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The robustness and optimality of response surface designs
Journal of Statistical Planning and Inference 3 (1979) 249-253. @ North-HaIland Publishing Company
Received 6 April 1977; revised Recommended by V. Fedorov
manuscript received16
December 1977
Abs~ct: Recent work on extended optimality criteria for robust designs is applied to response surface problems. Methods of calculation are described and the criteria illustrated with several exampfes. The extended criteria discriminate among designs equivalent by other criteria. Key words: Robustness; cvtimal Design; Missing Values; Matrix Compounds.
various optimality criteria have been used in the study and design of experiments. Typically these crit&a assume that all observations will have equal weight in a ieast-squares analysis. Recently, Box and Draper (1975) introduced a criterion for the construction of designs to minimize the effect of spurious observations. This criterion was motivated by considering the effect of a single outlier. The literature on missing value estimation is some evidence that missing observations or outliers can occur in well-designed experiments and that more than single outliers may occur. For a poor desigg the loss of a small number of observations may yield a singular design matrix from which no parameters may be estimated., It is of some interest therefore to develop robust &signs which guard against this. In a previous paper [Herzberg and Andrews, 1976) two measures of the robustness of designs were introduced: (i) the probability of breakdown; and (ii) the expected precision, i.e. the expected val of the generalized variance, p being the number
that in situations where
D.F. Atuhws und A.M. Hedwg
250
Wolfowitx (1960)) or the robust criterion (af Box and Draper fail to distinguish, (i) and (ii) can iden6fy the: preferred designs.
Let
y(x) = ‘XB + Q, where y(r) is an n x 1 vector of observations whose it;tr.element is y(q), X is an n x p matrix whose ith row is f’(q) = cfi(xi), . . . , fp(Xi)} a p-dimensional vector of the fundions in the model, the Xi being the experimental design points, 6 is a p x 1 vector of unknown paralmeters and E is an n )( 1 vector of independent errors. In order IO estimate 6, the method of least squares is used. One criterion for the design of experiments, known as D-optimal.@ is the maximization of IX’Xl, this being equivalent to the minirrization of the generarfzed variance of 6, the leaist squares estimate of 6. To model the possible iloss of observatlions, Her&erg and Andrews (1976) considered the value of /X’IVXl ‘lp, where: .IIG= diag{&) with &=
0
with probability a(x),
1 with probability
1 -a(x),
ahe value zero being associated with a missing observation. Thus QI(x) is the Iprobabir’?y of losing an o*&servation at x, the loss’es at different points being iindependcnt. Similar formulations may be used fo- outliers or non-Gaussian distr butilonc (Her&erg and Andrews, 1976). Ar kinsfon ( 19713)and others have used as a measure of efficiency of designs ~~
1 Ix’x/“” =--?I p!d!“” ’
(2)
where
is the Fisher information matrix of the appropriate D-optimal design esigns in which the design region and the model, measure. IFor the compari and thus p, zre identical is the same for any value of ~1,and it is oxaly sary to compare (11 x and Drapler (197
traduced a criterion for the constructiin of designs to reduce the effect of a possible outlier in the least squares estimatic? of icted response function. For the comparison qf designs, the value of the n variance of the variances of the estimated response at the design
The robustness and optimal@ of response surface designs
251
3 7 x7.1 (V&I), is used, implying the smaller the value of (3) the better or more robust the design. Note that this is purely an assessment of robustness and is independent of the efhciency of the design. An attempt to minimize (3) may lead to very inefficient designs. The probability of breakdown of a design pr()X’lD“Xl= 0)
(4
(Her&erg and Audrews, 1976) may be used as a measure of robustness to compare designs., the smaller the value the more robust the design. Another measure introduced by Herzberg and Andrews (1976)l was E(IX’lB~x~“P).
