Quality Technology &
Quantitative Management
Vol. 12, No. 1, pp. 13-28, 2015
QTQM
© ICAQM 2015
Experiments for Multi-Stage Processes
John Tyssedal1,* and Murat Kulahci2,3
1
Department of Mathematical Sciences, The Norwegian University of Science and Technology, Trondheim, Norway
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
3
Department of Business Administration, Technology and Social Sciences,
Luleå University of Technology, Luleå, Sweden
2
(Received October 2013, accepted February 2014)
______________________________________________________________________
Abstract: Multi-stage processes are very common in both process and manufacturing industries. In this
article we present a methodology for designing experiments for multi-stage processes. Typically in these
situations the design is expected to involve many factors from different stages. To minimize the required
number of experimental runs, we suggest using mirror image pairs of experiments at each stage following
the first. As the design criterion, we consider their projectivity and mainly focus on projectivity P  3
designs. We provide the methodology for generating these designs for processes with any number of
stages and also show how to identify and estimate the effects. Both regular and non-regular designs are
considered as base designs in generating the overall design.
Keywords: Alias structure, nonregular designs, projective properties, regular designs, restriction on
randomization, two-level designs.
______________________________________________________________________
1. Introduction
I
n most industrial processes, the raw materials go through a series of stages before the
final product is obtained. It is also often the case that the quality characteristics of the
final product are affected by several factors at each stage. These relationships can be
effectively determined via carefully designed experiments. The common practice in this
case is to design separate sets of experiments for each stage. The main reason for this is that
a more holistic approach in which the entire process is considered at once will most likely
involve many factors which will further imply an excessive increase in the total number of
experiments required. However the major drawback of running separate experiments
independently for each stage is the fact that this strategy will fail to capture the often
important interaction effects between factors from different stages. We will in this article
provide a methodology for designing experiments for the entire process that will enable the
practitioners to properly explore these relationships with the minimum number of runs.
In the execution of the experiments for multi-stage processes, another important issue
to consider is the randomization of the runs. In fact, a complete randomization will often
lead to overwhelmingly large costs. Thus certain restrictions should be enforced on
randomization. As depicted in Figure 1, a solution is to run the experiments in a splitting
fashion. That is, the output from the first experiment in a stage is split into smaller pieces
and used in the subsequent stage. This is in execution quite similar to the designs used in
split-plot experimentation [7]. In fact a typical split-plot experiment can be considered as an
experiment for a two-stage process. Similarly a split-split-plot design can be considered as a
design for a three-stage process.
*
Corresponding author. E-mail: [email protected]
14
Tyssedal and Kulahci
Figure 1. Experiments for a Multi-stage Process.
The earlier studies in split-plot experimentation can be found in agriculture [29] and in
manufacturing [1, 20]. Some recent work can be found in [2, 3, 4, 5, 6, 13]. These studies
were mainly concerned with two-level fractional factorial split plot designs. On the other
hand in [15] and [16] the general methodology for the use of Plackett and Burman (PB)
designs [23] in split-plot experiments is provided. Further work [28] provide the
methodology for designing split-plot designs of high projectivity based on regular and
nonregular designs using only two mirror image pairs at the sub-plot level for each level
combination of the whole-plot factors. This allows for keeping the total number of
experiments as small as possible while achieving desirable projectivity of 3 or higher.
Split-plot designs and their extensions for processes with more than two-stages are
characterized by having a nested unit structure. An example of a two-level screening design
with such a structure, a four stage split-split-split-plot design for identifying factors causing
rancidity of stored meat loaf, is given in [11]. D-optimal designs of split-split-plot
experiments are considered in [14]. The purpose of this paper is to extend the ideas
presented in [28] to two-level experiments for multi-stage processes using mirror image
pairs at each stage except the first. We obtain similar appealing properties. As for split-plot
designs with mirror image pairs as sub-plots (SPMIP designs) the identification of and
inference about active effects can be broken down to as many steps as there are stages and
can in most cases be performed by ordinary least square which saves computational time
and effort. In addition their projective properties can be quite attractive compared to their
run sizes making them very appealing for screening purposes. To our knowledge this way
of constructing experiments for multi-stage processes and the discussion on the possible
obtainable simplifications in the analysis have never been addressed in the literature before.
Similarly we believe that multi-stage experiments constructed from nonregular designs have
never been proposed previously.
Designs with a crossed unit structure have also been studied and used for multistage
processes. Examples are strip-strip-block and split–lot designs. For construction and
analysis of these we refer to [6, 10, 21, 24-25].
This paper is organized as follows: In Section 2 we first explain the general structure of
multi-stage split plot designs with mirror image pairs. Thereafter in Section 3 a short
introduction to the concept of projectivity and its general usefulness is given before we in
Section 4 and Section 5 show how multi-stage split-plot designs of projectivity P  3 can
be constructed from both regular and nonregular designs. In Section 6 we then proceed to
show that for these designs the generalized least squares estimates equals the ordinary least
squares estimates, develop the structure of aliasing between main effects and two-factor
interactions and show how the identification of active factors and the estimation of the
corresponding effects can be done in k steps using ordinary least squares (OLS) in each
step. Concluding remarks are given in Section 7.
Experiments for Multi-Stage Processes
15
2. Multi-stage Split-plot Designs with Mirror Image Pairs, MSPMIP DESIGNS
The major concern in a holistic approach to designing experiments for multi-stage
processes is the number of experiments that are needed to accommodate the increasing
number of factors as the number of stages increases. In fact the smallest set of experiments
will be achieved if only two experiments are run at each stage for each experiment from the
previous stage. A possibility is to run these two experiments as mirror image pairs. We will
call the new class of designs obtained by this construction multi-stage split-plot designs with
mirror image pairs, abbreviated as MSPMIP designs. Figure 2 shows an illustration of a
three-stage MSPMIP design. Now let
1 1 
B
.
1 1
Then using the standard convention letting -1 denote the low level of a factor and 1 the
high level, a MSPMIP design for two stages can be obtained from
B B 
 B B .


