Subset selection in two-factor experiments using randomization restricted designs

ELSEVIER

Journal of Statistical Planning and Inference 62 (1997) 339 363

journal of statistical planning and inference

Subset selection in two-factor experiments using randomization restricted designs T h o m a s J. Santner a'*, G u o h u a Pan b aDepartment of Statistics, Ohio State University, Columbus, OH 43210, USA bDepartment of Mathematical Sciences, Oakland University, Rochester, MI 48309, USA

Received 1 December 1994; revised 6 February 1996

Abstract This paper studies subset selection procedures for screening in two-factor treatment designs that employ either a split-plot or strip-plot randomization restricted experimental design laid out in blocks. The goal is to select a subset of treatment combinations associated with the larges~ mean. In the split-plot design, it is assumed that the block effects, the confounding effects (whole-plot error) and the measurement errors are normally distributed. None of the selection procedures developed depend on the block variances. Subset selection procedures are given tbr both the case of additive and non-additive factors and for a variety of circumstances concerning the confounding effect and measurement error variances. In particular, procedures are given for (1) known confounding effect and measurement error variances (2) unknown measurement error variance but known confounding effect (3) unknown confounding effect and measurement error variances. The constants required to implement the procedures are shown to be obtainable from available FORTRAN programs and tables. Generalization to the case of strip-plot randomization restriction is considered. © 1997 Elsevier Science B.V. Keywords: Ranking and selection; Split-plot design; Strip-plot design; Two-way layout; Optimal design; Least favorable configuration; Subset selection approach; Restricted randomization

1. Introduction In multi-factor experiments, the identification of a treatment combination associated with the largest mean is an important selection and screening goal. For completely randomized experiments, the procedure of Gupta (1956, 1965) can be applied to select a subset of treatment combinations containing the combination with largest mean; his procedure was initially developed to select a subset of levels, in a one-way layout, containing the level with greatest mean. If the factors of a CR experiment are additive, the procedure of Bechhofer and Dunnett (1987) is a more efficient subset selection method for the same goal. * Corresponding author. Tel.: 614-292-3593; fax: 614-292-2096; e-mail: [email protected]. 0378-3758/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved Pl1S0378-3758(96)00195-4

340

72£ Santner, G. Pan~Journal of Statistical Planning and Inference 62 (1997) 339 363

Other selection goals have been considered in the literature. For example, Wu and Cheung (1994) present, in a two-factor experiment, a procedure that selects, simultaneously for each level of the row factor, the level of the column factor with the greatest mean. Pan and Santner (1996) consider a two-factor experiment in a quality improvement setting where one factor is a manufacturing design variable and the second factor is a noise variable. For each fixed manufacturing design, consider the 'profile' of mean responses corresponding to different levels of the noise variable. They study several goals related to choosing the manufacturing design with the profile of mean responses which exhibits the least variation. Recently, Box and Jones (1992), Cantell and Ramirez (1994), and Gregory and Taam (1996) considered testing formulations of such problems for split-plot experimental designs. This paper considers the first goal above, in developing statistical procedures for selecting a subset of the treatment combinations containing the treatment combination having the largest mean when the experiment has been conducted, for convenience or out of necessity, using a randomization restricted design. For specificity, we focus much of the discussion on a two-factor experiment that has been conducted using a split-plot design laid out in blocks. Section 5 discusses extensions to other randomization restricted designs for which such an analysis is possible. To state this problem more precisely, we introduce the following notation and terminology. We refer to the factors as row and column factors, and denote them by R and C with numbers of levels r and c, respectively. There are two models that must be analyzed separately when considering selection of the treatment combination with the largest mean: the first model is when the factors are additive and the second is when they are non-additive . If #ik denotes the mean response for the ith level of the row factor and kth level of the column factor and the means #ik are decomposed as (1.1)

#ik = # + Ri + Ck + (RC)ik r

C

subject to the identifiability conditions ~ i = l R i = 0 = ~ k = l Ck and ~ = I ( R C ) i ~ r R C)ik, then additivity occurs if =i 0 k = ~i=1( (1.2)

#ik = # + Ri + Ck

for all i, k. Non-additivity occurs when not all of the (RC)i k in (1.1) are zero. When the row and column factors are non-additive, our goal is to identify the treatment combination associated with #[rcl where #[1]

~< " ' " ---< # I r e ]

denote the r × c ordered treatment means. When additivity holds, the largest mean is associated with the treatment levels having row and column main effects R[~] and C[c], respectively, where R[~]~-.-~
and

C[1]~<-..~
(1.3)

denote the ordered row and column main effects, respectively. We refer to the level associated with R[r] as the 'best' level of the row factor and the level associated with

T.J Santner, G. Pan~Journal of Statistical Planning and lnJi,rence 62 (1997) 339 363

341

C[c] as the 'best' level of column factor. Our goal is to select simultaneously the treatment levels associated with the best levels of the row and column factors. When additivity holds, certain efficiencies occur in the statistical subset selection procedures proposed below since the data can be made to 'work twice' - - once to identify a subset of row main effects and a second time to identify a subset of column main effects. In split-plot (or other randomization restricted) designs the observations for different treatment combinations are correlated and hence the BLUEs of the /~ik are correlated. Therefore the screening and selection procedures depend on the covariance structure of the BLUEs. In particular, let Yi)k denote the response when the ith level of the row factor and the kth level of the column factor are used in the jth block. We assume that the row factor is the 'whole-plot' factor with a separate randomization of the levels o f the column factor for each level of the row factor. We also assume that the experiment can be run in b complete blocks, each containing all possible r × c treatment combinations. We use the usual model

Y,i/k =/3/+/~i~ + e~i + ~/~

(1.4)

for l<.i<~r, l<~j<~b and l<~k<~c where the effects {[J/}j, {{:)ij}i,/, and {ci:k}~./~ are mutually independent with

[~/ i.i.d. N ( 0 , ~ )

for l<~j<<.h,

~oi/ i.i.d. N(0,¢2,)

for l<~i<~r and l<~j<~b,

8iik i.i.d. N(0, a 2)

for l ~ i < ~ r ,

l~j~b

and l<~k<~c

(Milliken and Johnson, 1984). The interpretation of these quantities is that [~/ is the effect the jth block, r~)ij is the (potential) confounding effect for the ith level of the row factor specific to the jth block due to lack o f randomization, and auk is the measurement error. In the additive model, (1.4) reduces to

Yi/k - It + fij -~ Ri ~- {Oij -~- Ck -~- ~:Uk~

( 1.5 )

In this paper we will use the BLUEs of/lik and their joint distribution under both the additive and non-additive models; these estimators are stated in Section 2. These distribution results are used to analyze the subset selection procedures we propose for the additive and non-additive models in Sections 3 and 4, respectively. Section 5 gives examples and comments on other randomization restricted designs.

