NATIONAL BUREAU OF STANDARDS REPORT

1602

SOME RELATIONS AMONG THE BLOCKS OF SYMMETRICAL

GROUP DIVISIBLE DESIGNS

by

W, S. CONNOR

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. As tin. Acting Director

THE fcJATIONAl BUREAU OF STANDARDS

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title. A brief description of the activities, and of the resultant reports and publications,

appears on the inside of the back cover of this report.

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2. OPTICS AMD METROLOGY. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

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Missile Instrumentation. Technics! Services. Combustion.
NBS PROJECT NBS REPORT

1163-11-1107 17 April 1952 1*02

SOME RELATIONS AMOWG TE1 BLOCKS OF SYMMETRICAL

GROUP DIVISIBLE DISIGHS

by

W* S e Connor

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

This report is issue

in any form, either ii frost the Office of th

tion, reprinting, or reproduction

permission in writing is obtained

Washington 25, D. C.

FOKEWORD

-fjie combinatorial problem which is studied in

this paper is of interest in the design of experiments,,

tise paper considers the way in which the blocks

of a symmetrical Group Biyisifole Incomplete block design 9 with parameters t 9 uf! xi

g p

and _JI «,* oosiaeoted

by common Structural and cfiara©tcr.r.3.sti©

aitrises are defined, tad a relation among the® is

exhibited* It is shown that if a solution exists *©?

a giwen set of parameters which are such that r # Ap

p and ra «*• and *21 *

*" «A

g relatively

primee t£en the interchange of Sleeks Vith treatments

yields a ©elution which corresponds to the given set

of parameters* v

<| 3 H0 Curtiss

Chief? national Applied

lathem®tie's haborator ies

Ao ?<, hstin

Acting Director

Mationnl Bureau of Standards
.

.

•

'
mm rilatiohs mom tbs wuocm of symmetrical

OEOGP DIVISIBLE BISIGNS

io Connor « N&tiosaal Bureau of

1 3 BVWAAWfo It im well known that if ©very pair of

ireaiseat© in a syaitietrlcal Balanced iseojaplei® block design

o®c«8*@ in A blocks* then ©very- two Mocks of the design haw

A tr@ata@ats in common 0 Io this paper it will b© shown that

a somewhat similar property holdsi for syssetrieal Group Divisible

desiga®* la the eoursw of the investigation there will fe#

introduced certain Matrices which mi*# of intrinsic inter©®?: »

2, IMfBODUCTIOSo Sosa® of the eonblaatcrial properties

of Group Divisible! laebsple&e block design® were considered

in |l|o Her© we shall a@c4 the definition of Group Divisible

design® cad the three gl»*i into which they fall* A®

incoaplete block design with v treatments ©acts replicated r

tin®© 1b b block© of sis® k is said to b© Group Divisible CGB) •

if the treatment® ®s® be divided last© a groups* esefc with

a treatments* m that the treatsents belonging to the earn®

group occur together 1® blocks cad the treatsents belonging

to different groups occur together In A* block®*

The three exhaustive and mutually exclusive ©las©®@ into which

th© GD d®@ig-n® fall ©re as follow® s

Cm) Singular 00 designs eharaeteriaed by r »^a ®- 0$

Cb) Seal -regular GS designs characterised by r > G*

%M© work wn© begun while th® author was at th® University

of North €^r©lina„
<
- 2 -

'fk - « 0 g and

(c> Ragwlas" CIS design® characterised by r - A, > O s rk - >0,

In this pap@? v# ©hall ©ttsdy ela®@@@ Cb» and C<§)) for

the mfmmtwiml m~m® 9 that im v the ©a©® wb#n r « k, ow

@qwival©tttly, b « ¥ c

So YHB IMCIPMCi M.P gTRfJCggBAL MATEX CBS B la |2j thsrs

m© dtfinad the structural s^atris for Balanced i»@©npl®te fel©@.k

design® 0 W@ sg©w shall define the i®ci<s8®®e© satrix-p sad two

structural matrices for Giv fesigaso

L-t-t as e©nsid<ir first ths incidence maty is of © G© design*

©li 0 0 0 0 0 Ei

$

C3oi:

X

%1 © © o © O ®¥b

|

where the rows represent tre&tseat® , tin© ©©lam®© represent

blbcka, E&d njj • 1 or 0 &©@®yding ©s th@ i-th treatment does

v # or d®@@ not occur in the j»th block » From the conditions

satisfied by the design it is @a©y to see that

h

(3 0 2$ *L» ® y # Ci 83 1» oo:>

8 »K

J-l J

and

b

1Z !»ijOuj - * 3. OB* * 4 (S.SJ
*
•» S

aceoy^ing mm th© i-tb &adl u-th (i # u} da l»@loiag or

do not fe@l©ffig to the group 0 'Throughout th® pap®y lot m adopt th® ooav@®tioa that

th® a(w~l) o 1, a(w<-l) ^ 2 S ° ° ° e ssw ©Mil to

th© w-th groap , Cw®l „ ° ° 0

» anK ftioa

C3o4) Ml 9

~A © « ° 3 A a o

o o

o o

o o

3

B

; B ® ° 0 ° A

wh®y® th® ©X@m@ni@ off th® sse subaatfflx A ay® y is th® pyiaoipal

di&goaal &ad X<, ©l*®*h©y® t aa«8 th® ®3.@»©ffits off th© a ss a imtuisatyiis

B ay© Xj, ®v®yy®h@y@o Off HI 0 coatsis® v ® mu yew® m®d

COXOBSffiSo

KO® ©boos® mif t < h blocks of th© tfcslga , Lot th©

@?ste&fcyis£ off 1 vhioh c©yy®spoads to th#®® t block® fe@ d@a©t®d

by K0

c L®t m fe@ th® aunbsy ©ff tyoata^ats cosmos to th® j-th

aod u«tb sh©©©a blocks, Qj u « l»2t e'*°»t)« *?h©a th® t z t

©ysa®@tyi@ ®a,tris£

CS.5> S* - s'h£ - (BJu»

1® d@ffia®d to h® th© iatcrscotioa atyactayal aatris off th® t

ghosoa blocks o tt© j»tb yo® ©y ®©lc®a off ®©yy«sp©od® to th©
gait

«:•; to®

. .•*. |:

:? )$ { M? ;’,j: = ;

• • " ’

m

- - .-• ' . V f
#

4

Jrth blo$k t$m ®vmm®&±v® ®i@«©ats of th© j *>th

wow ot aoluain giv« th® mmh®w of txmtvmnt® whi®h this !s>!©@k

fet@ in ®owmm with th® 1st, 2ad 9

o ««,

8 t~th ehmm hlmkm0 ¥# asst shall mmthm straatnvsl wteii, Ltt

dsaoi® th© mislwr ©f twmtwmt® twem ths w«th g?r©»p \wM®h

hloipks. J and u h&v© la eonos* rfh©i®

<M>

and

b.7)

w*l

6ju * "jo

M

®4 -5 65 IS ( W~I jj

fei th® Matrix

(3,8)

l 1 i

|

11 ®22 ° ° ” *44

2 2 „2

11 mm *

0 o

" * *tt|

O C O m rM ,.M | ll ®22 0 o , •**(

9

th® protfnet matrix

(3,9) S°

t

- G^Gt » /

$ph@r© th® ©lensnt in th© .1 -th r©$p and u-th <g©lu®». Is th© ©u?s of
I

y -‘S l

..

.