$)
This can be extended to be comparable with (2) to
The larger the value of (2) or (6) the more efficient or more robust the design. When at(x) is zero, (2) and (6) are equivalent. It will be shown in Section 3 that E(jX’D2XI) is not a good discriminator among designs. IIn Section 4, a few examples are given which show how the measure n-1E(IX’D2X11’*‘)is used to choose a robust design and how in some cases it points to the selection of a diffeaealt design from that itndicated by (2) or (3). In Section 3, it is shown how (5) m;\y be easily calculated.
Let ’ = X(X’X)-‘X’=’
iuij~,
the n x n matrix of variances and covariances of the estimated responses at the
design points, and let
where ij.
s.
1
2 = diag(cl2;).
with (k = i, j9. . . , l),
&z
1 otherwke. interes: that IC.
D.F. Andtews OF&A.84 Htnl=!!i
252 Let
A be an arb&rary p x p matrix and w be an arbitrary p X 1 vealor~ Then the identities, assurkng al\ terms are defbed,
well-~
may b combined to show that
where ij...lmVa denotes the (II, k)th element of ii..aimV*Therefore, I
Ix ij ...rmD2XJ=(X’ii...rD2XJ(I-u...lV’,) (7) say. Thus, when the ith observation is missing from the experiment
when the ith $nd jth observations are missing,
IPij==c+ Vi,)(l-- Vj,)-- v;; when the ith, jth and Ith observations are missing
etc. The vafues Of are diagonal elements of the compounds of the matrix 1-Y :=W, say. :For example, the &I are the diagonal elements of the 3rd compound of W. The elements I&... are determinants of order usually less than that af the left hland side of (7). Now W may be written as &j..
.
where is an orthogonal matrix and = diag{pi},an n x PZdiagonal matrix. Since is idempotent, the elements on the diagonal of are 0 or 1. Thus PI pi = ?I- p* i-1
IF s deno’l..esthe 8th compound of a matrix, then and
253
.
Therefore,
$ &*.*.* ‘-”(n-:p)_l $T!,~ s! (n-p-s)!’ iCjs
h..
%%#{
where s also denotes the number of subscripts ;sn R. Consider now ti(IX’D2X)Y). If of(x)= 01for all x, then
~(tX’Wxp) == (1 -ay +
=
(x’xl” +ar(l -a)‘+’ c (X:D’XI*’ i
c
a2(1- tKy- . &D2xp +
nf
&
-
i=O
l
l
l
U,n--i%,
1
(9) I
where
To = lX’Xl”,
Tl
:=To
ci R;?,
T3=To c R&c...
T2=7'ozR&
i#j
i
i#j#l i
Since it is with relatively small designs that robustness is important, not many observations need to be missing before 1X$... D”Xl ot R,.. . are zero and do not have to be calculated, ‘When T = 1, the Ti simplify considerably. i.e. (i=o,...,J%-p).
(10)
The ratio Tr/To might be considered as a measure of robustness (i = 1, . . . , n - p). However, substitution of (10) into (9) gives E{IX’D2XI)=IXX(n&i(l-a)“-i(n;P)=IX’XI(l-4# i=O
when ty(x) = cy. Thus if two or more designs for a particular problem have the same value of IX’Xlj(11) will not discriminate among them. Zor the comparison of designs consisting of different numbers of design points, (8) may be divided by the number of design points. This yields the average amount of information obtkned from each design point. This normalization has been used by Box and Draper (1975) and Atkinson (1973). When ar(x) is not constant, the calculation of {S) is more complicated.
