6.5: Quality Control Example

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Example - Fully Nested Random Effects Model

The temperature of a process in a manufacturing industry is critical to quality control. The researchers want to characterize the sources of this variability. They choose 4 plants and 4 operators within each plant, look at 4 shifts for each operator, and then measure temperature for each of the three batches used in production.

Collected data was read into SAS and proc mixed procedure was used to obtain the ANOVA model.

Show SAS Code

data fullnest;
input Temp Plant Operator Shift Batch;
datalines;
477    1    1    1    1
472    1    1    1    2
481    1    1    1    3
478    1    1    2    1
475    1    1    2    2
474    1    1    2    3
472    1    1    3    1
475    1    1    3    2
468    1    1    3    3
482    1    1    4    1
477    1    1    4    2
474    1    1    4    3
471    1    2    1    1
474    1    2    1    2
470    1    2    1    3
479    1    2    2    1
482    1    2    2    2
477    1    2    2    3
470    1    2    3    1
477    1    2    3    2
483    1    2    3    3
480    1    2    4    1
473    1    2    4    2
478    1    2    4    3
475    1    3    1    1
472    1    3    1    2
470    1    3    1    3
460    1    3    2    1
469    1    3    2    2
472    1    3    2    3
477    1    3    3    1
483    1    3    3    2
475    1    3    3    3
476    1    3    4    1
480    1    3    4    2
471    1    3    4    3
465    1    4    1    1
464    1    4    1    2
471    1    4    1    3
477    1    4    2    1
475    1    4    2    2
471    1    4    2    3
481    1    4    3    1
477    1    4    3    2
475    1    4    3    3
470    1    4    4    1
475    1    4    4    2
474    1    4    4    3
484    2    1    1    1
477    2    1    1    2
481    2    1    1    3
477    2    1    2    1
482    2    1    2    2
481    2    1    2    3
479    2    1    3    1
477    2    1    3    2
482    2    1    3    3
477    2    1    4    1
470    2    1    4    2
479    2    1    4    3
472    2    2    1    1
475    2    2    1    2
475    2    2    1    3
472    2    2    2    1
475    2    2    2    2
470    2    2    2    3
472    2    2    3    1
477    2    2    3    2
475    2    2    3    3
482    2    2    4    1
477    2    2    4    2
483    2    2    4    3
485    2    3    1    1
481    2    3    1    2
477    2    3    1    3
482    2    3    2    1
483    2    3    2    2
485    2    3    2    3
477    2    3    3    1
476    2    3    3    2
481    2    3    3    3
479    2    3    4    1
476    2    3    4    2
485    2    3    4    3
477    2    4    1    1
475    2    4    1    2
476    2    4    1    3
476    2    4    2    1
471    2    4    2    2
472    2    4    2    3
475    2    4    3    1
475    2    4    3    2
472    2    4    3    3
481    2    4    4    1
470    2    4    4    2
472    2    4    4    3
475    3    1    1    1
470    3    1    1    2
469    3    1    1    3
477    3    1    2    1
471    3    1    2    2
474    3    1    2    3
469    3    1    3    1
473    3    1    3    2
468    3    1    3    3
477    3    1    4    1
475    3    1    4    2
473    3    1    4    3
470    3    2    1    1
466    3    2    1    2
468    3    2    1    3
471    3    2    2    1
473    3    2    2    2
476    3    2    2    3
478    3    2    3    1
480    3    2    3    2
474    3    2    3    3
477    3    2    4    1
471    3    2    4    2
469    3    2    4    3
466    3    3    1    1
465    3    3    1    2
471    3    3    1    3
473    3    3    2    1
475    3    3    2    2
478    3    3    2    3
471    3    3    3    1
469    3    3    3    2
471    3    3    3    3
475    3    3    4    1
477    3    3    4    2
472    3    3    4    3
469    3    4    1    1
471    3    4    1    2
468    3    4    1    3
473    3    4    2    1
475    3    4    2    2
473    3    4    2    3
477    3    4    3    1
470    3    4    3    2
469    3    4    3    3
463    3    4    4    1
471    3    4    4    2
469    3    4    4    3
484    4    1    1    1
477    4    1    1    2
480    4    1    1    3
476    4    1    2    1
475    4    1    2    2
474    4    1    2    3
475    4    1    3    1
470    4    1    3    2
469    4    1    3    3
481    4    1    4    1
476    4    1    4    2
472    4    1    4    3
469    4    2    1    1
475    4    2    1    2
479    4    2    1    3
482    4    2    2    1
483    4    2    2    2
479    4    2    2    3
477    4    2    3    1
479    4    2    3    2
475    4    2    3    3
472    4    2    4    1
476    4    2    4    2
479    4    2    4    3
470    4    3    1    1
481    4    3    1    2
481    4    3    1    3
475    4    3    2    1
470    4    3    2    2
475    4    3    2    3
469    4    3    3    1
477    4    3    3    2
482    4    3    3    3
485    4    3    4    1
479    4    3    4    2
474    4    3    4    3
469    4    4    1    1
473    4    4    1    2
475    4    4    1    3
477    4    4    2    1
473    4    4    2    2
471    4    4    2    3
470    4    4    3    1
468    4    4    3    2
474    4    4    3    3
483    4    4    4    1
477    4    4    4    2
476    4    4    4    3
;
proc mixed data=fullnest covtest method=type3;
class Plant Operator Shift Batch;
model temp=;
random plant operator(plant) shift(plant operator) ;
run;