2. Preliminary results This section proves several distribution theory results required in Sections 3 5. We introduce the notation 2

2

7/~ = a/~/a~; and

2 ~;¥, = a,}/o-~-

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T.J. Santner, G. Pan~Journal o f Statistical Planning and Inference 62 (1997) 339-363

to simplify their statement. We use the standard notation in which a bar over the quantity and a dot replacing one or more o f its subscripts means that an average has been computed with respect to that subscript; for example, ~.k = ~ b = l Yijk/b. Lastly, we let 1~ = (1 . . . . . l y denote the n × 1 (column) vector with common element unity and similarly let 0, denote the n × 1 zero vector. Section 2.1 discusses results for the additive model (l.2)/(1.4) and Section 2.2 considers the non-additive model (1.4).

2.1.

The additive m o d e l

The goal o f this section is to determine the joint distribution of the BLUE o f (R1 R . . . . . . Rr-1 - Rr, C1 - Cc . . . . . C~-I - C~). This difference appears in the determination o f the least favorable configuration of additive subset selection procedure introduced in Section 3. The BLUEs o f the differences R i - Rr for 1 ~
Theorem 2.1. The e s t i m a t o r ~ = (Y1..,..., Yr.., Y-.I,.-.,Y..c)' is the B L U E o f (l~ + R~ . . . . . It + Rr, p + C~ . . . . . p + Co)' under M o d e l (1.2)/(1.4).

The joint distribution o f ~ can be determined by a straightforward calculation.

Theorem 2.2. Under M o d e l (1.2)/(1.4) ~ Nr+c

m,

Z*

where

~*m_ [ Z~I ZT21 with b l o c k structure c79 + e?~o + 1 C

ZT1 =

79

79 c? 9 + c7~o + 1

"""

79

C

c7~ + c?~ + 1 7~

=

rc79 + c7o + 1 FC

79

l r l"

"

C

rXr

m

=

72J. Santner. G. Pan/Journal of Statistical Plannin9 and b~erence 62 (1997) 339 363

343

and r?fl + 7,> r

F

r

rTl~ + 7oJ

rTlJ + "A,, + 1

r 7 fi + Z,,

r

F

r

rTl~ + 7~,

rT~ + 7,,~

r?/s + 7,~>+ 1

r

r

r

X~2 =

CX{

Aside from a multiplier of a2/b, our goal for this subsection can be restated as that of determining the joint distribution of ( W1 - W~. . . . . W~_ i - W,.,Zl - Zc . . . . . Z,._ i - Z< ). where W is r x 1, Z is c x 1, and (W,Z)

~ N(0,X*)

(2.1 "~

with X* defined in Theorem 2.2. Theorem 2.3. I f (W1 . . . . . W~, Zl . . . . . Z~ ) is distributed as in Eq. (2.1) then P{Wi-

W~<~w

=P{Wi-

for l~i

< r; Z k - Z , . < . z

W~<~w f o r l<~i < r} × P { Z k -

for l<~k < e} Z, <~z f o r l ~ k

< c}.

(2.2)

Proof. It suffices to show that Cov(Wi- W~,Z~-Zc)=O

for all l~
and

l~
(2.3)

A (necessary and) sufficient condition for (2.3) is that there exist constants d l . . . . . de-I so that Ck = ~,: + Iv x dk for l~," 1). It is straightforward to verify that X* satisfy this condition, fZ The two marginal probabilities on the right-hand side of Eq. (2.2) can be evaluated in terms of equicorrelated random variables with correlation ½; quadrant probabilities for the latter are widely tabled (Bechhofer and Dunnett, 1988; Dunnett, 1989). This relationship can be compactly stated in terms of the quantities

p

Corollary 2.4.

and

24,

I f (Wi . . . . , W~) is distrihuted as in Theorem 2.3, and ( Ui . . . . . U,.-I ) has the multivariate n o r m a l distribution with zero means, unit variances and c o m m o n

344

T.,L Santner, G. Pan~Journal o f Statistical Planning and Inference 62 (1997) 339-363

correlation 1, then for l ~ < i < r } = P { U i ~ w / p l

P{Wi-W~4w

for l<~i < r}.

Also, P{Zk-Zc<~z

for l ~ k

< c}=P{Vk


for l<~k < c}

where (Z1 .... ,Zc) is distributed as in Theorem 2.3 and (V1 ..... Vc_~ ) has the multi1 variate normal distribution with zero means, unit variances and common correlation ~.

Notice that the joint distribution of (W1 -

Wr,...,

Wr-1 -

Wr,Zl -

Zc . . . . . Z e - I

- Zc)

does not depend on the block effect 7/~. Intuitively, the block effects 'wash out' in the differences W,. - W~ and Zk - Z c . For this reason, none of the procedures developed in Section 3 depend on 7/~Remark 2.1. Generalized linear model theory suggests the following estimator of a~2 based on the BLUE of #ik under (1.5), Pik = Y//..+Y..k - Y . . . Set ~jk = Pik for 1 <~i~r, 1 <~j4b, 1 <~k<<.c and let Y be the rbc × 1 vector obtained by arranging the Y,jk in lexicographic order. Also let Y be the rbc × 1 vector obtained by arranging the Yijk 2 2 2 2 in lexicographic order and let a~S~ = %Y-~(~/~,Y~,) be the covariance matrix of Y. Then S2 = [(Y - Y ) ' Z ~ t ( Y -

Y ) ] l ( r b c - r - c + 1),

(2.5)

is an unbiased estimator of a 2 which is independent of Y and has the distribution determined by (rbc -- r -- c + 1) S v/~7~ 2 2 ~ )~2r b c - r - c + l " In practice, 7/¢ and 7oJ are not likely to be known and hence we focus on the ANOVA estimator of o 2 b

S2s = ~ E ~ (Yijk- Yi.k- Y.j. + Y...)2/v

(2.6)

i=1 j = l k - I

based on v = (c - 1)(rb - 1) degrees of freedom that is provided by standard statistical software. The estimator S2s corresponds to the residual row in the ANOVA. It does not require knowledge of 2/~ or 7~o. It is unbiased and (c - 1)(rb - 1)Sr2ms/~r2 has a chi-square distribution with ( c - 1 ) ( r b - 1) degrees of freedom (d.f.) independent of Y. However, the chi-square associated with $2~ has r ( b - 1) fewer d.f. than S2. 2.2. The non-additive model

Now assume that the non-additive model (1.1)/(1.4) holds. The goal of this subsection is to present the joint distribution of the BLUE of Pij - I&c for (i,j) ~ (r, c). This set of differences appears in the determination of the least favorable configuration of

T.J. Santner. G. Pan!Journal of Statistical Planninq and Inierence 62 (1997) 339 363

345

the non-additive procedure. The first result provides the BLUE of ~; its proof is given in the Appendix. Theorem 2.5. T h e e s t i m a t o r ~ = (Yt.i . . . . . Yi.c, Y2q,. . ., Y2 . . . . . . . Yr.l . . . . . Yr <.)' is the B L U E o f # = (/~11 . . . . . P lc,/~21 . . . . . # 2 c , - . . , IZ~l. . . . . IXr,.)' under (1.4). Fur ther

(2

"~ ~ N~<.

