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RECOVERY OF INTERBLOCK INFORMATION
by
Anis I. Kanjo, M.S.
Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute
in candidacy for the degree of
DOCTOR OF PHILOSOPHY
in
Statistics
May 1965
Blacksburg, Virginia
1. INTRODUCTION • .
- 2 -
TABLE OF CONTENTS
Page
4
2. THE CONVENTIONAL METHOD OF COMBINING ESTIMATES
3.
IN INCOMPLETE BLOCK DESIGNS. . . • . • • .
THEOREM 1. . . • • • • . 3.1 Remarks •.•.. 3.2 use of Theorem 1 .• . . . . . . .
7
14 20 22
4. LEMMA 1. . . . 4.1 corollary .
24 26
5. LE.MMA 2. . . . • . . . . . • • . . . • 5.1 Corollary 1 .••.•.• 5.2 Corollary 2 ..•..•••.
6. SOME RELATIONS IN B.I.B. DESIGNS •
28 30 30
31
7. THEOREM 2. . • . . . . . . . . . • . . 33 7.1 Exact Results for Theorem 2 • • • • • • • 42
7.1.1 Method for computing F . . . . . • • . 44 7.2 Application of Theorem 1 in B.I.B. Designs. • 46 7.3 Procedure for Recovery of Inter-block
Information in B.I.B. Designs • • . . • • 49 7. 4 Worked Example. • . • . . • . . • . • • . • • 51
8. P.B.I.B. WITH TWO ASSOCIATE CLASSES •• 8.1 8.2 8.3 8.4 8.5 8.6
Definitions and Useful Relations ••• Variances and Covariances . Sigh of the Quantity C-C' .••.... Theorem 3 • . . . . . . • Special Case. • • • . . . ..••.... Recovery of Inter-block Information in Group Divisible Designs • • • . . . . . . 8.6.1 Recovery in Regular Group Divisibles .
8.6.1.1 Study of the Ratio (V-C/(V-C+n(C-C')] ..•..•
8.6.1.2 Application of Special Case
56 56 59 63 65 70
75 76
81
of Theorem 3. . . . • • . . . 88
8.7
8.8
8.9
8.10
- 3 -
TABLE OF CONTENTS (cont.) Page
8.6.2 Analysis for Singular and Semi-regular G.D. Designs • • • • • • • • • • • • • 93 8.6.2.1 Inter-estimates and Recovery
of Inter-information in Singular G.D. • • • • • • • • 97
8.6.2.2 Inter-estimates and Recovery of Inter-block Information in Semi-regular G.D.
Recovery of Inter-block Information in LS Type. . . . . . . . . . . . . . . . . . . . . 8.7.1 Within Comparisons • • • • •• 8. 7. 2 Among Comparisons. • . • • • • • • • • Triangular P.B.I.B. Designs •..••..•• 8.8.1 Definition and Comment on the,Singu-
larity of the Inter-Analysis Model • . 8.8.2 Application of Theorem 3 When V(z.)
is of the Form v-c+p(C-C') ••. "!- •• 8.8.3 Recovery of Inter-block Information
in Triangular Designs •••••••.. 8.8.3.1 Study of the Ratio
[V-C+p(C-C')]/(v~c) . 8.8.3.2 Combined Estimates for
. . Triangular Designs •...•.
Cyclic P.B.I.B. Designs with Two Associate Classes . . . . . . . . . . . . . . . . . . . 8.9.1 Definition ••••••.••••••• 8.9.2 Recovery of Inter-block Information
in Cyclic Designs ••••••••• 8.9.2.l Combined Estimates for
Cyclic Designs •.•••• General Procedure for Recovery of Inter-block Information in P.B.I.B. with Two Associate Classes • • • • • • • . • •
99
102 110 112 115
115
119
123
129
131
132 132
133
143
144 8.11 comment on the Numerical Methods Used in
Table VII Through Table X • • . • • . . . 9. SUMMARY AND DISCUSSION •
147
151
154
155
157
158
10. ACKNOWLEDGEMENTS • . . . . . . . . . 11. TABLE OF REFERENCES. . . . . . . . . 12. VITA .
TABLES ••
- 4 -
1. INTRODUCTION
The problem of the recovery of inter-block information
in incomplete block designs was recognized by Yates (10) in
1939. The idea is to combine the two independent intra- and
inter-block estimates in order to increase the accuracy of
our estimation of the treatment effects, and consequently
to recover some of the efficiency that was lost by reducing
the number of plots in a block from v, the number of treat-
ments, to some number k ( v.
The best linear combination of the intra-block and
inter-block estimates is:
Intra-variancexinter-estimate+Inter-variancexintra-estimate Intra-variance + Inter-variance
However this combined estimate is merely theoretical since
there is, in practice, no knowledge about the exact inter-
and intra-variances. A reasonable way of overcoming this
difficulty is to use a random weight which can be computed
from the data of our experiment.
Yates suggests in (10) a rather complicated expression
for the combined estimate in the B.I.B. designs. Later in
1960, v. Seshadri (8) has shown that Yates' combined estimate
- 5 -
is unbiased and that it is uniformly better (better in the
sense of less variance) than the intra-block estimator only
~ when 2 a is smaller than r Although no specific results
were available about the case when cr.2 b > .£._ ~ nevertheless cr2 ;..v '
Seshadri's work indicates clearly that Yates' combined
estimator is not uniformly better than the intra-block cr.2
estimate when ~ ) .r . a ,,v
In (7) Rao, C.R., has established what we shall call
the conventional method of combining estimates in any incom-
plete block design. This method was adopted by Bose, R.C.,
in (1) and is the method pursued in practice almost every-
where.
Graybill in (4) suggests a combined estimator which is
uniformly better than the intra-estimate provided severe
restrictions are placed on the size of the experiment.
Seshadri, V., in (9) combines two estimates x and u in
the balanced case in such a way that the combined estimate
is uniformly better than both x and u provided that v ) 5.
It should be noted that he misnames x and u the inter- and
intra-treatment estimates, respectively; in fact, his x and
u are special linear functions of the observations as they
- 6 -
were previously defined in (3) by Graybill and Weeks. This
article, however, is suggestive and indicates that something
can be done which permits an effective utilization of the
idea of the recovery of inter-block inf orm.ation even when v
is small.
In general, one can say that there is, so far, no
practical solution to the problem without severe restric-
tions on the size of the experiment, and no solution at all
for a clear and precise answer to the question of how much
is recovered. In fact, regardless of how large the experi-
ment is, the experimenter applying the methodology available
to him now cannot be sure that he is really improving the
accuracy of his estimation.
In this dissertation, after giving a brief critique of
the conventional method of recovering the inter-block
information, a practical solution which avoids the handicaps
of the conventional method in B.I.B. designs will be dis-
cussed. This practical solution is extended to the P.B.I.B.
designs with two associate classes. Finally, an exhaustive
enumeration of the amount of recovery achieved or a lower
bound of it is given.
- 7 -
2. THE CONVENTIONAL METHOD OF COMBINING ESTIMATES
IN INCOMPLETE BLOCK DESIGNS
The most widely used method is that of Rao (7), where
the intra- and inter-normal equations are combined in the
same way that two independent estimates are usually combined,
that is by first weighing inversely by the variance of each
and then obtaining a least squares joint solution. Bose (1)
gives the resulting combined intra- and inter-block estimate
T. of treatment t. as J.. J..
where
1 w =2 a '
1 w' = -----cr2+kcr~ '
P. = wQ. + w'Q~ J.. J.. J..
' ( 2. 1)
(2.2)
(2.3)
' (2.4)
Q. and Q'. are the adjusted yields in the intra- and inter-i J..
analysis respectively,
and,
c .li.+r/\jW
dj = .6.+rHW+r2 w2
w' w=--w-w'
' (j = 1,2) (2.5)
(2.6)
- 8 -
r, k, A1 , A2 , c 1 , c 2 , 6, and Hare defined in reference (1),
and they will be defined explicitly in the sequel.
In practice, estimates for cr2 , a2 are usually obtained b
from the analysis of variance table, namely:
and
..... 2 2 cr = s e
where s: is the error mean square and S~ is the blocks ..... .....
adjusted mean square. Since (j2 and ~ obtained from the
analysis of variance table are consistent estimates for a2
and cr~, these estimates would be good enough to represent
the unknown parameters a2 and ~ if estimated from a large
experiment.
The questions now arise: {a} How reliable is the
substitution of these estimates into a complicated expression
as {l)? .....
(b) If a and
ment, do the estimates
,... ~are ,... 1 w = """2
0
estimated from a large experi-
and ...... 1 w = S+k~ possess the
"'-..... same reliability? (c) Are
..... w W=~ w-w• and
..... cj6+r/\jW dj = 6+r~+r2~
(j=l,2), consistent estimators? One way of answering these
questions is to find the variance of T. after substituting J.
- 9 -
in (1), w, w', d1 , d 2 by their estimates. A casual glance
at this expression is enough to show that this is not
mathematically feasible. It should also be noted that by
taking P. = w Q. + w'Q'., a new source of sampling variation J. J. 1
is being introduced into the adjusted yield, and this is of
special significance for small v.
To show that at least, theoretically, the combined
estimator obtained by the Rao method is better than the best
linear combination of the intra-block and inter-block esti-
mates, we will compare the theoretical variances of each.
be:
The theoretical variance of T. will be shown later to J.
k(v-l}-n1d1-n2d 2 V ( T ) = -----=-...;;;;;.,._---......-...
i vr[w(k-l)+w'] (2.7)
Letting [w(k-l)+w'] = Tj and substituting (2.5) into (2.7),
one obtains
( ) k(v-1} VT. =
1 vr71
n 1 c 16+r/\1W
vr71 6+rHW+r 2 w2 ~ c 26+rp..2w vr71 6+rHW+r2 W2
. (2.8)
By substituting (2.6) into (2.8) and simplifying
VrTj V(T.) 1
k(v-1) =
n 1 c 16(w-w' ) 2 +rr..1w• (w-w')
(VrT) 6 (w-w') 2 +rHw' (w-w') +r2 w' 2
n 2 c 26(w-w' ) 2 +rt..2w• (w-w') <2 · 9 > +-- } vrT) 6(w-w' ) 2 +rHw' (w-w' )+r2 w' 2 •
- 10 -
Now by collecting like terms, (2.9) may be written as
V(T.) 1
= k(v-1)
VrT}
(n1c 1+n2c 2)6(w-w') 2 +rw'(w-w')(n1A1+n2A2 )
vrT}[6(w-w' ) 2 +rHw' (w-w' )+r2 w' 2 ]
(2.10) w'
u = -;;;- ' and since
V(T.) = k(v-1) 1 vrT}
w6(1-u) 2 +rau(l-u} vrT}[6(1-u) 2 +rHu(l-u}+r2 u2 ] • <2 ·11 )
Since
w' U -- -- -w
1 =--2 = 0:
l+k ~ CJ
1 l+kR
the following inequalities are obtained
l<R<co and
Now by substituting for TJ, after some manipulation (2.11)
becomes
VWV(T.) 1 = k(v-1)
r (k-l+u) r(k-l+u)[(6+r2 -rH)u2 +{rH-26)u+6]
= k{v-1)[6+r2 -rH)u2 +(rH-26}u+6]-{ffi6-ra}u2 +(2m6-ra)u-p6 r(k-lfu)[(6+r2 -rH)u2 +(rH-26)u+6]
(2.12)
Finally, letting A= 6+r2 -rH,
,
VWV(T.} ].
= [Ak(v-l)-w6+ra]u2 +[k(v-l)(rH-26+2p6-ra]u+k(v-1)6-p6 r(k-l+u)[Au2 +(rH-2A)u+6]
(2.13)
- 11 -
A A Now if the intra-estimate t. and the inter-estimate t'
1 i
of t. are combined, the best linear theoretical estimate is 1
A A A A t . v ( t '. ) +t '. v ( t . )
A 1 1 1 1
-r i = V ( t '. ) +V ( t . ) 1 1
and A A
V(t.)V(t'.) A 1 1
v ('r i > = v ( £ '. } +v (£ . ) 1 1
It will be shown later that:
where
A
V(t.) = 1
A
V( t '.) = 1
k(v-l)-n1c 1-n2c 2 avw
k(v-l)-n1c 1-n2c 2 rvw'
6c.-rA. 6c.-rA. c' = J J = J J
j 6-rH+r2 A
(2.14)
(2.15)
' (2 .16)
' (2.17)
(2.18)
- 12 -
Now from (2.18)
= 6cp-ra A
Substituting (2.20) into (2.19),
6cp-ra A [k(v-1) -A ][k(v-1)-~]
V( T.) = ----------------1 6~ra ' avw[k(v-1) -A J+rvw'[k(v-1)-~]
= [Ak(v-l)-6cp+ra][k(v-l)-w] avw[Ak(v-l)-6~+ra]+rvw'A[k(v-l)-~]
Then
A
vwV(T.) l
= [Ak(v-l)-6cp+ra][k(v-l)-cp] rA[k(v-l)-~]u+a[Ak(v-l)-6~+ra]
(2.20)
(2.21)
(2.22)
A A numerical study of the difference y = vw[var(T.)- Var(-r.)]
l l
has been conducted for the 68 Regular Group divisible
experiments listed in reference (1). The range of u from
0 to l~k is divided into ten equal intervals and eleven
numerical values have been computed for the difference y, A
and for the percentage difference S = y/[v · V(T.)]. l
The results show that y is always negative, and the
difference in absolute value increases as the ratio
- 13 -
~ ~ gets closer to one. cr
~ The difference is zero when~~ CD. cr
Although the percentage difference is very small, this shows
that the Rao method is theoretically better than obtaining,
first, a separate intra- and inter-estimate, and then com.bin-
ing. 1 The results for S = ~[percentage gain due to Rao cr method] appear in Table 1.
In brief, one can say that, theoretically, the Rao
method has some desirable properties. It constitutes the
best way, knownso far, for getting a linear combined esti-
mate; it is, relatively, simple to apply; and it is very
general. But, the formula it produces contains a rather
complicated function of unknown parameters which have to be
estimated. Thus, the desirable theoretical properties of
Rao's method are not likely to stand up in practice. Many
valid questions arise about its reliability in practice, and
it appears that the only amnesty it has is that it is very
difficult to show mathematically if it is good or not.
- 14 -
3. THEOREM 1
In this section a theorem will be proved concerning a
new method of combining two independent unbiased estimates.
It will also be shown how much improvement one can hope to
get by combining two independent estimates. Finally, it
will be shown how much of the improvement this new method
utilizes.
Consider then independent parameters T1 , T2 , ... , Tn
and suppose that for each T. there exist two independent l
unbiased estimates U. and X., where U. ~ N(T., p cr2) and l i i i
X. /"\ N(T., p ! 0' 2 ). Suppose also that independently of the i i
X 'sand u. 's, there exist two unbiased estimates s 2 and s' 2 i l
02 0•2 for 0 2 J 0 I 2 respectively' where s 2 () X2 f and S I 2 /'"\ X2 f I '
then:
(1) the combined estimate:
T. i
9(m-2}s 2 = u. + ~-'-'._____..._~-i m I (X.-U.)2
. 1 J J J=
(X. -U.) , l i
m<
is unbiased and uniformly better than u., and i
n ' (3.1)
- 15 -
(2) the combined estimate:
A.
'T. =X. + J. J.
e' (m-2) s '2
2: {U.-X.) 2 . 1 J J J=
(U.-X.) , J. J.
m( n ' (3.2)
is unbiased and uniformly better than X .• 8 and 81 are constants. J.
Proof: • • • ' x i S2' S I 2 n
are independent, and E(U.) = E(X.) = -r., i = 1, 2, ... , n, J. J. J.
x.-u. E(;,) = T. + 9(m-2)E(s 2 )·E--=1-=1--
J. i m (3.3)
I (X .-u .) 2 j=l J J
Let X.-U. = z., then z. ~ N(O, V(U. )+V(X.)). The joint J. J. J. J. J. J.
distribution of z 1 , z 2 , ... , zm is of the form:
m -B Z z~
j=l J , -oo(z.(+oo, (3.4) J.
an even function of z 1 , z 2 , ... , z ; the expectation, there-m
fore, of any odd function of z 1 , z 2 , . . . , z is zero. m If z . J.
belongs to the subset z 1 , z 2 , ... , zm, then
z. E
J. = 0 m
I z2 j=l J
- 16 -
A A
Hence E(T.) = T. and T. is the unbiased estimate for T .. l. 1. l. l.
Now
A A 2 V(T.) = E(T.-T.)
1. l. l. =Efcu.--r.) + l i 1
e(m-2)s 2
m I (X .-u .)2 j=l J J
2
2
(x.-u. l i ij (3.5)
s 2 (X. -U. J = E(U.--r.) 2 +92 (m-2} 2 ·E ~~-=1'--~1;......_
s 2 (X.-U.){U.-T.) l. l. l. l.
1. l. m I (X .-u .J2 '=l J J
z2
+2e (m-2) ·E m I (U .-x .)2 j=l J J
4 i = V(U. )+92 (m-2) 2 ·E(s } ·E---=::,._-+ 29(m-2J ·E(s2 ) ·E i m
Z. (U. -T.) 1. l. l.
~z~ where
and
z~ E
1.
m c I z~)2 j=l J
1
c I z~) 2 . 1 J J=
2 2 2 =-E
zl +z2 + ... +zm m m ( I z~) 2
j=l J
J
because of the symmetry of the Z, IS i 1.
that is,
z~ z2 l.
=E k k 1, E = m m J
( I z~) 2 < I z::) 2 j=l J j=l J
j=l J
J (3.6)
{3.7}
2, ... ' m.
J
Hence:
E
Then
E
- 17 -
m
z~ Iz~ 1 1 j=l J 1 1 - . E . E m m m m 2>
, ( I z~) 2 ( I z~) 2
j=l J j=l J j=l J
1 1 1 = --------------------~ m(m-2) (V(U.)+V(X.)J
1 1 m • E (V(U.)+V(X.))X2
1 1 m
z. ( u. -'[.) 1 1 1
z. 1
Consider now the multivariate normal vector
and the variance covariance matrix is:
. (3.8)
z.(U.-T.) 1 1 1
(3.9)
,
-18 -
N
N
r-1 N
N
V\I
r-1 .-I
r-1 N
N
N
II
--r-1 ~ -
0 0
0 ~ -
·r-1 ::::> -
0 :>
... ...
0 ...
-...
·r-1 ...
0 ~
--
·r-1 ~
::::> ...
--
:> 0
·r-1 I
::::> -:>
... ... ...
0
... ... -
0 ·r-1
x -~ ...
... -
0 0
0 ·r-1
::::> -
... :>
...
--
·r-1 ...
·r-1 ...
::::> ::::>
-0
-0
:> :> I
II -.-I + e [/'(]~
-r-1 + e -
- 19 -
Since the conditional mean of uilz1 , z 2 , ... , zm is:
-1 = T . + ( 0, 0, ... , -V ( U. } , 0, ... , 0} ( ( V ( U. ) +V (X. ) } I ] · Z i i i i m -
= T. + (o,o, ... , ( > ( >,o, ... ,o}z = :i. VU. +V X.
i i
-V(U.) i
then
Substituting (3.10) into (3.9) one obtains:
E
=
=
z. (U. -T.) i i i
~z2 j=l J
-V(U.) i
V(U.) +V(X.) i i
-V(U.} i
E zl ... zm
m(V(U. )+V(X.)) i i
'
-V(U.) i
z2 i
V(U.) +V(X.} i i
z~ -V(U.} i i
= m(V(U.) +V(X.)) m
I z~ i i
. 1 J J=
'
making use of the symmetry of z 1 , z 2 , ••• J z . n
V(U.}·z. i i
T. -i V ( U. } +V (X. } '
i i
(3.10}
2 2 2
E zl+z2+ ... +zm
J m
Iz~ . 1 J J=
(3.11)
Substituting
(3.6), (3.8), and (3.11} into (3.5), one obtains
- 20 -
A e2a4Cm-2> 2Cf+2) 2 V(Ti) = V(U.) + - 2 ea VCU; >Cm-2)
' 1 fm(m-2)(V(U.)+V(X.)) m[V(U. )+V(X.)] 1 1 1 l.
e2Cm-2)(f+2)[V(U.)] 2 2 e(rn-2 )[V(U.)] V(U.) + l. - 2 1 = ' l. 2 p fm[V(Ui) + V(X.)] pm[V(U.)+V(X.)]