Consider the model
DJ;= Att&w# and A.M. Ifikmh
254 the
’
design region being -1 s x, s 1 (i = 1,2). Let the designs consist of the points (19l), (1, -lb, (-1,119 t-1, -11, (1901, (-1, O), (0,11,(0, -3,
with no points at (0,O). For Design I, let aO= Q, for Design II, let Ito= 1 and for = 2. Table 1 gives Wues of S(\X!D‘%~““) and B(/~~%\“‘% 0.02 0.05, and 0.1, values of af and A for DeG@s I, IXand III, As. Q increases, tile measures E(lXl’@Xlrraj for each cr(x) . and tr$ increase leading to the choice of Design 1x1for the tit mc5as?xeand Design I for the second measae. Measures E(~D2X~““,/n and A point to the choice of Design II, lit is inter&in ?o note that for E(~D2X13t6)jn, Design II is the best for all at, but 8s a increases, Design III becomes better than Design I. CIeiarly the probability of breakdown of the design decreases as no incrtxuk Consider the+model E{~(~j}=e~+rP~x~9e*X~$.e~%~+e~r_X:+o*~X~+e~~X~
(13
+e,,~,~*+et3X*~3+e23X*X3, the design
points
regionbeing
-lSx+l
(i = 1,2,3). Let the designs consist of the
(1~1,f),(1,1,,-1),(1,-1,1),(--1,1,1),(1,-1,-1), (-41,
-4, (-1, -1, l), (-1, -1, -ii, (LO, 01, (-l,O,O),
:0, -LO), (0, LO), (O,O,-11, (O,O, 11, Table 1
Compatin of values of E(jX’D2X1”6, E(~X’iB2XI”6)/n,a: and A far igns I, II and iII for the model
and various vale
Design 1 (n, =t0)
0.01 Q.02 0.05
,
100
7f a(x) = a
a 0
x
3.434 3.572 3.507 3.303
Design II (n, = 1) Design III (n, = 2) E((XWX1 l’6) 4.160 4.094 4.028 3.826 3.4?6 E(IX’D2XJ”‘),‘n
4.478 4,409 4.400 4.129 3.757
255
Comparisons af values &S(~‘D2X~1’xo),E(pD2X11’1a)/n, C$ and A x 1QO far Designs 1, 2 and 3 for the made!
-s1
*
-
D@&p 1 (tip 01
Design 2 (no = 1)
6.483
6.707 6.594 6.4150 6,130 5,513 4.099 2.531 1.193 0,394
iir 0 O&l
02 WI5 0.1 0.2 0,3 0.4 OS
6.369 6.258 5.863 5.131 3.455 1.861 0.757 0,217
4 Ax100
Design 3 (n, = 2)
*=
6,880 6,765 6.650 6.298 5,692 4.343 2.825 1.431 OS13
*
E((X’WXl”““)ln
a 0 0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5
B(lX’l[s2Xl”f9
.’
0.463 0.455 0.446 0.419 0.366 0.247 0.133 0.054 0.015
0.447 0.440 0.432 0.409 0.368 0.273 0.169 0,080 0.026
0.430 0.423 0.416 0.394 0.356 0.271 0,177 0.089 0.032
0.010 97.59
0.024 94.24
0,035 90.62
with ycOpoints at (0,0,O). For Design 1, let )tO= 0, for Design 2, let ylo= 1 anb for I1’lO)ln Design 3 let no = 2. Table 2 gives values of E(IX’D*XI*‘*“)and E(( for u(x)=O, 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, values of a,” and A for I”‘“) for each CY(X) and Designs 1,2 and 3, As ytoincreases the measures E rst measure and Design I C$ increase leading to the choice of Design 3 for t for the second measure. As a!(x) increases the Design with the largeit value of t it is important to take i “Xll’lo)/~ changes showing E(I ign 1. For these designs Measure A leads to the choice of breakdown is ahnost the same. Consider the model
I
E@rwxp}/(s4)~
(y =
1*
and awious values of a(x) = a --
-
Design B Eilx’d2x\ls4}“3 Et~WXlW
a
Q 0.01 Q.02 0.5 0.1
pints
1 0.937 0.975 0.935 0.865
1 0.97 0,941 0.857 0.729
of support
1 0.97 0.941 0.857 0.729
1 0.988 a975 0.938 0.875
equispaced on the unit circle with two replications at each point,
Design B having six equispaced points on the unit circle. The values p = 3 and n E=6 are ftxed for both designs. the values of iX’Xi*”and at are also the same for both designs, namely (54)*‘3and 0. Thus (2) and (3) are not ahle to distinguish between the two designs. However, (5) divided by (§4)*‘3 shows clearly that Desi B is mare robwt as is shown in Table 3. Table 3 gives corresponding a’( 1 -a)&-‘T;/ln (7 = I, 4) for the two designs for a = 0, 0.01, Box and Draper (W75, p. 350” remark that their measure of robusltness is equivalent to arranging that the maximum value of 1015 - ii1:=Isi - p/n1 is minimal. They claim that ‘this is the G-efficiency criterion used in the selection of a design to control the variance of the 9’s’, j; being the estimated value of the response function at a design point. Consider lthe following simpk example. Let E{jr(x)} = e,+ e,x, and let the two designs. be 4,
(9 (ii)
-1, i, 11,
-_ z -4, 29
3, 3
r both designs all the Vii=j (i = 1,. . . ,6) and IVii- ii1= 0. But only Design (i> is ptimai. It is also easy to show that Design (i) is more robust for all the other meacures discussed.