#x, ~7" (TlJlrcl'r<. + 7<,,(1,.l'~) ~b I<~ + I,.< )

)

.

b

Corollary 2.6 gives the distribution of a set of random variables which appear in the expressions for the probability o f correct selection derived in Section 4. Corollary 2.6. U n d e r (1.4)

It21 7

llrc

~rc--I --~rc f o l l o w s the (rc - 1 ) dimensional multivariate n o r m a l distribution with m e a n

~/11- ~/rc / ]t21 7 ]lrc

[2re-- 1 -- I-lrc

and covariance m a t r i x ozX./b where

I2 = 2(7,,,~21 +

~2),

(2.7)

lcltc

~l~lc ~

""

~l l ~ l c ~

OcO~ I

~1 l c l ~ !

l c l c!

"'"

1 ! ~lcl~.

0C0!c

•

.

,

'

-i

.

.

OcO:. i

11,1! c

llcl'c

-'"

lcl~.

Oc--lO!c

Oc 10;

"'"

Oc lO!c

l

Oc--lOc_ 1 (rc--1)×(rc

1)

contains r rows and r columns o f block submatrices and ~

''-

1

...

Ii ' l

1

1

2 1 •

1

(2.8) (re-

I)×(rc--1)

346

TJ. Sanmer, G. Pan~Journal o f StatLvtical Plannin 9 and Inference 62 (1997) 339-363

The result in Corollary 2.6 is symmetric in the subtracted variable in the following sense. Let P-ik be the r e - 1 random vector obtained by deleting ~ik from ~ and similarly for laik. Then

~--ik -- lrc-l"fiik ~ N r c - l ( ! a _ i k -- lrc--l#ik,ff, i)

where ~i = 2a~(7,o2;l,i 2 +

~2)/b, where

l/clc

1 t ~lclc

• - •

O~Orc_

l

I

1 t ~1~1~

t l~lc

• " "

OcO~-

1

~lclc

oc_,o' 1 t

~lclc

l/lc

1

• ..

¢

lclc

0~_10~_

1

O~0/c_,

t

O~_ l 0'~ 1 t 21~1~

"o. 1

t

~lclc

t

½1clc

• "

OcO'c-

1

l/cl~

(rc--1)×(rc--1)

and has the zero matrix in the ith block of rows and columns and '~Y'2 is defined by (2.8). One immediate consequence of Corollary 2.6 is that the joint distribution of the differences f i i j - #~c does not depend on a~ and thus the selection and screening procedures described in Section 4 are independent of block effects• The probabilities of correct selection for the procedures introduced in Section 4, require the evaluation of equi-coordinate probabilities of the form P{Ul < u,...,U~_l < u}

(2.9)

where (/-.71..... Urc-1 ) ~ N~_I(0~_I,X)

(2.10)

and 2; is defined in Eq. (2.7)• While the probability (2.9) involves a simple region, the entries of 2; involves blocks having three different off-diagonal values: those involving 27,0 + 1, those involving 7,o + 1 and those involving 1. This block covariance structure of 2; makes (2.10) difficult to compute. In principle (2.9) can be determined from the following representation. Let D 1 . . . . . D(r-l)c, El ..... Er-1, FI ..... Pc-l, G and H be independent and identically distributed standard normal random variables. It can be checked that the vector

72Z Santner, G. Pan~Journal o/ Statistical Planning and lnJerence 62 (1997) 339 363

347

Di - xf~o~E1 - V/~o~G - H

D~ x/~,,El ~G ( D''+I-x/~Ez-x/~)G-H

\ Dc+c

~.jE2

- x/~o~G

D(r-2)c+l - x / ~ E r - I \

H )o × 1

H

c×l

- ~/coG-- H

D(~ 1)c - x / ~ E ~ - ~ - ~ , , G

- H

c×

1

F,,-1 - H )(o-L 1×1 has the Nr,._l(0~c_j,Z) distribution. Conditioning on H, G and E1 .... ,E,._) gives the following integral representation of the quadrant probability (2.9) P { U i < u . . . . . Ure-1 < U} = [__

--

*°(u

+ g~

+ ex/~/~)O(e) de

d~

× ~,,,-l(u +

k)4)(~)4~(h)dgdh,

(2.11)

In many cases the integral (2.11) will be difficult to compute and so we also provide several single and double integral lower bounds for (2.9) in the Appendix that can be either determined from existing programs and tables or can be programmed easily. Remark 2.2. While the analog of S~ can be developed for this model, it also depends on 7l~ and ?,.~. Thus we restrict attention to the usual ANOVA moment estimator (see pp. 299 of Milliken and Johnson 1984, for example). We denote this estimator by SZs; it is used to perform hypothesis tests for split-plot designs. The estimator S~s is unbiased, and r ( c - 1)(b - 1)S~ms/a 2 2~ h a s a chi-square distribution with r ( c - l)(b 1) degrees of freedom independent of Y (again with r(b - 1) fewer degrees of freedom than the analog of St,2).

3. Subset selection procedures for the additive model This section proposes procedures for the additive model (1.5) to select a subset containing the best level of the row treatment factor and a subset containing the best level of the column treatment factor. Throughout we assume that b blocks have been

348

T.J. Santner, G. Pan~Journal o f Statistical Planniny and Inference 62 (1997) 339-363

used in a split-plot experiment with r levels of the whole-plot treatment R and c levels of the split-plot treatment C. Goal 3.1. To select simultaneously a subset of the row levels containing the level associated with R[r] and a subset of the column levels containing the level associated with C[c]. We let CS denote the event that Goal 3.1 is satisfied. Procedures are developed for three cases. Section 3.1 analyzes the case when (~r~o,cr~) is known, Section 3.2 assumes a~2 is known while %2 is unknown and Section 3.3 allows both ao,2 and cruzto be unknown. All the procedures proposed in this section are defined in terms of 'yardsticks' which are multiples of the standard deviations of differences of appropriate treatment means. Let Pl and P2 be defined by Eq. (2.4). From Corollary 2.2, the variance of the BLUE of the whole-plot treatment difference Ri, - R i 2 is Var(Yil

.. -

Y//2 - • )

2 2 2 = 2(~ro,/b + f f 2 / b c ) = ~repl/b

for any il ~ i2 with 1 ~
3.1. Subset selection when (~r~, ~2) is known We seek to determine a procedure that satisfies the following probability requirement. Confidence requirement: For specified P* (1/rc < P* < 1 ) we require that

P{CS I I*} >P*

(3.1)

whenever/* satisfies (1.2). We propose the following class of subset selection procedures: Procedure RAt Include level i of the row factor in the selected subset if Yi.. > Y[r].. - hRa~pl/v~

(3.2)

and include level k of the column factor if

Y..k > Y..tcl - h c ~ W z / v ~ where hR and hc are determined by (3.6).

