A Step-Down Lenth Method for Analyzing
Unreplicated Factorial Designs
Kenny Q. Ye, Michael Hamada, C. F. J. Wu
Journal of Quality Technology, Apr 2001, Vol. 33, pg. 140, 13 pgs
報告者：陳怡綾
Outline
1.
2.
3.
4.
5.
6.
Introduction
A Step-Down Lenth Method
An example
A Synthetic Example
A Simulation Study
Conclusions
1.Introduction
A commonly used method to identify active effects from such experiments is the
half-normal plot.
To overcome the subjectivity of using the half-normal, many formal testing
methods have been proposed. In this paper, we propose a step-down version of
the Lenth method called “step-down Lenth method”.
It is compared via simulation with the original Lenth method and with stepwise
methods proposed by Venter and Steel (1998). It is shown that the step-down
Lenth method is better than the original Lenth method and Venter and Steel stepdown method.
IER: the proportion of inactive individual effects declared active
EER(experimentwise error rate): the error rate of at least one inactive
effect being declared active.
2. A Step-Down Lenth Method
Lenth Method
P254
Let ˆ1 ,ˆ2 , ,ˆI denoted I mutually orthogonal estimated factorial effects (i.e., the contrasts
of a factorial design ). Assuming that there are only a few active effects, Lenth (1989) uses a
pseudo standard error ( PSE ) to estimate the standard deviation of ˆ :
i
PSE  1.5  medianˆ 2.5 s ˆi
i
0
where s0  1.5  median ˆi .
Lenth (1989) then calculates statistics by dividing the ˆi by PSE, which we will refer to as
ˆ
Lenth statistics : t Lenth,i   i .
PSE
He suggests using a t  distributi on with I/3 degrees of freedom as an
approximat ion to the t
reference distributi on.
Lenth , i
The EER critical value at significan ce level  with I contrasts the (1   ) 100% of the
ˆi
distributi on under the null hypothesis H 0 : 1   2     I  0.
PSE
Once the effect correspond ing to the largest absolute contrast is declared active, say  I ,
max
it is natural to test the largest absolute contrast of the remaining I  1 contrasts.
The correspndi ng critical values, however, should be calculated under the null hypothesis
H 0 : 1   2     I 1  0.
This method can be repeated until the effect correspond ing to the largest absolute
contrast of the remaining contrast is not declared active.
Step-Down Lenth Method
We propose a step  down version of the Lenth mothod for controllin g EER .
Let ˆ
(1)
 ˆ
( 2)
   ˆ
the test statistics ti 
ˆ
(I )
(i )
PSEi
be the order statistics of I absolute contrasts. Obtain
, where PSEi is the pseudo standard error of ˆ(1) ,ˆ( 2) , ,ˆ(i ) ,
the signed contrasts correspond ing to the absolute contrasts ˆ , ˆ
(1)
( 2)
, , ˆ .
(i )
Let Ci denote the EER critical value at significan ce level  of the original
Lenth method with i contrasts. If ti  Ci for all i  I  k , then the largest k
factorial effect are declared active.
Values of Ci for i between 4 and 35 are obtained by simulation and listed in Table 1.
3.A Example
P246
Montgomery(1991) presents a unreplicated
experiment using a 2 4 design to study the
filtration rate of a pressure vessel.
The four factors are temperature (A), pressure
(B), concentration of formaldehyde ,
(C) ,and the stirring rate (D)
 C15
0.1  3.51
 C14
0.1  3.56
13
 C0.1  3.53
 C12
0.1  3.60
11
 C0.1  3.56
 C10
0.1  3.63
 C15
0.1  3.51
 C14
0.1  3.56
13
 C0.1  3.53
 C12
0.1  3.60
11
 C0.1  3.56
 C10
0.1  3.63
In this example, the original Lenth statistics and step-down Lenth
statistics happened to be the same for all the effects calculated,
because the PSE remained the same in each step.
4.A Synthetic Example
The example was generated to illustrate the power of the step - down
Lenth method.
The fifteen contrasts used in the example are randomly generated from fifteen
independen t normal distributi ons N ( i ,1),  i , i  1,2  ,15, where 1   2     8  0
are inactive effects and  9  1, 10  1.5, 11  2, 12  3, 13  3.5, 14  4, and  5  5
are active effects.
Since there is no large difference
between the absolute
contrasts,using the half-normal
plot might not detect any effects.
Table 5 shows the true effects, the
chosen set of contrasts and the
correspond ing Lenth statistics .
 C15
0.1  3.51
 C14
0.1  3.56
 C15
0.1  3.51
 C14
0.1  3.56
 C12
0.1  3.60
 C13
0.1  3.53
 C11
0.1  3.56
 C10
0.1  3.63
This example shows that when the magnitudes of the active effects vary from small to
large, both the Lenth method and half-normal plots may fail to detect them.
5.A Simulation Study
We performed a simulation study to compare four methods, which are the stepdown Lenth method, the original Lenth method,the fix RMS scaling step-down
method, and the sequential RME scaling step-up method. The last two
methods were proposed and studied by Venter and Steel(1998).
The test statistics used by Venter and Steel are similar to our step-down Lenth
i
statistics, expect that the root mean square(RMS) ((1 / i ) (2k ) ) 2 replaces PSE.
k 1
The fix RMS scaling step-down method uses the same set of small absolute
contrasts to calculate the RMS at each step.The the sequential RME scaling
step-up method uses a different set of contrasts to calculate the RMS at each
step.
Our the simulation study considers the case of 15 contrasts under the
following six configurations:
C1 : 1    14  0,15  #
C2 : 1    12  0,13  14  15  #
C3 : 1    10  0,11    15  #
C4 : 1     8  0, 9    15  #
C5 : 1    12  0,13  # ,14  2# ,15  3#
C6 : 1    10  0,11  # ,12  2# ,13  3# ,14  4# ,15  5#
Here , #  0 is referred to as the " spacing" . Three figures of merit are calculated to
compare the four methods :
1.EER, the fraction of experiment s in which at least one inactive effect is declared
active.
2.Power, the expected fraction of active effects that are declared active.
For the RMS scaling stepwise methods, we first used l  7 as the lower bound on the
number of inactive effect.
Figure 2 shows the EER of the four methods.
It can be seen that the EER of the three stepwise methods is closer to
the nominal level 0.1 than the original Lenth method.
Figure 3 shows the power of the different methods.
Except for C4, the power of the four methods is very similar,
with the step - up methods being slightly better.
In every case, the step  down Lenth method outperforms the original Lenth method.
6.Conclusions
In this paper, we propose a step-down version of the Lenth method for
identifying active effects in unreplicated experiments. It controls the EER
closer to the nominal significance level than the original Lenth method
does.
In particular, when there are a moderate number of active effects whose
values range from small to large, the original Lenth method and halfnormal plot tend to miss them.
The performances of the RMS stepwise methods and step-down Lenth
method are comparable in most cases.
The End