(1)
It should be noted that this design can accommodate 1 factor at the first stage and 2 factors
at the second stage.
Stage 1


Stage 2
















Stage 3
























Figure 2. A three-stage MSPMIP design with 1, 2 and 4 factors in stages
one, two and three respectively. Only the signs are used for the levels.
For a three-stage process we can have
B B  B B  

 
 
 B B  B B  .
B B 
B B  




  B  B 
 B  B  
(2)
In this design, we can allocate 1, 2 and 4 factors in stages one, two and three respectively.
If we now let D be any design matrix for a two-level design expanded with a column
of +1’s and use it as the ”basis” design, matrix representations for two-, three- and
four-stage processes constructed from D by successive doubling, can be found in Figure 3.
It can then be shown that for a given D that can accommodate up to p factors, the
design in Figure 3 can accommodate up to p factors at the first stage, p  1 factors at
second stage, 2( p  1) factors at stage 3 and 4( p  1) factors at stage 4. In general designs
constructed in this manner can accommodate up to 2 k 2  p  1 factors at stage k where
k2.
16
Tyssedal and Kulahci
To further generalize the above procedure to any MSPMIP design and allow for
different factorial designs to be used at each stage, we can write any k -stage design as a
two-stage design in the following way.
(a) two-stage
(b) three-stage
(c) four-stage
Figure 3. The matrix representations for the designs for multi-stage processes.
 D*
D*k   *k 1
 Dk 1
Dk 
.
 Dk 
(3)
Here D*k 1 consists of the design columns for a k  1 stage process and  Dk Dk 
consists of the design columns at stage k . It should be noted that Dk can be any factorial
design. The same decomposition can also be performed for the previous stages except the
first. Provided that the smallest possible design is employed at the first stage, MSPMIP
designs offer designs with the least amount of total number of experiments possible. The
drawback of this is the fact that if many factors are to be considered in each stage, the alias
structure may become quite cumbersome. For the rest of this paper we will only consider
MSPMIP designs constructed from orthogonal two-level designs. The aliasing between
main effects and two-factor interactions can then be simplified using designs of projectivity
P  3 or higher. In the next section we will define projectivity and explain the usefulness
of exploring projective properties of MSPMIP designs.
3. Projective Properties and MSPMIP Designs
According to the definition of projectivity for two-level designs given in [8], a N  k
design with N runs and k factors each at two levels is said to be of projectivity P and
is called a ( N , k, P ) screen if every subset of P factors out of possible k contains a
complete 2 P factorial design, possibly with some points replicated.
The projective properties of a design are considered most important for screening
experiments where only a few out of many factors considered are assumed to be active. For
two-level fractional factorial designs, i.e. 1 2 p , p  0,2,, k  1 fractions of 2 k factorials
also called regular designs, there is a close relationship between the concepts of resolution
and projectivity. In fact for two-level fractional factorial designs with p  1 , P  R  1
where R is the resolution.
Another class of two-level designs is the nonregular designs i.e. those orthogonal
two-level designs that do not belong to the 2 k  p family. These designs apparently exist for
every N a multiple of four, N  12 , and thereby provide a lot more design alternatives than
the 2 k  p fractional factorials. For many of these nonregular designs the projectivity P is
greater or equal to 3 while their resolution is also 3. Therefore design properties of
nonregular designs are not well described in terms of resolution while the concept of
Experiments for Multi-Stage Processes
17
projectivity is useful for both regular and nonregular designs. Besides its more general
applicability, projectivity P ensures that all main effects and all interactions of any P
factors can be estimated with no bias if the other factors are inert and projectivity P  3
implies that it is possible to de-alias main effects from two-factor interactions.
4. Construction of MSPMIP Designs from Regular Designs
Consider the four stage design in Figure 3 and let i  1 1 and 1   1 1 . It may
then be given the following representation.
i
i
i
i
i
i
i
i
1
2 12
1
  i 1
1
1
1
1
1
1
1
i
 i
i
 i
i
 i
i
1
1
1
1
1
1
1
3
13
i
1
i
i
1
i
1
i
i