(3.3)

T.J. Sanmer, G. Pan~Journal of Statistical Planning and InJerence 62 (1997) 339 363

349

We show that any configuration p satisfying homogeneity of main effects, i.e., R1 . . . . .

Rr

and

C1 . . . . .

Cc

(3.4)

is least favorable for Procedure RA1. Theorem 3.1. Under Model (1.2)/(1.4), P { C S

p} attains its minimum whenever #

satL~:fies (3.4). Proof. Let Y(O.. denote the sample mean associated with level R[i] of the row factor and E.(,) be the sample mean associated with level C[k] of the column factor. Then P{CSIp} = P

-Y(r).. > Y(i)..

hRpxa~, ~/~ for i = 1. . . . . r -

Y~.(c) > Y..(k)

hcpxaa x/b fork=

1 and

1. . . . . c - l j

= P{(Y(r).. -- R(r)) > (Y(i).. - R(i)) + (R(i) - R(r)) hRPl 0~:

for i =

1. . . . . r - - l a n d

(Y..(c) - C(c)) > (Y-.~ - C(k)) + (C(,~ - C(c))

hcp2a~.

=P

Wi-Wr<

for k = 1. . . . . c -

1I

x/b(R(,.) - R(i)) + h R p l for i . . . .1,.

r--I

Z k - Z c < x/b(C(c) - C(k)) + hcp2 for k = 1.... ,c ('i t

> ~ P { W , - - W , . < h n p l for i = l , . . . , r Zk-Z~.
for k =

l ..... c-

and

1~

J

1 and 1}

= P{CS I P* } where p* satisfies Rl . . . . . Rr and (71 . . . . . distributed as in Corollary 2.3.

Co, and (Wi . . . . . Wr, ZI . . . . . Z~) is

To achieve the design requirement (3.1), hR and hc must satisfy P{ Wi - W~ < hRpl for l <~i < r and Z k - Z , . < h c P 2

for l ~ < k < c }

=P*. (3.5)

From Theorem 2.3 and Corollary 2.4, (3.5) is equivalent to P { U i < h R for l ~ < i < r }

x P { V , < h e for l ~ k < c }

=P*

(3.6)

350

T.J. Santner, G. Pan~Journal of Statistical Plannin9 and Inference 62 (1997) 339-363

where (U1 . . . . . Ur-l ) and (V1. . . . . Vc-1 ) are r - 1 dimensional and ( e - 1)-dimensional multivariate normal, respectively, with zero means, unit variances and common correlation ~. 1 Theoretically, any (hR, hc) satisfying Eq. (3.6) produces a valid selection procedure. O f course, the choice of (hR, hc) must be determined before looking at the data. Two choices of (hR, hc) which are computationally simple to implement are the following.

Method 1: Equal length row and column factors Set hR = hR = h where h solves P{U,.
xP{V~ < h for l ~ < k < c } = P * .

This method produces a selection procedure with the 'yardsticks' having the same number of standard deviations for treatment differences in the row and column factors. The value h can be computed by trial and error.

Method 2." Equal (least favorable) selection probabilities for R and C Choose hR and hc independently to satisfy

P { U i < h R for l ~ < i < r } = v @ - 2 = P { V k < h c for l ~ < k < c } .

(3.7)

Thus hR and hc can again be determined from standard tables or programs.

2 is known and ~r~ 2 is' unknown 3.2. Subset selection when cr~ We seek to determine a procedure that satisfies the following probability requirement. Confidence requirement: For specified P* (1/rc < P* < 1) we require that P{CS [/l, a 2} >~P*

(3.8)

whenever/~ satisfies (1.2) and a~ > 0. We propose the following class of procedures to achieve (3.8): Procedure RA2 : Include level i of the row factor in the selected subset if Y,... > 7[r].. - hRSrmspl/v~.

(3.9)

and include level k of the column factor if Y...,,, > Y...tcl - hcS, m s P 2 / v ~

(3.10)

where h8 and hc are determined by (3.12) and 52s is the ANOVA estimator (2.6).

T.J. Santner, G. Pan~Journal of Statistical Planning and Inference 62 (1997) 339 363

351

By an argument similar to that in the proof of Theorem 3.1, it can be shown that P { C S I ~ , o "2} for RA2 is minimized by the equal main effects configuration (3.4) and is independent of o-~ 2 under this condition. Calculation of the probability of correct selection under this configuration shows that hR and hc must be chosen so that

p { Wi_W,.
s, l ~ < i < r ;

Zk-Z~.< hcP2Srms, 1-<.k
From Corollary 2.4, Eq. (3.11) is equivalent to

P s < h R for l<.i
=

(3.12}

where (c - 1)(rb - 1) $2 "" )d(c-'21)(~b-I~ is independent of (UI . . . . . U r - I ) and (VI .... Vc ~) and the latter are independent multivariate normal vectors having dimensions r - 1 and c - 1, respectively, with zero means, unit variances and common correlation L This follows, in part, from the facts that (c 1)(rb 2 2 "~ is independent of (W1 . . . . . Wr) and (Z1 .... ,Ze) (Remark 2.1). It can be shown that the left-hand side of (3.12) is equal to +~,o

) ]4)(y) dy × .

o~ 4~c-1

y + V/(C- 1 ) ( r b - 1) q~(y)dy

f,.(t)dt

(3.13)

where 4~(.) and 4)(') are the cumulative distribution function and the density function of the standard normal distribution, respectively, v = (c - 1)(rb - 1 ), and L,() is the density function of the chi-square distribution with v degrees of freedom. A conservative, easily implemented procedure can be derived that uses existing tables or FORTRAN programs for the equicoordinate critical points of the central multivariate t-distribution with constant correlation ½. From Lemma 2.2.1 of Tong (1980) (wilh g x(x, y, s) = I [x/s <~u] and g2(X,y, s) = I [y/s <<.v]) we have

p{Ui-
l~
I<.k
>~P { ~
l<~k
Therefore by setting P

~

---

we obtain conservative hR and

= x/P - £ = P

hc

< h c for l ~ < k < c

,

(3.14)

from critical points of the multivariate t-distribution.