In the SAS code, notice that there are no terms on the right-hand side of the model statement. This is because SAS uses the model statement to specify fixed effects only. The random statement is used to specify the random effects. The proc mixed procedure will perform the fully nested random effects model as specified above, and produces the following output:

Type 3 Analysis of Variance
Source	DF	Sum of Squares	Mean Square	Expected Mean Square	Error Term	Error DF	F Value	Pr > F
Plant	3	731.515625	243.838542	Var(Residual) + 3 Var(Shift(Plant*Operato)) + 12 Var(Operator(Plant)) + 48 Var(Plant)	MS(Operator(Plant))	12	5.85	0.0106
Operator(Plant)	12	499.812500	41.651042	Var(Residual) + 3 Var(Shift(Plant*Operato)) + 12 Var(Operator(Plant))	MS(Shift(Plant*Operato))	48	1.30	0.2483
Shift(Plant*Operato)	48	1534.916667	31.977431	Var(Residual) + 3 Var(Shift(Plant*Operato))	MS(Residual)	128	2.58	<.0001
Residual	128	1588.000000	12.406250	Var(Residual)	.	.	.	.


Covariance Parameter Estimates
Cov Parm	Estimate	Standard Error	Z Value	Pr Z
Plant	4.2122	4.1629	1.01	0.3116
Operator(Plant)	0.8061	1.5178	0.53	0.5953
Shift(Plant*Operato)	6.5237	2.2364	2.92	0.0035
Residual	12.4063	1.5508	8.00	<.0001

The largest (and significant) variance components are: (1) the shift within a plant × operator combination and (2) the batch-to-batch variation within the shift (the residual).

Note that the Covariance Parameter Estimates here are in fact the variance components. SAS does not express the variance components as percentages in this procedure, but by summing the variance components for all sources to serve as the denominator, each source can be expressed as a percentage.

Because this type of model is so commonly employed, SAS also offers two other procedures to obtain the variance components results: proc varcomp (which stands for variance components) and proc nested.

The equivalent code for these procedures is as follows:

The `proc varcomp`:

proc varcomp data=fullnest;
class Plant Operator Shift Batch;
model temp= plant operator(plant) shift(plant operator);
run;

Note that the model statement for proc varcomp differs from the mixed procedure, in that proc varcomp assumes that the factors listed in the model statement are random effects.