1
i
1 
1
i
1
i
1
i
1
i
1
i
1
i
1
i
1
1
i



i
23 123
1
1
4
14
24 124
34 134
234 1234
i
i

i

i
1
i
1
i
1
i
1
i
1
i
1
i
1 
1
i
1
i
1
i
1 
1
i
1
i
1
i
1 .
i
i

i

i
1
i
1
i
1
i
1
1
i
1
i
1
i
 1
1
i
1
i
1
i
 1
1
i
1
i
1
i
1
1


(4)


In (4), the column headings represent the main effects and interactions in a 2 4 design.
Following the arguments from the previous section, we can see that this design can
accommodate up to one, two, four and eight factors in stages 1, 2, 3 and 4 respectively. We
also notice that the representation in (4) can be used to construct a design in three stages
with three factors at stage 1 (allocated in columns labeled 1, 2 and 12), four factors at stage
2 (in columns 3, 13, 23 and 123) and eight factors at stage 3 (in columns 4, 14, 24, 124, 34,
134, 234 and 1234) as well as a two-stage design with seven factors at stage 1 (1,2,12, 3, 13,
23, 123). In general the possible splitting of a saturated regular two level designs into k
stages can easily be accomplished by in each splitting considering it as a two-stage design as
illustrated in (3). Since the decomposition given in (3) also applies for the previous stages
we can write
 D*
Dk-i 
i  0,1,, k - 2,
(5)
D*k-i   *k-i -1
,
 D k-i -1 Dk-i 
where D*k-i -1 consists of the design columns for all stages up to k - i  1 and
 Dk i Dk i  consists of the design columns at stage k  i . The maximum possible
numbers of factors on stage k is then N 2 where N is the total number of runs.
Similarly the maximum number of factor columns at stage k  1 can be found by treating
D*k 1 as a two-stage design in N 2 runs and so on. The alias structure for these designs is
identical to the alias structure of the corresponding fractional factorial.
In the following we will concentrate on designs of projectivity P  3 . The procedure
given above also applies to these designs. In [28] it is shown that the maximum number of
factors in each stage for a two stage P  3 design is N 4 . To generalize the notation
introduced in [28], we will define the designs for a k-stage process using the notation
 N , n1 , n2 ,, nk , P  where N is the total number of experiments, ni is the number of
factors in stage i and P is the projectivity. It is then for instance possible to construct a
(16, 4, 4, 3) screen for a two-stage process. If we want to construct a design for a three-stage
process of projectivity P  3 , we can use the (8, 2, 2, 3) screen as D*2 in (3) and construct
18
Tyssedal and Kulahci
a (16, 2, 2, 4, 3) screen. Similarly, a P  3 design for a four-stage process is then possible
with four factors at stage 4 and two factors at stage 3. We will then be left with two factors
for stages 1 and 2 yielding a (16, 1, 1, 2, 4, 3) screen for a four-stage process.
In general for a three-stage process with total number of experiments N, a
N , N 23 , N 23 , N 22 ,3 screen can be obtained using this construction technique. In this
design up to N 23 factors can be allocated in each of the first two stages and up to N 2 2
in the last stage to ensure a projectivity P  3 design. To further generalize this to k-stage
processes, we can have N , N 2 k , N 2 k , N 2 k 1 ,, N 22 , 3 screens following the same
design construction methodology. In Tables 1 and 2, we summarize design possibilities and
factor allocations for 16, 32 and 64 runs experiments for three-stage and four-stage
processes. These tables correspond to the maximum allowable number of factors at each
stage in order to have a design of projectivity P  3 .