;
- s -

produets of th® au®b®r of tr#ata®»t@ which its© j*th @ho@@m

block sad th® u~th ©h©s@a block ooatain froo each group* W@

4 e 511 CBARACTBRISTIC MTE11, W® shall define aa aaalogu©

of the characteristic aairis which was developed for ®§ii.ia®@©d

incomplete block designs is |S)U For tbs remainder of the

paper* encept for th® last section, w® ©hall restrict w#

attention to the regular GB designs*

Let fho @©Ius$as of H be permute! @© that tho first t

columns correspond to th® t chosen blocks e The® let the

incidence matrix be extended by adjoining t new roes, so that

th@ elements ©f th® j-th ®ij©iss@d w&m m® ser© , except for th®

j-th which im unity* If® thus got

wh&ofe- 1^ is th® miris; of order t, and 0 i® th® t s (b~t)

ser© matrix* Th®®

C4ol>

J

1

lsJVe,re--

(4 © S )

@^@luati©® of I H|N| | lend® to
: p $ :

•

I

‘

’

'

/

'

>

S
- 6 -

<4. 3) | - C*krft*U*-A 1 >*“^tek-vA 8 > B"t"1 IC*i ,

s wfc®r@ th® typical ®l®a®at of is

(4 c 4 > e

j u « (rk-v A s > Cffk©

j u

4- A$&® }

m

* <>i-X 8 )<rk I sii®as,- •*.*’) . u-1

*whm@ « Cr-Tl^k) or -®j.aS according a® j » u os? j ^ u.

The aa&irisg is d®£ is@d as tlx® characterimt Ac matrix of th® t

otopgga blocks0 Th® j-th tow or t2s® j-th column of corresponds

to th© j-th Mock of th® d@@ign„

W®> observe that ths characteristic Matrix is related to

th® two structural matrices is described ia the following

theorem0 Theorea 4 0 lo For the regular GB designs there exists a

(1,1) correspondence ®m©ag thie ©l@®@ni© of th® intersection

v 4* structural mtfis s^, th® group structural Matrix Sj, and th©

csfe&rscteristi© Matrix C^o This correspondent® is gi?©R fey

® rk (rk-v A n

I [ Cr~ A £> X^-Sg 3 * s* C A ? - A ) 8^

\

4> A

g

k

s (r~ A ?

wh@r® l

g

is th® identity matrix of order t, % is th® singular
i
t s t Matrix all of wteo>®@ ar® ©siifcy 9 $m$ &h® otte?

quastitibs ®r@ scalars*

f©r tbs particular ©as© wh©sa r « &, t&@ vain® of

!%Mj | a® gi¥#2a by (4*3) reduces to

1*1*1 1 - r"a(t_1 ) (r»-vA ;s )»-*-llct i ,

wfe@r® the typical @l®n@at of G$ is

a

C4«6) r* Cr®-^

W@ ©ball state am anmlogu® of Tl&eorea 30 1 of (2j 0 Th@

proof is ®@ for that. theorem.

Theorem 4,2 0 If C| i§ th® characteristic Matrix of

®®4 of t blocks ©fe®@©a fro® a r@gulfis? design with pasweteg1 ©

^,fe 8 r,k,m,® 5 h ls and th#n

Ci) lCt l > 0 if t<fe~F,

Cii) |C$f a 0 if ®s®d

is a perfect integral ®qm^@

s » 1HEQUALITIB3 OH @j u FOR RBSULAR 8TRMBTR3CAL OP pBSjGWS o Lft t®l* Than @in@@'th@ factor outsid® of JCjJ in' (4 0 §) i©

positive, it follows from Theorem 4 0 2 that |C

X | - 0 o Hence from

»li) r-2 <t-l> (s._ * 1 )

2v-b-"(r*_» A j

M-bOW-1
*

- .

, I
® ©«

& A oa A

3 3

CSo4|

FffOi C5«3)

Tte>r©y Solo For a regular syaasotrieai G© $@©iga the autfbar

of ir^atasaais <g©is§®oi& to two bl©®k® satlsfios tfe® ia©qa&liti®@

) < s

wh®u Th® ia©q^liti@© as1 ® r$wor8m& «h®o A < A

i

o
'

- •

-

. I
** 9 *”>

S , Til BLOCK STRUCTURE FOR REGULAR S7S3SIETRXCAL O© ©ESXGM8

WHEN r*-v A ^ AND Ai~ MS RELATIWELW PRIMEo W® n@@iS to

®onmid@r th@ 4i@triteration of the tr©at«®st« ^o©tai®©d ia

initial tel©@k ®ssoag tte© ®th®% bloeksi, Lot fe@ &h© numb®*?

