,\; k = 1,\dots ,r,\; \mu _ y$ is an unknown constant, and $P_ i, O_ j, (PO)_{ij}, E_{ijk}$ are jointly independent normal random variables with means of zero and variances $\mr{Var}(P), \mr{Var}(O), \mr{Var}(PO),$ and $\mr{Var}(E)$, respectively. The corresponding SAS statements are as follows: 

proc varcomp method=grr;
   class P O;
   model y = P|O;
run;

The first random effect in the MODEL statement is assumed to be the "Part" effect and the second is "Operator."

The ANOVA table for the preceding model is shown in Table 108.2.

Table 108.2: GRR Analysis of Variance

Source

DF

Mean Square

Expected Mean Square

Parts(P)

$p-1$

$S^2_ P$

$\mr{Var}(E) + r \mr{Var}(PO) + o r \mr{Var}(P)$

Operators(O)

$o-1$

$S^2_ O$

$\mr{Var}(E) + r \mr{Var}(PO) + p r \mr{Var}(O)$

P$\times $O

$(p-1)(o-1)$

$S^2_{PO}$

$\mr{Var}(E) + r \mr{Var}(PO)$

Error(E)

$p o (r-1)$

$S^2_ E$

$\mr{Var}(E)$


The gauge R&R parameters of interest are given in Table 108.3 in terms of $\mr{Var}(P), \mr{Var}(O), \mr{Var}(PO),$ and $\mr{Var}(E).$

Table 108.3: Gauge R&R Parameters

Parameter

Formula

Mean of population of measurements

$\mu _ y = \bar{y}_{\cdot \cdot \cdot } = \Sigma _{ijk} y_{ijk}/por$

Variance of the monitored process

$\gamma _ P = \mr{Var}(P)$

Variance of the measurement system

$\gamma _ M = \mr{Var}(O) + \mr{Var}(PO) + \mr{Var}(E)$

Total variance of the response variable

$\gamma _ y = \mr{Var}(y) = \gamma _ P + \gamma _ M$

Ratio of process variance to measurement variance

$\gamma _ R = \gamma _ P/\gamma _ M$

Proportion of total variance due to the process

$\rho _ P = \gamma _ P/\gamma _ y = \frac{\gamma _ R}{1 + \gamma _ R}$

Proportion of total variance due to the measurement

$\rho _ M = \gamma _ M/\gamma _ y = 1 - \rho _ P$

Signal-to-noise ratio

$\mr{SNR} = \sqrt {2\times \gamma _ R}$

Discrimination ratio

$\mr{DR} = 1 +2\gamma _ R$


For a one-way model, $\gamma _ M = \mr{Var}(E),$ and for a two-way model with no interaction, $\gamma _ M = \mr{Var}(O) + \mr{Var}(E).$

If you use the SPECLIMITS option to give specification limits, the two parameters in Table 108.4 will also be estimated and displayed.

Table 108.4: Gauge R&R Parameters Related to Specification Limits

Parameter

Formula

Precision-to-tolerance ratio

$\mr{PTR(LSL,USL,k)} = k\sqrt {\gamma _ M}/(\mr{USL}- \mr{LSL})$

Process capability ratio

$\mr{Cp(LSL,USL,k)} = (\mr{USL} - \mr{LSL})/(k\sqrt {\gamma _ P})$


Here, USL and LSL are the specification limits, and the value k corresponds to the number of standard deviations between the "natural" tolerance limits of a normal process.

If you use the RATIO option, the ratios in Table 108.5 will also be estimated and displayed.

Table 108.5: Gauge R&R Ratios

Ratio

Formula

Ratio of process variance to total variance

$\mr{Var}(P)/\gamma _ y$

Ratio of operator variance to total variance

$\mr{Var}(O)/\gamma _ y$

Ratio of process by operator variance to total variance

$\mr{Var}(PO)/\gamma _ y$

Ratio of process variance to residual variance

$\mr{Var}(P)/\mr{Var}(E)$

Ratio of operator variance to residual variance

$\mr{Var}(O)/\mr{Var}(E)$

Ratio of process by operator variance to residual variance

$\mr{Var}(PO)/\mr{Var}(E)$