Variance Components Method

The GAGE Application

Variance Components Method

As an alternative to the average and range method, you can use the variance components method, which is a more powerful statistical technique for analyzing measurement error and other sources of variation in a gage study. Until recently, this method was underutilized for gage studies because it is computationally more difficult to carry out than the average and range method.

Moreover, the language of variance components analysis is alien to most engineers. To overcome this obstacle, the GAGE report for the variance components method is displayed in the same format as that of the average and range method. This format is a modification of the gage repeatability and reproducibility report given in Measurement Systems Analysis Reference Manual (1990). Figure A.5 is a sample GAGE report using the variance components method.

As with the average and range method, calculations for the variance components method are based upon a specified multiple of $\sigma$ , where the multiple $\nu$ can be 4, 5.15, or 6.

The advantages of this method versus the average and range method are:

Variance components are estimated more efficiently in a statistical sense.
More information can be obtained, such as the interaction between operators and parts (see Figure A.5).
There are fewer restrictions on the data.

                   Variance Components Method

    Test ID: Gasket         Performed By:
    Date:    09/18/98       John Smith

    Part No. & Name:   Gasket
    Characteristics:
    Specification:     0.6-1.0 mm

    Gage Name: Thickness
    Gage No.:  X-2034
    Gage Type: 0-10 mm


      MEASUREMENT UNIT ANALYSIS       % PROCESS VARIATION
    Repeatability
        EV  =      0.1662               % EV   =  15.77 %
    Reproducibility
        AV  =      0.1483               % AV   =  14.06 %
    Part x Condition
        IV  =      0.2423               % IV   =  22.98 %
    Gage R&R
        R&R =      0.3291               % R&R  =  31.21 %
    Part Variation
        PV  =      1.0016               % PV   =  95.00 %
    Total Variation
        TV  =      1.0543


         Results are based upon predicting 5.15 sigma.
    (99.0% of the area under the normal distribution curve)

Figure A.5: Variance Components Method Sample Report

The variance components method in the GAGE application uses the MIXED procedure in SAS/STAT software. The MIXED procedure fits mixed linear models, which are a generalization of the standard linear model used in the GLM procedure. Refer to SAS/STAT User's Guide for further information on PROC MIXED.

When there is only one operator, PART is a random effect. The MIXED procedure estimates the variance component for PART and for the residual variance (equipment variation) using restricted maximum likelihood (REML).

   proc mixed;
      class part;
      model meas=;
      random part;
   run;

When there is more than one operator, there are three random effects: OPERATOR, PART, and OPERATOR*PART. The MIXED procedure uses REML to estimate variance components for these and for the residual variance (equipment variation).

   proc mixed;
      class operator part;
      model meas=;
      random operator part operator*part;
   run;

The MIXED procedure creates a table of covariance parameter estimates, including

$\sigma_{EV}^2$	the variance component due to equipment variation
$\sigma_{AV}^2$	the variance component due to operator variation
$\sigma_{IV}^2$	the variance component due to the interaction of operators and parts
$\sigma_{PV}^2$	the variance component due to part variation

From these estimates, repeatability (EV), reproducibility (AV), the interaction of operators and parts (IV), and part variation (PV) are calculated.

$EV & = & \nu \sqrt{\sigma_{EV}^2} \ AV & = & \nu \sqrt{\sigma_{AV}^2} \ IV & = & \nu \sqrt{\sigma_{IV}^2} \ PV & = & \nu \sqrt{\sigma_{PV}^2}$

When using the variance components method, the measure of gage repeatability and reproducibility has another component, the interaction term.

$R\&R = \sqrt{(EV)^2 + (AV)^2 + (IV)^2}$

Total variation is calculated similar to the average and range method.

$TV = \sqrt{(R\&R)^2 + (PV)^2}$

The results you get using the variance components method will differ slightly from those you get using the average and range method. This is because the variance components method is more precise, and the variance components method incorporates an interaction term in the measure of gage R&R.

As with the average and range method, the right-hand side of the report can be a percent of process variation or a percent of tolerance. %EV, %AV, %IV, %R&R, and %PV are calculated similar to the average and range method.

The variance components method is more flexible than the average and range method in terms of the data that it can handle. Data for the average and range method should be balanced with the same number of parts and trials for each operator in the study. For example, if your study is composed of two operators, two trials, and ten parts, each operator should have 20 measurements. If the measurement for operator one, trial two, part three is missing, the average and range method cannot compute the gage measures. However, the variance components method can handle such missing data.

The average and range method also requires that a minimum number of parts be collected depending on the number of operators and the number of trials. Otherwise, the estimates will be imprecise. This is another situation where the variance components method can be used.

Note: The flexibility of the variance components method does not imply that you should not use locally recommended procedures for setting up and collecting data for gage studies.

Only a subset of the capabilities of PROC MIXED is used in the GAGE application. The procedure is capable of analyzing much more sophisticated statistical models. For example, you could fit an extended model to study the variability among several gages.

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