Analysis of Experiments with Random Effects
Example 1 – Variation in looms in textile manufacturing
How much variation in fabric strength is due to the fact different looms are used to
produce the fabric? To answer this question plant engineers randomly sampled four
looms (from many) at the plant and tested the fabric strength of n = 4 fabric samples from
each. The data entered into JMP is shown below.
The random effects model for these data is given by:
y ij     i   ij i  1,2,3,4 (looms) and j  1,2,3,4 (replicates)
where we assume,
 i ~ N (0,  2 ) and  ij ~ N (0,  2 )
This implies that the total variation in the fabric strengths is
2
Var( yij )   Total
  2   2
We want to estimate both variance components given the data, and look at what
percentage of the variation of the total variation can be attributed to the fact that different
looms are used produce fabric in the factory.
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Analysis in JMP
Select Analyze > Fit Model and put Strength in the Y box and Loom in the effects in
model box. The critical step is to highlight Loom in the model effects box and select
Random Effect from the Attributes pull-down menu as shown below.
We have to methods of estimation at our
disposal. The E(MS) approach which is used
extensively in the text, and the REML
(restricted maximum likelihood) method which
is discussed in section 13-7.3 of your text. For
now we will concern ourselves with the E(MS)
method.
Results of E(MS) Method for Estimating Variance Components
p-value = .0002  Variation in the response due
to looms is statistically significant.
E(MS) Table
2
E ( MS Loom )   2  4 Loom
E ( MS Error )   2
2
ˆ Loom
 6.96
ˆ  1.90
2
% variation due to looms = 78.6%
% variation due to error = 21.4%
MS Loom  MS Error 29.72  1.90

 6.96
n
4
This estimate comes directly from the E(MS)
above, thus the name of the estimation method.
2
ˆ Loom

If we use the REML method we basically the same estimate along with a 95% CI
2
for  Loom
.
The CI is too wide to be useful! To get precise estimates of variance components much
largersample sizes are needed.
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Confidence Interval for Percent of Variation Due to an Effect (pgs. 489-490)
100(1 - )% CI for
 2
is given by:
 2   2
2
L
U
 2  2 
1 L    1 U
where
L


1  MS Treatment
1
1  MS Treatment
1
 1 and U  
 1


n  MS E
F / 2,a 1, N  a
n  MS E
F1( / 2),a 1, N  a


Note: For unequal replicates replace n by the n o whose formula is given by equation
(13-9) on pg. 487.
Constructing a 95% CI for
 2
   2
for the loom data we first need the F-quantiles which
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can be calculated using either the file F-quantile Calculator.JMP on the class server or
by using tables in the F-table in the appendix of your text. To find the upper F-quantile
using the table you have to use the fact that
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(see pg. 490 for an example)
F1( / 2),a 1, N a 
F / 2, N a ,a 1
Thus F / 2, a 1, N  a  F.025,3,12  4.4742 and F1( / 2), a 1, N  a  F.975,3,12  .06975
Thus we have,
L
1  29.73 1
1  29.73 1


 1  .625 and U  
 1  55.633

4  1.90 4.47 
4  1.90 .06975 
which gives,
2
2
.625
55.633
and finally .38  2  2  .98 .
 2  2 
1  .625     1  55.633
  
So looms account for between 38% and 98% of the total variation in fabric strength.
Again this interval is very wide because of the small number of replicates, however we
certainly know that loom to loom variation is not negligible.
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Factorial Experiments with Random Effects
Example 1: Gage R & R Study
Schematic
2
 part
2
2
 repeat
  error
or  2
2
 operator
2
 operator
* part
We set up the spreadsheet in JMP as
shown to the left. We need one column to
denote operator (1,2,or,3), one column for
part being measured (1 – 20), and the
reading or measurement made. The trial
column will not be used in the actual
analysis. Each operator measured each
part twice and trial denotes which
measurement it is 1st or 2nd.
To conduct the analysis in JMP set up the effects as you would for a two-factor factorial
design making sure to change each effect to random as shown below.
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Output from E(MS) approach to estimating the variance components.
2
Notice that the estimate for  operator
* part is
negative! This is impossible because
variances by definition are positive!
However, we see that the interaction between
operator and part is not significant (p =
.8614). The E(MS) approach for estimating
variance components will often times lead to
negative estimates for NON-SIGNIFICANT
effects. The best thing to do here is drop the
non-significant terms and re-run the model or
use the REML approach which will not give
negative variance component estimates,
because it will essentially drop the effect
yielding negative variance component
estimates from the model.
E(MS) method estimates after dropping the operator*part interaction.
2
REML estimates of variance components, notice  operator
* part is zeroed out.
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The Gage R & R study here produced results that indicate almost all of the variation
came from part to part variation,  92% . The fact different operators were used
accounted for less the one-tenth of a percent of the total variation and the repeatability
variance component was approximately 8%. This measurement system seems good, now
we can focus on eliminating part to part variability by using designed experiments to
identify potential sources of that variation.
Example 2 – Turbine Experiment (Example 13-7 pg. 507)
This is a three factor experiment with factor A (Gas temperature) is fixed and factors B
(operator) & C (pressure gauge) are random.
Fitting the model we set up as we would for a three-factor factorial experiment and
change all effects involving B and/or C to random effects as shown below.
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The E(MS) differ from those derived on the board because JMP uses the unrestricted
approach when handling random interaction effects. We can still use the restricted
approach, however we have to do the testing and variance component estimation “by
hand” using the Mean Squares returned by JMP.
Which approach is best to use? If we are willing to carefully write out the E(MS) by
hand it does not take to much extra effort analyze using the restricted model with JMP. If
you are not willing to painstakingly write out the E(MS) for your mixed model then I
would use the unrestricted approach and go with all of the results returned by JMP.
Many statistical software packages offer the use of either the restricted or unrestricted
approach, e.g. MINITAB.
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