1 l. l.
e ( m-2 ) [ V CU . ) ] 2 [eCf+2) = V(U.) + l. - 2] • 1 pf mp [ V ( U . ) + V ( X . ) ]
l. l.
The value of .e which makes the second term above the most
negative is e_pf or e_ f substitutin_ g for ! one obtains: -1+2, 'P-1+2, p
A
V('t'.) l.
2 2 [V(Ui)]2 = V(Ui)-(l-m){l-1+2)V(U.)+V(X.) ~ V(Ui)'
1 1
provided m.::_2.
(3.12)
The proof of the second part follows similarly.
3.1 Remarks
(a) The best linear unbiased combined estimate T· of l.
T • is: 1
"[. = l.
U.V(X.)+X.V(U.) l.-1 l. 1
V(U.)+V(X.) 1 l.
with the minimum variance:
... V(T.)
1
V(Ui)•V(Xi) = = V(U. )+V(X.)
1 1
V(U.) -1
[V(Ui)]2
V(U.)+V(X.) 1 1
(~.1.1)
- 21 -
This shows that the most improvement one can hope to get by
combining the two independent estimates of Ti' namely Ui and
[V(U.)] 2 X. is the quantity ___ i __ _ • It should be noticed that
1 V(U.)+V(X.) l l
the above d . (m-2)f suggeste estimate recovers of the most
improvement possible
· · 2 m(f+2) [V(U.)]
l. • The fraction
V(U. )+V(X.) l. l.
(m-2)f _ 2 2 -+ 1 when m and f are large. (1--) (1--)
m(f+2) m f +2 It is obvious now that the larger m is, the better
new estimate will be. This means that we should take m
whenever it is possible.
(b) The variance of the combined estimate T! is: l.
the
= n
V(T!) = V(X.) -1 J_
(m-2)f'[V(X.)J 2 l.
• (3.1.2) (f'+2)m[V(U.)+V(X.)]
l. l.
Subtracting (3.1.2) from (3.12),
V(T.) - V(T!) = [V(U.)-V(X.)] l. J_ l. l.
+ m-2 m[V(U. )+V(X. )]
l. l.
f'[V(X.)] 2 l.
f'+2
f[V(U.)] 2 • l. •• f +2
~
Simplifying the R.H.S., the sign of V(Ti) - V(Ti) is the same
as the sign of:
- 22 -
m(f 1 +2Xf+2)[V2CU.)-V2(X.)]+(m-2)[f'(f+2)V2CX.)-f(f 1 +2)V 2(U.)] 1 1 1 1
= 2v2cu.)[mf 1 +ff 1 +2f+2m]-2V2CX)[mf+ff 1 +2f 1 +2m] 1 1
= 2(m+f)(f 1 +2)V 2(U.)-2(m+f 1 )(f+2)V2(X.) 1 1
Thus:
" " V(T.) > V(T!), 1 1
if v2cui>
> (f 1 +m){f+2) and V2 (X.) ( f +m )( f ' + 2 )
1
v2 (U.) ( f ' +m )( f + 2 ) if 1 < v2 (X.) •
(f+m)(f'+2) 1
V(t.) < V(-r!), 1 1
3.2 Use of Theorem 1
This theorem could be utilized for obtaining better
estimates for a number n of independent comparisons between
the treatments in any incomplete block design, provided that
each comparison has the same variance, and the number of
such comparisons is more than 2.
It could also be used to combine the estimates of the
same treatment from two independent similar experiments,
provided that the interaction over time and location is
negligible.
Also, if one is in doubt about the homogeneity assump-
tion when a randomized block design has been utilized, the
blocks could be divided into two homogeneous_ groups, and
- 23 -
then obtain estimates for the v-1 comparisons among treat-
ments within each group. Then combine these estimates by
theorem 1. The gain in the accuracy of this estimate
increases as the heterogeneity between the two groups
increases. Again, the block by treatment interaction must
be negligible.
- 24 -
4. LEMMA 1
If two independent random variables X and Y have density
functions:
f (X) =
f (Y} =
Then:
Proof:
where
Let x --
1 al x a 1+1
rca1+1}131
1 a2 y a 2+1
r(a2+1)132
= r(al+a2+2-r)
r(al+l)I'(a2+1)
e
e
-x/13 1 dx x > o, 131 > 0,
( 4 .1) -y/13 2 dy y > o, 132 > 0.
(4.2)
Sooroo 1 al -x/131 a2 -y /132 = Ct. x e y e dxdy 0 (x+y) r (4.3)
_y_ - z A - 2 -' 1-'2
,
then
= Ct.
Now let
a 1+1 a +l = Ct. t)l ~2 2
. e -z
2
- 25 -
then:
uz2 --1-u 1 (l-u}2 dudz2 e
Let z 2 = (l~u)v , u = u , then:
e
uz2 1-u
(4.5)
- 26 -
-uv du e ( l-u) 2 ( 1-u) dv,
a +l a2+1 s<D a1+a2-r+l dv sl 1 f3 1 -v = Ct. f32 v e 1 r
0 0 [ f31 u+f32 ( 1-u) ]
al a2 du . u (1-u) ,
a +l a 2+1 sl 1 al a2 = Ct. f3 1 f32 r(al+a2+2-r} r u (1-u} du , 1 O [ f31 u+f32 ( 1-u} ]
r(al+a2+2-r) = r(al+l)I'(a2+1)
al a2 (' 1 _..;;;;u __ (~l"'""-..-u...._}__ du
~ [ f31 u+f32 ( 1-u} ] r (4.6}
4.1 Corollary
Since (32 ) 0, 0 { u ( 1, and as Stlllling f31 ) f32 , then:
or (4.1.1)
- 27 -
Then
and
,
which can be written respectively as:
(4.1.2)
and
(4.1. 3)
- 28 -
5. LEMMA 2
Let z1, z2, ••• , zn have the joint multivariate normal
distribution, with mean vector b!:..' = Q and variance covariance
matrix ~ with the diagonal element (a) and the element (b)
otherwise. Then ~ z2 /..; i can be considered as equivalent to
i=l
the sum of two independent, Gamma distributed, random
variables.
Proof: The characteristic function of ~ z2 /..; i
i=l 1
= sCD ... fCD l~I !!2 e9Z'Z e -CD -CD ( 2'1T) 2
- 1.[z'~-lZ-29Z'Z] 2 - - --e dZ
1 ~· [~- 1-29I]Z
1~1 n 2 fCD ... foo e z= - dZ = ( 2'1T) 2 -CD -CD
n 1 1
(2'1T}2 1~1 2 1~-112 1 = 1 . n = 1 =
,
J
1 I 2:-1-29! 12 ( 2'1T) 2 I L:-1-29! 12 I I-292: 12
is:
,
(5.1)
- 29 -
Since the inverse of an (nxn) two element matrix, a along
the diagonal and b otherwise is an (nxn) two element matrix,
d along the diagonal and f off diagonal where:
d _ -[(n-2)b+a] - b 2 (n-l}-a[(n-2)b+a] ' (5.2)
and
b f = b 2 (n-l}-a[(n-2}b+a] ( 5 . 3 >
It is known also that the determinant of such a matrix is:
n-1 6. = ( a-b) ( a-b+nb) ( 5. 4}
Substituting (5.4) into (5.1),
and then
n-1 1 -- -= [l-2(a-b)e] 2 [l-2(a+nb-b)e] 2
n
cpe ( l z~) = . 1 1 1=
1 1 n-1 ----------------
[ 1-2 ( a-b} 9] 2 [l-2(a+nb-b)e] 2
,
This means that ~> can be considered as the sum of two . 1 1 1.=
independent random variables X and Y, where:
n-3 2
n-1 x e-x/2(a-b) dx f(X) = 1
[ 2 (a-b) J-2-r<n21 >
x) O, a-b) O ,
' (5.5)
- 30 -
and
1 1 f(Y) = -1----1-- y 2 -y/2{a+nb-b} e dy
2 2 ( a+nb-b} 2 r ( ~} {5.6) y ) 0, a +nb-b ) 0
5.1 Corollary 1
Using Lenuna 2, one can write:
1 1 E("'z2.) = E{-)
LJ X+Y = E(J:.{l+y}-1] ( E J:. = __ 1 __
x x x (n-3}(a-b) ' l.
i.e.,
1 1 E(L:z~) < (n-3){a-b)
l.
(5.7}
5.2 Corollary 2
Applying the results of Le nun a 1,
n-3 f31 2 {a-b) al --- = 2 ' '
-1 f32 2(a+nb-b) a2 ' = 2
(5.2.1)
Substituting (5.2.1) into {4.6),
1 r 1 r E ("'z2) = E(-)
LJ X+Y i
n-3 1 sl u 2 (1-u) 2 r du
0 [-2nbu+2(a+nb-b)]
(5.2.2)
- 31 -
6. SOME RELATIONS IN B.I.B. DESIGNS
A
Let t. denote the intra-block estimate of the i-th 1.
A
treatment effect and let t' denote the corresponding inter-i
block estimate, then
A k{v-1} (J2 V(t.) = vl = 7w2 1. ( 6. 1)
A k(v-1} cr'2 V(t '.) = v2 = 1. v(r-f..} ' (6.2)
A. A
cov(t., t.) 1. J
for all i ,¢ j ' (6.3)
and
"' "' -k cr'2 cov(t '., t '.) = c = 1. J 2 v(r-f..) for all i ,¢ j (6.4}
The following relations hold:
c1v2 - v1c2 = 0 ' (6.5)
vl + (v-l)C = 0 1 ' (6.6}
and
(6.7)
Then
v - c = - vc ' ( 6. 8)
where v = vl + v2 '
- 32 -
and
v =---v - 1 (6.9)
The intra-estimate of the i-th treatment is given in (6) as:
A = Qi ti rE J (6.10)
where Q. is the adjusted treatment total, and the efficiency, l.
(k-l)v E = k(v-1} {6.11)
The inter-estimate of the i-th treatment is given in (11) as:
A Q~ t I = _..;;;;l. __
i r(l-E) J (6.12)
where
Q'. = Y. - Q. - ry l. l. . l. ••
(6.13)
Y. is the total yield of the i-th treatment, and y is l. •
the overall mean.
- 33 -
7. THEOREM 2
In B.I.B. designs, the combined estimate:
T. 1
A k(v-4)s 2 = ti + ___ ...__v__._ .... l ____ _
:Av(f+2) I c-t:-t.) 2 . 1 1 1 1=
A A
( t '. -t. ) , 1 1
( 7 .1)
is unbiased for the i-th treatment effect t., and uniformly 1
better than intra-block estimate provided v ) 4, where s 2
and f are the error sum of squares and the error degrees of
freedom, respectively.
where
Proof: Consider first the estimate:
T. 1
"' 9Bs 2 = t. +-~;;..;;;;... __ _ J. n I (t'.-t .)2
j=l J J
A A
(t'.-t.) J. J.
, n < v , ( 7. 2)
9 _ k(v-1) - f..v2 ,
A
the coefficient of cr2 in var(t.), s 2 J.
is the error mean square, and B is to be determined later.
Now
A
E(-r.) = J.
z. t. + 9Bcr2 E (-=1 -)
i !z~ j=l J
, ( 7. 3)
A A A A
where zi =ti - ti. Assuming ti~ N(ti, v 1 ), and ti ~N(ti,v2 )
for i = 1,2, ... ,n, the zi ~N(O, V), where V = v 1 + v 2 ,
- 34 -
then A A A A
cov(z.,z.) = E(z.z.) = E[(t'.-t.)-(t.-t.}][(t'.-t.)-(t.-t.)] 1 J 1J 11 11 J J J J
A A A A A A A A = E(t'.-t.)(t'.-t.) + E(t.-t.}(t.-t.} = cov(t'.,t'.} + cov(t.,t.)
J. 1 J J 1 1 J J 1 J J. J
= c ( 7. 4)
Now z 1 ,z2 , ... ,zn have the multivariate normal distribu-
tion with mean vector !!. = Q, and variance covariance matrix
~ with the element V along the diagonal and the element C nxn
otherwise. Now one can write the density function as:
, -oo ( z. ( +oo . 1
This is an even function and the expectation of any odd
function of z. 's in zero. Since 1
its expectation is zero. Going A
E ( -r . ) = t . . Hence 1 1.
A ,... 9Bs 2 2 V(-r.) = E((t.-t.) +---------- z.J 1 1 1 n 1
Iz~ . 1 J ]=
z. J. is an n
Iz~ j=l J
back to ( 7. 3)
,
+ 29BE (s 2 ) ·E
odd function,
one can write
( 7. 5)
A
z. (t. -t.) 1. 1 1.
- 35 -
Since
= f(f+2)cr4 f+2 cr4 f 2 = -f- '
(7.5) may be written as
A
V(T.) J.
z2 i
n '\"""' ( L z~ > 2 . 1 J J=
By symmetry of the Z, Is, we can write:
i.e.,
E
Now,
J
z~ L. 1 =-E
n n ( l z~}2 j=l J
A
E z. (t. -t.)
J. 1 1
n
l z~ j=l J 1 =-E n n ( l z':) 2
j=l J
Consider now the multivariate vector
1 n l z':
j=l J
A
z. ( t. -t.} J. J. J.
(7.6)
(7.7)
( 7. 8)
... , z. 1 , z. 1 , ... , z] with mean vector i- 1+ n
,
- 36 -
[t., O, o, ... , O] and variance covariance matrix: 1.
-vl =
-cl (n+l) (n+l)
-v 1 -c 1
where Z is as defined before. Now
-c 1
=
,
z12
( 7. 9)
-1 where Z is a two-element matrix with diagonal elements d
and off-diagonal elements f_, where:
d -[ {n-2) c+v] = C2 (n-l)-V[(n-2)C+V] ,
and ( 7 .10)
f c = c 2 (n-l)-V[(n-2)C+V]
Now
d f f f
f =
f f d
, ... ,the same ... ]
'
- 37 -
But
=
=
=
-V d - (n-l}C f vl[(n-2}C+V]-(n-l}ClC
= 1 1
v 1 (n-l}c-c1 (n-l}C+V1 (V-C} (n-l}C2 -(n-l)VC-V(V-C} =
(n-l)C(V1-c1 )+v1v-v1 (c1+c2) (n-l)C(C-V)+V(C-V)
(n-l)c(v1-c1 )+v1v-v1c 1-v2c 1 {C-V} [ (n-1) c+v]
c 2 (n-l)-V[(n-2)C+V]
V (V-C)-(n-l)C(V -C) 1 1 r (n-l)C(C-V}+V(-V+C)
(since C =
,
using (6.5}. Thus
=
(n-l}C(V1-c1 }+V(V1-c1 )
(C-V)[(n-l)C+V]
c-v
,
,
,
( 7. 11}
(7.12)
-1 To compute the remaining elements ,in the nxl row L:12L: , one
needs to compute only
= -Ci-[(n-2)C+V]+(n-2)C}-v1c (n-l)C2 -V[(n-2)C+V]
vc1-v1c v 1c 1+v2c 1-v1c 1-v1c 2 = (n-l)C2 -V[(n-2)C+V] = (n-l)C2 -V[(n-2)C+V]
= (n-l)C2 -V((n-2)C+V) = O '
- 38 -
i.e.,
Now (7.9) can be written
-1 = L.122: z v1-c1
= ( c-v ' 0' 0' ... '0)
= v1-c1
c-v v -c 1 1 z = -
i v-c z. 1
Substituting (7.7) and (7.14) into (7.6),
V2 B2 (f+2) A 1 1
V('r.) = V + ----- E --=--- 2V BE 1 1 nf n - 1
I z~ . 1 J J=
z2 i
It should be noted that 9a2 = v1 . Utilizing again the
synnnetry property, z?
E -=1- = 1. . ' Iz2
j=l J
hence: n
( 7 .13)
z. 1 1-
z. 1 1+
z n
( 7 .14)
( 7 .15)
- 39 -
( 7 . 16)
Using (5.7) with a= V, b = c,
1 1 E -- ( (n-3) (V-C}
~z~ j=l J
(7.17)
Now one can write
{ 7 .18}
We would like the quantity V~B2 {f+2) V1B{V1-c1 }
n(n-3)f{V-C) - 2 n{V-C) to
be negative. First we notice by (6.8) that
k k v-c = -vc = -v(C +c ) = ~ a2 + ~- cr' 2 which is always 1 2 ~v r-A
positive: if, in addition n ) 3, then the above quadratic in
B is negative whenever
The value of B which makes it the most negative is
B = f{V1-c1 } {n-3)
v1 {f+2)
- 40 -
From (6.9)
fv{n-3} B = (v-1) (f+2)
Now substituting Bin (7.18},
A v~ f+2 v 2 f 2 (n-3) 2 v ( '! ) < v + -[-~;..._ ________ __. _____ ____ i 1 V-C nf(n-3) (v-1} 2 (f+2} 2
i.e.,
2v n(v-1)
( 7. 19)
vf (n-3) (v-1) (f+2}]
{7.20)
It is obvious that one should take n as large as possible,
but since n is strictly less than v, take n e v-1. It
should be noted that one cannot take n = v, for then the
matrix Z is singular. Hence
V(~.) < V _ v~f(v-4} i 1 (v-1) 3 (f+2)
v2 __ l
v-c
As was mentioned previously, the quantity v2
1
(7.21)
represents
the utmost improvement possible over the intra-block
variance. Now one can write
v~ c v~ c = -[1 + -] = -c1 + -) using (6.8}. v v-c v -vc ,
- 41 -
v2 v2 v-1 Then 1 1 and substituting into (7.21), = ' v-c v1+v2 v
fv{v-4} v2
A < vl 1 V('r.) (7.22)
1. {v-1) 2 (f+2) Vl+V2
The upper bound of the variance of the proposed estimate is
uniformly better than the intra-block variance for any
B.I.B. design when v) 4. Substituting n = v-1 into (7.19)
one obtains B = fv(v-4} (f+2) (v-1} ' and then from (7.2),
~ . = t . + ---'e~v"""'('""'v'""'--4~)'""'f"'"'s_2 ___ _ i i v-1
A A (t '.-t.)
(f+2) (v-1} I (t '.-t. }2
j=l J J
1. 1.
and recalling that 9 = k(v-1) t..v2 or ev
v-1
"' "' k{v-4JS2 Ti =ti + v-1
f..v(f+2} l (t'.-t .) 2 j=l J J
k AV
A A
(t :-t.) 1. l
'
( 7. 23)
where S 2 = fs 2 is the error sum of squares in the intra-
analysis. From (7.22} it is seen that the ratio of improve-
ment or the recovery ratio is:
_ fv(v-4} Dl - (v-1) 2 (f+2) ( 7. 24)
- 42 -
7.1 Exact Results for Theorem 2
Using lenuna 2 one can get a fairly exact value for
1 E " 2 LJZ.
l
Here a = V, b = C, n = v-1, so,
v-4 1
1 __ r_( v_;_3_>_ 51 ___ u==2=_(.._l_-_u ..... )_-_2 __ du E(v-1 ) = I'(v-l)I'(J:.} O 2(v-l)C(l-u)+2(V-C)
\ 2 2 L z2: . 1 l l=
but since V-C = -vc, one obtains
v-4 1 -- -1 __ __;2;;;.__ ___ sl u 2 (1-u) 2 du
E( ) = ( } v-1 ( ) (v-2 ..!:.) 0 2(v-l)C 1-u -2vC \ v-3 f3 -2-, 2 L z2:
. 1 l J.=
v-4 1
1 sl 2 (1-u) 2 u = v-2 ..!:.) l+(v-l)u -C(v-3) 13(-2-, 0 2
The problem now is to evaluate the quantity,
v-4 1
u 2 (1-u) 2 l+(v-l)u du
du
for different values of v. Later a method of computing this
quantity will be given. Now, one may write
-F (7.1.1) C(v-3)
- 43 -
Substituting (7.1.1) into (7.16), and taking n = v-1, one
obtains
A VfB2 (f+2)F V(Ti} = vl + -cf(v-l}(v-3)
but v-c = -vc and v v-1 or
= -C(v-1) J
V(~.) = V + V~B [B(f+2)F i 1 -(v-l}C f(v-3)
Since C is negative, one needs
/ < 2f (v-3} O ' B. F(v-1) (f+2)
The required B is
- _f_.(_v_-_3..._) __ B = F(v-1) (f+2)
2V l B ( V l -Cl}
(v-1} (V-C}
_2_] v-1
Substituting (7.1.3) into (7.1.2}, one may write
v2 v2 v2
'
(7.1.2)
(7.1.3}
since 1 1 1 -- - -- = -=---v-c -vc v +V 1 2 ' then (7.1.4) becomes
B v-1
v2 1
v1+v2 (7.1.5)
The combined estimate in (7.2} becomes, after substituting,
- 44 -
the value of e:
"' A 9Bs 2 (''I A ) T. = t. + t.-t. l. l. v-1 l. l.