It
has been shown how the consideration of several measures of robulstness for e ~x~~rim~nt~!r’schoir2 of a desi
7% mbuhtas
and optima&y
ofreqmnse surface desitw
257
E:(iX’D@X/‘Y) of any design relatively easily. It is very interesting that it is not possible to discruninate among designs having equal values of IX’Xlwhen y = 1. Several measures oP robustness introduced by the authors and others are calculated for various designs. What is still needed, however, if it is possible, is a theoretical comp&on of robust designs. These the authors hope to investigate in the future,
We are grateful to Mrs. L. Lapczalc for help with the calculations and to the National Research Council of Canada for its continued support.
Atkinson, AC. (1973). Multifactor seconci order designs for cuboidal regions. Biometrika 60, 15-19. Box, GEE and Draper, NE%.(1975). Robust designs. &iomefri&a62, 347-352. Her&erg, A.M. and Andrews, D,F. (1976). Some considerations in the optimal design of experiments ‘n non-optimal situations. J. Roy. §?atis~$0~. Ser. B 38, 284-289. Kiefer, J. and Wolfowitz, J. (1960). The equivalence:of two extremum problems. Canad. J. Math. 12, 363-366.
Received 6 April 1977; revised Recommended by V. Fedorov
manuscript received16
December 1977
Abs~ct: Recent work on extended optimality criteria for robust designs is applied to response surface problems. Methods of calculation are described and the criteria illustrated with several exampfes. The extended criteria discriminate among designs equivalent by other criteria. Key words: Robustness; cvtimal Design; Missing Values; Matrix Compounds.
various optimality criteria have been used in the study and design of experiments. Typically these crit&a assume that all observations will have equal weight in a ieast-squares analysis. Recently, Box and Draper (1975) introduced a criterion for the construction of designs to minimize the effect of spurious observations. This criterion was motivated by considering the effect of a single outlier. The literature on missing value estimation is some evidence that missing observations or outliers can occur in well-designed experiments and that more than single outliers may occur. For a poor desigg the loss of a small number of observations may yield a singular design matrix from which no parameters may be estimated., It is of some interest therefore to develop robust &signs which guard against this. In a previous paper [Herzberg and Andrews, 1976) two measures of the robustness of designs were introduced: (i) the probability of breakdown; and (ii) the expected precision, i.e. the expected val of the generalized variance, p being the number
that in situations where
D.F. Atuhws und A.M. Hedwg
250
Wolfowitx (1960)) or the robust criterion (af Box and Draper fail to distinguish, (i) and (ii) can iden6fy the: preferred designs.