352

T.J. Santner, G. Pan~Journal of Statistical Planning and Inference 62 (1997) 339-363

Remark 3.1. If 7/~ is known, Procedure RA2 c a n be improved (in the sense of increasing the d.f. associated the estimator of a~) by substituting Sv given in Eq. (2.5) for Srms in (3.9), (3.10) and (3.11) and substituting ( r b c - r - c + 1) for ( c - 1 ) ( r b - 1) in all the formulas in this subsection. 2 and ~r~ 2 are unknown 3.3. Subset selection when both a~o

We wish to determine a procedure that satisfies the following probability requirement.

Confidence requirement: For specified P* (1/rc < P* < 1) we require that P{CS i/t, a2, a~} ~>P*

(3.15)

2 2 whenever p satisfies (1.2), ao~ > 0 and ~r~ > 0.

We propose the following class of subset selection procedures:

Procedure RA3 Include level i of the row factor in the selected subset if Y

>

-- heS +w V /Cb

and include level k of the column factor if Y..k > Y..[c] - hcSrmsXf2/rb where hR and hc are selected to satisfy Eq. (3.16), S2rmsis the ANOVA estimator (2.6), and

S:2w : b )< ~ ~ (-Yi.k -- Yi.. -- Y"k "Jr-Y...)2/o i=1 k = l

based on v = (b - 1)(r - 1) d.f. is the ANOVA estimator of c x %2 +a~. The estimator S;+w 2 corresponds to the block by whole-plot treatment row for the fitted model (1.5). The values of hR and hc required to implement Procedure RA3 are any solutions of the following equation: P{Ui < h e for l ~ < i < r } × P { ~ < h c for l ~ < j < c } = P *

(3.16)

where (UI . . . . . Ur-I) has the ( r - 1)-dimensional multivariate t-distribution with zero mean vector, unit variances, common correlation ½ and ( b - 1 ) ( r - 1) d.f. and (V1..... Vc-1) has the ( c - 1)-dimensional multivariate t-distribution with zero mean vector, unit variances, common correlation ½ and ( c - 1 ) ( r b - 1) d.f. As in the previous sections, Eq. (3.16) does not uniquely determine he and hc. Both methods suggested above for choosing he and hc to satisfy (3.16) can also be

T.J. Santner. G. Pan~Journal o]'Statistical Plannin9 and InJerence 62 (1997) 339 363

353

employed here. One method is to select hR = h e ; the second is to require equal selection probabilities, i.e., P { U i < h R for l < ~ i < r } = v ~ * = P { V j < h c

for l ~ < j < c } .

(3.17)

4. Subset selection procedures for the non-additive model This section treats the non-additive case of (1.4). It proposes procedures fbr selecting a subset of the treatment combinations so that the subset contains the treatmenl combination associated with #[rc] with a prespecified probability. As in Section 3, we suppose that b blocks have been used in the split-plot experiment with r levels of the whole-plot factor and c levels of the split-plot factor. Goal 4.1. To select a subset of the treatment combinations containing the treatment combination associated with #[rd. Section 4.1 proposes a procedure for the case when (o-2,,~r}) is known, and Section 4.2 assumes a(,, 2 is known while cry, 2 is unknown. In all the subsections we will use d = x / ~ as the 'yardstick' when defining selection procedures. This is because Corollary 2.6 states that the BLUE of t~i,k, - #i2k2, Yiil.k, -- ~2"k2, satisfies -/ (Yco 4- 1 )o-2d 2 Var(gi, k, - ~2.k~) = 2a2/b = a~d 2

for il 5g i2, for i, 7~ i2.

%

4.i. Subset selection when (a,,~, 2 a~: 2 ) is known

We seek to determine a procedure that satisfies the following probability requirement.

Confidence requirement: For given P* (1/rc < P* < 1) specified prior to the start of experimentation, we require that P{CS I ~}>~P*

(4.1)

tbr all ~. We consider the following class of subset selection procedures:

Procedure RNI: Select treatment combination with level i of the factor R and level k of the factor C if g.k > max{Y//*.e*: l < . i * ~ r ,

l~
(4.2)

where h satisfies (4.3) (or h satisfies (4.4) for a conservative procedure). We now prove that homogeneity configurations are least favorable for Procedure R,vl.

354

T.J. Sanmer, G. Pan~Journal of Statistical Plannin 9 and Inference 62 (1997) 339 363

Theorem 4.1. U n d e r M o d e l (1.4), P{CS I/t} f o r P r o c e d u r e R N l a t t a i n s its m i n i m u m w h e n p[1] . . . . .

P(rcl.

Proof. Paralleling the notation introduced in Section 3.1 we let ~(q) denote that ~ij associated with the qth ordered treatment combination mean #[q] for q = 1,..., rc. Also let (U1,..., Urc-1) have the multivariate normal distribution stated in Eq. (2.10). Then P{CS

) p } ~-~ e{~(rc) > ~(q) - haffe, q = 1 . . . . . rc -

1}

= P{~fi(rc) - #Ire] > P(q) -- /2[q] -~ (,//[q] -- ]~[rc] ) -- hda~,

1}

q = 1. . . . . r c -

p [ v~[(~(q) - ~lql) - (~(rc~ - Ulr~)]

(

O"e

<

= 1,

.,rc

J

) P { U q "( v / 2 h , q = 1 . . . . . r c -- 1}

= P{CS]~*}

where p* satisfies /~11 . . . . .

P[rc¿-

[]

Notice that Procedure RN1 requires the knowledge of yo, since the covariance matrix of (U1 . . . . . Ur~_~ ), proportional to (2.7), depends on 2o,. Thus an exact procedure uses h to be the solution of P(Ui < v~h,

i = 1,...,rc-

1) = P*.

(4.3)

to achieve the requirement (4.1) where (UI . . . . . U ~ _ j ) has the multivariate normal distribution (2.10). A conservative procedure uses (4.4)

h = u/x~2

for any u that makes the right sides of any of Eqs. ( A . 3 ) - ( A . 5 ) equal to P*. 4.2.

S u b s e t s e l e c t i o n w h e n 7o~ is k n o w n

a n d a 2 is u n k n o w n

We seek to determine a procedure that satisfies the following extension of (4.1).

Confidence requirement: For specified P * ( 1 / r c < P* < 1) we require that P{CS I # , a 2 } >~P* for all /~ and a 2 > O.