,
Icf'.-t.) 2 j=l J J
"' k(v-l)Bs 2 A A = t. + (t~-t.)
l. v-1 l. l.
)\V2 I ( t '. -t.) 2 j=l J J
(7.1.6)
From (7.1.5) the recovery ratio is
D2 = B/(v-1) (7.1.7)
The estimate in (7.1.6) can be applied whenever v) 3. The
values of B and n2 appear in Table II for the 58 B.I.B.
designs listed in Reference (2).
7.1.l* Method for computing F
The hypergeometric series,
(a) (b) n n F{a,b; c; z}
n ! (c) n
n z -, {7.1.8}
converges to the definite integral [see Reference (12)]:
1 Sl I b-1 c-b-1 -a
= j3{b, c-b) 0 u ( 1-u) ( 1-uz) du (7.1.9}
*I am indebted to Dr. L. R. Shenton who introduced me to
this method.
- 45 -
Now one may write:
v-4 1 -F = _ ___.l~- sl ...;;;u'"---2-"'-'( l~-_..u;.;,.<.)_2
f3 ( v; 2 , ~ ) 0 1 + ( v-1) u du ( 7. 1.10)
Comparing (7.1.9} and (7.1.10), we get:
b-1 = v-4 c-b-1 1 1-v 1 = z = , a = ' 2 ' 2 '
or b v-2 v-1 1-v 1 --- c --- z = a = 2 , 2 , '
One needs to evaluate, therefore, the quantity,
v-2 v-1 = F(v-2, v-1 ) F(l, - 2-: - 2-: 1-v) 2 l; - 2-: 1-v ' (7.1.11}
due to the synunetry of a and b in the hypergeornetric form
above. Now F(b,ljc;z) can be put in the form of Gauss's
continued fraction as follows [see Reference (13}]:
where h 2p-l = b+p-1 c+2p-2 ,
1 -
= p c+2p-l '
, (7.1.12)
p = 1,2,3, ...
(7.1.13)
- 46 -
The convergence is very rapid and it is far easier to compute
than the usual Simpson rule, in this case. It is to be
noted that the terms required to assure an accuracy up to
the sixth decimal place, range from 4 terms for v = 91 to
14 terms for v = 4.
7.2 Application of Theorem 1 in B.I.B. Designs
Consider the vxv matrix
1 1 1 1
1 -1
1 1 -2
1 1 1 -3
Ml = , (7.2.1)
1 1 1 1 (2-v)
1 1 1 1 1 (1-v)
and let M be the orthogonal matrix we get by normalizing Ml.
Consider the two
u
vectors
=
0
ul
u2
u v-1
of
=
comparisons :
M
A
tl A
t2
A
t v
= A
(7.2.2) M_t ,
and
x =
In general,
0
xl
x2
x v-1
- 47 -
= M
A
t' 1
A
t' 2
A
t' v
A = M t'
v(u) = v[ 1 (t1+t2+ ... +£m-mtm+l)] m ./m(m+l)
(7.2.3)
,
. (7.2.4}
This result is independent of m, that is, all U. 's, where l
i = 1,2, ... ,v-l, have the same variance, namely v1-c1 .
Similarly, V(X.) = v1•-c• for i = 1,2, ... ,v-l. Therefore, l 1
theorem 1 is applicable for combining the two comparisons
U. and X., and their combined estimate is: l l
but
A
T. = l
k v1-c1 = - cr2 AV ,
A
u. + l
9(v-3)s~ v-1 I (X.-U .)2 j=l J J
k
(X.-U.) l l
hence p = AV , and
1' . l
= u. + 1 l v-
fk(v-3}s 2 (X.-U.}
l l
( f + 2 ) AV I {X . - u . ) 2 . 1 J J J=
(7.2.5)
- 48 -
It should be noted that
v-1 \ (X. - u . ) 2 = (X-U) I (X-U} = -~ J J - - - -
J=l
/'\ A A A ( t I -!) I M IM(.!_ I -.!_}
v A A A. A
= (.!_I -.~J I Ct I-.!_) = I ( t '.-t. > 2. j=l J J
A
Let .I. = =M
then (7.2.5) can be written in vector notation as
"' f k(v-3) s 2 MT = Mt + - v
(f+2)}.v I (t'.-t.) 2
j=l J J
Since M'M = I, one can write
"' fk(v-3)s 2 T. = t. + ____ ....__ __ l. 1. v
(f+2)"Av I (t'.-t .) 2
j=l J J
A A M (!.I -,.E_) (7.2.6)
A A
( t '.-t.) ' 1. 1.
i=l, 2, 3 ... v.
(7.2.7)
=MT _,
This is applicable when v ) 3, and the recovery ratio here is
D = (v-3l f 3 (v-lX£+2)
(7.2.8)
The results for n1 as given in (7.2.4), and n3 above
appear in Table II. The results show that n2 and n3 are
always better than n1 ; the three ratios approach practically
the same value as v becomes relatively large. n2 and n3
- 49 -
are practically the same, but n3 is,
, always better than n2 .
7.3 Procedure for Recovery of Inter-block Information in
B. LB. Designs
Compute:
(1)
(2)
(3)
Y. l..
y . . J
y
= ~Ii .. y .. ' . 1 1J 1J J=
v
= l 0. ,y .. , . 1 1J 1J 1=
= lo .. Y .. . . . l.J l.J 1J
i=l,2,3 ... v. (Total yield of
i-th treatment) .
j =l, 2' ... 'b. (Total yield of
j-th block.}
(Grand total. }
(4) Q. Y. 1 (Sum of block totals in which treat-= k
(5)
(6)
1 l.
rnent i occurs.),
i = 1,2, ... ,v (Adjusted treatment totals.)
Q. A l. ti - rE ' i=l, 2' ... 'v . (The intra-estimate,
SST
v
= lt.Q. . 1 1 l 1=
(k-l)v E = k(v-1} . )
(Treatment sum of squares.)
(7)
( 8)
( 9)
(10)
( 11)
(12)
-50-
= 1 1 b s2 (2: o .. y2 .. - SST - k 2: Y~). (Error mean f ij lJ lJ j=l J
square, f = bk-b-v+l.)
~- + q~ = Y. - 1 y ' l l l• v •• i=l,2, ••• ,v. (Sum of
intra- and inter-adjusted totals.)
Q! A' ,... l t. - t. -= --=--l l r(l-E)
y A A
l: (t.-t.) 2 • i=l l l
J = fk(v-3)s 2
Q. l
rE
y_ "' I\ (f+2)~v ~(t.-t. ) 2 i=l l l
E(Q! + Qi)-Qi = l rE(l-E) 'i=l,2, ••• ,v.
" "' "' The combined estimate T.= t. + J(t.-t.) , l l 1. 1.
i=l,2, ..• ,v. This estimate recovers the ratio n3 of the utmost possible
recovery, and n3 appears in Table II.
-51-
7.4 Worked Example
An experiment with 6 treatments in blocks of 2 will
be worked out. The data is taken from Reference (2),
page 444; the treatment estimates obtained in (2) by applying
the
- 52 -
conventional method are to be compared with the results of
the new method.
Rep. I
( 1) 7 (2) 17
(3) 26 (4) 25
( 5) 33 (6) 29
Rep. III
(1) 10
(2) 26
(3) 24
(4) 25
(6) 37
(5) 26
Block Rep. II
totals
24 ( 1) 17 (3) 27
51 (2) 23 (5) 27
62 (4) 29 (6) 30
Block totals
Rep. IV
35 (1) 25
63 (2) 25
50 (3) 34
Rep. V Block totals
(1) 11
(2) 24
(4) 26
(6) 27
(3) 21
(5) 32
38
45
58
(5} 40
(4) 34
(6) 32
Block totals
44
50
59
Block totals
65
59
66
In this experiment v = 6, k = 2, r = 5, b = 15, ).. = 1, E = .6.
- 53 -
The following table is to be set up.
Treat. A A A A
Y. Q. t. Q.+Q'. t'-t T. T. No. l.. l. l l l. i i l. l.
1 70 -33.0 -11.00 -58.17 -1. 59 -11.31 -11. 2
2 115 - 5.5 - 1.83 -13.17 -2.00 - 2.23 - 2.1
3 132 4.0 1. 33 3.83 -1.42 1. 05 1.1
4 139 8.0 2.67 10.83 -1. 25 2.42 2.5
' 5 158 15.5 5.17 29.83 2.00 5.57 5.5
6 155 11. 0 3.67 26.83 4.25 4.52 4.4
0 0 0 0 0
For Treatment 1, for instance, the following steps
should be carried out.
Step 1: Treatment total= Yl. = 7 + 17 + ... + 11 = 70 .
Step 2: 1 01 = Yli - 2 (sum of block totals in which
Treatment 1 appears},
= 70 - ~(24 + 44 + ... + 38) = -33
Step 3: 01 -33 rE = 5(.6) = -ll '
where E = Efficiency of the design (k-l)v = k(v-1)
(2-1)·6 = = 2·5
Step 4: 1
Ql + Q' = y - -(y 1 1. v ) = 70 - 769 = -58.17 ' 6
where Y =Grand Total= 24 + 51 + ... + 58 = 769
. 6.
Step 5: Q'
1 r(l-E)
- 54 -
Ql = E(Ql+Ql)-Ql rE rE(l-E}
= .6(-58.17)+33 = -1.59 5(.6)(.4)
J
After computing the same quantities for all treatments, one
should compute the following quantities:
Sum squares of blocks (unadjusted)
1 (769) 2 = 2<242 + 512 + ... + 582) - 30 J
= 1051. 5
Sum squares of treatments (adjusted)
v \" = iJt.Q. =
. 1 1. 1. 1.=
= 520.2
J
Error mean square = b 1 (Total sum of squares - Sum rv-v- + 1
squares of treatments adjusted - Sum squares of blocks
unadjusted) J
= 1~(1649 - 520.2 - 1051.5) = 7.73
= (-1.59) 2 + (-2) 2 + ... + (4.25} 2 = 32.2
-55-
J = fk(v-J)s 2
(f+2)fw ~(t!-t.)2 l..J l l .
i=l
now the combined estimate ~- is: l
" " "' ,.. ~-= t.+ J(t.-t.) l l l J.
= 10 • 2 • 3 • ( 7 • 73 ) 12 • 1 • 6 • ( J 2 • 2 )
For the first treatment, for instance:
" ~l = -11 + ( .20005) (-1.59) = -11.31
=
- 56 -
8. P.B.I.B. WITH TWO ASSOCIATE CLASSES
8.1 Definitions and Useful Relations
An incomplete block design is said to be partially
balanced with two associate classes if it satisfies the
following requirements:
(i) The experimental material is divided into b blocks
of k units each, different treatments being applied to the
units in the same block.
(ii) There are v()k) treatments each of which occurs
in r blocks.
(iii) There can be established a relation of associa-
tion between any two treatments satisfying the following
requirements:
(a) Two treatments are either first associates or
second associates.
(b)
(i = 1,2).
Each treatment has exactly n. i-th associates l
(c) Given any two treatments, which are i-th associates,
the number of treatments common to the j-th associate of the
. i first and the k-th associate of the second is pjk and is
independent of the pair of treatments we start with. Also
i i pjk = pkj (i,j,k = 1,2).
- 57 -
(iv) Two treatments which are i-th associates occur
together in exactly A. blocks (i = 1,2}. J.
The munbers v, r, k, b, n 1 , n2 , J..1 , and J..2 are called
parameters of the first kind, whereas the numbers p~k
(i,j,k = 1,2) are called the parameters of the second kind.
The following relations between the parameters are
known to hold:
vr = bk ,
n 1 J..1 +n2J..2 = r(k-1) ,
l 1 P21+P22 = n2 '
nl+n2 = 1 1
Pi1+P12
2 2 pll+pl2
1 nlpl2 =
v-1 ,
n -1 = 1 ,
= nl , (8.1.1)
2 n2pll ,
In the analysis of such designs, T. is defined as the J.
total of the observations for the i-th treatment, B. as the J
sum of the k observations from the j-th block. Qi denotes
the adjusted yield for the i-th treatment obtained by
subtracting from T., the sum of the block averages for those 1
blocks in which the i-th treatment occurs. Also, s1 (Qi)
denotes the sum of the adjusted yields for all the first
associates of the i-th treatment, and G denotes the total of
all N observations.
- 58 -
R. C. Bose defines the constants 6, H, c1 , c 2 by the
relations:
where,
a = r(k-1) , 1 f = P12 ' g = p2
12
(8.1.2)
(8.1.3)
(8.1.4)
(8.1.5)
(8.1.6)
In the intra-block analysis, the best linear estimate
" t. of the treatment effect t. is given by, l l
k-c " 2 t. = Q. +
i a i
c -c 1 2 s (Q.)
a 1 i (8.1.7)
The variance of the intra-estimate of the difference
between two treatment effects is given by:
2(k-c .) " " V(t.-t ) = cr2
i u a , (8.1.8}
where the treatments i and u are j-th associates (j=l,2).
In the inter-block analysis where only the block totals
" are used, M. Zelen (11) gives the best linear estimate t'. l
of the treatment effect t. as: l
where:
and
In this case,
A
t' i
o: 1.
c '. J
A
- 59 -
k-c2• = Q'. + r 1.
c'-c' 1 2 r
T. Q. rG =
1. 1. N J
c.6-r"}... = ] ]
6-rH+r2 '
2(k-c'.} A
(J • 2 V(t'.-t') = 1. u r
J (8.1.9}
(8.1.10)
(j=l, 2} . (8.1.11)
' (8.1.12}
where the treatments i and u are j-th associates (j=l,2),
and where
cr' 2 = (J2 + ko:2 b ( 8 .1. 13)
cr2 is the error variance in the intra-model, and ~ is the
variance of the block effect in the random or inter-model.
8.2 Variances and Covariances
From the restriction the
A A A
V( 2 t.) = . 1 1. 1.=
v V(t.) + v(v-1) Cov(t.,t.) = O 1. 1. J
0, or:
(8.2.1}
Since every treatment has n 1 first associates and n 2 second
associates, and n 1+n2 = v-1, then:
- 60 -
or (8.2.2)
A A
where t .. is the j-th associate oft., (j=l,2). From 1] 1
(8.1.8) for j = 1,2:
A A A A V(t.} + V(t. 1 } - 2 Cov(t.,t. 1 } =
1 1 1 1 a
and
A A
Cov(ti,ti2 ) = c2 , then (8.2.3) may be written as:
and
or
v1 + n 1c1 + n 2c2 = o ,
v -1
- 2C 2
k-c 1 - -- cr2 a
k-c = __ 2 cr2 a
,
,
,
(8.2.3)
(8.2.4}
(8.2.5}
(8.2.6)
- 61 -
By subtracting (8.2.6) from (8.2.5) and (8.2.5) from
{8.2.4), one obtains:
' (8.2.7)
and
{8.2.8)
Remembering that n 1 +n2+1 = v, (8.2.7) and {8.2.8) may be
written as:
, and
c = c2(nl+l)-nlcl-k cr2 2 av
Substituting (8.2.9) into (8.2.5), one obtains:
or
= k-c1 + c 1 (n2+1)-n2c 2-k
=
a av
vk-vc1 +c1 (n2+l)-n2c 2-k
av =
(8.2.9)
(8.2.10)
av
(8.2.11)
Since o, where t''s are the inter-estimates for i
'
- 62 -
treatments, one obtains, in exactly the same way:
V' k{v-l)-n1cl-n2c2
= 1 vr
C' ci{n2+1)-n2c2-k
= 1 vr
C' {n1+l)c;2-n1ci-k
= 2 vr
where A A
cr'2
cr'2
cr'2
' (8.2.12)
(8.2.13)
, {8.2.14)
A A
= Cov(t'.,t'. 2 ), and 1 1
t '. . is the j-th associate of t '.,, {j=l, 2). Let 1J 1
v =V 1 + V' 1 , (8.2.15)
c = c 1 + C' 1 , (8.2.16)
and
C' = c 2 + C' 2 (8.2.17)
In the combined analysis of Rao, the·variance of the
difference between two treatments is given in Reference (1)
as:
2(k-d.) V(T.-T.} = 1 J r[ w' +w(k-1)] , (j=l,2) (8.2.18)
Also, here
v
V[ ! T.] LT. = 0 = 0 , , . 1 1 . 1 1 1= 1=
or as in (8.2.2),
- 63 -
{8.2.19), together with the two equations resulted from
(8.2.18) for j=l,2, give a system of three equations of
three unknowns. The work is exactly parallel to that of the
intra-estimates, and
k(v-1)-n d -n d V(T ) = 1 1 2 2
i vr[w{k-l)+w']
as it was mentioned in (2.7).
8.3 Sign of the Quantity C-C'
By subtracting (8.2.10} from (8.2.9),
= c 1 (n1+n2+1}-c 2 (n1+n2+1)
av
(8.2.20)
'
'
= ' by (8.1.4) and {8.1.5),
=----Ma '
Similarly, by subtracting (8.2.14) from (8.2.13),
C'-C' = 1 2
=
ci(n1+n2+1)-c2(n1 +n2+1)
vr
c'-c' 1 2 r '
'
{8.3.1)
(8.3.2)
- 64 -
where cl, c2, cr' 2 are as defined in (8.1.11) and (8.1.13).
Substituting for cl, c2, (8.3.2) may be written,
Now by adding (8.3.4) and (8.3.1) one obtains:
C-C' 6(c1-c2)-r(A1-A2 )
02 + -----------r (6-rH+r2)
A1-A2 [6a(A1-A2 J/k6]-r(A1-A2 ) = --= a2 + __ ____;;; ______________ ____;;; __ k6 r(6-rH+r2 )
A1-A2 a(A1-A2 )-kr(A1-A2 ) = a2 + -----------
k.6 rk(6-rH+r2 )
and since a = rk-r,
Al-)..2 C-C' = --= k6
Substituting for cr' 2 from (8.1.13),
c-c'
,
1 02 - ----6-rH+r2
,
,
~] '
(8.3.4)
'
(8.3.5)
(A ).. ) [---=r ..... (=H.._-.-.r ..... )_ 02 - --=1-..,,.. = 1- 2 k6(6-rH+r2 ) 6-rH+r2 a2 J . (8.3.6)
Now if
6 ) 0, 6- rH+r2 ) O, H ) r , (8.3. 7)
then (8.3.6) is of the form:
- 65 -
C-C' = (1'1-A2 )[negative quantity] ,
i.e.,
C-C' > 0 if Al < A2 , (8.3.8)
C-C' < 0 if Al > A2
It is to be noticed that the above conditions in (8.3.7} are
satisfied in every design listed in R.C. Bose "Tables of
p.b. i.b. designs with two associate classes", Reference (1).