Table 1. Designs of Projectivity P  3 for a three-stage process with the maximum
number of columns. Factor allocation is done according to the column labels of the
corresponding full factorial design augmented with interaction columns.
Total
Number
of Runs
8
16
32
64
Design
(8,1, 1, 2, 3)
(16, 2, 2, 4, 3)
(32, 4, 4, 8, 3)
(64, 8, 8, 16, 3)
Factor Allocation
Stage 1
Stage 2
Stage 3
1
1,2
1,2,3,123
1,2,3,123,4,
124,134,234
2
3,123
4,124,134,234
5,125,135,145,
235,245,345,12345
3,123
4,124,134,234
5,125,135,145,235, 245, 345,12345
6,126,136,146,156,236,246,256,346,356
,456,12346,12356,12456,13456,23456
Table 2. Designs of Projectivity P  3 for a four-stage process with the maximum number
of columns. Factor allocation is done according to the column labels of the corresponding
full factorial design augmented with interaction columns.
Total
Number
of Runs
16
Design
Factor Allocation
Stage 1
Stage 2
Stage 3
(16,1,1,2,4,3)
1
2
3,123
32
(32,2,2,4,8,3)
1,2
3,123
4,124,134,234
64
(64,4,4,8,16,3)
1,2,3,
123
4,124,
134,234
5,125,135,145,235,
245,345,12345
Stage 4
4,124,134,234
5,125,135,145,235,
245,345,12345
6,126,136,146,156,236,246,
256,346,356,456,12346,
12356,12456,13456,23456
MSPMIP designs with projectivity P  3 can also be obtained following this
construction methodology starting with a two-stage P  3 design. A list of such designs is
given in [28]. For instance there exists a (16, 3, 2, 4) design and thereby one can construct
(16, 2, 1, 2, 4) or (16, 1, 2, 2, 4) three-stages designs or a (16, 1, 1, 1, 2, 4) four-stage design.
If at any stage less number of factors is to be considered, the designs provided in
Tables 1 and 2 can still be used by only allocating the necessary amount of factors in any of
the columns suggested in these tables. However other choices of designs may lead to
designs with less aliasing or a more favourable alias pattern for the given experimental
situation. In Appendix A, tables of projectivity P =3 designs are provided for N  16 and
32 (Table A1) and also for projectivity P  4 designs for N  16 , 32 and 64 (Table A2).
These are constructed from tables for two-stage designs provided in [28] following the
procedure presented above.
However as the number of stages increases, the total number of experiments needed
gets large. Furthermore there will be more restrictions on the number of factors that can be
allocated in each stage to ensure the desired projectivity. Therefore MSPMIP designs based
Experiments for Multi-Stage Processes
19
on nonregular designs offer more appealing alternatives.
5. Construction of MSPMIP Designs from Nonregular Designs
The best known nonregular designs are the Plackett and Burman (PB) designs with
the number of runs N  2 k . Classification of PB designs with respect to projectivity
P  2 , P  3 and P  4 can be found in [8] and [26]. Most nonregular PB designs are in
fact  N , N  1,3  screens from which  2 N , N  1, N  1,3  screens can be constructed for
two-stage processes using the technique of doubling described below. The exceptions are for
the number of runs N  40, 56, 88 or 96 where only N  2 of the main effects columns
can be used. From these designs,  2 N , N  2, N  2,3  screens can be constructed for
two-stage processes. The  24, 11, 11, 3  screen obtained by doubling the 12-run PB
design (PB12) is given below
PB12 
 PB12
(6)
 PB12  PB12  .


This design is of projectivity 3 and can accommodate up to 11 factors in the first stage
allocated to the columns 1 to 11, and up to 11 factors in the second stage that can be
allocated to the last 11 columns.
As in the case of regular designs, MSPMIP designs can be generated for processes
with more than two stages. For a three-stage process for example, the following
 48,11,11,22,3  screen can be obtained as
 PB12

 PB12
 PB12

 PB12
PB12
 PB12
PB12
 PB12
 PB12
 PB12

 PB12

 PB12
PB12  

 PB12  
.
PB12  

 PB12  
(7)
In general a MSPMIP design for a k -stage process can be obtained from a
N ,(( N 2 k 1 )  1),(( N 2 k 1 )  1),(( N 2 k 1 )  1)21 ,,(( N 2 k 1 )  1)2 k 2 ,3
screen
con-structed by successively doubling a nonregular two-level design of projectivity P  3 . As it
can be seen from this general representation, nonregular designs offer better alternatives
compared to regular designs in terms of the number of factors allowed in each stage while
still ensuring a projectivity P  3 design.


As mentioned in [28] for two-stage cases, the construction methodology given in (6)
may lead to designs where some two-factor interactions are fully aliased with other
two-factor interactions which may complicate the identification of active effects. This can
be avoided using distinct columns of the same nonregular design at different stages as for
example the following design for a three-stage process.
Stage 1
i

i
i

 i
S1
S1
 
S1
 
S1
Stage 2
Stage 3
i   S3 i  
 S2
 S2 i   S3 i   ,

 

 S2


i
S3 i  

 

 S2 i   S3 i  
(8)
where S1 S2 S3  PB12 . It should be noted that the columns  i i i i  and
i i i i  are added for the stages two and three respectively. This implies that the
20
Tyssedal and Kulahci
total number of factors that can be allocated in this design is 13. We represent this design as
a (48, n1 , n2 , n3 ,3) n1  n2  n3 13 screen. In general, it can be shown that we have
( N , n1 , n2 , n3 ,3) n1  n2  n3 n 32 screens where a n run nonregular P  3 design is used as the
base design and N  4  n . In the same way we can generalize this representation to
k -stage processes by constructing (2 k 1  n, n1 , n2 ,, nk ,3) n1  n2  nk n  k 2 screens.
6. Inference for MSPMIP Designs
Let the columns for the main effects and interactions be given in X and let Σ be the
covariance matrix for the responses. The generalized least squares (GLS) estimator and the
OLS estimator for the coefficients are given by β *  ( XΣ 1 X) 1 XΣ 1Y and
βˆ  ( XX) -1 XY respectively. For SPMIP designs for two-stage processes as given in [28] it
follows from the results given in [19] that these two estimators are equal, see also [27]. It is
also explained in [27] how inference about and identification of active effects can be done
in two separate steps when SPMIP designs are used. These results also generalize to the
MSPMIP designs.
Result: Let X contain the columns for the main effects and interactions to be estimated for
a three stage MSPMIP design. Furthermore assume that the error terms on different stages
are uncorrelated and that the error terms in stage i , i  1,2,, k are identically and
independently distributed with expectation 0 and variance  i2 . Then the generalized least
squares estimates of the coefficients are identical to the ordinary least squares estimates.
Proof. The covariance matrix for the response in a three-stage process can be written as the
N  N matrix
C 0 0 0 
0 C 0 0
.
V
0 0  0


