# The VARCOMP Procedure

## Example 97.2 Using the GRR Method

In this example from Houf and Burman (1988), the response variable is the thermal performance of a module measured in Celsius degrees per watt. Each of three operators measures 10 parts three times. It is assumed that parts and operators are selected at random from larger populations. The following statements produce Output 97.2.1.

```data Houf;
input a b y @@;
datalines;
1  1 37    1  1 38    1 1 37
1  2 41    1  2 41    1 2 40
1  3 41    1  3 42    1 3 41
2  1 42    2  1 41    2 1 43
2  2 42    2  2 42    2 2 42
2  3 43    2  3 42    2 3 43
3  1 30    3  1 31    3 1 31
3  2 31    3  2 31    3 2 31
3  3 29    3  3 30    3 3 28
4  1 42    4  1 43    4 1 42
4  2 43    4  2 43    4 2 43
4  3 42    4  3 42    4 3 42
5  1 28    5  1 30    5 1 29
5  2 29    5  2 30    5 2 29
5  3 31    5  3 29    5 3 29
6  1 42    6  1 42    6 1 43
6  2 45    6  2 45    6 2 45
6  3 44    6  3 46    6 3 45
7  1 25    7  1 26    7 1 27
7  2 28    7  2 28    7 2 30
7  3 29    7  3 27    7 3 27
8  1 40    8  1 40    8 1 40
8  2 43    8  2 42    8 2 42
8  3 43    8  3 43    8 3 41
9  1 25    9  1 25    9 1 25
9  2 27    9  2 29    9 2 28
9  3 26    9  3 26    9 3 26
10 1 35   10  1 34   10 1 34
10 2 35   10  2 35   10 2 34
10 3 35   10  3 34   10 3 35
;
```
```proc varcomp data=Houf method=grr (speclimits=(18,58) ratio);
class a b;
model y=a|b/cl;
run;
```

You specify METHOD=GRR in this example to drive the VARCOMP procedure to produce a gauge repeatability and reproducibility analysis. With the option speclimits=(18 58), the parameters PTR and Cp are estimated and displayed. With the RATIO option, certain additional ratios of variance components are also estimated and displayed. Finally, the CL= option in the MODEL statement specifies that estimates of GRR quantities should have the corresponding confidence limits.

Output 97.2.1 Class Level Information Using Method=GRR
Variance Components Estimation Procedure

Class Level Information
Class Levels Values
a 10 1 2 3 4 5 6 7 8 9 10
b 3 1 2 3

 Number of Observations Read 90 90

Dependent Variable: y

The "Class Level Information" table in Output 97.2.1 displays the levels of each variable specified in the CLASS statement.

Output 97.2.2 Analysis of Variance Using Method=GRR
GRR Analysis of Variance
Source DF Sum of Squares Mean Square Expected Mean Square
a 9 3935.955556 437.328395 Var(Error) + 3 Var(a*b) + 9 Var(a)
b 2 39.266667 19.633333 Var(Error) + 3 Var(a*b) + 30 Var(b)
a*b 18 48.511111 2.695062 Var(Error) + 3 Var(a*b)
Error 60 30.666667 0.511111 Var(Error)
Corrected Total 89 4054.400000

The GRR analysis of variance in Output 97.2.2 is the same as for the Type I analysis when the design is balanced.

Finally, the estimates of the GRR parameters of interest and their confidence limits are displayed in Output 97.2.3.

Output 97.2.3 Parameter Estimates Using Method=GRR
GRR Estimates
Parameter Estimate 95% Confidence Limits
Mu Y 35.80000 30.49477 41.10523
Var(a) 48.29259 22.69452 161.63918
Var(b) 0.56461 0.07296 25.75077
Var(a*b) 0.72798 0.33273 1.79272
Var(Error) 0.51111 0.36816 0.75754
Gamma Y 50.09630 24.48844 166.22217
Gamma P 48.29259 22.69452 161.63918
Gamma M 1.80370 1.20623 27.01724
Gamma R 26.77413 1.69168 105.60895
SNR 7.31767 1.83939 14.53334
PTR(18,58,6) 0.20145 0.16474 0.77967
Cp(18,58,6) 0.95933 0.52437 1.39942
DR 54.54825 4.38336 212.21791
Rho P 0.96400 0.62848 0.99062
Rho M 0.03600 0.0093801 0.37152
Var(a)/Gamma Y 0.96400 0.62848 0.99062
Var(b)/Gamma Y 0.01127 0.0008700 0.34151
Var(a*b)/Gamma Y 0.01453 0.0027083 0.04744
Var(a)/Var(Error) 94.48551 40.19199 327.32469
Var(b)/Var(Error) 1.10467 0.13662 50.37744
Var(a*b)/Var(Error) 1.42432 0.55232 3.74691

You can draw the following inferences from the results of the analysis. Most of the variation is due to differences between parts because of the relative larger value of Gamma R. The measurement system is nearly inadequate because the PTR exceeds 20%. However, the measurement system is of value in monitoring the process since the SNR is greater than five. Refer to Burdick, Borror, and Montgomery (2003) for more information about interpreting gauge R&R studies.

The confidence limits in Output 97.2.3 are based on large-sample asymptotic approximation. You can alternatively compute more accurate and usually smaller confidence intervals by using CL=GCL for generalized confidence limits. The following statements produce Output 97.2.4:

```proc varcomp data=Houf method=grr (speclimits=(18,58) ratio) seed=104;
class a b;
model y=a|b/cl=gcl;
run;
```

Output 97.2.4 Generalized Confidence Limits
Variance Components Estimation Procedure

GRR Estimates
Parameter Estimate 95% Generalized Confidence
Limits
Mu Y 35.80000 30.48351 41.31148
Var(a) 48.29259 22.79316 168.91421
Var(b) 0.56461 0.07157 24.28846
Var(a*b) 0.72798 0.33476 1.75806
Var(Error) 0.51111 0.36816 0.75754
Gamma Y 50.09630 25.47092 180.85535
Gamma P 48.29259 22.79316 168.91421
Gamma M 1.80370 1.18494 25.76890
Gamma R 26.77413 1.91286 87.60026
SNR 7.31767 1.95594 13.23633
PTR(18,58,6) 0.20145 0.16328 0.76145
Cp(18,58,6) 0.95933 0.51295 1.39639
DR 54.54825 4.82572 176.20052
Rho P 0.96400 0.65669 0.98871
Rho M 0.03600 0.01129 0.34331
Var(a)/Gamma Y 0.96400 0.65669 0.98871
Var(b)/Gamma Y 0.01127 0.0010082 0.32122
Var(a*b)/Gamma Y 0.01453 0.0032088 0.04300
Var(a)/Var(Error) 94.48551 40.44585 336.50782
Var(b)/Var(Error) 1.10467 0.12886 47.19043
Var(a*b)/Var(Error) 1.42432 0.55232 3.74691

Note that the generalized confidence interval widths from Output 97.2.4 for parameters and DR are 85.7 and 171.4, respectively. These widths are much shorter than the MLS-based widths, which are 103.9 and 207.8 from Output 97.2.3.

In general, the GCL method provides a more accurate confidence interval with a shorter interval width than the MLS method. However, as discussed in the section Generalized Confidence Limits, they are computationally intensive and somewhat nondeterministic, because they are based on an underlying Monte Carlo simulation.