8.4 Theorem 3
As a generalization to Theorem 1, consider the t
independent parameters T1 , T2 , ... , Tt and suppose that for
each T. there exist two independent unbiased estimates U. i i
and X. , where U. ~ N ( T . , V. = 9 . 0 2 ) and X. /""'\ N ( T . , V ~ ) V. } • i i i i i i i i i
Suppose also that independently of the X. 's, U. 's there i i
exists an unbiased estimate s 2 for 0 2 where s 2 ~ X2 a2 then f ,
we can determine a known constant B so that the unbiased
combined estimate,
"' T. i
9 Bs 2 i = u i + -t-"------
I (X .-u .) 2 . 1 J J J= j;ti
(X.-U.) i i
has a variance less than V(U.) whenever t) 5. i
(8.4.1)
- 66 -
Proof:
A 1 E(T.) = 'T. + e .Bcr2 E -t--=--- • E (X . - U . ) = T .
1. l. l. l (X .-U.) 2
j~i J J
l. l. l.
z2 V(;,) = V(U.) + 9~B2E(s 4 )·E ____ i __ + 29.BE(s 2 )·E
l. l. l. t l. ( l z~) 2
j=l J j~i
where z . = X . - U . J z . = X. - U. . J J J l. l. l.
e~cr4B2 (f+2) V(~.) = V + _i. __ f ___ E(z~) ·E --1--
1. i l. t
but
( ~ z~)2 . . J J l.
· E[z.(U.-T.)] 1 1 1
(8.4.2)
Z, ( U. -T. ) l. l. l.
t
lz~ . 1 J J= j~i
(8.4.3)
E(z~) = E[ (X.-T. )-(U.-T.) ] 2 = E(X.-T. } 2 +E(U.-T. ) 2 = v.+V'. J 1 1 1 l. l. 1 l. l. l. l. l.
substituting into (8.4.3)J
A
V(T.) = V. l. l.
V~B2 (f+2)(V.+V~) l + 1 1 l. E ---='---
f t ( l z~) 2 •...J. J
J:r-1.
1 + 2V.BE-t--E[(X.-U.)(U.-T.)]
l. l. l. l. l.
lz~ j~i J
(8.4.4)
- 67 -
Let
¢. = v.+V'. = V(z.}, i i i i
i=l,2,3 ... t , (8.4.5}
then
V~B2 (f+2) ¢. l l V( T . } = v. + i f i E --=--- + 2V. BE ( 2 ) ( -v. )
i i t i i ( l z::} 2 z2 j~i J j~i j
,
V~B(f+2}¢. l l = V. + i i E ---- - 2V~BE -----
i f t i t ( l z~)2 l z~ j~i J j~i J
(8.4.6)
It should be noticed that zj A N(O, ¢j). Let zj = /¢j · Yj ,
then, Y. /'\ N(O, 1), .t.Y"J. """X(t-l) , and, J Jri
t
lz~ = •..J.. J Jri
Now one can write:
_........;;;;! __ ( E __ ........;;;;! ____ = ___;l;;;.....__ E ( 2 1 } 2 E t . t ¢2. X
\ \ min. (t-1} c L z::; l 2 ¢2 . c L Y~ > 2 j~i J min. j~i J
1 = ---,,..----------¢2. (t-3)(t-5) min.
,
(8.4.7)
(8.4.8)
- 68 -
and
1 = --=---qi max.
1 1 . E -X-2-=--- = -qi---='-(-t---3-}
(t-1) max.
(8.4.9)
Using (8.4.8) and (8.4.9), we can write (8.4.6) as:
qi (t-3} . (8.4.10) max.
In (8.4.10) we want the quadratic in B, in the last two
terms to be negative, i.e.,
or
V~B B(f+2)qi. 2 i [ i t-3 fqi2 . (t-5) min.
qi max.
2fqi2 . (t-5) 0 ( ( min.
B (f+2}qi . . qi
B opt.
i max.
fqi2 . (t-5) min. = --==------( f+2) qi . • qi i max.
] < 0 ,
, t > 5 .
' (8.4.11)
where B t is the value of B which minimizes the above op .
quadratic.
Since qi. ( qi , one can write: i :- max.
B = f(t-5) . (qimin.) 2 f +2 qi
, (8.4.12) max.
as an admissible B, i.e., within the above range of B which
"' makes V(T.) ( V(U.). i i
- 69 -
In the incomplete block designs where both inter- and
intra-block estimates are available and independent, the
~ variance is of the form a 1 a2 + a 2 ~. Let a2 = R, and know-
ing that ~ should never be less than a2 in any reasonable
incomplete block design,
<P • min. <P max. ' (8.4.13)
where a 1 , a 2 , y1 , y2 are known in terms of the parameters of
the design.
F(R) is a hyperbolic function; it is monotonically
decreasing if a 2y1 ( a 1Y2, and monotonically increasing if
a2yl) al Y2·
Now if a2y1 ( a1Y2 , then
F(cx:>) ( F(R) ( F(l)
and from (8.4.13),
<P • min. <P max.
In this case,
= F(R) ) F(ro)
B = f(t-5) f +2
F(l) ( F(R) ( F(a>)
, (8.4.14)
(8.4.15)
(8.4.16}
' (8.4.17)
- 70 -
and from (8.4.13},
<P . min. <P max.
= F(R) ) F(l} =
In this case
B = f(t-5) f +2
(8.4.18)
(8.4.19)
In both cases B can be computed from the parameters of our
design.
8.5 Special Case
When there are only two different variances, let t 1 of
the z's have variance <P1 and let the remaining t-t1 have
variance <P2 , then (8.4.7) becomes:
t I z~ = <Plx2 (tl-1) +<P2X2 (t-tl) .;:: <Plx2 ( v 1) +¢2X2 ( v 2> j~i J
t
j~izj = ¢1X2(t1)+¢2X2(t-t1-ll = ¢1X2(v1)+¢2X2(v2)
when V(zi} = <P2
(8.5.1)
(8.5.2}
It should be noticed that v1 and v 2 assume different values
in the two cases, but v 1+v 2 = t-1 always.
- 71 -
In both cases let us assume that
where
f{X) =
and
h
Iz~ = . 1 J J= j~i
1 x ~v -1
1
Applying lemma 1, we have
-y/2<!>2 e dy ,
thus substituting into {4.6) we get:
, {8.5.3)
x > o,
(8.5.4)
y > o,
{8.5.5)
du ,
du {8.5.6}
Suppose that one is able to find a lower bound L and an
upper bound P for the ratio which are relatively not
- 72 -
far from each other, then one can write:
~v -1 ~v -1 Sl u 1 (1-u) 2
0 [u+L{l-u) ] 2 du '
, {8.5.7)
and
~v -1 ~v -1 Sl u 1 {1-u) 2 u+P(l-u)
0 du '
(8.5.8)
Accounting for (8.5.3), and substituting (8.5.7} and (8.5.8)
into (8.4.6}:
,
(8.5.9)
Assuming t ) 5, the second term is negative when
2f (t-5} G1 <l\ 0 ( B ( (f+2 )@.G (8.5.10)
' ' l. 2
The value B which makes it the most negative is: opt.
B opt. =
- 73 -
f<I>1G1 (t-5)
<I>iG2 (f+2) (8.5.11)
Since <Pi is either <I>1 or <I>2 in this special case, we have
B opt. = fG1 (t-5)
(f+2)G2 ' when <I> = <I> i 1 (8.5.12)
However, (8.5.11) depends on the variances <I>1 , <I>2 when
<Pi = <I>2 ; to avoid this, we go back to (8.5.6) and take <I>2
outside the integral instead of <I>1 to get:
= du .(8.5.13)
¢2 1 1 The lower and upper bounds for <Pl are P and L respectively,
hence using (8.5.13):
r(~v 1+v 2-2) ~l J,;v 1-1 ~v -1
E(x!Y)2 2
< ;~ u + (1-u}
4q,~ r ( ~v 1 > r ( ~v 2 ) r (1-u)]
r(~v 1+v 2-2)G2 G' 2 = 4<I>~ r ( ~v 1) r C ~v 2 ) = (t-3) (t-5)f3(~v 1 , ~v 2> q,~
~v -1 2 (1-u)
+ (1-u)
du ,
(8.5.14)
du '
(8.5.15)
- 74 -
Substituting in (8.5.7) for the case, where ¢i = ¢ 2 :
V~B B(f+2)G2 = v. + ]. [ - 2G I ]
i (t-3)~(~v 1 , ~v 2 )¢2 f(t-5) 1
2V2 BG' i 1
Assuming t ) 5, the second term is negative when
and
2f (t-5)Gi 0 < B < ------(f+2)G2
B opt. = f (t-5)Gi
(f+2) G2
'
(8.5.16)
(8.5.17)
(8.5.18)
,
This is independent of the ¢'s, and (8.5.18) will be m.ed.when
¢i = ¢ 2 . Substituting (8.5.12) into (8.5.9), we get for the
case ¢i = ¢ 1 :
or
But
A V~fGl(t-5) V(Ti) < vi + (f+2}G2(t-3)~(~vl, ~v2)¢1 [G1- 2G1]
A
V(T} < V. ].
v2 v;; i ].
¢ 1 = v. +v~ ]. ].
f(t-5}G~
is the utmost possible recovery.
recovery ratio is at least:
,
(8.5.19)
Thus the
- 75 -
f(t-5)G~ D = ~~~~~~-='--~~~~-
( f + 2) G 2 ( t - 3 )~(~v1, ~v2) (8.5.20)
Similarly substituting (8.5.18) into (8.5.16}, we get for
the case ¢. = cP : l 2
A V~f (t-5)Gi V(Ti) <vi+ (t-3)~(~vl, ~v2)cP2(f+2)G2[Gi- 2GiJ ,
Again v2 v2 ......!. = _.;;;:i_ cri2 v.+v~
l l
(8.5.21)
is the utmost possible recovery, and the
recovery ratio is at least:
(8.5.22)
8.6 Recovery of Inter-block Information in Group Divisible
Designs
In this case v = mn, and the treatments can be divided
into m groups of n treatments each, such that any two treat-
ments of the same group are first associates, while two
treatments of different groups are second associates. The
association scheme can be displayed by placing the treat-
ments in a rectangular of m rows and n columns, where each
- 76 -
row of n treatments constitutes a group. Clearly:
Further:
or
Also:
n = n-1 1 , n = n(m-1)
2
(n-l)A1 + n(m-l)A2 = r(k-1}
,
,
n-1 \
n(m-2)}
(8.6.1)
(8.6.2)
(8.6.3)
(8.6.4)
Bose and Connor have shown that the following inequalities
hold in group divisible (G.D.) designs:
r ) /.. - 1 , rk - f.. v) O 2-
They have divided the G.D. into three subclasses:
(i) Singular (S) if r = t..1 ,
(8.6.5)
(ii) Semi-regular (SR) if r ) t..1 and rk - t..2v = O,
(iii) Regular (R) if r) t..1 and rk - t..2v) 0.
8.6.1 Recovery in Regular Group Divisibles
In regular G.D. designs we define the mn x mn matrix
M1 as follows:
1 2
n-1
n
2n-2
m(n
-1)
m(n
-1)+
1
mn
-2
mn
-1
mn
1
1 2
3. . .
n
~
1 -1
1 1
-2
. . . . . . 1
1 1 ••• (-n
+l)
. .
. .
.
1 1
1 ... 1
1 1
1 ... 1
. .
. .
. 1
1 1 ...
1
1 1
1 ... 1
]._ 1
1 ... 1
2
1 2
3 ... n
1 -1
1 1
-2
. .
. .
. . 1
1 1
. .. (-n+
l}
. .
. .
.
-1 -1
-1
... -1
1 1
1 ... 1
. .
. .
. .
1 1
1 ... 1
1 1
1 ... 1
1 1
1 ... 1
3 m
-1
m
1 2
3 ... n ... 1
2 3
n 1
2 3 .•
e n -
. .
. .
. .
. .
. .
. .
. .
. .
.
1 -1
1 1
-2
. .
. .
. .
1 1
1 . . .
(-n+
l}
1 -1
1 1
-2
. .
. .
. 1
1 1 ... (-n
+l)
-2 -2
-2
... -2
. .
. .
. .
. .
. .
. .
. .
. .
. 1
1 1 ...
1 . . . (-m
+2
) (-m+
2) ... (-m
+2
}
1 1
1 .... 1 . . .
1 1
1 . ..
1 (-m
+l} ... (-m
+l)
1 1
1 ... 1 . . .
1 1
1 . . .
1 1
. . . 1
-
- 78 -
Define the following contrasts between the rnn intra-block
A A A
estimates of the treatments t 1 ,t2 , ... ,tv:
ul
u2 u = = M
A
tl A
t2
A
t nm
A
(8.6.6) = Mt '
where M is the same as M1 after normalizing its rows.
Define also the similar contrasts between the inter-block
A A
estimates of the treatments, namely tl, t I: v
x2 A x = = Mt' = M .(8.6.7)
A
t' nm
0
The problem now is to combine u and x to get new estimates A
'r l :I.. = It should be noted that U = O, X = O, and nm nm
the corresponding combined estimate is assumed to be zero,
A
i.e., 'r = 0. nm
- 79 -
For the variance-covariance matrix of both U and X, it
is noticed first that U. is uncorrelated with u. for i ~ j; J. J
also X. is uncorrelated with X. for i ~ j, and the u. 's and J. J J.
X. 's are independent of each other for every i, j, by virtue J
of the well known fact that the inter-block estimates are
independent from the intra-block estimates.
"' "' It should also be noted that in the vector tort' we
are grouping the treatments according to group divisible
association plan, i.e., the first n are the first group (or
row} in the plan, the next n treatments are the second row
in the association plan, etc. ; moreover, . it should be noted
that the contrasts included in the matrix M are of two kinds,
within group contrasts and among group contrasts. It has
been shown in (7.2.4) that all within contrasts have exactly
the same variance, namely v1-c1 , where v1 is the variance of
the intra-treatment estimate and c1 is the covariance between
two first associate intra-estimates. For the variance of
the contrasts among groups, let us take a general one,
U 2 , say. Then mn-
- 80 -
1 = n(m- 2 ) (m-l)(n(m-2)(m-l)V1+(n-l)c1 (n(m-2)+n(m-2)2]
,
' (8.6.8)
where c 2 is the covariance between two second associate
intra-estimates. Now V(U 2 ) is independent of m, which mn-
means that all among contrasts have the same variance.
Similarly, one can deduce the variances of the X's, it
contrasts, where Vi is the variance of the inter-treatment
estimate, and cl, c2 are the covariances between two, first
or second associate inter-estimates respectively.
Consider now the vector:
z =
z mn-1
= x - u
z. and z. are independent for i ~ j. l J
, (8.6.9)
- 81 -
For within group z's, one has
For among group z's, one has
= V-C + n(C-C') , (8.6.11}
where V, c, C' are defined in (8.2.15) through (8.2.17}.
From (8.3.8), it is seen that
v-c > (V-C) + n(C-C') , if Al) A2 ,
and
v-c < {V-C) + n{C-C') , if Al ( /\2 .
This will give rise to two divisions of the problem:
(i) Regular G.D. with A1 ) "A2 ,
(ii) Regular G.D. with Al ( A2 .
8.6.1.1 Study of the Ratio (V-C)/(V-C+n(C-C')]
Subtracting (8.2.9) from (8.2.11),
k-c1 - -- cr2
a
(8.6.12)
,
(8.6.14}
- 82 -
Subtracting (8.2.13) from (8.2.12},
Adding
V'-C' 1 1
(8.6.14)
kv-cl{n1+n2+1) cr'2 = vr
and {8.6.15},
k-c
k-c' 1 cr'2 = r
k-c' v-c = v -c + V'-C' =--1 CJ2 + 1 cr'2
1 l 1 1 a r
By (8.3.5) one can write:
V-C + n(C-C') k-c n(A1-A2 ) k-c'
= [--1 + ] cr2 + [ 1 a k6 r
J
. (8.6.15}
(8.6.16)
(8.6.17)
1 From {8.6.4) one obtains f = p 12 = O, g = p~2 = n-1, and
substituting in (8.1.2),
k 2 6 = {a+A1 )(a+A2 } +(A -A )(-a-A )(n-1) 1 2 1 J
and using {8.6.3), one can write
{8.6.18}
Substituting f and gin (8.1.3),
kH = (2a+A1 +A2 ) - n(A1-A2 ) + A1-A2 ,
= a+A1 + [a+A1-n{A1-A2)J ,
- 83 -
and using (8.6.3), one obtains
Substituting f and gin (8.1.4),
by (8.6.3}:
Subtracting (8.1.5) from (8.1.4),
or
ak()..1-)..2} cl-c2 = k 2 6 '
and using (8.6.18), one obtains:
ka{)..l-).2} cl-c2 - )..2v{a+)..1 )
Now using (8.6.18) and (8.6.20),
k-c 1
k 2 6-Mc1 a)..2v A2Vk = = = k 2 6 a aM aM
k ---a+)..l
A2Vk = :A2v(a+:A1 }
(8.6.19)
(8.6.20)
(8.6.21)
,
(8.6.22)
- 84 -
(8.6.22} indicates that in regular G.D., k-c1 ) 0 or k) c 1 .
Let A= 6-rH+r2 , then
kA = M - rkH + r 2 k ' and using (8.6.18) and (8.6.19), one obtains
(r-)\1 ) (rk-P.2v) . (8.6.23)
k
Since r) A1 , rk) )\2v in regular G.D., one can say that
A = 6-rH+r2 ) 0 always in this class of designs.