 0 0 0 C
where
 12   22   32

 12   22
 12
 12


2
2
2
2
2
2
2
1   2   3
1
1
 1   2

C
.
2
2
2
2
2
2
2
1
1
1   2   3
1   2 


 12
 12
 12   22
 12   22   32 

C-1 0
0
0 


-1
0
0 
 0 C
-1
.
We also have V  

 O  


 0
0
0 C-1 
Let  2   12   22 and 1  1 1 . Then C can be written as:
 Σ
 2   32
2 
 12 11
where

Σ
C 22


,
2
2
 2   32 
Σ 2 
 1 11
 
has the form of the C matrix for the corresponding two-stage process. From [18] p. 459, it
can be shown that
Experiments for Multi-Stage Processes
21
 Σ21  k1 11
k2 11 
C 
,
Σ21  k1 11
 k2 11
1
where k1 and k 2 are two constants. In [19] it is shown that for the least squares
estimates to be equal to the generalized least squares estimates, Xy  0 should imply
XV 1y  0 , where X is the design matrix. Following the notation in [17] any column in
X may be written as x1  x 2  x 3 where  is the Kronecker product. It should be
noted that the elements in x2  x3 are either equal to 1 or the elements in at least one of
x 2 and x 3 sum to zero.
Case 1. The elements in x2  x3 are all equal to 1. Thereby x2  x3 = 114 and
 x1  x2  x3  V 1y  x1  1C1 y . The elements in C-1 will column wise sum to the
same constant, say a (see also [17]). Therefore


 x  1C  y  a  x  1 y  a  x  x  x  y  0
1
1
1
1
2
if
3
 x1  x2  x3  y  0 .
Case 2. The elements in at least one of x 2 and x 3 sum to zero. Then also the elements
in x2  x3 sum to zero. If the elements in x 3 sum to zero, the two first elements in
x2  x3 as well as the two last ones also sum to zero. Let the two first elements in x2  x3
be x . For the two stage process we know [17] that if x is a 2  1 vector with elements
that sum to zero, x  21 y  0 if Xy  0 and obviously k1 x11  k2 x11  0 and the
result follows. If the elements in x3 are equal to 1 then x2  x3 =  x3  x3  or
  x3  x3  . In both cases  x2  x3  C1  a1  x2  x3  where a1 is a constant and
 x1  x2  x3  V 1y  x1   x2  x3  C1 y  a1  x1  x2  x3  y  0 if  x1  x2  x3  y
0.


The general proof for a k -stage process can then be fulfilled by induction in exactly
the same way assuming the result is true for the  k  1 -stage process and from that derive
that it also must be true for a k -stage process.
In MSPMIP designs, one concern is about the potentially excessive confounding
among the effects. To assess the degree of aliasing in these designs we will use the alias
matrix originally introduced in [9]. Suppose we fit the regression model E( Y )  X1β1 but
the true expectation is given by E( Y)  X1β1  X2 β2 . The expected value of the OLS
estimator βˆ 1  ( X1 X1 ) 1 X1 Y is then E βˆ 1  βˆ 1  ( X1 X1 ) 1 X1 X2 βˆ 2 . The matrix
A  ( X1 X1 ) 1 X1 X 2 is the alias matrix showing to what extent an assumed model
potentially will be biased by additional effects in X2 . The equivalence of the GLS and
OLS estimators makes the expression for the OLS alias matrix also valid for the effects
from MSPMIP designs.
 
In order to derive an expression for A in a k -stage process, we again write the
design matrix as in (3). The matrix of two-factor interactions can then be written as:


 D*k 1  D*k 1
X2  
 D*  D*
k 1
 k 1


 D
  D
*
k 1
 D
 D  D
 Dk
*
k 1
k


 Dk 
,

k  Dk 
k
(9)
where D*k 1  D*k 1 represents the matrix with the two-factor interaction columns
between the factors in D*k 1 , D*k 1  Dk represents the matrix with the two-factor
interaction columns between the factors in D*k 1 and Dk , and Dk  Dk represents the
matrix with the two-factor interaction columns between the factors in Dk .