Let
y(x) = ‘XB + Q, where y(r) is an n x 1 vector of observations whose it;tr.element is y(q), X is an n x p matrix whose ith row is f’(q) = cfi(xi), . . . , fp(Xi)} a p-dimensional vector of the fundions in the model, the Xi being the experimental design points, 6 is a p x 1 vector of unknown paralmeters and E is an n )( 1 vector of independent errors. In order IO estimate 6, the method of least squares is used. One criterion for the design of experiments, known as D-optimal.@ is the maximization of IX’Xl, this being equivalent to the minirrization of the generarfzed variance of 6, the leaist squares estimate of 6. To model the possible iloss of observatlions, Her&erg and Andrews (1976) considered the value of /X’IVXl ‘lp, where: .IIG= diag{&) with &=
0
with probability a(x),
1 with probability
1 -a(x),
ahe value zero being associated with a missing observation. Thus QI(x) is the Iprobabir’?y of losing an o*&servation at x, the loss’es at different points being iindependcnt. Similar formulations may be used fo- outliers or non-Gaussian distr butilonc (Her&erg and Andrews, 1976). Ar kinsfon ( 19713)and others have used as a measure of efficiency of designs ~~
1 Ix’x/“” =--?I p!d!“” ’
(2)
where
is the Fisher information matrix of the appropriate D-optimal design esigns in which the design region and the model, measure. IFor the compari and thus p, zre identical is the same for any value of ~1,and it is oxaly sary to compare (11 x and Drapler (197
traduced a criterion for the constructiin of designs to reduce the effect of a possible outlier in the least squares estimatic? of icted response function. For the comparison qf designs, the value of the n variance of the variances of the estimated response at the design
The robustness and optimal@ of response surface designs
251
3 7 x7.1 (V&I), is used, implying the smaller the value of (3) the better or more robust the design. Note that this is purely an assessment of robustness and is independent of the efhciency of the design. An attempt to minimize (3) may lead to very inefficient designs. The probability of breakdown of a design pr()X’lD“Xl= 0)
(4
(Her&erg and Audrews, 1976) may be used as a measure of robustness to compare designs., the smaller the value the more robust the design. Another measure introduced by Herzberg and Andrews (1976)l was E(IX’lB~x~“P).
$)
This can be extended to be comparable with (2) to
The larger the value of (2) or (6) the more efficient or more robust the design. When at(x) is zero, (2) and (6) are equivalent. It will be shown in Section 3 that E(jX’D2XI) is not a good discriminator among designs. IIn Section 4, a few examples are given which show how the measure n-1E(IX’D2X11’*‘)is used to choose a robust design and how in some cases it points to the selection of a diffeaealt design from that itndicated by (2) or (3). In Section 3, it is shown how (5) m;\y be easily calculated.
Let ’ = X(X’X)-‘X’=’
iuij~,
the n x n matrix of variances and covariances of the estimated responses at the
design points, and let
where ij.
s.
1
2 = diag(cl2;).
with (k = i, j9. . . , l),
&z
1 otherwke. interes: that IC.
D.F. Andtews OF&A.84 Htnl=!!i
252 Let
A be an arb&rary p x p matrix and w be an arbitrary p X 1 vealor~ Then the identities, assurkng al\ terms are defbed,
well-~
may b combined to show that
where ij...lmVa denotes the (II, k)th element of ii..aimV*Therefore, I
Ix ij ...rmD2XJ=(X’ii...rD2XJ(I-u...lV’,) (7) say. Thus, when the ith observation is missing from the experiment
when the ith $nd jth observations are missing,
IPij==c+ Vi,)(l-- Vj,)-- v;; when the ith, jth and Ith observations are missing
etc. The vafues Of are diagonal elements of the compounds of the matrix 1-Y :=W, say. :For example, the &I are the diagonal elements of the 3rd compound of W. The elements I&... are determinants of order usually less than that af the left hland side of (7). Now W may be written as &j..
.
where is an orthogonal matrix and = diag{pi},an n x PZdiagonal matrix. Since is idempotent, the elements on the diagonal of are 0 or 1. Thus PI pi = ?I- p* i-1
IF s deno’l..esthe 8th compound of a matrix, then and
253
.
Therefore,
$ &*.*.* ‘-”(n-:p)_l $T!,~ s! (n-p-s)!’ iCjs
h..