Also, from (8.1.11),
k-c' __ l r
k - -r
,
and using (8.6.20) and (8.6.23), one obtains
k-cl k A1A2v-kr)\1
=
-r r r k(r-A1 )(rk-/\2v)
k(r-)\1 )(rk-)\2v)+k)\1 (rk-)\2v}
r(r-A.1 }(rk-A2v)
,
= (8.6.24)
Substituting (8.6.22) and {8.6.24) into {8.6.16), one can
write
v-c = (8.6.25)
- 85 -
From (8.6.18),
n('Al-/\2) nk('Al-/\2) nk{/\1-/\2) = k 2 6 = k6 /\2v{a+/\l)
,
and using (8.6.3), one obtains
n(/\1-/\2) k{a+/\1-/\2v} k k = =-- -
k6 /\2v(a+A1 } )\2V a+/\l (8.6.26)
From (8.6.23), one can write
n(/\1-)..2) nk(A.1-/..2) nk()..1-/..2) = nk2 A = kA (r-)\1 ){rk-f..2v) . (8.6.27)
Substituting (8.6.22), (8.6.24}, (8.6.26), and (8.6.27) into
{8.6.17),
V-C + n(c-c')
and by using (8.6.3), one obtains
v-c + n(C-C'} ,
(8.6.28)
From (8.6.25) and (8.6.28), one can write:
- 86 -
k a2 + k cr'2 a+'/\l r-'/\l v-c __ ___;.___;;;_____ = ---------------------
v - c + n (c-c') k cr2 + k er. 2 "A2v rk-'/\2v
Substituting {8.1.13) into (8.6.29),
( 1 1)2 k 2 + er +--a: V-C a+'/\l r-'/\l r-'/\l b -------"----- = ____________ ___;;;; _________ __,;;;; _____ ~
v-c+n(c-c') 1 1 k ( "A2 v + rk- '/\2 v) a2 + rk- )\2 v er~
Let 0:2 b 2 = R ) 1, then, cr
where
r
,
'
'
1 132 = J
(8.6.29}
J
(8.6.30)
(8.6.31)
d.F(R) = dR
- 87 -
~l(a2+~2R)-~2(al+~lR)
(a2+~2R)2
Substituting from (8.6.31), one can write,
'
r
r(a+A1 )-r"A2v r(a+).1-f,.2v) = ~~~~~----~--=-~~~ = ~~~~~~~~~~~ A2v(r-/,.1 }(a+f,.1 )(rk-/,.2v) A2v(r-A1 )(a+/,.1 )(rk-A2v)
by (8.6.3),
'
'
nr(A1-"A2 ) = ~~~~~~~~~~~
f,. 2v(r-i\1 ) (a+/,.1 ) (rk-i\2v) (8.6.32)
Since r-/,.1 ) O, rk-/,.2 ) 0 in regular G.D., one can say
that
Thus, F(R)
d.F(R) dR is
< 0
is monotonically
monotonically decreasing if
F(l) < F(R) < F (ro)
F (ro) < F(R) < F(l)
(8.6.33)
if i\l < i\2
increasing if "'1 > "'2 ' and
"1 < i\2 • This means that
when "'1 > )\2 (8.6.34)
when "1 < A2
- 88 -
Substituting from (8.6.31),
F(l} = = = A2v(rk-A2v} (rk+A1} (r-A1 ) (a+A1 ) (r+A2v} '
(8.6.35}
and
F(oo) 131 rk-A2v
-- -- -!32 r-A 1
(8.6.36}
For the inverse ratio G(R) 1 v-c+n ( c-c ' ) = = F(R} v-c ' one has
G(l) 1 (r-A1}(a+A1 ) (r+A2v} = = F(l} A2v(rk-A2v} (rk+A1 } (8.6.37}
and
1 r-A G(oo} 1 = = F (oo) rk-A2v (8.6.38}
Also,
G(oo} < G(R} < G(l} if Al) A2 ' (8.6.39)
G(l) < G(R} < G(oo} if Al ( ).2
8.6.1.2 Application of Special Case of Theorem 3
It should be noticed that the results of the special
case are exactly what one desires here, where w1 = V-C,
w2 = V-C + n(C-C'). Let:
- 89 -
L = (8.6.40}
Then for within comparisons with variance w1 = v-c, one has
v1 = m(n-1)-1, v2 = m-1. Thus for:
( i} "I. > "I. /\1 /\2:
' (8.6.41)
and G2 in (8.5.7) becomes:
sl ~(v-m-3) ~(m-3) G2
_ u {1-u} du - 0 [u+L(l-u)] 2 (8.6.42)
Also Gl in (8.5.8) becomes:
sl ~(v-m-3) ~(m-3) Gl
_ u (1-u} du - O u + P(l-u) (8.6.43)
And for:
(ii) "Al < "'2=
G2 and G1 in (8.5.7) and (8.5.8) become:
Sl ~(v-m-3) ~(m-3) _ u (1-u) G2 - 0 [u+P(l-u) ]2 du ' (8.6.44}
- 90 -
and
Sl ~(v-m-3) ~(m-3) G = u (1-uJ du
1 O u + L(l-u)
For among comparisons with variance w2
one has v1 = m(n-1), v2 = m-2, and for:
;
G' 2 in (8.5.14) becomes:
= sl ~(v-m-2) (l-uJ~(m-4)
G' u du 1 2 0 [- u + (1-u)] 2
p
= P2 \1 u~(v-m-2) (l-uJ~(m-4) ~ [u + P(l-u)] 2 du
G' in (8.5.15) becomes: 1
Gl = sl u~(:-m-2) (l-u)~(m-4) du ,
0 L u + (1-u)
,
(8.6.45)
= V-C + n(C-C'),
(8.6.46}
(8.6.47)
fl ~(v-m-2) ~(m-4) = L u <( 1-r> du (8.6.48} u + L 1-u
- 91 -
And for:
(ii) Al ( 1-.2:
CD 1 1.<-1.< L ~2 - p i (8.6.49}
G' 2 and G]_ become, in this case:
G' = Il u ~(v~m-2) ~l-u}~(m-4) du 2 ,
[- u + (l-u)] 2 L
= L2 sl u~{v-m-2) ~l-ul~(m-4}
du 0 [u + L(l-u)] 2 , (8.6.50)
and
G' = sl u~<:-m-2) (l-u}~{m-4)
du 1 , 0 P u + {1-u)
= p Il ~(v-m-2) ~(m-4) u ( 1-ul
u + P(l-u) du (8.6.51}
One notices that the integrals for the within comparisons
can be evaluated for m ) 2, but for among comparisons one
must have m) 2. The combining constant B and a conservative
lower bound of the recovery ratio Dare given in (8.5.12)
and (8.5.20} for the within comparisons; and for the among
comparisons Band Dare given in (8.5.18) and (8.5.22},
respectively. The required integrals have been evaluated
- 92 -
and the corresponding values for B and D have been computed
for 62 regular G.D. designs that appear in Bose's "Tables
for P.B.I.B". The results are listed in Tables III and IV
for the within and'arnong comparisons respectively. The
combined estimate, as given in (8.4.1) is
A
T. = U. + 1. 1. v-1
9 Bs 2 i (X.-U.)
1. 1. ' (8.6.52)
I (X.-U. )2 j=l J J j~i
where 9. is the coefficient of cr2 in V(U.). 1. 1.
For within comparisons, one has, using (8.2.5} and
(8.6.22):
k-c V(U.) = V -C = l cr2 =
i 1 1 a , (8.6.53)
hence 9. 1.
k --- and the combined estimate in (8.6.52)
becomes:
"' kBs 2 Ti = ui + ~~---v---1---------
(a+t..1) .l (X.-U.} 2
j=l J J j~i
(X.-U.) .(8.6.54) 1. 1.
For among comparisons, one has, using (8.6.8), (8.2.5),
(8.3.1}, (8.6.22), and (8.6.26}:
V(U.) J.
- 93 -
,
,
{8.6.55}
hence and the combined estimate in (8.6.52)
becomes:
(X.-U-.) (8.6.56) J. J.
The B's in (8.6.54) and (8.6.56) are the combining constants
to be obtained from Tables III and IV, respectively.
8.6.2 Analysis for Singular and Semi-regular G.D. Designs
The formula (8.1.9) of M. Zelen for the inter-block
estimate is not applicable in the two subclasses, singular
G.D., and semi-regular G.D. designs because the quantity
6-rH+r2 = 0 for both. An inter-analysis for those subclasses
will now be given.
It is known that in the inter-block analysis it is
assumed that the block effects b are random variables, s
uncorrelated with each other and with the plot errors e .. 's, l.JS
having mean zero and {unknown) variance 0~. Let the total
- 94 -
of blocks be denoted by B, s = 1,2, ... ,b, and the effect of s
the treatment in the i-th row and j-th column in the associa-
tion scheme rectangle by t .. , where i = 1, 2, ... ,m, j = 1, 2, ... ,n, lJ
mn - v. One can then express the yield y,. as: lJS
Yi. J's = µ + t .. + b + s:.. , lJ s lJ s (8.6.57)
where y,. is defined only when the treatment (ij) occurs in . lJS
the block s. The total of the s-th block is:
where
B = kµ + s
6.. = lJS
m n I Io .. t .. . 1 . 1 l.JS lJ l= J=
+ kb s + ~ !o. e.. , . 1 . 1 lJ s l.J s l.= J=
(8.6.58)
1 if (ij)-th treatment occurs in s-th block,
0 otherwise
Here E(B) = kµ + .Z.Z 5 .. t .. and V(B) = k(cr2 + ka2 ). One S ij lJS lJ S b
obtains the normal equations by minimizing the quantity,
I <a -kµ - 2: 1 o . . t .. l 2 -1 S . . lJS lJ s= i J
The normal equations, therefore, are:
b
-2k I (B -kµ - lo .. f ~ . ) = o 1 S . . lJS lJ
s= iJ
b
e It .. . . lJ lJ
'
- \ 5 .. (B -~ s~l lJS S l A I ) 6.,.,t.,., -9=0
l J s l J i'j I
(8.6.59)
(8.6.60)
(8.6.61)
- 95 -
b
Noticing that Io .. S=l l.JS
= r, ! !6. . 1 . 1 l.J s l.= J=
= kJ and simplifying
(8.6.60)J and (8.6.6l)J one can write
b
kllo .. t'..=kG 1 . . l.J s l.J s= l.J ' (8.6.62)
b A \ A
kr µ + /_,; ( 2: 0 . , 0 , I • 1 ) t : 1 • I = • I • I S l.J S l. J S l. J
\ o .. B s~l l.JS S
+ 9 . ( 8. 6. 63) l. J
A A - _Q_ Let 2:Z2: o .. t'.. = 0 , then (8.6.62) givesµ - bk, and sij l.JS l.J
substituting in (8.6.63) one obtains:
" 0 .. 0. I • I }t: I • I = B .. l.JS l. J S l. J l.J. rG + n = Q' + n b 't::7 • • 't::7 l.J J
i I j I (8.6.64) b
where B .. l.J . = \ o .. B
sf'l l.J s s = the sum of block totals in which
treatment (ij) occurs. If ij = i'j' then
z o .. o., ., = z o~. = z o .. = r J thus (8.6.64} becomes S l.JS l. J S S l.JS S l.JS
"'• rt .. + l.J I (Io .. o .... )t: ... = i'j'~ij S l.JS l. JS l. J Q~j + 9 '
or
rt : . + l cl 0 , , 0 , • I } t : • I + l l cl 0 • • 0 • I • I . } t : I • I =Q ~ • +9 J l.J ·•..J· l.JSl.JS l.J ·•~· , 1 l.JSl.JS l.J l.J J r J s l. rl. J s i'=i
C8.6.65}
but,
- 96 -
I6ijs6ij 's =the number of blocks in which treatment s
(ij) and treatment (ij'), where j~j' occur together= Al.
Also,
I 6 . . 6. , . , = the number of blocks in which treatment lJS l J S s
(ij) and treatment (i'j') occur together, where i~i' = A2 .
Hence (8.6.65) becomes:
rt : . + l A1t '. . , + l l A2t : , . , = Q ~ . + e . ( 8. 6. 66) lJ , I , lJ , I.../., , I l J lJ J =J l rl J
Summing over all treatments one obtains:
r(sum of all treatments) + n 1A1 (sum of all treatments)
+ n 2A2 (sum of all treatments) = O + ve , (8.6.67)
i.e., ve = O or e = O. Using again the assumption that
the sum of all treatments = O, one can write:
I If: ... =-If: .. • 1...J • •I l J •I lJ i .rl J J
' (8.6.68}
i.e., the sum of all treatments except the i-th row= - the
sum of the i-th row. Hence (8.6.66) takes the following
form for all treatments in the i-th row:
Q'. . ' lJ j = 1, 2, ... , n ,
- 97 -
or
A
t~ .• = Q~., l.J l.J
j = 1, 2, ... , n .
(8.6.69)
Now there are m equations similar to (8.6.69) for
i = 1,2,3 ... m. Equation (8.6.69) becomes in matrix notation,
for j = 1, 2, ... , n:
r-).. 2
)..1-)..2
)..1-)..2
r-).. 2
)..1-)..2
/\1-)..2
r-).. 2
A
tj_l
"'• ti2
A
t' in
=
Qj_l
Qj_2 (8.6.70)
Q'. in
8.6.2.1 Inter-estimates and Recovery of Inter-information
in-Singular G.D.
In this case r = ).. and the coefficient matrix in 1
(8.6.70) is of rank one. Out of each group or row of treat-
ments, one can in fact obtain an inter-estimate of one
selected treatment and impose arbitrary values for the
remaining n-1 treatments. This is natural, due to the fact
that one can construct singular G.D. designs by stretching
every treatment in a B.I.B. design to become a group of n
treatments or a row in the association scheme. Assume that
- 98 -
n-1 treatments in each row have the estimate zero. Then one
can write
i = 1,,2,,3 ... m (8.6.71}
"' and t ~ . , = 0 for j ' = 1,, 2,, ... ,, j-1,, j + 1, j +2,, ... ,, n. J.J
In this way one has inter-estimates for m treatments,,
one from each row. It should be noted that these estimates
must sum to zero,, and that each treatment is a second
associate of all the remaining m-1 treatments. It is
obvious that one can inter-estimate m-1 comparisons; we
shall choose the m-1 among comparisons defined in Section
8.6.1 to be inter- and intra-estimated. They have the same
variance,, as it was shown in (8.6.8),, so that theorem 1 is
applicable. From formula (3.1), where mis m-1, one obtains:
"' 1'". = J_
9(m-3)s 2 u i + -m---1-----
L (X.-U .}2 j=l J J
(X.-U.) J_ J_
This combined estimate is applicable form 3.
was shown before in (8.6.55). Hence
(8.6.72)
pf is the .e=-, p f +2
= -1L CY2 f..2v
,, as
- 99 -
(8.6.72) becomes:
A
'L . J_ = u. + 1 i m-
fk(m-3)s 2
~ f + 2) "'2 v I (X. - u.) 2
j=l J J
(X.-U.) J_ J_
The recovery ratio, from theorem 1, is then:
D _ -c~-L (m-3) -( 2+f)(m-l)
where f = rv - v - b + 1.
,
(8.6.73)
(8.6.74)
The choice of the among comparisons is justified here
by the fact that the efficiency E of any comparison between
first associate treatments is unity, as R. c. Bose's tables
for P.B.I.B. indicate.
8.6.2.2 Inter-estimates and Recovery of Inter-block Infor-
mation in Semi-Regular G.D.
In this case rk = J..2v, and by (8.6.3) one can write:
or
or
(8.6.75)
- 100 -
The matrix equation in (8.6.70) becomes:
-nl(t..1-/\2) "1-"2 )\1-"2
Al-)..2 -nl()..l-A2) "1-)\2
"' tf.1
/'-.
ti2
/'-.
t' in
0 i1
Qi_2 =
(8.6.76)
The coefficient matrix in (8.6.76) is of rank n-1, i.e., out
of each group or row of treatments, one can in fact inter-
estimate n-1 treatments and impose arbitrary value for the
remaining one. Applying the restriction O, and
using (8.6.75) one can write (8.6.69) as:
' or
n
-n(1..1-A2>fij + o,1-A2> .I1t:ij = 0 ij J= ' or
/'-.
-n(/\l-/...2)t ~. = Q~. ' J.J 1J i.e.,
/'-.
t ~ . J.J ' (8.6.77)
j = 1,2,3 ... n-l
- 101 -
It is obvious that one can inter-estimate n-1 compari-
sons within each row of treatments; in all one can inter-
estimate m(n-1) comparisons. we shall choose the m(n-1)
within comparisons defined in Section 8.6.1 to be inter- and
intra-estimated. They have the same variance, namely v1-c1
for intra-estimate and Vl-cl for inter-estimate, so that
theorem 1 is applicable. This gives the following combined
estimate:
"' 'LI = J.
u. + J.
9(mn-m-2)s 2
m(n-1) I (X.-U.)2 . 1 J J J=
(X. -U.) J. J.
(8.6. 78)
This combined estimate is applicable for mn-m-2 = v-m-2 ) 0
> 2 "•pf . h .. or v-m . Q-r+2, p is t e coefficient f 2 . o a in k-C
1 V(U.)=V-C =-----1. 1 1 a
0 2 = _k_ 0 2 as it was shown in a+)... , 1
(8.6.53). k Hence P = , and then (8.6. 78) becomes a+)...l
"' 'T . l.
k(v-m-2)s 2 = u. + i v-m
(a+/\l)l (X.-U.)2 .. 1 J J J=
The recovery ratio is,
___ .I_" (v-m-2) D -( 2+ f)(v-m)
where f = rv-v-b+l .
(X. -U.) l. l.
f ..,.._ f +2
(8.6.79)
(8.6.80)
- 102 -
The choice of the within comparisons is justified by
the fact that the efficiency E1 of a comparison between first
associate treatments is always less than E2 , the efficiency
of a comparison between second associate treatments, as
R. C. Bose's tables for P.B.I.B. indicate.
8.7 Recovery of Inter-block Information in LS Type
In this case v = n 2 , and the treatments can be arranged
in an mm square association scheme so that two treatments
are first associates if they occur together in the same row
or in the same column, and they are second associates other-
wise. Such a design will be said to belong to the sub-type
L2 of the Latin Square type design denoted by LS. We also
have designs with n 2 treatments belonging to the sub-type L3
of the Latin Square type design. In this case it is possible
to form an nxn square array and to impose a Latin Square
with n letters on this array, so that any two treatments are
first associates if they occur in the same row or column of
the array or correspond to the same letter, and are second
associates otherwise.
The following relations hold for the sub-type L. of LS ].
designs (i=2, 3).
n1 = i(n-1} ,
- 103 -
n = (n-l)(n-i+l) 2
( i 2 -3i+n
(i-1) (n-i+l)
(i-1) (n-i+l})
(n-i}{n-i+l)
(i(i-1)
p2 = i (11-l}
i(n-i) )
(n-i) 2 +i-2 ,
Consider now the following n2 x n2 matrix M1 ,
(8.7.1)
(8.7.2)
M = 1
1
2
n-1
•
n(n-1) n(n-1)+1
n 2 -l
n2
1 -1
1 • 1
.
1 . 1
1 -
2
-1 1 -2 . . . . 1 ...
. .
1 ... • . 1 . . . 1 . . .
n n+l n+2
. -n+l
1 -1
1 1 -2 . . . . . 1 1 ...
. • . . . .
1 -1 -1 . . . • . . • . •
1 1 1 . . . 1 1 1 • • •
2n
. . -n+l
. . . . .
-1 . . • . •
1 . . . 1 . • .
n(n-1)+1 n(n-1)+2
. • • . . . 1 -1
1 1 . . . . . . . . . 1 1
• • • • . . -n+l -n+l
1 1
. •
-2 . . . . . .
• . • • • . . .
n2
.
. . -n+l
•
-n+l 1
I-' 0 .j::>.
- 105 -
Let M be the matrix M1 after normalizing its rows, then M is
orthogonal. Define:
u =
Also define
X=
A A
u n(n-1)
u n(n-1)+1
u 2 1 n -
0
x n(n-1)
x n(n-1)+1
x 2 1 n -
0
=M
=M
A
t 2 n
A
t' 2
A
t' n2
(8.7.3) A
=Mt
A
=Mt' , (8.7.4)
where t. and t' are the intra- and inter-estimates of the l. i
A
i-th treatment effect, respectively. The treatments in..!:. or
€' are arranged according to the rows of the association
scheme, i.e., the first n are the treatments of the first
row, the next n treatments are those of the second row, etc.
- 106 -
Since the treatments in the same group (or row) are first
associates, we have in similarity with (7.2.4) that:
V ( U j ) = V 1 - C 1 , for j = 1, 2 , . . . , n ( n-1) . ( 8 . 7 . 5 )
Thus all within comparisons have the same variance, namely
For the among comparisons, let us compute for L. 1
sub-type (i=2,3):
1 A A A A
V(U ( ) ) = V(/ [(t1+ ... +t )+(t 1+ ... +t2 ) n n-1 +m nm(m+l) n n+ n
+ ... -m(t 1+ ... +t ( lJ})} , nm+ n m+
1 = V1 + nm(m+l) c1 [n(n-l)m+n(i-l)m(m-l)+n(n-l}m2 -2nm2 (i-l))
1 + nm(m+l} c2 [nm(m-l)(n-i+l)-2m2 n(n-i+l}] ,
nmc1 = v1 + nm(m+l)[n-l+(i-l)(m-l)+m(n-l)-2m(i-l)]
c2nm{n-i+l} + { l) (m-l-2m) , nm m+
cl = V 1 + m+l [ (n-1) (m+l) +(i-1) (m-l-2m)] - c2 (n-i+l} ,
- 107 -
This is independent of m, which indicates that all among
comparisons have the same variance, namely:
V(Uj) = vl-cl + (n-i+l)(Cl-C2) ,
where j = n(n-1)+1, . . . , (8.7.7)
and (i=2,3) according to the design being of L2 or L3 sub-
types.
The inter-variances are similarly:
v (X . ) = v I - c I J 1 1 , i = 1,2, ... ,n(n-1}
v(xj) = v1-ci + (n-i+l)(ci-c2) ,
j = n(n-1)+1, ... ,n2 -l,
, (8.7.8)
(8.7.9)
and (i=2,3).