22
Tyssedal and Kulahci
We will now assume that interactions of order higher than two can be neglected and
restrict ourselves to the case where the matrix of main effects columns, X 1 , is orthogonal.
Then  X1 X1   NI and following the procedure in [28], we have after some rearranging


   
2  D k-1 Dk 1  Dk 1
A=
N
0



.


Dk  Dk 1  D k 

Dk 1  D k  Dk 
0

0
To elaborate further we use (5) and write
 D*k 2 Dk -1 
*
Dk 1   *
,
 Dk 2  Dk -1 
which gives for A 
 



0
  2 D k 2 Dk 2  Dk 2 

  

0
  2 Dk 2  D k 1  Dk 1  

 


  D 

   Dk -2 
k
-1


 Dk  Dk  
      Dk  Dk  
  - D k -1 

   Dk -2 




2 
A = 
N
2 D  D  D 
0

k -2
k -1
 k 1




0
0





0
0






(10)

(11)




0




0



.
0






  Dk -2 
 
 Dk  
  Dk   
  D 


  k -2 
 



  Dk -1 
 
 Dk  
  Dk   
 - D 
 

  k -1 
 
0
(12)
To investigate the alias structure given in (12), let  i , j  represent a two factor
interaction between a factor from stage i and a factor from stage j . It can then be seen
that a main effect from stage k is possibly aliased with two-factor interactions of the type
i , k  where i  k  1, k  2,,1 . In fact it can be shown that except for the first stage, a
main effect from stage i is possibly aliased with two-factor interactions of the type  i , j 
where j  i  1, i  2,,1 and with two-factor interactions of the type  l , l  where
l  i  1, i  2,, k . The main effects from the first stage are possibly aliased with two-factor
interactions of type  i , i  where i  1,, k  1, k .
For example the alias matrix for a three-stage process is given by

2 D1 D D
1 1


2 

A
0
N 


0




2 D1  D2 D2 
0
0
 D1 
 D   D3 D3 
 1
0
 D2 
  D   D3 D3  2 D2  D1D2 
 2
0
0
0
0
 D 

D3   1 D3 
  D1 






 . (13)
0


 D 

D3   2 D3  
   D2 


0
Experiments for Multi-Stage Processes
23
Equation (13) reveals that there is a possible aliasing between main effects from stage
one and two factor interactions of the type (1,1), (2,2) and (3,3), between main effects from
stage two and two factor interactions of the type (1,2) and (3,3) and between main effects
from stage three and two factor interactions of the type (1,3) and (2,3).
However in most of the cases discussed here, the MSPMIP design is constructed from
a base design D and the factor columns on a given stage can be split up further. For instance

in Figure 3, the design factors on stage 2 are  D D D D and the design factors on
stage 3 are
 D D  D  D 
 D D D D  .


A closer inspection of the multi-stage processes in Figure 3 also reveals that the sign
pattern for the base design D in a k-stage design is the same as for an ordinary 2 k -1
design. As a result, a specific interaction between two factors on the same stage can only be
aliased with main effects on exactly one of the stages and two-factor interactions aliased
with main effects on different stages are orthogonal. A further simplification is achieved if
the columns on each stage are constructed from different main effects columns of the same
design as in (8). Then the main effects on a specific stage after the first can only be aliased
with two-factor interactions between factors on that stage and the first stage.
For the cases given above, define X *   D1* , D*2 ,, D*k  where Di* contains the
columns for the main effects at stage i and their possibly aliased two-factor interactions to
be estimated as explained after equation (12). From the discussion above it follows that
Di*D*j  0, i, j =1,2,,k, i  j . The covariance matrix for the GLS estimator of the
coefficients, β̂ , can be derived the same way as for SPMIP designs in [27]. With Y the
vector of observations, the GLS estimator is equal to the OLS estimator
ˆ  ( X * X * ) 1 X * Y and, hence Cov(ˆ )  ( X *X * ) 1 ( X * VX * )( X *X * ) 1 , where V =
Cov  Y  . Now following the same procedure as in [27], V can be written as:
k 1
V    k2-i M k -i ,
i 0
where M k  I N N and in general
M k -i
 B k -i
 0