of telocto anoag tb© r@a®ini©g (b-l) blo@ki§ wfei@te j ir@a$»@nt§

in ©o®88oe witte Bj_ 0 Th@n flora th© definition of the design v©

obtain

<*.1:

js n 4 « b«l » v-1 , and &*» j

1 j®0

JT « r<k-X£ • r(r-l) 0 j sCJ k

Also ©oa®itar M ® Ju jU^Do* , wtei©te i© twi©@ th© auasb®^

of p^irii of tF@st»®ats of Bj wteiete lio among the otte®gr fol©©ks§ 0 M i® given fey

© m

<«.a> m - X ®ii <*Tx~1,< ^i'15 * X * ®ii *u <*,-*> , W“1 2£.W®1

S]?4®

From C3 0 T5; <5.2},

as. (6.3) X •u<*Xl-l> " M)Aj, cod

n

C®o 4 ]) ^

» (ra-Dra

*,wrt 11 11 *

H®ac@

I! - «B-1 )C A1 h>1 _1) * (a.x)(a)( A H A _1} (60 5)
(

mmm m m u - s

- - -m m m ij |oe \ m i @ t m . m mi « >

si - ® ; Mm • 8 1 1 l i 10

^

:

Iasi - M J .

; Jg*Jt !l •

' $<& M 9 . t&Si :

• 10

. g

- ' ?! « ..

0 \

ix

m

.

I

1.

' ;

' ^ t

- ,

' -

- ‘

. _
- 10 -

Rqiv ©©gMsidtet?

k

C8,6> B J[L 0~A,)CJ~A > sa 4 j ®0 1 * 3

From C6 a l)» CS 0 5>, usd C©oS> mm obtain

CioTI 8 • 0 .

Hen©# thm following l©s^a 0 Lfmma 6,1, If for a regular GO design Sj

denotes the number of feloek® ^hteh have J treatments is @©sse©i?! 5

with a given initial Meek, tbmn

k

Ho® let r

s -vA and A.-A be relatively prim©* It

8

A

8

follow® from (§ 0 S> that eanssot lie in the ©p@as interval (AjjA K

Them ©v#ry ter® of B is 'positive ©r wo« But ©ino® B ® ®„

every t@r® must b@ sero, W© thus get

Theorem 6 n lo If for a regular ©yametrieal 60 design

r*-vA and Ai-A are r©l$tiv@ly prisss®, then umj two blocks

s have ©Ith©r A , or A treatments in eoono&»

1 1

We further observe that even if r$~vA and A.-

S

A

.2 sot relatively prim®, it still rmf not h® possible to ehoose

thv elements of Gfr of (3 , 8 > , subjeet to the restr lotions of

* OuT is <7ct A, cT , (3,7) and (6 0 3), @w@h that s . is integral^ Consider* for
A • $v - $< : V" i v Y

:

:

(

; ••

•

- 1

- 3,’; ...

' • . r -

•'
11

©xaapl®, ih© GD design with parameters v®h®45, r«k®9, aa®3, ia-*1S,

A^®s e A ®1. SeCoFo ©f &*8d A •- A i® 2„ It

! 3 * S

is elear that tin© only positive integer® which satisfy <3*71

m

and C5«2) are 1,1, and 7« But th©n w© must have either £ jj uu

§1 oi* i§„ which correspond respectively to A, and A 0 a §

Mow assume that the condition of theorem 6 J is net, or

ssos9® generally, that positive integers do sot exist which sa@@t

ths restriction© of (3.7), C5 a 2| and l<mmm 6„1 ®nd imply valued

of »j u other than A ^ and A « Then fro® C8»l)» w© obtain

C««8)

whence

CS,9D

n % n

Va

® v-1,

X “

*» r<r*

3 3

a „ « a-1, and

*1

a ^

• C»-l>a,

$

@® that with t to a®y initial block B

?