%%#{
where s also denotes the number of subscripts ;sn R. Consider now ti(IX’D2X)Y). If of(x)= 01for all x, then
~(tX’Wxp) == (1 -ay +
=
(x’xl” +ar(l -a)‘+’ c (X:D’XI*’ i
c
a2(1- tKy- . &D2xp +
nf
&
-
i=O
l
l
l
U,n--i%,
1
(9) I
where
To = lX’Xl”,
Tl
:=To
ci R;?,
T3=To c R&c...
T2=7'ozR&
i#j
i
i#j#l i
Since it is with relatively small designs that robustness is important, not many observations need to be missing before 1X$... D”Xl ot R,.. . are zero and do not have to be calculated, ‘When T = 1, the Ti simplify considerably. i.e. (i=o,...,J%-p).
(10)
The ratio Tr/To might be considered as a measure of robustness (i = 1, . . . , n - p). However, substitution of (10) into (9) gives E{IX’D2XI)=IXX(n&i(l-a)“-i(n;P)=IX’XI(l-4# i=O
when ty(x) = cy. Thus if two or more designs for a particular problem have the same value of IX’Xlj(11) will not discriminate among them. Zor the comparison of designs consisting of different numbers of design points, (8) may be divided by the number of design points. This yields the average amount of information obtkned from each design point. This normalization has been used by Box and Draper (1975) and Atkinson (1973). When ar(x) is not constant, the calculation of {S) is more complicated.
Consider the model
DJ;= Att&w# and A.M. Ifikmh
254 the
’
design region being -1 s x, s 1 (i = 1,2). Let the designs consist of the points (19l), (1, -lb, (-1,119 t-1, -11, (1901, (-1, O), (0,11,(0, -3,
with no points at (0,O). For Design I, let aO= Q, for Design II, let Ito= 1 and for = 2. Table 1 gives Wues of S(\X!D‘%~““) and B(/~~%\“‘% 0.02 0.05, and 0.1, values of af and A for DeG@s I, IXand III, As. Q increases, tile measures E(lXl’@Xlrraj for each cr(x) . and tr$ increase leading to the choice of Design 1x1for the tit mc5as?xeand Design I for the second measae. Measures E(~D2X~““,/n and A point to the choice of Design II, lit is inter&in ?o note that for E(~D2X13t6)jn, Design II is the best for all at, but 8s a increases, Design III becomes better than Design I. CIeiarly the probability of breakdown of the design decreases as no incrtxuk Consider the+model E{~(~j}=e~+rP~x~9e*X~$.e~%~+e~r_X:+o*~X~+e~~X~
(13
+e,,~,~*+et3X*~3+e23X*X3, the design
points
regionbeing
-lSx+l
(i = 1,2,3). Let the designs consist of the
(1~1,f),(1,1,,-1),(1,-1,1),(--1,1,1),(1,-1,-1), (-41,
-4, (-1, -1, l), (-1, -1, -ii, (LO, 01, (-l,O,O),
:0, -LO), (0, LO), (O,O,-11, (O,O, 11, Table 1
Compatin of values of E(jX’D2X1”6, E(~X’iB2XI”6)/n,a: and A far igns I, II and iII for the model
and various vale
Design 1 (n, =t0)
0.01 Q.02 0.05
,
100
7f a(x) = a
a 0
x
3.434 3.572 3.507 3.303
Design II (n, = 1) Design III (n, = 2) E((XWX1 l’6) 4.160 4.094 4.028 3.826 3.4?6 E(IX’D2XJ”‘),‘n
4.478 4,409 4.400 4.129 3.757
255
Comparisons af values &S(~‘D2X~1’xo),E(pD2X11’1a)/n, C$ and A x 1QO far Designs 1, 2 and 3 for the made!
-s1
*
-
D@&p 1 (tip 01
Design 2 (no = 1)
6.483
6.707 6.594 6.4150 6,130 5,513 4.099 2.531 1.193 0,394
iir 0 O&l
02 WI5 0.1 0.2 0,3 0.4 OS
6.369 6.258 5.863 5.131 3.455 1.861 0.757 0,217
4 Ax100
Design 3 (n, = 2)
*=
6,880 6,765 6.650 6.298 5,692 4.343 2.825 1.431 OS13
*
E((X’WXl”““)ln
a 0 0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5
B(lX’l[s2Xl”f9
.’