 0

 0
0
B k -i
0
0
0
0 
0
0 
,
 0 

0 B k -i 
where B k -i  2i  2i matrices of 1’s. From this we can derive the following expression for
Cov βˆ . Cov βˆ 





2 

k 





* *
D1  D1
0

0


1
0

* *
D 2 D 2

0
0

0
1
0
0


0

* *
Dk Dk








  2 2 
k -1 





1





* *
D1  D1



1
0
0


0
0
0
0

*
*
D k 1 D k 1
0
0

1


k 1 2
    2 1

0


0








* *
D1  D1


1
0
0
0
0
0
0
0




0
0
0



0

.
24
Tyssedal and Kulahci
Assuming interactions of order higher than two are negligible, this shows that inference
about the effects for a k -stage process can be done in k separate steps, where we in each
step consider the main effects for the factors at the respective stage and their possibly
aliased two-factor interactions. In [27] it is also shown how the identification of active
factors for a two-stage process can be done in two separate steps. Writing a k -stage design
as a two-stage design as in (3) and thereafter successively using (5), it follows that the same
identification procedure generalizes to the k -stage process. Starting with stage k and
working backwards we can for each stage perform a search procedure based on OLS to
identify the respective active effects among the main effects for that stage and their possibly
aliased two-factor interactions.
In some applications it is not realistic to assume that the error terms at a given stage
have identical variances. V is still a block diagonal matrix but the blocks need not to be
identical and the OLS and GLS estimates may differ. It is expected that the occurrence of
unequal variances is more likely to happen for the first stages of the experimentation. If the
error terms within a certain stage have equal variances and this is also true for all the
following stages, the OLS estimates will still equal the GLS estimates for all main effects
and their possibly aliased two-factor interactions on that stage and the succeeding ones. The
generalization of the two stage procedure given [27] also applies to these stages. To treat the
more general situation we refer to [22].
7. Concluding Remarks
Generating designs for multi-stage processes where in each stage the experiments are
run as mirror image trials ensures that the total number of experiments can be minimized.
Saturated fractional factorials have a structure that fits well into the MSPMIP scheme.
However, in order to have designs of projectivity P  3 , there will be strict restrictions on
the number of factors allowed at each stage. Nonregular designs such as Plackett Burman
designs with the number of runs N  2 k for instance are therefore very appealing
alternatives as base designs, offering a wide variety of choices and being able to
accommodate considerably more number of factors compared to their regular counterparts.
Especially designs created by using different columns in a nonregular designs, the
(2 k 1  n, n1 , n2 ,, nk ,3) n1  n2  nk n  k 2 designs, has great flexibility and desirable alias
structure, avoiding full aliasing between two-factor interactions. As for two-level split-plot
mirror image pairs designs, the GLS estimates and OLS estimates of the effects in
MSPMIP designs are identical using the standard assumptions, and in most cases the
identification of active factors and estimation of the corresponding effects can be done in
k steps using OLS in each step.
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2.
3.
4.
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26
Tyssedal and Kulahci
Appendix A
Table A1. A list of P  3 MSPMIP design for N =16 and 32. The notation
 N , x , y,, nk ,3 s m means that x , y, can take any number in the set 1,2,, s as
long as their sum is m . Factor allocation can be done assigning 1,, x in Stage 1,
x  1,, x  y in Stage 2 and so on.
Total
Number
of Runs
16
32
Factor allocation
Design
(16,2,2,4,3)
(16,1,1,2,4,4)
(32,1,1,8,3)
Stage 1
Stage 2
1,2
1
1
3,123
2
2
(32,1,4,5,3)
1
(32,2,3,5,3)
1,2
(32,3,2,5,3)
1,2,3
4,24,34,
1234
4, 34,
1234
4,1234
(32,2,3,6,3)
1,2
(32,3,2,7,3)
1,2,3
(32,4,1,8,3)
(32,2,4,8,3)
1,2,3,
123
1,2
(32,3,4,8,3)
1,2,3
(32,4,2,4,3)
1,2,3,
123
1,2,3,
123
1,2,3,
123
1
1
(32,x,y,4,3)s=3