, thes*© are (n-1]

other blocks which hav© A treatment® in eosasson with it,

1

and Cn-1) a other block® which haw© A treatments in ©oissmon

3

with it, * Fro® (5*3) w® @@@ that

C6ol03)

w w

'll ®JJ

©„_ © 4Jt ® r ^aw®l

isplie® that block® 1 ssid j haw® A

^ treatment® in common, and
. j I

... . ' - Sit I - • I/' ^ 3.-. is -
- 12 r

conversely , But tlten fro® C5o23> and 16,10), it follows that

» n

C6.u> 2~ efj - 2,(8*!'*

W®J r®l

whi©fe implies that » sj^, 0 °

° ,1>K S@s@€^,

if blocks E| Bj have A

j

treatments is common, aad blocks

% and have A^ treatments in ®©stob» than B. and B^ hm@

A 1b common <> W@ thus have

Theorem 6.2, if for © regular symmetrical @© design

r

3-v^ and ®r© relatively prime, then the bloeks

fall into m group© of & Mock© each, which as*© ©uch that any

tw Mock© fro® the earn© group mntmin A treatment® in

©©©moss and any two blocks fro® different groups ©©ntaia A

a treetarnts in @obbooo

As te® to@@is indicated this theorem could h® stated

somewhat mow® generally. .

To THE SEMI-REGULAR CLASS , For this ®Im® i t mSLMtamir

rim flj that Sj — ^- -K „ e-eonsteat r for ^©nc© the

©bov® theory do®® not apply. , Sfe ©hall giv© a ©isspl® example

which demonstrates for snail v that there do exist

or A for ©cm© j and u.

a solutions in which Sj w ^ A|_

Consider the G0 design with parameters r^k®4 f

ffl~4 5 b®2, A-# « 0, sind A ® 2, On® ©olutiosa is

1 ‘ s

t

^/7 <2^
I $i : SI 8 t t / ' as

>; t

V

j

... V * ' a fh ' k MM

m m s a .

:

; v A'

• aa- m& I • a ti . . .

&gj- : n

’• ' &<-

.

A ;

• a

a - %q %K® ®I®* s s - I : I - ® . A-

- M item - 'snM i a a.

.

,

' AAAA-A:.' ASSS;.' ' ! <- ? A A^A

‘

. I . . . t 111 ?€

- -A'' . I

; S I . \

t g » «®

''

C.:AIAA£A:A Aj. -

' •: ‘ 1 !

- ;

:

:

' v \- -

" • A V

' ; a :

a a: . V - . •

: : aa , * a; . ?

. f« ~m ; km e » i : I

ll

: ' •: ' A V :: :

- AA

' a... "v A

"1

#
I

- 13 -

I 1 1 1 1 0

s © 0 0 © 1

! 1 1 © 0 0

1 0 0 1 1 1

1 © 1 1 0 1

1 © © 1 0

1 0 1 0 0

l 0 1 © 1 1

whieh Isa© tte© property that th©

ef 1 M@§1§ ©aeh, which are sue]

group tew aero trf^tiMts is ©<

0 0 cl

I. 1 l\

9 ' H\ 1 0 0 1} 0 10 ; i o 1

} 110

o o i

i

felqeks hrcisis up into 4 group©

i that two fel&mkm in the mm)

and 'any two l>S-oek® fro©

diffident groups haw© Si treatneats im eomsen.

Amtkm* solution is

H

m

1111

b'o oo

1 0 4) I

0 110

© 1 0 i

1 0 10

0 0 I 1 110 0

0 0 0 0 I 1111

1 0 0 1 I 0 110

0 i o i I 10 10

0 0 11

1 1 0 oj

9

vhiek im mmk, that any initial hloek has 1 treato&at 1 ® ©osson

sash of three Meek®, 1 treats©®*® in eoatais® with each &f

three il&ete, and 3 treat©©®is in ©ommm with 0a© blo#k„

I® ao'-t shall obtain inequalities for the number of

‘treatments s. common to any. two blooks of a svsaietrieal ju

seal-regular GO design. Since for a semi-regular GS design

r k ~ it follows that a* - Xx * n (% 2 - > x ), from which

&© obtain the following lemma 0

14 -

Lgaaaa T-, 3U for a seni*regular OD design, it. .•:.$* necessary

thi& .A t > Ji |

o Hew 1st r « k 0 Choose any two blocks and 1st the'