Let:
Then,
and
z = x - u (8.7.10)
V ( Z . ) = V (X . ) + V ( U . ) J J J J
= v1-ci + vi-c1 = v-c, j = 1,2 ... n(n-l) ,
(8.7.11)
= V-C + (n-i+l) (C-C') , j = n(n-l)+l, ... ,n2 -l ' (8.7.12)
where V, c, C' are as defined in (8.2.15) through (8.2.17).
- 108 -
As it was shown in (8.6.16), one can write in this case:
k-c k-c' v-c = 1 er2 + __ l er'2
a r (8.7.12)
By (8.3.5) we can write:
V-C + (n-i+l}(C-C') (8.7.13)
k-c' (n-i+l) 0..1-i-.2) + [ 1 + ] 12 ~----=-~ er r k(.6.-rH+r2 )
One notices that the special case of theorem 3 is applicable
here where w1 = V-C, w2 = v-c + (n-i+l)(C-C'), and
a er2+a er' 2 = ~~~l=--~~2=--~~~~ (al+f3l}cr2+(a2+f32}er'2
(8.7.14)
Substituting for er' 2 from (8.1.13}, one obtains
¢1 ( a 1 +a2 ) er2 +ka2 er~
¢2 (al+f31+a2+f32}er2+k(a2+f32)er~ , (8.7.15)
where: k-c 1 k-c' 1
al = a ' a2 = r ' (8.7.16)
(n-i+l} ().,1-i-.2} - (n-i+l) (A -).. }
f31 f32 1 2
= = k(.6.-rH+r2 } k.6. ,
- 109 -
Remembering that 0:2 b ---;;r = R) 1, one can write (8.7.15) as:
CJ
dF(R) dR
' (8.7.17)
(8.7.18)
For the LS type designs listed in Bose's tables a 1 , a 2 ,
6, and (6-rH+r2 ) are all positive, hence the sign of
Substituting ~l' a 1 , ~2 , a2 from (8.7.16), one obtains
(n-i+l)a2 (n-i+l)a1 = (Al-f..2)[ k6 + k(6~rH+r2 )]
(8.7.19)
Thus, the sign of dF{R} is the same as that of J..1-t..2 , i.e., dR
F(R) is monotonically
F(R) is monotonically
Consequently one can write:
F(l) < F(R) < F(ro)
F(a::>) ( F(R) ( F(l)
increasing if
decreasing if
when Al ) A2
when 1'1 < t.. 2
/..1)
Al <
,
For the inverse ratio G(R} = 1 F(R) '
one has:
f..2 ,
A2
(8.7.19)
- 110 -
G(CXJ) < G(R) < G(l) if )..1) )..2 ,
G(l) < G(R) < G(<D) if Al< )..2
Then let
L = G(l) 1 1 + /31 +(k+l} /32
= F(l} = a 1+(k+l)a2 ,
and
p = G(<D) 1 1 /32
= = +-F(CXJ) a2
8.7.1 Within Comparisons
Since v1 = v-n-1, v2 = n-1, then:
( i) "\ > "\ /\1 /\2 :
<P p < <P2 < L
. 1 . , using (8.7.20).
G2 in (8.5.7) becomes:
-sl u~(v-n-3) (l-u)~(n-3) du G2 - 0 [u + P(l-u)] 2
Also G1 in (8.5.8) becomes:
Sl ~(v-n-3) ~(n-3) G = u (1-u} du
1 O u + L(l-u)
(8.7.20)
(8.7.21)
(8.7.22)
(8.7.23)
(8.7.24)
- 111 -
and
, using (8.7.20) ,
and
_ 51 u~(v-n-3) (l-u}~(n-3} G du 2 - 0 [u + L(l-u)] 2 , (8.7.25)
= sl u~(v-n-3) (l-u)~(n-3) du Gl O u + P(l-u) (8.7.26}
The above integrals can be evaluated for n ) 2. The
combining constant B and a conservative lower bound D of
the recovery ratio are given in (8.5.12) and (8.5.20).
These integrals have been evaluated and B and D has been
computed for all designs listed in reference (1), except in
LS6 through LS9 where the inter-model is singular. It
should be noted that LS16 through LS20 are of L3 sub-type of
LS type designs, and the remaining designs are of L2 sub-type.
The results are listed in Table v. The combined estimate, as given in (8.4.1}, is:
"' 't" • 1.
9.Bs2 = u + _ __,;;;;;;1.'-----
i v-1 .l (X.-U. ) 2 . 1 J J J= j~i
(X.-U.) 1. 1.
, i = 1,2, ... ,n(n-l)
(8.7.27)
By {8. 2. 5),
k-c 1
a
and then {8.7.27) becomes,
A
- 112 -
0"2 , so e. = 1.
k-c 1 a J
·r. = U. + 1. 1.
(k-c1 )Bs2
v-1 (X. -U.) , 1. 1.
i = 1,2, ... ,n(n-l)
al (X.-U .) 2
j=l J J j1'i
8.7.2 Among Comparisons
Here v1 = v-n, v2 = n-2, and for:
' using (8.7.19).
G' in (8.5.14) becomes: 2
G~ = sl u!.,z(v~m-2) (l-u)~(n-4) du
0 [Lu+ (l-u)] 2
= L2 sl u~{v-n-2) (l-u)!.,z(n-4) du
0 [u + L(l-u}] 2
J
(8.7.28)
(8.7.29}
G' in (8.5.15} becomes: 1
G{ = sl u ~(v~n-2} -u+ 0 p
- 113 -
(l-u}~(n-4) du
(1-u)
= p sl u~(v-n-2} (l-u)~(n-4) du O u + P(l-u)
'
' using (8.7.19), hence
G' = sl u~(v-:-2) (l-u}~(n-4} du 2 O [- u + (l-u)] 2
p
= p2 sl u~(v-n-2) (l-u)~(n-4} du 0 [u + P(l-u)] 2 J
and
Sl ~(v-n-2} ~(n-4) G' = u (1-u) du
1 1 0 L u + (1-u)
= L sl u~(v-n-2) (l-u}~(n-4} du O u + L(l-u}
(8.7.30)
(8.7.31)
(8.7.32}
For the convergence of the above integrals n) 2. The
combining constant B and a conservative lower bound D of the
recovery ratio are given in (8.5.18) and (8.5.22} respectively.
The above integrals along with B and D are evaluated and
- 114 -
listed in Table VI for the same set of designs mentioned in
Section 8. 7 .1.
The combined estimate is
Q Bs 2
~ = U + i (X U ) i i v-1 i- i I (X.-U.)2 j=l J J
, i = n(n-l)+l, ... ,n2 -l
(8.7.33)
j~i
By (8.2.5) and {8.3.1), one can write:
hence:
for L 2 sub-type,
and
k-c1 (n-2)(A1-A2 ) 9 . = -- + ------------
1. a k6. for L3 sub-type.
Accordingly,
[k6.(k-c1 )+(n-l)a(A1-A2 )]Bs 2
Ti = ui + v-1 (Xi-Ui) , ak.6. I (X .-U.) 2
j=l J J (8.7.34)
j~i i = n{n-l)+l, ... ,n2 -l,
for L2 sub-type, or LSl through LSS and LSlO through LSlS as
listed in reference (1), and
- 115 -
A
'r. = u. l. l.
,
(8.7.35)
i = n(n-l)+l, ... ,n2 -l,
for L3 sub-type, or LS16 through LS20 as listed in reference
( 1).
8.8 Triangular P.B.I.B. Designs
8.8.l Definition and Conunent on the Singularity of the
Inter-analysis Model
In triangular designs the number of treatments can be
n(n-1) expressed as v = 2 , and the association scheme is an
array of n rows and n columns with the following properties:
(i) The positions in the principal diagonal are left
blank.
(ii) The n(n-1)/2 positions above the principal diagonal
are filled by the numbers 1,2, ... , n(n-1)/2 corresponding to
the treatments.
(iii) The n(n-1)/2 positions below the principal
diagonal are filled so that the array is synunetrical about
the principal diagonal.
- 116 -
(iv) For any treatment i the first associates are
exactly those treatments which lie in.the same row (or in
the same column) as i.
The following relations hold
n1 = 2n-4 , n 2 = (n-2)(n-3)/2, (8.8.l)
pl= c-2 -3
n-3 )
(n-3)(n-4)/2
,
{8.8.2)
p2 = ( 4
2n-8
2n-8 )
(n-4)(n-5)/2
The parameters v, r, k, b, A1 , A2 , and n, along with the
constants c 1 , c 2 , ~, and H, are given for 36 triangular
designs in reference (1).
Reference {6) gives a set of three equations with three
unknowns for each treatment resulting from Rao's combined
analysis. These equations are:
aTj - ~AlGjl - ~A2Gj 2 = Pj , (8.8.3)
l. l. -~AlnlTj + (a-~A1Pll-~A2Pl2)Gjl
- {~AlP~l+~A2P~2)Gj2 = ~pjl J
, (8.8.4)
and
where
and
- 117 -
1 1 -f3A2n2Tj - (f3AlP2l+f3A2P22)Gjl
+ (a-~A1P~ 1-f3A2P~ 2 )Gj 2 =
k-1 w' a= r[w - + -]
k k ' w-w' f3 =--
k
l:P.2 . J J
'
' (8.8.5)
(8.8.6)
Gjl is the total of treatments which are first asso-
ciates to the j-th treatment,
Gj 2 is the total of treatments which are second asso-
ciates to the j-th treatment,
(8.8.7)
l:P. 1 is the Sl.lltl of P.'s for the treatments which are j J J
first associates to the j-th treatment,
l:P. 2 is the Sl.lltl of P.'s for the treatments which are j J J
second associates of treatment j.
From the definitions of Gjl and Gj 2 , one can write the
further relation,
(8.8.8)
If one lets w = 0 in (8.8.3} through (8.8.7), the
corresponding normal equations for the inter-analysis above
- 118 -
are obtained:
J (8.8.9)
A 1 1 nlAltJ + (r+A1P11+A2P12)Gjl + (A1P~1+A2P~2)Gj2 = ZQJl ,
(8.8.10)
"'• ( l. 1 ) ( 2 2 ) n2A2tj + A1P21+A1P22 Gjl + r+A1P21+A2P22 Gj2 = ZOJ2 '
(8.8.11}
and
(8.8.12}
Substituting the p~k's from (8.8.2) into (8.8.9)
through (8.8.12), one obtains for the triangular designs:
and
A
rtj + AlGjl + A2Gj 2 = Oj J
(2n-4)A1tj + [r+A1 (n-2)+A2 (n-3)]Gjl
+ [4A1+A2 (2n-8)]Gj 2 = IQJl ,
~(n-2)(n-3)A2tj + [A1 (n-3)+~A2 (n-3)(n-4)]Gjl
(8.8.13)
(8.8.14)
{8.8.15)
(8.8.16}
Computing Gj 2 from (8.8.16) and substituting into (8.8.13}
and (8.8.14), one can write:
- 119 -
and
[2n-4}A1-4A1-(2n-8)A2Jtj
+ [r+A1 (n-2)-4A1+A2 (n-3)-A2 (2n-8)]Gjl =
Simplifying the above equations,
, (8.8.17)
and
"' (2n-8}(A1-A2}tj + [(r-A1)+(n-S)(A1-A2)]Gjl = 1QJl
(8.8.18)
This set of two equations has no unique solution when the
coefficients matrix is singular, i.e.,
= (r-A1)(r-~2 ) + (n-5)(r-A2)(A1-A2)- 2(n-4)(A1-A2) 2 = O (8.8.20)
8.8.2 Application of Theorem 3 When V(z.} is of the Form i;
v-c+p cc-c • >
The formula (8.4.7) in theorem 3 can be written here as:
t
z2 = I j~i
[v-c+p j (c-c ·) Jx~ (8.8.21)
Suppose that p = O for v 1
v1 + v2 = t-1, then:
- 120 -
Z, I SJ l.
t
and p ) O for v 2 Z, I SJ l.
where
l [ cv-cJ +p. cc-c · > Jx~ j=v +l J
. (8.8.22}
j~f
In the case Al) A2 and in view of (8.3.8}, (8.8.22) can be
written as:
and
t
~ z:: > . . J J l.
(V-C)X2 + [(v-c}+p . (C-C 1 )Jx2 v1 min. v2
+ 1/1 x2 = x + Y 2 v 2 1 , (8.8.23}
(V-C)X2 + [v-c+p (C-C')]X2 v1 max. v2
= 1/1 x2 + 1/1 x2 = x + Y 1 v1 3 v2 2 , (8.8.24)
From (8.8.23) and (8.8.24) one can develop the formula
(8.4.6) in theorem 3 as follows:
V~B2 (f+2)¢. l l V(~.} = V + 1 1 E ------- - 2V~BE _.......___
l. i f t l. t ( l z2) 2 l z~ j~i j~i J
(8.8.25}
- 121 -
From (8.5.3) and (8.5.6), one can write:
du
!av -1 !av -1 1 sl u 1 ( 1-u) 2 < ~ (v-4)(v-6)~(!av 1 , ~v 2 )t~ 0 [u + L(l-u)] 2
'
where L is a lower bound of t 3/t1 .
Again using (8.5.6),
~v -1 !av -1 1 sl u 1 ( 1-u) 2
( (v-3)~(~v 1 , ~v 2 )t1 0 u + P(l-u)
,
where P is an upper bound of t 2/t1 .
(8.8.26)
du
(8.8.27)
Substituting (8.8.26) and (8.8.27) into (8.8.25),
(v-4)~(~v 1 ,~v 2 )t1
(8.8.28}
It is required that
and
B opt. =
- 122 -
,
fG1 (v-6Jip-1 (f+2)G2ct>i
When ct> . = ,µ- , 1 i.e., ct> . = v-c, (8.8.30) becomes: l. l.
B fG1 (v-6)
= opt. (f+2)G2
Substituting (8.8.31) into (8.8.28),
fG1 (v-6) (-G ) (f+2)G2 1
fG~(v-6) =V -----=-------i (v-4)G2 (£+2)~(~v 1 ,~v 2 >
,
so that the recovery ratio is at least
fG~(v-6) D - -----'""'-------- (f+2)G2~C~v 1 ,~v 2 )(v-4)
(8.8.29)
(8.8.30)
(8.8.31)
J
(8.8.32)
(8.8.33)
When ct>i ~ ip-1 , i.e., V(zi) = V-C+p(C-C') with p) O, then
£G1 (v-6)ip-1 £G1 (v-6) Bopt. ) (f+2),P2G2 ) (f+2)G2P , (8.8.34)
,
-123 -
1/12 where P is an upper bound of ~
1/11
B = fGl (v-6)
(f+2)G2P
Thus one can take
In view of (8.8.30), one can write (8.8.28} as:
(8.8.35)
fG1 (v-6)1/fl (f+2)G2¢i - 2Gl] ,
fG~(v-6)
(f+2)G2P
fG~(v-6)L v~
<vi - (v~4)(f+2)~(~v1 ,~v 2 )G2P · ¢i
Hence the recovery ratio is at least:
fG~(v-6)L D = ~~~~-=-~~~~~~-
( v - 4) ( f + 2 )~(~v1,~v2)G2P
,
,
(8.8.37)
If Al ( A2 , then p . in (8.8.23) and p in (8.8.24) . min. max.
exchange positions.
8.8.3 Recovery of Inter-block Information in Triangular
Designs
The designs to be discussed here, are those in which
the inter-analysis is not singular. Out of the 36 designs
- 124 -
listed in (1), one finds, applying the formula (8.8.20),
that the inter-analysis is not singular in the following
designs:
T6, T7, T8, Tl2, Tl3, Tl4, Tl8, Tl9, in which v = 10,
and in
T23, T24, T30, in which v = 15.
(a) For designs with v = 10, consider the matrix:
1 2 3 4 5 6 7 8 9 10
1 -1
1 1 -2
1 1 1 -3 ---------- --------- ----------
1 -1
1 1 -2 Ml - ---------- ---------- ---------- (8.8.38)
1 -1
1 1 -2 ---------- --------- ---------3 3 3 3 -4 -4 -4
3 3 3 3 3 3 3 -7 -7 -7
1 1 1 1 1 1 1 1 1 1
Let M be the orthogonal matrix obtained by normalizing the
rows of M1 .
- 125 -
Let
A
ul tl A
u2 t2 u = =M = Mt , {8.8.39)
u A
v-1 t v 0
and
xl A
t' 1 A
x2 t' 2 A x = = M = M t' , {8.8.40)
x A
v-1 t' v
0
A A where t., t' are the intra- and inter-estimates of the i-th
l. i
treatment effect t., respectively. The treatments are l.
arranged in the vector t arithmetically from 1 to v. From
the construction of M1 in {8.8.38), it should be noted that
the treatments have been divided into three groups; the
first contains tl, t2, t3, t4; the second contains ts, t6,
t7; the treatments ta, t9, tlO form the third group. The
first seven rows in Ml are comparisons within groups; the
next two comparisons are among groups. It is to be noticed
that the treatments in the same group are mutually first
- 126 -
associates. It is simple to verify that for the within
comparisons:
V{U.) = v -c i = 1,2, ... ,7 1 1 1 J J (8.8.41}
V{X.) = v•-c• i = 1,2, ... ,7 1 1 1 ' (8.8.42)
For among comparisons:
V(U8 ) = v -c 1 1 12 +-7 {Cl-C2) ' (8.8.43}
V{U9 ) = v -c 9 (Cl-C2} +- ' 1 1 7 (8.8.44)
V(X8 } = V'-C' 1 1 12 +-7 (C 1-C I}
1 2 (8.8.45)
and
9 V(X) = v·-c· + - (C'-C') 9 1 1 7 1 2 (8.8.46)
Let
z = x - u ' (8.8.47)
then
V(Z.) = V(U.) + V(X.) 1 1 1
(8.8.48)
and using (8.8.41) through (8.8.46), one can write:
V(z8 ) = v-c 12 +-7 (C-C I} J (8.8.52)
and
V(Z9 ) = v-c 9 +-7 (C-C I} (8.8.53)
- 127 -
It should be noted that V(Z.) has been expressed in the l.
general form V-C+p(C-C'), where p = 0 for z1 through z7 , and
2 7 for z8 and z9 , respectively.
(b) Similarly for designs in which v = 15, consider
the matrix:
1 2 3 4 5
1 -1
1 1 -2
1 1 1 -3
1 1 1 1 -4
4 4 4 4 4
2 2 2 2 2
1 1 1 1 1
6 7 8 9
1 -1
1 1 -2
1 1 1 -3
-5 -5 -5 -5
2 2 2 2
1 1 1 1
10 11 12
1 -1
1 1 -2
1 1 1
-3 -3 -3
1 1 1
13 14 15
1 -1
1 1 -2
-1 -1 -1 -3 -3 -3
1 1 1
(8.8.54)
Let M be the orthogonal matrix obtained from M1 by normalizing
its rows, and let U and X be as defined in (8.8.39) and
(8.8.40), respectively. From the construction of M1 in
- 128 -
(8.8.S4), it should be noted that the lS treatments have
been divided into four groups. The first group contains t 1
through ts, the second group contains t 6 through t 9 , the
treatments t 10 through t 12 form the third group, and the
fourth group contains t 13 through t 1 S. The first 11 rows
of M1 are comparisons within groups; the next three com-
parisons are among groups. The treatments within the same
group are mutually first associates. For within comparisons:
and
For
and
V(X ) = V'-C' i 1 1 '
i = 1,2, ... ,11
i=l,2, ... ,11
among comparisons:
V(Ul2) v1-c1 8
(Cl-C2) = +-3
V(Ul3} = v1-c1 + (Cl-C2)
V(Ul4) 7
(Cl-C2) = v1-c1 +- i 3
V{X12} = V'-C' 1 1 + .§. { C I -c 1 )
3 1 2 '
V(X13) = V'-C' 1 1 + {C '-C'} 1 1 '
V(Xl4) V'-C' 7 (C '-CI) = +-1 1 3 1 2
' (8.8.SS)
(8.8.56)
(8.8.S7)
(8.8.S8)
(8.8.59)
(8.8.60)
(8.8.61)
(8.8.62)
- 129 -
Taking into account (8.8.47) and (8.8.48), and using (8.8.55)
through (8.8.62}, one can write:
V(Z.) = v-c i = 1,2, ... ,11 . (8.8.63) 1
, I
V(Zl2) = v-c 8 +-3 (C-C I) , (8.8.64)
V(Zl3) = v-c + (C-C I) (8.8.65)
and
V(Zl4) = v-c 7 (8.8.66) + - (C-C') 3
It is to be noted also that V(Z.) has been expressed in 1
the general form V-C+p(C-C'), where p = 0 for z1 through z11,
8 7 and p = 3 , 1, 3 for z12 , z13 , z14, respectively.