(4.5)

T.J. Sanmer, G. Pan~Journal

of Statistical Planniml and In/erence 62 (1997) 339 363

355

We propose the following class of statistical procedures: Procedure RN2: Select treatment combination (i,k) if ~.k > m a x { ~ * 4 * : 1 <.i* <~r, 1 ~ k * -<.c} - hdSn~

(4.6)

where S ~ is defined in Remark 2,2 and h satisfies (4.7) (or h satisfies (4.8) where u* is defined by any of ( 4 . 9 ) - ( 4 . 1 1 ) for a conservative procedure). By an argument similar to that in the proof homogeneity configurations #Ill . . . . . ff[rd and that P{CS I/~,a~} is independent of a~2 for under the least favorable configuration is given

of Theorem 4.1, it can be shown that are least favorable for Procedure R.~,2 such configurations. The P{CS I/l, a2} by the left-hand side of

P{Ui < x/bhdS, i = 1. . . . . rc - l} =: P*

(4.7)

where r ( c - 1 ) ( b - 1 )S 2 ~ X2(c- l)(b- 1) is independent of (U1 . . . . . U,.c-1 ). This probability can be written as a four-fold integral using Eq. (2.11 ). However, due to its complexity we focus on conservative procedures that make use of existing tables and FORTRAN programs. Conditioning on S and applying Theorem A. l we have the following conservative choice of constant:

(4.8)

h = u*/'~d

where u* is determined by

Jo

(7

"

(v)(1 +7o,)

c,.,-~

u +x

)

d4)(x)

~)c--1 ~

U*-[-X d@(x)

)

(4.9)

x J ; ( y ) dy = P*, or

/o {i_.S +"

,,

xq)c-l(u'~+w)d+(x)deb(w)}J~,(,)dy=P

(4.10)

*,

or

+

oo

q~c-i

t, V T

+x

~b(x)dx- 1 fv(y)dy

-

P*.

(4.1))

where v = r ( c - 1 ) ( b - 1 ), and f,,(.) is the density function of the chi-square distribution 2 and d~2 are both unknown, approximate procedures with v degrees of freedom. When a<,> can be obtained by substituting the estimated 7,, into the procedures in this subsection.

T.J. Santner, G. Pan~Journalof Statistical Planning and Inference 62 (1997) 339-363

356

5. E x a m p l e

and discussion

5.1. Example W e illustrate the m e t h o d o l o g y p r o p o s e d in this paper by applying the subset selection procedure ~A3 to analyze a generic e x p e r i m e n t with r --- 3 levels o f a w h o l e - p l o t ( r o w ) factor R and c = 4 levels o f a split-plot ( c o l u m n ) factor C run in b = 3 blocks. This particular procedure assumes no information regarding the m a g n i t u d e o f the variance 2 , but does assume additive factors. o f the c o n f o u n d i n g factor, ~r~o Suppose that Tables 1-3 list the ~jk and the marginal sample m e a n s for R and C, respectively. estimate (2.6) o f a~2

W e apply the p r o c e d u r e R~3 with P * = 0.80. The A N O V A c o r r e s p o n d i n g the M o d e l (1.5) is b

2

Srm S =

~-~(Yijk i=1 j=l

--

Yi.k

_

Y.j.+Y...)2/v

6.11

=

(2.47) 2

k=l

Table 1 Observations from a split-plot experiment Level of C

Block 1

Block 2

Block 3

1

2

3

4

1

2

3

4

1

2

3

4

30 34 29

35 41 26

37 38 33

36 42 36

28 31 31

32 36 30

40 42 32

41 40 40

31 35 32

37 40 34

41 39 39

40 44 45

Level of R 1

2 3

Table 2 Mean responses fii.. for the three levels of the whole-plot factor Level of R 1

2

3

35.67

38.50

33.92

Table 3 Mean responses 32.k for the four levels of the split-plot factor Level of C 200

225

250

275

31.22

34.56

37.89

40.44

T.J. Sanmer, G. Pan~Journal of Statistical Planning and lnj~,rence 62 (1997) 339-363

357

with v = (c - 1)(rb - 1) = (4 - 1)(3 x 3 - l) = 24 degrees of freedom. Similarly, s~+w = 9.07 based on (b - 1)(r - l) = 4 d.f. By trial and error we find that hR = 1.91 and hc = 1.77 satisfy Eq. (3.17), i.e., P { U ~ < I . 9 1 for l ~ < i ~ < 2 } = ~ = 0 . 8 9 4 = P { b < l . 7 7

for l~
where U = ( U 1 , U2) has the multivariate t-distribution with 4 d,f., V =(V1, V2, b~) has the multivariate t-distribution with 24 d.f. and both U and V have zero mean vectors, i unit variances and common correlations ~. Hence we select those levels of R for which ¥~ > 3 8 . 5 0 -

1.91x 9x/-K~.07x 2 x / ~ ( ~ = 3 8 . 5 0 - 2 . 3 5 = 3 6 . 1 5

and those C for which y./. > 4 0 . 4 4 - 1.77 x ~

x X / ~

= 4 0 . 4 4 - 2.06 = 38.38.

Thus procedure RA3 selects the subset containing the single treatment combination consisting o f level 2 o f R and level 4 o f C.

5.2. Discussion The techniques described in this paper can be extended in several ways. One ex.tension is to allow other types of randomization restrictions and a second is to allow treatment designs with three or more factors. To illustrate the former, we mention one natural randomization restriction in some detail, that o f the strip-plot experimental design. For specificity, consider an agricultural experiment in which there is a randomization restriction in space. A strip-plot design run in b complete blocks has experimental units arranged in b rectangles, each rectangle having r rows corresponding to the r levels of the row factor and c columns corresponding to c levels of the column factor. The levels of the row factor are randomly assigned to all of the experimental units in a row, and the levels of the column factor are randomly assigned to all o f the experimental units in a column. Thus there are two potential sources of confounding - - that due to rows and that due to columns. The two confounding sources are embodied in the standard model

~/k = I~i~ + fij + wij + t/jk + ~:ij~r

(5)

which represents the response for the ith level of the row factor and kth level of the column factor in the jth replication for 1 <<.i<-K.r, 1 <~j<~b and 1 <~ik<<.c. In this model, wq is the potential confounding term associated with the rows; rljk is the potential confounding term associated with the columns; fij is the block effect and ~:~1~ is tile measurement error. We assume that the effects {&}j, {wij}i,y, {tljk}j,~, and {~:Vk}i,j,k are mutually independent with

flj i.i.d. N(O,a~)

for l < ~ j ~ b ,

358

T.Z Santner, G. Pan/Journal of Statistical Planning and Inference 62 (1997) 339-363 wi/ i.i.d. N(0, a 2)

for l<~i<~r and l<~j<~b,

tljk i.i.d. N(0, a 2)

for l<.j<~b and l<,k<_c,

eijk i.i.d. N(0, a f )

for l<~i<~r, l<~j<~b, and l<~k<~c.