0.463 0.455 0.446 0.419 0.366 0.247 0.133 0.054 0.015
0.447 0.440 0.432 0.409 0.368 0.273 0.169 0,080 0.026
0.430 0.423 0.416 0.394 0.356 0.271 0,177 0.089 0.032
0.010 97.59
0.024 94.24
0,035 90.62
with ycOpoints at (0,0,O). For Design 1, let )tO= 0, for Design 2, let ylo= 1 anb for I1’lO)ln Design 3 let no = 2. Table 2 gives values of E(IX’D*XI*‘*“)and E(( for u(x)=O, 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, values of a,” and A for I”‘“) for each CY(X) and Designs 1,2 and 3, As ytoincreases the measures E rst measure and Design I C$ increase leading to the choice of Design 3 for t for the second measure. As a!(x) increases the Design with the largeit value of t it is important to take i “Xll’lo)/~ changes showing E(I ign 1. For these designs Measure A leads to the choice of breakdown is ahnost the same. Consider the model
I
E@rwxp}/(s4)~
(y =
1*
and awious values of a(x) = a --
-
Design B Eilx’d2x\ls4}“3 Et~WXlW
a
Q 0.01 Q.02 0.5 0.1
pints
1 0.937 0.975 0.935 0.865
1 0.97 0,941 0.857 0.729
of support
1 0.97 0.941 0.857 0.729
1 0.988 a975 0.938 0.875
equispaced on the unit circle with two replications at each point,
Design B having six equispaced points on the unit circle. The values p = 3 and n E=6 are ftxed for both designs. the values of iX’Xi*”and at are also the same for both designs, namely (54)*‘3and 0. Thus (2) and (3) are not ahle to distinguish between the two designs. However, (5) divided by (§4)*‘3 shows clearly that Desi B is mare robwt as is shown in Table 3. Table 3 gives corresponding a’( 1 -a)&-‘T;/ln (7 = I, 4) for the two designs for a = 0, 0.01, Box and Draper (W75, p. 350” remark that their measure of robusltness is equivalent to arranging that the maximum value of 1015 - ii1:=Isi - p/n1 is minimal. They claim that ‘this is the G-efficiency criterion used in the selection of a design to control the variance of the 9’s’, j; being the estimated value of the response function at a design point. Consider lthe following simpk example. Let E{jr(x)} = e,+ e,x, and let the two designs. be 4,
(9 (ii)
-1, i, 11,
-_ z -4, 29
3, 3
r both designs all the Vii=j (i = 1,. . . ,6) and IVii- ii1= 0. But only Design (i> is ptimai. It is also easy to show that Design (i) is more robust for all the other meacures discussed.
It
has been shown how the consideration of several measures of robulstness for e ~x~~rim~nt~!r’schoir2 of a desi
7% mbuhtas
and optima&y
ofreqmnse surface desitw
257
E:(iX’D@X/‘Y) of any design relatively easily. It is very interesting that it is not possible to discruninate among designs having equal values of IX’Xlwhen y = 1. Several measures oP robustness introduced by the authors and others are calculated for various designs. What is still needed, however, if it is possible, is a theoretical comp&on of robust designs. These the authors hope to investigate in the future,
We are grateful to Mrs. L. Lapczalc for help with the calculations and to the National Research Council of Canada for its continued support.
Atkinson, AC. (1973). Multifactor seconci order designs for cuboidal regions. Biometrika 60, 15-19. Box, GEE and Draper, NE%.(1975). Robust designs. &iomefri&a62, 347-352. Her&erg, A.M. and Andrews, D,F. (1976). Some considerations in the optimal design of experiments ‘n non-optimal situations. J. Roy. §?atis~$0~. Ser. B 38, 284-289. Kiefer, J. and Wolfowitz, J. (1960). The equivalence:of two extremum problems. Canad. J. Math. 12, 363-366.