Partial Output:


MIVQUE(0) Estimates
Variance Component	Temp
Var(Plant)	4.21224
Var(Operator(Plant))	0.80613
Var(Shift(Plant*Operato))	6.52373
Var(Error)	12.40625

Note that, even in this procedure we will have to use the sum for a total and calculate the percentages ourselves.

The `proc nested`

On the other hand, the proc nested procedure will provide the full output including the percentages:

proc nested data=fullnest;
class plant operator shift;
var temp;
run;

Partial Output:


Nested Random Effects Analysis of Variance for Variable Temp
Variance Source	DF	Sum of Squares	F Value	Pr > F	Error Term	Mean Square	Variance Component	Percent of Total
Total	191	4354.244792				22.797093	23.948351	100.0000
Plant	3	731.515625	5.85	0.0106	Operator	243.838542	4.212240	17.5889
Operator	12	499.812500	1.30	0.2483	Shift	41.651042	0.806134	3.3661
Shift	48	1534.916667	2.58	<.0001	Error	31.977431	6.523727	27.2408
Error	128	1588.000000				12.406250	12.406250	51.8042

Calculation of the Variance Components

From the SAS output, we get the EMS coefficients. We can use those to compute the estimated variance components.

Source	MS	EMS	Variance Components	% Variation
Plant	243.84	\(\sigma_{\epsilon}^{2} + 3 \sigma_{\gamma}^{2} + 12 \sigma_{\beta}^{2} + 48 \sigma_{\alpha}^{2}\)	4.21	17.58
Operator(Plant)	41.65	\(\sigma_{\epsilon}^{2} + 3 \sigma_{\gamma}^{2} + 12 \sigma_{\beta}^{2}\)	0.806	3.37
Shift(Plant × Operator)	31.98	\(\sigma_{\epsilon}^{2} + 3 \sigma_{\gamma}^{2}\)	6.52	27.24
Residual	12.41	\(\sigma_{\epsilon}^{2}\)	12.41	51.80
		Total	23.95

One can show that MS is an unbiased estimator for EMS (using the properties of Method of Moments estimates). With that, we can algebraically solve for each variance component. Start at the bottom of the table and work up the hierarchy.

First of all, the estimated variance component for the Residuals is given: \[\mathbf{12.41} = \hat{\sigma}_{\text{error}}^{2} = \hat{\sigma}_{\epsilon}^{2} \nonumber\]

Then we can use this information and subtract it from the Shift(Plant × Operator) MS to get: \[\begin{aligned} 31.98 &= \hat{\sigma}_{\epsilon}^{2} + 3 \hat{\sigma}_{\gamma \text{ or Shift(Plant} \times \text{Operator)}}^{2} \\[4pt] \hat{\sigma}_{\gamma}^{2} &= \frac{31.98 - 12.41}{3} = \mathbf{6.52} \end{aligned}\]

Similarly, we use what we know for Error and Shift(Plant × Operator) and subtract it from the Operator(Plant) MS to get:

\[\begin{aligned} 41.65 &= \hat{\sigma}_{\epsilon}^{2} + 3 \hat{\sigma}_{\gamma}^{2} + 12 \hat{\sigma}_{\beta \text{ or Operator(Plant)}}^{2} \\ &= 31.98 + 12 \hat{\sigma}_{\beta}^{2} \\[4pt] \sigma_{\beta}^{2} &= \frac{41.65 - 31.98}{12} \\ &= \mathbf{0.806} \end{aligned}\]

Our total = 12.41 + 6.52 + 0.806 + 4.21 = 23.95

Then, dividing each variance component by the total (in this case 23.95) gives the % values shown in the output from SAS proc nested.

Example - Fully Nested Random Effects Model

The proc varcomp:

Partial Output:

The proc nested

Partial Output:

Calculation of the Variance Components

The `proc varcomp`:

The `proc nested`