The terms of the covariance matrix of the BLUEs of the main effects in the additive model for the #ij involve additive terms in the extra unknown variance component a~2; the same is true of the terms of the covariance matrix of the BLUEs of the /~6 in the non-additive model. This is the critical feature that permits the construction of theoretically analyzable selection procedures for this design. The details of this construction will not be given because of its similarity to the procedures provided in Sections 2 - 4 . The second extension mentioned above is to allow more than two factors. It is possible to construct selection procedures which have three or more factors allowing for a hierarchical randomization restriction. For example, in a three-factor experiment with factors R, C and L ('row', 'column' and 'layer' factors), we could force all levels of C be run once a level of R is fixed and then force all levels of L to be run once the level of C is determined. Under additivity, selection procedures for the best R, C and L levels could be developed for a balanced experiment run according to such a design. In conclusion, balanced randomization restricted designs occur often in practice. Meaningfully formulated subset selection procedures can be constructed for the data from such experiments in either the case that the factors are known to be additive or not.

Acknowledgements The authors wish to thank the referees for their detailed comments on an earlier version of this paper.

Appendix A Proof of Theorem 2.1. It is straightforward to verify that ~ is the ordinary least squares estimator (OLSE) of m under (1.2)-(1.4). We apply Theorem 2 of Zyskind (1967) which gives equivalent conditions under which the OLSE of a set of linear combinations of parameters is the BLUE of those quantities. One such condition requires the identification of the projection matrix onto the space spanned by the mean of the data. Define P =

Ir ® (lbcl~c) +

(lrbl'~b) ® Ic

roc

For an arbitrary real matrix W, let cg(W) denote the linear space spanned by the columns of W. It is easy to calculate that p2 = p and P~ = P; therefore P is a projection

72J. Sanmer, G. Pan~Journal of Statistical Planning and InJerence 62 (1997) 339 363

359

matrix. Furthermore P is the projection matrix onto ~ ( X ) , where X is the design matrix for (1.5), since P maps ~b~ onto ~ ( X ) , this follows from r a n k ( P ) = trace(P) -r + c - 1 = rank(X), and the fact that the columns of P are linear combinations of the columns of X fig(P) C_~ ( X ) ) . A straightforward calculation shows that the covariance of Y satisfies PZ~ = S:P.

and thus the OLSE of m is also its BLUE by Zyskind's characterization.

IN

Proof of Corollary 2.4. Let rl2 _ C7l~ + c7o~ + 1 c '

)q =

] I,'2

cYl~ cTls + c7~..~+ I

If H=qt((l

- 2 2 ) b'2 x L , - ) q x It)

and X ~N(0r+l,lr+l), then it is straightforward to verify that Therefore P{Wi-W~
i = 1 ..... r - l }

= P{~-

T,. < w , i = 1 ..... r - l }

=P{r/l@l-22(Xi-Xr)
= P

=P

Ui < rll

{

Ui<--,

w

/47- ]

i--1 ..... r -

T =HX

~

Nr(0, S~t).

1}

-2~-' i = 1 .... , r - 1

--/~1)

i = 1 .... , r - I

}.

L5

Pl

Proof of Theorem 2.5. It is straightforward to verify that ~ is the ordinary least squares estimator (OLSE) of p under the general non-additive model (1.1)-(1.4). We again apply Theorem 2 of Zyskind (1967). It is straightforward to check that P = ~11rG((lbl~)®I,.) is the projection onto the nonadditive column space X since p 2 = p , p , ~ p , r a n k ( P ) = trace(P) = rc = r a n k ( X ) , and ~ ( P ) C ~ ( X ) where ~ ( W ) denotes the space spanned by the columns of the matrix W. Lastly a straightforward calculation shows that P2~z ----2~zP. Thus the OLSE o f p is also its BLUE by Zyskind (1967).

360

T.J. Santner, G. Pan~Journalof Statistical Planning and Inference 62 (1997) 339-363

A.1. Lower bounds for the probability (2.11) Theorem A.I. Suppose that (UI . . . . . Urc--]) has the joint distribution given in Eq. (2.10), then (A.2), {

P { U I < U . . . . . Urc-l

(A.3)>~

(A.1)

(A.4), (a.5),

where f

q~C(u+ V/1 + yo)y + x/-~x)qb(x) dx

)r,

qb(y) dy

oo

+

/7

qbc-t(u + x ) ~ ( x ) d x - l

(A.2)

oo

(]~(r--1)c U-~- x / ~ x + y <,-~

q~c_l(u+ vl(o(x)~(y)dxdy "

v/1 + 70,

.,-00

+c~ ~ ( r - - l ) c

U

x)dx

+ X

x

~/1 + 7,, +oo q~(r-1)e

J_

(

u

+x

(x)dx +

/'-

(A.3)

q:-~(u + x)dp(x)dx

(A.4)

q~c-l(u + y)dp(y)dy - 1.

(A.5)

Note that the single integrals in (A.2), (A.4) and (A.5) are of the form ~oo q~m--1 (Y + v) q~(y) dy,

f_

O<3

which can be obtained from the widely available tables and programs of the equicoordinate critical point of the multivariate normal distribution using the relationship

P{ZI <~c, Z2 <~c..... Zm_l ~
f

~m-l(z + cv~)cb(z)dz

(A.6)

C2~

(Bechhofer and Dunnett, 1988; Dunnett, 1989). Here the vector (ZI . . . . . Zm-1) has the ( m - 1)-variate multivariate normal distribution with mean vector zero, unit variances, and common correlation ½. While more difficult to compute, the double integrals in (A.3) and (A.2) can be found by using standard quadrature routines such as those available in IMSL.

Proof of Theorem A.I. Bound (A.2) is obtained by applying the Bonferroni inequality to (UI ..... Urn-l) as P { U I < u . . . . . Urc_ 1 < u } ~ P { U

I < u,..., U(r_l) c < u}

+P{U(r-t)c+l < u..... Urc-t < u} - 1.