(32,x,y,8,3)s=3
(32,x,y,5,3)s=4
(32,x,y,8,3)s=4
(32,4,2,8,3)
(32,4,4,8,3)
(32,1,1,1,4,3)
(32,1,1,1,8,3)
4,34,
1234
4,1234
4
4,134,
124,234
4,134,
124,234
4,124
4, 124
Stage 3
4,124,134,234
3,123
4,124,134,234
5,35,45,1345,2345,
1245,1235,125
5,45,1235,12345
5,45,125,1345,
2345,135,1245,235
5, 1235,1245,
1345,2345
5,12345,125,345,
135,245,145,235
5,1235,1245,
1345,2345
5,1235,1245,
1345,2345
5,1235,1245,
1345,2345
5,125,135,145,
235,245
5,125,135,145,
235,245,345
5,12345,125,345,
135,245,145,235
5,12345,125,345,
135,245,145,235
5,12345,125,345,
135,245,145,235
5,1345,2345,125
4,124,
134,234
2
2
5,12345,125,345,
135,245,145,235
5,12345,125,345,
135,245,145,235
3
3
(32,x,y,z,5,3)s=4
(32,x,y,z,8,3)s=4
(32,1,1,3,5,3)
(32,1,2,2,5,3)
(32,2,1,2,5,3)
(32,2,2,1,5,3)
(32,1,1,4,8,3)
1
1
1,2
1,2
1
2
3,23
3
3,123
2
4,34,1234
4,1234
4,1234
4
4,134,124, 234
(32,1,2,4,8,3)
1
3,123
4,124,134,234
(32,2,2,4,8,3)
1,2
3,123
4,124,134,234
2
3,123
(32,1,1,2,4,8,3) 1
Stage 4
5,45,1235,12345
5,45,125,1345,2345,
135,1245,235
5,1235,1245,1345,2345
5,12345,125,345,
135,245,145,235
5,1235,1245,1345,2345
5,1235,1245,1345,2345
5,1235,1245,1345,2345
5,125,1245,1345,2345
5,12345,125,135,
245,145,235,345
5,12345,125,135,
245,145,235,345
5,12345,125,345,
135,245,145,235
4,124,134,234
5,12345,125,135,
245,145,235,345
Experiments for Multi-Stage Processes
27
Table A2. A list of P  4 MSPMIP designs for N =16, 32 and 64. The notation
 N , x , y,, nk ,3s m means that x , y, can take any number in the set 1,2,, s as
long as their sum is m . Factor allocation can be done assigning 1,, x in Stage 1,
x  1,, x  y in Stage 2 and so on.
Total
Number
of Runs
16
32
64
Design
Factor allocation
Stage 1 Stage 2
4,1234
(16, x, y, 2, 4)s=3
(16, 1, 1, 1, 2,4)
1
2
(32, x, y, 3, 4)s=3
(32, 3, 2, 1, 4)
(32, 1, 4, 1, 4)
1,2,3
1
4,1234
4,24,34,
1234
(32, x, y, 2, 5)s=4
(32, 1, 1, 4, 5)
1
2
1
2
1
1
1
2
2
2
(64, x, y, 3, 4)s=5
(64, 1, 1, 5, 5)
1
2
(64, x, y, 3, 5)s=4
(64, 2, 4, 1, 5)
1,2
1,2,3,4
5,45,35,
12345
5,12345
1
2
(32, 1, 1, 1, 3, 4)
(32, x, y, 2, 1, 4)s=3
(32, x, y, z, 2, 5)s=4
(32, 1, 1, 1, 2, 1, 4)
(32, 1, 1, 1, 1, 2, 5)
(64, 1, 1, 6, 4)
(64, x, y, 5, 4)s=3
(64, x, y, 4, 4)s=4
(64, 4, 2, 1, 5)
(64, x, y, 4, 6)s=3
(64, x, y, 2, 6)s=5
(64, 1, 1, 1, 5, 4)
Stage 3
3
Stage 4
Stage 5
4,1234
4,1234
5, 45, 12345
5
5
5,12345
5,35,45,
12345
3
4,1234
3
3
6,36,46,56,
1346,23456
6,46,56,1456
23456
6,56,1236,
23456
6,1236,3456
6,56,46,36,
123456
6,56,123456
6
6
6,56,46,
123456
6,123456
3
(64, x ,y, z, 4, 4)s=4
(64, x, y, z, 3, 4)s=5
(64, x, y, z, 2, 4)s=6
(64, x, y, z, 3, 5)s=4
(64, x, y, 2, 1, 5)s=4
(64, 1, 1, 1, 1, 4, 4)
5,45, 12345
5
5,12345
4,1234
4
5
5,12345
6,56,46,1456,
23456
6,56,1236,
23456
6,1236,3456
6, 1236
6,56,12346
6
6,56,1236,
23456
(64, x, y, z, v, 3, 4)s=5
(64, x, y, z, 2, 1, 5)s=4
(64, x, y, z, v, 2, 6)s=5
(64, 1, 1, 1, 1, 1, 3, 4)
1
2
3
4
6,1236,
3456
6,1236
6,1236
6,56,
12346
6
6,12345
5
(64, 1, 1, 1, 1, 2, 2, 4)
(64, 1, 1, 1, 2, 1, 2, 4)
(64, 1, 1, 1, 1, 2, 1, 5)
(64, 1, 1, 1, 1, 1, 2, 6)
1
1
1
1
2
2
2
2
3
3
3
3
4
4,1234
4
4
5,12345
5
5,12345
5
(64, x, y, z, 2, 2, 4)s=4
(64, x, y, 2, 1, 2, 4)s=3
(64, 1,1, 1, 1, 3, 5)
1
2
Stage 6
4,1234
3
5, 12345
5
4
5,12345
6,1236,
3456
6,1236
6,1236
6
6,123456
28
Tyssedal and Kulahci
Authors’ Biographies:
John Tyssedal is an Associate Professor at the Department of Mathematical Sciences at
the Norwegian University of Science and Technology. He serves as an Associate Editor for
QTQM. His research interests are in design of experiments and time series analysis.
Murat Kulahci is an Associate Professor in the Department of Applied Mathematics and
Computer Science at the Technical University of Denmark and in the Department of
Business Administration, Technology and Social Sciences at Luleå University of
Technology in Sweden. His research focuses on design of physical and computer
experiments, statistical process control, time series analysis and forecasting, and financial
engineering. He has presented his work in international conferences, and published over 50
articles in archival journals. He is the co-author of two books on time series analysis and
forecasting.