8.8.3.1 Study of the Ratio [V-C+p(C-C')]/(V-C)
Now
V-C+p ( c-c I ) = 1 + p c-c I v-c v-c (8.8.67)
By (8.2.16) and (8.2.17},
C-C' = C +c' C C' = (C -C } + (C'-C') 1 1 - 2- 2 1 2 1 2 (8.8.68}
Substituting (8.2.7} and {8.3.2) into (8.8.68), one obtains:
C-C' = c -c
1 2 2 (J + a
C 1 -C I
1 2 (8.8.69} r
Substituting (8.8.69) and (8.6.16) into (8.8.67}, one
- 130 -
obtains:
cl-c2 c'-c' a2 + 1 2 cr'2
V-C+p (C-C I) 1 a r = + p v-c k-c k-c' 1 cr2 1 CJ I 2 +--
a r
and using (8.1.13},
= 1 + p k-c ( 1
a
c'-c' + 1 2)cr2
r k-c'
+ 1) cr2 r
k(c'-c') + 1 2
r k(k-c.i}
+ cr.2 r b
0:2
Let __g = R) 1, then (8.8. 70} becomes: cr2
v-c+p C c-c' J v-c
where:
c -c 1 2
al = a
k-c 1 a2 = + a
and Cl-' c2 were
+
,
c'-c' k(c'-c') 1 2
131 1 2 = r , r
k-c' k(k-cl) 1 132 = r ' r
defined in (8.1.11}.
F(R) is always between the limits:
F(l) 1 + p a 1 +t\
= a2+132
,
and
F(oo) 1 + 131
= p-132
,
'
,
cr.2 b . (8.8. 70)
(8. 8. 71)
(8.8.72)
(8.8.73)
(8.8.74)
- 131 -
8.8.3.2 Combined Estimates for Triangular Designs
The results of Section (8.8.2) are applicable in the
triangular subclass for combining the comparisons U 's and i
X. 's. l.
given
where
Thus
(a} For within comparisons, the combined estimate
in (8.4.1) is:
A 9.Bs 2
u. + l. (X. -U.) T. = l. l. t l. l. I {X.-U .)2
j=l J J
(8.8.75}
j~i
Q. is the coefficient of (j2 in V(U.} . l. l.
Using {8.2.5},
k-C V(U.) = v1-c1 =--1 cr2
l. a (8.8.76}
k-c Q = 1 for every within comparison, and (8.8.75)
i a
becomes finally:
A
T. = U. + l. l.
{k-c1 )Bs 2
(X.-U.) l. l.
, (8.8.77) t
a I {X.-U .) 2 •..J.. J J Jr-1
where k, c 1 , a=rk-r, are known parameters, and B is a
constant to be taken for a specific design from Table VII.
A conservative lower bound D of the ratio of recovery
achieved is given for each design in the same table.
- 132 -
(b) For among comparisons, the variance of u. is of J.
the general form:
By (8.2.5) and {8.2.7), one can write {8.8.78) as:
k-c V{U.)
J. [ l = + p. a l.
c 1-c {k-pl..c2 )+(pl..-l)c1 2] cr2 = -----=-------=---= cr2 a a
Thus
{k-p.c2)+{p.-l)c1 e. = i i i a
and {8.8.75) becomes:
T. = l.
{8.8.79)
, (8.8.80)
{8.8.81)
B is given for each design in Table VIII. Also in the same
table, a lower bound D of the recovery ratio is given. The
p. 's are given in Table XI. l.
8.9 Cyclic P.B.I.B. Designs with Two Associate Classes
8.9.l Definition
A non-group divisible partially balanced incomplete
block design is called cyclic if the set of first associates
of the i-th treatment is obtained by adding i-1 to the
numbers in the set of first associates of the first treatment
- 133 -
and subtracting v, whenever the sum exceeds v, where v is
the number of treatments in the design. In giving the
association scheme of such a design, it is, therefore,
sufficient to give the first associates of the first treat-
ment. The parameters and plans of these designs are given
in reference (1). The inter-analysis in all ten designs
listed is not singular.
8.9.2 Recovery of Inter-block Information in cyclic Designs
Following a similar approach to that in traingular
designs, the treatments in each design will be grouped into
a number of groups, with the treatments falling in one group
being mutually first associates. An orthogonal matrix M
will be defined in each case. The vectors U and X will be
defined as:
"' u = M t J {8.9.1)
and
"' x = M t' {8.9.2)
The treatments t 1 , t 2, ... ,tv are arranged in the vector
"' "' t or~· according to the grouping plan; that is, the first
number of treatments are those of the first group, followed
by treatments of the second group, etc. A vector,
- 134 -
, (8.9.3)
will be considered, and the variances of z. 's will be l.
computed. Finally, the two linear functions U. and X. will l. l.
be combined, i = 1,2, ... ,v-l. The U and X are both zero's v v A
and their combined estimate T will be assumed to be zero v also.
(a} For cyclic designs cl through c4, where v = 13,
consider the 13 x 13 matrix:
i.e.'
1 3 8
1 -1
1 1 -2
-------1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
2 4 9
1 -1
1 1 -2
-----------1 -1 -1
1 1 1
1 1 1
1 1 1
1 1 1
5 7 12
1 -1
1 1 -2
----------2 -2 -2
1 1 1
1 1 1
1 1 1
6 11 13 10
1 -1
1 1 -2 ---------- -----
-3 -3 -3
1 1 1 -12
1 1 1 1
(8.9.4)
The grouping plan appears on the top of the matrix,
the first group includes tl, t3, ta; the second group
- 135 -
includes t 2 , t 4 , t 9 ; the third group includes t 5 , t 7 , t 12 ;
the fourth group includes t 6 , t 11 , t 13 ; and the treatment
t 10 alone forms the fifth group. Normalizing the rows of M1 ,
the orthogonal matrices M that appear in (8.9.1) and (8.9.2) A A
are obtained. The colwnn vectors, ..t_ in (8.9.1} and t' in
(8.9.2), are in this case,
(8.9.5)
and
(8.9.6)
It should be noted that the number of within comparisons is
eight and that of among comparisons is four.
and
The variances of Z. 's are as follows: 1.
V(Z.) 1.
V(ZlO}
V(Zll)
V(Zl2}
= V(U.} + V(X.) = 1. 1.
V C + V'-C' = 1- 1 1 1 v-c
i=l,2, ... ,8 . I
'
= V(UlO} + V(XlO) = v-c + 2 (C-C I)
= V(Ull} + V(Xll) = v-c 3 + - (C-C') 2
V(Ul2) + V(Xl2) 1 (C-C I) = = v-c +-2
' (8.9.7)
(8.9.8)
' (8.9.9)
' (8.9.10)
(8.9.11)
- 136 -
(b) For cyclic designs cs through c7, where v = 17,
consider the 17 x 17 matrix:
1 4 7
1 -1
2 5 8 3 6 9 10 13 16 11 14 17 12 15
1 1 -2
1 -1
1 1 -2
1 -1
1 1 -2
1 -1
1 1 -2
1 -1
1 1 -2
1 -1 ------- --------- --------- ---------- --------- --------1 1 1 -1 -1 -1
1 1 1 1 1 1 -2 -2 -2
1 1 1 1 1 1 1 1 1 -3 -3 -3
1 1 1 1 1 1 1 1 1 1 1 1 -4 -4 -4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 -15
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(8.9.12)
The grouping plan put in a rectangle where each row
represents a group is:
1
2
3
10
11
4
5
6
13
14
12 15
7
8
9
16
17
(8.9.13)
-15
1
- 137 -
The variances of Z. 's are: J.
V(Z.) J.
= v-c , i = 1, 2, ... , 11; (8.9.14}
V(Zl2) = v-c 7 (C-C'} (8.9.15) +- , 3
V(Zl3) = v-c 7 (C-C') (8.9.16) +- , 3
V(Zl4) = v-c 5 (C-C') (8.9.17) +- , 6
V(ZlS) = v-c 43 (C-C') (8.9.18) + 30 ,
and
V(Zl6) = v-c 16 + 15 (C-C I) (8.9.19)
(c) For cyclic designs c8 and c9, where v = 29,
consider the 29 x 29 matrix:
1 2 6 7 3 4 8 9 10 11 15 16 17 18 22 23 19 20 24 25
1 -1 1 1 -2 1 1 1 -3
1 -1 1 1 -2 1 1 1 -3
1 -1 1 1 -2 1 1 1 -3
1 -1 1 1 -2 1 1 1 -3
1 -1 1 1 -2 1 1 1 -3
---------- ----------- ------------ ------------ ---------------------- ------------ ------------ ------------ ---------------------- ----------- ------------ ------------ ------------1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -2 -2 -2 -2 1 1 1 1 1 1 1 1 1 1 1 1 -3 -3 -3 -3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -4 -4 -4 -4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5 12 21 28
1 -1 1 1 -2 1 1 1 -3 ------------
------------------------
-5 -5 -5 -5
5 5 5 5 1 1 1 1
13 14
--------1 -1 --------
--------
1 1 1 1
-24 -24 1 1
26 27
----------------
1 -1 --------
-1 -1 1 1
-24 -24 1 1
(8.9.20)
29
-4 -24
1
...... w ro
- 139 -
The grouping plan put in a rectangle where each row represents
a group is:
1 2 6 7
3 4 8 9
10 11 15 16
17 18 22 23
19 20 24 25 (8.9.21)
5 12 21 28
13 14
26 27
29
The variances of the Z. 's are: l.
V(Z.) = v-c , i = 1,2, ... ,20 l.
(8.9.22)
V(Z21) = v-c 5 (C-C I) +-2 (8.9.23}
V(Z22) = v-c 11 +-6 (C-C I) , (8.9.24)
V(Z23) = v-c 61 (C-C I) + 24 , (8.9.25)
V(Z24) = v-c 97 (C-C I) + 40 , (8.9.26)
V(Z25) = v-c 11 (C-C I) +- , 5 (8.9.27)
V(Z26) = v-c + (C-C I) , (8.9.28)
V(Z27) = v-c + (C-C I) , (8.9.29)
- 140 -
and
1 V(Z ) = V-C + - (C-C'} 28 2 (8.9.30}
(d} For cyclic design clO in which v = 37, consider
the 37 x 37 matrix:
- 142 -
The grouping plan put in a rectangle, where each row
represents a group, is:
1 2 5 12
3 4 7 14
6 9 10 13
8 11 15 18
16 17 20 27 (8.9.32} 19 23 26 30
21 24 25 28
22 29 32 33
31 34 35
36 37
The variances of z. 's are: J.
V{Z.) J.
= v-c , i = 1,2, ... ,27: (8.9.33}
V(Z28) = v-c 5 (C-C I) (8.9.34) +- , 2
V{Z29) = v-c 5 (C-C I) (8.9.35) +- , 3
V{Z30) = v-c 47 + 23 (C-C I) , (8.9.36)
V(Z31) = v-c 91 (C-C I) (8.9.37} + 40 ,
V{Z32) = v-c 11 (C-C I) (8.9.38) +- , 5
V(Z33) = v-c 37 {C-C I) (8.9.39) + 14 ,
V(Z34) = v-c 243 (C-C I) (8.9.40) + 112 ,
- 143 -
V(Z35) v-c 8 (C-C I} (8.9.41) = +-5 ,
and
V(Z36) v-c 3059 (C-C I) (8.9.42) = + 2960 .
8.9.2.1 Combined Estimates for eyclic Designs
It should be noted that all the variances of Z. 's have l.
been expressed in the general form V-C+p.(C-C'), where l.
pi) O. Hence the results of Section (8.8.2) and Section
(8.8.3.1) are applicable for combining U. 's and X. 's. Thus l. l.
the combined estimates in Section (8.8.3.2) are suitable for
the cyclics under study.
(a) For within comparisons, the combined estimate is
as in (8.8.77):
A
'r. = u. + l. l.
(k-c1 )Bs 2
(X. -U.) l. l.
(8.9.43) t
a~ (X.-U .) 2 . . J J J J.
where k, c 1 , a=rk-r, are known parameters, and B is a
constant to be taken from Table IX. A conservative lower
bound D of the ratio of recovery achieved is given for each
design in the same table.
- 144 -
(b} For among comparisons, the combined estimate is,
as in (8.8.81}:
A
T. 1. (8.9.44}
B is given for each design in Table X. Also in the same
table, a very conservative lower bound of the recovery ratio
D is given. The p. ·~ are given in Table XII. 1.
8.10 General Procedure for Recovery of Inter-block Informa-
tion in P.B.I.B. with Two Associate Classes
Compute:
1. Y. 1. .
total}.
2. y . j
total) .
3. y
- ~a. y. J . 1 1.J 1.J J= i = 1,2, ... ,v (i-th treatment
v
- lo .. y .. i=l 1.J 1.J
J j - 1,2, ... ,b; (j-th block
= Io .. y .. ; (grand total}. . . 1.J 1.J 1.J
4. Q. = Y. 1 - (sum of block totals in which treat-k 1. 1..
ment i occurs}, i = 1,2, ... ,v; (adjusted treatment totals}.
- 145 -
5. s1 (Qi) , i = 1,2, ... ,v; (the sum of Q's for all
treatments which are first associates of treatment i).
6. A k-c c 1-c2 t. = __ 2_ Q. + Sl(Ql..)
i a i a , i=l,2, ... ,v;
(the intra-treatment estimate).
y 7. o: = Y. l. l..
- Q. -l. v , i = 1,2, ... ,v.
9. k-c'
A 2 t: =--Q~ + l. r i
(the inter-treatment estimate).
10. A A
t ! - t. ' l. l. i=l,2, ... ,v
11. x - u = = M
i = 1,2, ... ,v;
A A
ti-t1
A A
t' -t v-1 v-1 A A t -t v v
A A = M(~ '"".'.:!=_} ;
where M is the orthogonal matrix defined in each case
previously.
12. s 2 = f1cI o .. y~ .-ssT . . l.J l.J l.J
square, f = bk-b-v+l).
(error mean
- 146 -
To combine X. and u., compute: J. J.
13.
14.
15.
v l (X.-U .)2 • j=l J J
v v l (X.-U .)2 = •...J.. J J Jrl.
l (X.-U .)2 j=l J J
- (X. -U. ) 2 • 1. 1.
9.Bs 2 J = ~---1.---~~~
lex .-u .) 2
j~i J J
J where 9. was defined previously 1.
in each case; B is a constant to be taken from the correspond-
ing table.
16. The combined estimate of U. and X., 1. J.
If the combined estimates of the treatments t 's them-i
selves are desired, one should compute:
T v
= M'
where M' is the transpose of M.
A
=E: M '.!.. J
- 147 -
8.11 Comment on the Numerical Methods Used in Table VII
Through Table X
In Section 7.1.1, a method of utilizing the Hypergeornetric
series and Gauss' continued fraction was discussed. In
Table VII through X, the same idea was applied for computing
E1 and E2, where:
and
~v -1 2 (1-u) P(l-u) du ' (8.11.1)
(8.11.2)
In view of (7.1.8), (7.1.9), and (7.1.11), (8.11.1} may be
written as:
=
Now,
v-2 P-1 F(~v 1 , 1, 2 , ---p-J
p
J
(8.11.3)
v-2 P-1 F(~v 1 , 1, ~2- , ---p-> could be put in the form of
(7.1.12) and evaluated.
- 148 -
Similarly, for E2 , one can write:
v-2 L-1 F(2, ~Vl' -2- ' -L-) = ~----___;;;;:..__,...;;;:.._----=--L 2
du ,
(8.11.4)
To evaluate v-2 F(2, ~v 1 , 2 L-1)
' L ' the following recursion
formula (see reference 13, page 363) is needed:
a(c-b)z F(a,b,c;z) = F(a,b+l,c+l;z) - c(c+l) F(a+l,b+l,c+2;z) ,
(8.11.5)
or
F(a+l,b+l,c+2;z) c (c+l} = ( b) [F(a,b+l,c+l;z)-F(a,b,c;z)] ; a c- z (8.11. 6)
putting a = 1, one gets:
c(c+l} F(2,b+l,c+2;z) = (c-b)z[F(l,b+l,c+l;z}-F(l,b,c;z)] .(8.11.7}
Comparing (8.11.4} and (8.11.7), one obtains:
and
_ v-2 c+2 - 2
L-1 z =--L
,
,
b = ~v -1 1
v-6 c =--2
,
(8.11.8}
- 149 -
Thus, substituting in (8.11.7),
v-6 v-4 -2-. -2-v-6 L-l[F(l,
<-2- - ~v1+1>L
v-4 L-1 ~v1, -2-; L>
- F(l k2v -1 , 1 , v-6 --. 2 I
L-1 ] -) L
. (8.11.9)
In view of (7.1.11) and the fact that v1+v 2 = v-2,
(8.11.9) may be written as:
v-2 L-1 F(~v2 , 2, - 2-; L) - (v-6)(v-4}·L [ v-4 L-1
4(~v -l)(L-1) F(~vl, l, -2-; -r:-> 2
v-6 L-1 - F(~v 1-1, 1, - 2-; L)] . (8.11.10)
Now, v-4 L-1 F(~vl, 1, -2-; L) CL l V-6 L-1) and F ~v 1- , 1, - 2-; L
be put in the form of Gauss' continued fraction as in
(7.1.12) and evaluated.
can
The above was progranuned for an IBM 1620 and used for
cyclic designs (Tables IX and X); however, as we notice from
(8.11.10), the above method is not applicable when ~v 2-l = 0
or v = 2, the case that was encountered in triangular 2
designs (Tables VII and VIII). Whenever this happened the
integrals,
- 150 -
kv k2v -1 1 2 1 2 = s _u __ ..._( l_-_u_,)....._ __ O u + P(l-u) du , (8.11.11}
and
~v -1 ~v -1 -sl u 1 (1-u) 2 G2 - 0 [u + L(l-u)] 2 du , (8 .11.12)
were evaluated by the Simpson rule. The accuracy of the
Simpson rule was listed in the tables; the accuracy of E1
or E2 was set to the sixth decimal place constantly.
- 151 -
9. SUMMARY AND DISCUSSION
In summary, this dissertation has brought up the follow-
ing points:
1. Under certain conditions, a new method of combining
two independent estimates has been given in theorem 1. This
new method has its immediate application in incomplete block
designs, in similar experiments, and in randomized block
designs with heterogeneous variances. The amount of improve-
ment obtained by this new method is very satisfactory,
compared with the utmost possible theoretical improvement.
2. A procedure for recovering the inter-block informa-
tion in balanced incomplete block designs was given which is
applicable in experiments as small as t = 4.
3. A generalization of theorem 1 was given which shows
that the idea of recovering the inter-block information can
be practically utilized in any incomplete block design with
seven treatments or more.
4. For a partially balanced incomplete block design
with two associate classes, a separate development was given
to each of its four subclasses, namely the group divisible
(G.D.), Latin square type (L2s type and L3s type), the
- 152 -
triangular, and the cyclic designs. A combined estimate was
given for each case, and a general procedure for recovery of
inter-block information in these subclasses was developed.
Because of the special nature of singular and semi-regular
G.D. designs, an inter-analysis was discussed for G.D. 's in
general and for these two cases in special, and a partial
utilization of the inter-information was made possible.