(A.7)

T..L Santner, G. Pan~Journal o f Statistical Planning and lnJerence 62 (1997) 339 363

361

The first probability can be calculated from the representation (S~ -

~S1

-

X/1 + 7,,,r, . . . . S~ - ~ S ,

- x / l + ),(,,T,S~ -- ~ $ 2

- . v / 1 + > o T , . . . , S , 2 - x/~,'~S2 - v/1 + 7,,,r, . . . . S ~ - ' - ~ S , . _ , - - X ~ l q- )'roT, . . . . S r - 1 - ~ - ~ S r - 1

- ~

÷ )'ro T )

for (UI < u .... , U(,._ 1)~.) where S~ . . . . . s,5-~,& . . . . . S r _ ~ , T are i.i.d, standard normal distributions. The second probability is standard. Both (A.3) and (A.4) are proved by applying Slepian's inequality to reduce the number of off-diagonal blocks in X from three to two. The three covariance elements in X are 27,~,+ 1, 7{,,+ 1 and 1. Bound (A.3) is obtained replacing each 2 > , + 1 in X by 7¢,, + 1. Specifically let (I~ . . . . . V,rc_l)~'.~Nrc_l(O,.c

(A.8)

l , - y'* )

where Z* =2(7o,I27 + Z2) , with 1

1 2

1

f

1

1

2

OcOc- 1

1

1

1

0~, 0 C_ 1

Oc- ~O'c

0~._ j O'c--|

OcOc- I

1

zT= 5

oc_~o; oc_,o;

!

(rc-- [ )×(rc--

[)

and X2 defined by (2.8). Then Z" and ,Y* have identical diagonal elements but each off-diagonal 2y,~lclc in 221 has been replaced by 7~olc1~ in I;*. Hence P { U l < u . . . . . U r ~ - I < U} > P { V ~ < u . . . . , V,.~_I < u}.

To evaluate the right-hand probability in (A.9), let $1 . . . . . S(,._I )c,S, T1 . . . . . Trc-1, T

have i.i.d, standard normal distributions. Then it straightforward to verify that

T(r l)c+l - T, .... Trc-1 - T)

(A.!))

362

TJ. Santner, G. Pan~Journal of Statistical Planning and Inference 62 (1997) 339-363

has the same distribution as (V1 .... , Vrc-l). Therefore

P { V~ < u .... , Vrc-l < u} = P { x / ~ S i

- v / ~ S + Ti - T < u, i = 1 . . . . . ( r -

Ti-T
i=(r-

1 ) c + l ..... rc-

1)c;

1}

and conditioning on S and T, we obtain (A.3). The bound (A.4) can be obtained by replacing each off-diagonal 1 in Z* by a 0 giving the variance-covariance matrix

Z'** = [ ( l + yo,)I(r-')c + ( l + yeo)l(r-)'cl~r-l)cOc-lO(r-),c

lc_,O(r-')cOc-'+lc-, l'c_ l ] "

A mean zero multivariate normal distribution with covariance matrix Z** is obtained as follows. Let

SI . . . . . S(r-1)c,S, Tl . . . . . T(rc-1), T be i.i.d. N(0, 1). Then (41

q- ~o,S1 - 4 1 Jr- "~coS. . . . . 4 1 q- ])wS(r-1)c - 4 1 -}- ycoS,

T{r-1)c+l - T, .... Trc-1 - T)' has multivariate distribution with zero mean and covariance matrix Z**. Thus P{Vl < u . . . . . V~_~ < u}

T(r-l)c+l -- T < u . . . . . Trc_ 1 - T < u}. Conditioning the right-hand side probability on S and T yields Eq. (A.4). Lastly, Eq. (A.5) is obtained by applying the Bonferroni inequality to (Vl . . . . . V~_~ ) in the proof of (A.3)

P{V~ < u . . . . . V~,._i < u} >~P{ Vl < u , . . . , V{~_,)c < u} -~P{V((r_l)c+l < g . . . . . V(r¢_l)
[]

(A.10)

The proof of the Theorem A.1 provides some insight into the relative accuracy of the bounds. If y~, = 0 the bound (A.3) is exact since all terms involving multiples of )'o, are zero. As ~o,--+cxz, the Bonferroni bound (A.2) is exact since the correlation I/(27o, + 1) converges to 0 which is the assumption under which (A.7) becomes an equality. Thus we conjecture that (A.3) is superior for small To, and (A.2) is superior for large 7o,. References Anderson, V.L. and R.A. McLean (1974). Design of Experiments: A Realistic Approach. Marcel Dekker, New York.

T.J. Santner, G. Pan/Journal of Statistical Planning and lnJerence 62 (1997) 339 363

363

Bechhofer, R.E. (1954). A single-sample multiple decision procedure for ranking means of normal populations with known variances. Ann. Math. Statist. 25, 16-39. Bechhofer, R.E. and C.W. Dunnett (1986). Two-stage selection of the best factor-level combination in multi-factor experiments: common unknown variance. In: C,E. McCullogh, S.J. Schwager, G. Casella, and S.R. Searle, Eds., Statistical Design: Theory and Practice. A Conf~'rence in Honor of Walter 72 Federer. Biometrics Unit, Cornell University, Ithaca, NY, 3 16. Bechhofer, R.E. and C.W. Dunnett (1987). Subset selection for normal means in multi-factor experiments. Comm. Statist. Theor. Meth. 16, 2277-2286. Bechhofer, R.E. and C.W. Dunnett (1988). Percentage points of multivariate Student t distributions. ln: Selected Tables in Mathematical Statistics, Vol. 11. American Math. Soc., Providence, RI. Bechhofer, R.E., T.J. Santner and D.M. Goldsman (1995). Designing ExperimentsJbr Statistical Selection, Screening, and Multiple Comparisons. Wiley, New York. Box, G.E.P. and S. Jones (1992). Split-plot designs for robust product experimentation. J. Applied Stati~t. 19, 3 26. Cantell, B. and J.G. Ramirez (1994). Robust design of a polysilicon deposition process using split-plot analysis. Quality Reliability Eng. lnternat. 10, 123-132. Dunnett, C.W. (1989). Multivariate normal probability integrals with product correlation structure. App Statist. 38, 564-579. Fabian, V. (1962). On multiple decision methods fbr ranking population means. Ann. Math. Statist. 33, 248-254. Gregory, W.L. and W. Taam (1996). A split-plot experiment in windshield fracture resistence test. Qualit~ Reliability Eng. lnternat. 12, 7%87. Gupta, S.S. (1956). On a Decision Rule for a Problem in Rankin 9 Means. Ph.D. Dissertation (Mimeo Set. No. 150). Institute of Statistics, Univ. of North Carolina, Chapel Hill, NC. Gupta, S.S. (1965). On some multiple decision (selection and ranking) rules. Technometrics 7, 225 245. Milliken, G. and D. Johnson (1984). The Analysis ~ Messy Data: Vol. 1, Experimental Data. Van Nostrand Reinhold, New York. Pan. G. and T.J. Santner (1997). Selection and ranking procedures for split-plot experiments in which a larger response is better, in preparation. Tong, Y.L. (1980). Probability Inequalities in Multivariate Distributions. Academic Press, New York. Wu, K.H. and S.H. Cheung (1994). Subset selection for normal means in a two-way design. Biomctrieal J 36, 165-175. Zyskind, G. (1967). On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models. Ann. Math. Statist. 38, 1092 1109.