.columns of M fee persutsd so that the first two ceLunas

correspond to. the chosen • blocks* Then to I affix 53 new

lungs, the w-~th of which contains iA a ~ ‘ in th®

rows which correspond to the treatments of the w~tti group,

Cw'**l s. %wj, and ser© elsewhere* Let the augmented matrix

fee demoted fey %« Mow form

a fclf h

3 -

w!i#re tg is the identity matrix of order 3 and § is she

2 x (bwa-'S) matrix all of whose elements are aero® Then

<7«»> |%X|| - «r*'uJ 4 -A 1 r'

l Cs-A 1

))^"3 |B s | >-3

where B® is a 2 .x 2 matrix with elements

(?a)

^11 * h22

m Cr^r A *- AiM-fc jA ^^**‘*9

hi2 bSl ~ Cr^iP^ i- Ai) * JU*a

«

As far Theorem 4 0 2 it is necessary that |NSK' | > C, and

since the factor outside of jB& | in (? c 2) is positive, it is

necessary that |BS j > ©« Hence, the following theorem:
'
- 15 -

theorem 7 0 1„ For a symmetrical semiHregnlar GD design,

the number of treatments common to two blocks,, Sj aa -satisfies

the inequalities

1 <

& r ®

o

r&

1 wish to express »y thanks £or Professor E 0 Co Bob©

for suggesting this problem

RSFlREieiS

| i *j E-5 Cc i8o$k© sad W0 So , €©®Ms$f&tog* $.m

Ptop^rti©® ©t

Grow# I®@osapl©t# Blo©k 0©@ige@ ’* Awml# of

HBthewiticcl subnitted for

til] w c So Cottfer, Jr* »

,#0a tbe Stfftioiii*'© of B&AmmM imomplmt®

BIcm'fc Saigas" , Ama&l© of statist tm , vpj 'ist. mm> , pp

,

*7.-71 ,—— — — • ———r* -*=%=•— !

\

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The National Bureau of Standards is the principal agency of

the Federal Government for fundamental and applied research in physics, mathematics, chemistry, and engineering. Its activities range

from the determination of physical constants and properties of ma- terials, the development and maintenance of the national standards

of measurement in the physical sciences, and the development of

methods and instruments of measurement, to the development of special

devices for the military and civilian agencies of the Government.

The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services,

and various scientific and technical advisory services.. A major

portion of the NBS work is performed for other government agencies,

particularly the Department of Defense and the Atomic Energy Commission. The functions of the National Bureau of Standards are set

forth in the Act of Congress, March 3, 1901, as amended by Congress

in Public Law 619, 1950. The scope of activities is suggested in

the listing of divisions and sections on the inside of the front

cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual

equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program.

Published papers appear either in the Bureau’s own series of publications or in the journals of professional and scientific societies.

The Bureau itself publishes three monthly periodicals, available

from the Government Printing Off ice : the Journal of Research, which

presents complete papers reporting technical investigations; the

Technical News Bulletin, which presents summary and preliminary re- ports on work in progress; and Basic Redio Propagation Predictions,

which provides data for determining the best frequencies to use for

radio communications throughout the world. There are also five

series of nonperiodical publications: the Applied Mathematics Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous Publications.

Information on the Bureau’s publications can be found in

NBS Circular 460, Publications of the National Bureau of Standards

( $1.00). Information on calibration services and fees can be found

in NBS Circular 483, Testing by the National Bureau of Standards

(25 cents). Both are available from the Government Printing Of fice

Inquiries regarding the Bureau’s reports and publications should be

addressed to the Office of Scientific Publications „ National Bureau

of Standards, Washington 25, D. C.
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