The essence of this work arises from the fact that so
far a method for recovering the inter-block information was
based on the common sense of the consistency property of an
estimate. Consequently, the applicability of the resulting
method would make sense only in big size experiments. The
resulting combined estimates are of so complicated expression
that a mathematical study of their merits or demerits is not
feasible, even in their domain of applicability. On the
contrary, this dissertation provides a general method by a
mathematical treatment. This work can be said to have the
following two merits:
1. It makes possible the utilization of the inter-block
information in small size experiments, as small as four treat-
ments in B.I.B. designs and as small as seven treatments in
P.B.I.B. designs. This is of special importance because of
- 153 -
the fact that the intra-block efficiency is in general lower
in small and moderate size designs. It follows that the
recovery is needed more in this sort of designs.
2. As a ratio of the utmost possible theoretical
recovery, either exactly or a lower bound of the amount of
the ratio of recovery is always computable; this ratio was
tabled for all B.I.B. designs listed in reference (2), and
P.B.I.B. designs listed in reference (1). The ratio of
recovery depends on the structure of the design; it always
increases with v, the number of treatments.
A glance at the tables, where these ratios are listed,
shows that the new methods, for B.I.B. and P.B.I.B. designs,
give good results where the old method is not applicable;
where the old method starts, hopefully, to be valid, the
ratio of recovery achieved by the new methods begins to
approach the theoretical value that can be achieved, assuming
the intra- and inter-variance are known
- 154 -
10. ACKNOWLEDGEMENTS
I wish to express my sincere thanks to Professor c. Y.
Kramer who introduced me to the subject and guided me during
the course of this work~ and to Professor B. Harshbarger for
his encouragement. Thanks are due also to Mrs. Lela
Barnhisel for her extreme care in typing this dissertation.
- 155 -
11. TABLE OF REFERENCES
1. Bose, R.C., W. H. Clatworthy, and S.S. Shrikhande, (1954). North Carolina Agricultural Experiment Station, Technical Bulletin No. 107.
2. Cochran and Cox, (1957). Experimental Design. New York, John Wiley and Sons.
3. Graybill and Weeks, (1959). "Combined inter- and intra-block information in balanced incomplete blocks. 11
A.M.S. 30, 799.
4. and Deal, (1959). "Combined unbiased estimators. 11
Biometrics, 15, 543.
5. Kramer, C.Y., (1957). 1'Examples of intra-block analysis for factorials in group divisible, partially balanced incomplete block designs. ti Biometrics, 13.
6. Kempthorne, o., (1952). Design of Experiments. New York, John Wiley and Sons.
7. Rao, C.R., (1947). tlGeneral method of analysis for incomplete block designs. ti J. A. S. A., 42, 541.
8. Seshadri, V., (1961). ''Estimation in the balanced incomplete block designs. " Unpublished Ph.D. thesis, Oklahoma State University.
9. , (1963). "Combined unbiased estimates." Biometrics, 19, 163.
10. Yates, (1937). tlThe recovery of inter-block information in balanced incomplete block designs. 11 Annals of Eugenics, 10, 317.
11. Zelen, M. (1957). tlThe analysis of incomplete block designs." J.A.S.A., 52, 204.
- 156 -
12. Bailey, W.N., (1935). Generalized Hypergeornetric Series. Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, 4.
13. Wall, H.S., (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc.
The vita has been removed from the scanned document
- 158 -
TABLES
Table I: ~ [percentage gain due to the Rao Method] C1
u 1 1 3 4 5 6 7 8 9 _l_ No. of O lO(l+k) S(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) l+k Design
Rl 0 .000 .001 .002 .003 .004 .005 .006 .008 .010 .012 R2 0 .000 .001 .001 .002 .002 .003 .004 .005 .006 .007 R3 0 .ooo .001 .001 .002 .003 .004 .005 .006 .007 .009 R4 0 .000 .ooo .001 .001 .002 .003 .003 .004 .oos .007 RS 0 .ooo .001 .002 .003 .003 .004 .006 .007 .008 .009 R6 0 .ooo .001 .002 .003 .003 .004 .006 .007 .008 .009 R7 0 .ooo .001 .001 .002 .002 .003 .004 .005 .006 .007 RB 0 .001 .001 .002 .002 .003 .004 .005 .006 .007 .008 R9 0 .000 .001 .001 .002 .002 .003 .004 .004 .005 .006 RlO 0 .001 .001 .002 .003 .004 .005 .005 .006 .007 .008 I-'
lJl Rll 0 .ooo .ooo .001 .001 .002 .003 .004 .004 .oos .006 i..o
Rl2 0 .ooo .ooo .001 .001 .002 .002 .003 .004 .• 004 .005 Rl3 0 .ooo .001 .001 .002 .003 .004 .004 .005 .007 .008 Rl4 0 .ooo .001 .001 .001 .002 .002 .003 .004 .004 .005 Rl5 0 .ooo .000 .ooo .001 .001 .001 .002 .002 .003 .003 Rl6 0 .ooo .000 .001 .001 .001 .002 .002 .003 .004 .004 Rl7 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .003 .004 Rl8 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .003 .004 Rl9 0 .ooo .ooo .001 .001 .001 .002 .002 .002 .003 .003 R20 0 .ooo .001 .001 .002 .002 .003 .004 .004 .005 .006 R21 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .004 .004 R22 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .003 .004 R23 0 .000 .ooo .ooo .001 .001 .001 .002 .002 .003 .003 R24 0 .ooo .ooo .ooo .001 .001 .001 .002 .002 .002 .003 R25 0 .ooo .ooo .ooo .001 .001 .001 .002 .002 .003 .003
'hllle It (can.t.iaued)
u l l 3 4 i 6 7 8 9 _l:__ O lO(l+k) S(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+lt) lO(l+k) lO(l+lt) l+lt
Ro. of Design U6 0 .ooo .ooo .ooo .001 .001 .001 .002 .002 .002 .003 ll27 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .OOl .004 ll28 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .003 .004 A29 0 .ooo .ooo .ooo .001. .001 .001 ~002 .002 .003 .003 lllO 0 .ooo .ooo .001 .001 .001 .002 .002 .003 .003 .004 llll 0 .ooo .ooo .ooo .001 .001 .001 .002 .002 .003 .003 ll32 0 .ooo .ooo .ooo .001 .001 .001 .002 .002 .002 .003 R33 0 .ooo .000 .001 .001 .001 .002 .002 .003 .003 .004 ll34 0 .ooo .ooo .001 .001 .001 .001 .002 .002 .002 .003 .... ll35 0 .ooo .ooo .001 .001 .002 .002 .003 .003 .004 .004 0\
0 lll6 0 .ooo .ooo .ooo .001 .001 .001 .001 .002 .002 .002 a37 0 .ooo .ooo .001 .001 .001 .001 .002 .002 .OOJ .003 lllS 0 .ooo .ooo ~000 .001 .001 .001 .002 .002 .002 .003 ll39 0 .ooo .ooo .ooo .ooo .OOl .001 .001 .002 .002 .002 a.40 0 .000 .ooo .001 .001 ~001 .002 .002 .003 .003 .004 •41 0 :ooo .ooo .ooo .ooo .001 .001 .001 .002 .002 .002 ll42 0 .ooo .ooo .ooo .ooo .001 .001 .001 .001 .002 .002 &43 0 .ooo .ooo .ooo .001 .001 .001 .001 .002 .002 .002 •44 0 .ooo .ooo .ooo .ooo .001 .001 .001 .002 .002 .002 a.45 0 .ooo .ooo .ooo .ooo .001 .001 .001 .OOJ. .001 .002 ll46 0 .ooo .ooo .ooo .ooo .ooo .001 .001 .001 .001 .002 &41 0 .ooo .ooo .ooo .001 .001 .001 .002 .002 .002 .003 8.48 0 .ooo .ooo .ooo .ooo .001 .001 .001 .001 .002 .002 ll49 0 .000 .ooo .ooo .ooo .ooo .001 .001 .001 .001 .001 llSO 0 .ooo .ooo .ooo .ooo .001 .001 .001 .001 .001 .002
Table I: (continued)
1 1 3 4 5 6 7 8 9 1 u O lO(l+k) S(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) lO(l+k) l+k
No. of Design
R51 0 .000 .000 .000 .000 .000 .000 .001 .001 .001 .001 R52 0 .000 .000 .000 .000 .000 .001 .001 .001 .001 .001 R53 0 .000 .000 .000 .000 .000 .001 .001 .001 .001 .001 R54 0 .ooo .000 .000 .000 .000 .001 .001 .001 .001 .001 R55 0 .ooo .000 .ooo .000 .000 .001 .001 .001 .001 .001 R56 0 .ooo .ooo .ooo .000 .000 .000 .001 .001 .001 .001 R57 0 .ooo .ooo .ooo .ooo .ooo .ooo .001 .001 .001 .001 R58 0 .ooo .000 .ooo .ooo .ooo .ooo .ooo .ooo .000 .ooo R59 0 .ooo .ooo .ooo .ooo .000 .ooo .ooo .000 .001 .001 I-' R60 0 .ooo .ooo .ooo .ooo .ooo .ooo .ooo .ooo .001 .001 O"I
I-' R61 0 .000 .ooo .ooo .ooo .ooo .ooo .ooo .ooo .001 .001 R62 0 .000 .ooo .ooo .ooo .ooo .ooo .ooo .ooo .001 .001 R63 0 .ooo .ooo .ooo .ooo .ooo .000 .ooo .ooo .000 .ooo R64 0 .ooo .ooo .ooo .ooo .ooo .ooo .ooo .ooo .ooo .ooo R65 0 .ooo .000 .ooo .ooo .ooo .ooo .ooo .ooo .000 .ooo R66 0 .ooo .ooo .ooo .ooo .ooo .ooo .000 .ooo .ooo .ooo R67 0 .000 .ooo .000 .ooo .ooo .ooo .ooo .ooo .ooo .ooo R68 0 .000 .ooo .ooo .ooo .ooo .ooo .ooo .ooo .ooo .000
Table II: Balanced Designs
s v r k b A f E F* B D2 D3 Dl
1 4 3 2 6 1 3 .67 .38017290 .526 .18 .20 o.oo 2 4 3 3 4 2 5 .89 II .626 .20 .24 o.oo 3 5 4 2 10 1 6 .62 .27639332 1.357 .34 .38 .23 4 5 6 3 10 3 16 .83 II 1.608 .40 .44 .28 5 5 4 4 5 3 11 .94 II 1.531 .38 .42 .26 6 6 5 2 15 1 10 .60 .21540524 2.321 .46 .50 .40 7 6 5 3 10 2 15 .80 II 2.458 .49 .53 .42 8 6 10 3 20 4 35 .80 II 2.635 .53 .57 .45 9 6 10 4 15 6 40 .90 II 2.653 .53 . 57 .46
10 6 5 5 6 4 19 .96 II 2.520 .so • 54 .43 11 7 6 2 21 1 15 .58 .17614569 3.339 .56 . 59 .51 I-' 12 7 3 3 7 1 8 .78 II 3.028 .so • 53 • 47 O"I
l\.l
13 7 4 4 7 2 15 • 88 II 3.339 .56 .59 .51 14 7 6 6 7 5 29 .97 II 3.541 .59 .62 .55 15 8 7 2 28 1 21 .57 .14899933 4.377 .63 .65 .60 16 8 7 4 14 3 35 .86 II 4.535 .65 • 68 .61 17 8 7 7 8 6 41 .98 II 4. 571 .65 .68 .62 18 9 8 2 36 1 28 .56 .12916656 5.419 .68 .70 .66 19 9 8 4 18 3 46 .84 II 5.565 .70 .72 .63 20 9 10 5 18 5 64 .90 II 5.631 .70 .73 .68 21 9 8 6 12 5 52 .94 II 5.591 .70 .72 .68 22 9 8 8 9 7 55 .98 II 5.603 .70 .72 .68 23 10 9 2 45 1 36 .56 .11405141 6.461 .72 .74 .70 24 10 9 3 30 2 51 .74 II 6.562 .73 .75 • 71 25 10 6 4 15 2 36 .83 II 6.461 .72 • 74 .70 26 10 9 5 18 4 63 .89 II 6.610 .73 .75 • 72 27 10 9 6 15 5 66 .93 II 6.619 .74 .75 .72 28 10 9 9 10 8 71 .99 II 6.633 • 74 . 76 .72
Table II: (continued)
s v r k b f.. f E F* B D2 D3 Dl
29 11 10 2 SS 1 4S .SS .10214480 7.499 • 7 s .77 .74 30 11 s s 11 2 34 .88 " 7.397 • 74 .76 .73 31 11 6 6 11 3 4S .92 II 7.499 .75 .7l .74 32 11 10 10 11 9 89 .99 II 7.660 .77 .78 .75 33 13 6 3 26 1 40 .72 .084S7048 9.384 .78 .79 .77 34 13 4 4 13 1 27 .81 II 9.174 .76 .7f!, .76 3S 13 9 9 13 6 92 .96 II 9.644 .80 .82 .80 36 lS 7 3 3S 1 S6 • 71 .07220328 11.462 • 82 • 84 .81 37 lS 7 7 15 3 76 .92 II 11.567 .83 • 84 .82 38 lS 8 8 15 4 91 .94 " 11.616 .83 .84 .82 39 16 6 6 16 2 6S .89 .06729477 12.494 .83 .84 .83 40 16 9 6 24 3 lOS .89 II 12.638 .84 .8S .84 41 16 10 10 16 6 129 .96 II 12.682 .8S .8S .84 42 19 9 3 S7 1 97 • 70 .OSS91668 1S.S76 • 87 • 87 .86 I-'
°' 43 19 9 9 19 4 134 .94 .OSS91668 lS.663 • 87 .88 .87 w 44 19 10 10 19 s 1S3 .95 II 15.692 .87 .88 .87 4S 21 10 3 70 1 120 .70 .05026207 17.613 • 88 .89 .88 46 21 s s 21 1 64 .84 II 17.364 • 87 • 87 .87 47 21 10 7 30 3 160 .90 II 17. 685 .88 .89 .88 48 25 8 4 so 1 126 • 78 .04181724 21.S78 .90 .90 .90 49 2S 9 9 2S 3 176 .93 II 21.674 .90 .91 .90 so 28 9 4 63 1 162 .78 .03714312 24.62S .91 .91 .91 51 28 9 7 36 2 189 .89 II 24.668 .91 .92 .91 S2 31 6 6 31 1 12S .86 .03341029 27. 496 .92 .92 .92 S3 31 10 10 31 3 249 .93 II 27.713 .92 .93 .92 S4 37 9 9 37 2 260 .91 .02782200 33.687 .94 .94 .93 SS 41 10 s 82 1 288 .82 .02S03212 37.690 .94 .94 .94 56 S7 8 8 57 1 343 .89 .Ol78687S S3.6S2 .96 .96 .96 S7 73 9 9 73 1 512 .90 .01389S09 69.697 .97 .97 .97 S8 91 10 10 91 1 729 .91 .01111421 87.73S .97 .98 .97
Table* III: Regular G.D., Within Comparisons
Lower Upper bound bound Error of Error of
s v of cti2/qi1 of cti2/cti1 Gl Gl G2 G2 B D
RS 8 2.7SOOOOOO 3.00000000 .21033814 -.00018116 .13394966 -.0001SS93 2.S7 .34 R6 8 2.75000000 3.00000000 .21033814 -.00018116 .13394966 -.0001SS93 2.99 .40 R7 RS 9 .1428S714 .16290727 .56841986 .00000717 .94426396 .00006088 1.63 .46 R9 9 .50000000 .S4901961 .46S43160 .000002os .S7522229 .000004S4 2.23 .S2 RlO 9 ., • 33333333 .39682S40 .49736833 .00000285 .69030269 .00001040 2.02 .so Rll 9 1.89393940 2.00000000 .31913S21 .OOOOOOS6 .27019144 .00000031 3.35 .S3 Rl2 9 .66666667 .70000000 .43977843 .00000160 .49866676 .000002S3 2.S4 .56 Rl3 9 2.78494620 3.00000000 .272399S6 .00000037 .20781181 .00000013 3.79 .S2 Rl4 10 3.80000000 4.00000000 .07240079 .00000000 .03680778 , • 00000000 7. S7 .SS I-'
O'I
RlS 12 .sooooooo .Sl8518S2 .09866736 .00000332 .13872044 .00000572 3.95 .66 ,j:::..
Rl6 12 1.S7407410 1.66666670 .OSS8126S .00000181 .04SS9198 .00000160 6.87 .6S Rl7 12 .66666667 .70000000 .08708004 .00000275 .10820428 .00000369 4.S8 .68 Rl8 12 .55S55556 .61191626 .27243S60 .00000000 .30782790 .00000000 S.08 .69 Rl9 12 R20 12 2.7SOOOOOO 3.00000000 .08072434 -.00013892 .OS994471 -.00014068 7.82 .64 R21 12 1. S7407410 1. 66666670 .OSS8126S .00000181 .045S9198 .00000160 7.14 .68 R22 12 l.439Sl610 1.50000000 .107312Sl -.00013718 .09716970 -.00013761 6.44 .70 R23 12 1.94708990 2.00000000 .21184402 -.00000000 .184744S2 -.00000000 6.71 • 71 R24 14 l.928S7140 2.00000000 .022S8870 -.00000000 .0162S612 -.00000000 10.40 .70 R25 14 l.428S7140 l.SOOOOOOO .02682109 -.00000000 .02291372 -.00000000 8.9S .72 R26 14 1.928S7140 2.00000000 .02258870 -.00000000 .01625612 -.00000000 10.81 • 7 3
*G1 and G2 are computed by the Simpson rule in Table III through Table VI.
Table III: (continued)
Lower Upper bound bound Error of Error of
s v of ¢2/cpl of ¢2/cpl Gl Gl G2 G2 B D
R27 15 3.80000000 4.00000000 .02249261 .00000000 .01406606 .00000002 13.52 .68 R28 15 l.86666670 2.00000000 .03139546 .00000000 .02615760 .00000000 10.35 .73 R29 15 .66666667 .70769231 .04453214 .00000000 .05079515 .00000000 7.66 .77 R30 15 3.80000000 4.00000000 .02249261 .00000000 .01406606 .00000002 14.02 .71 R31 15 3.35858590 3.50000000 .13773795 -.00000002 .11094994 -.00000006 10.89 .75 R32 15 .58333333 .64367816 .19287289 .00000000 .20999550 .00000000 8.05 .78 R33 15 .50000000 .56000000 .04709829 .00000000 .05832701 .00000000 7.08 .75 R34 15 1. 57948720 1.60000000 .03434892 .00000000 .02968222 .00000000 10.22 .79 R35 16 2.75000000 3.00000000 .04674152 -.00014124 .03722217 -.00014304 12.06 .73 R36 16 .50000000 .52884615. .07203230 -.00013834 .08256627 -.00013773 8.43 .79 I-'
°' R37 16 .33333333 .37096774 .07533001 -.00013816 .09176736 -.00013735 7.98 . 78 U1
R38 16 .63636364 .68573668 .06917094 -.00013852 .07650495 -.00013805 8.82 . 79 R39 16 1.93918920 2.00000000 .05368398 -~00014006 .04616141 -.00014109 11. 38 .79 R40 18 1.28395060 1. 33333333 .00613640 .00000000 .00550561 .00000000 13.04 . 80 R41 18 .77777778 .80459770 .00792662 .00000000 .00908759 .00000000 10.24 .81 R42 20 1.24285710 1. 28571430 .00294074 .00000000 .00267767 .00000000 15.08 . 83 R43 20 .37500000 .41826923 .04575421 -.00014053 .05238933 -.00013975 12.05 .84 R44 21 1. 42857140 1.50000000 .00418409 -.00000000 .00377673 -.00000000 16.31 .83 R45 24 4.83333333 5.00000000 .00286382 .00000123 .00174614 .00000127 28. 73 .80 R46 24 1. 69531250 1. 7 5000000 .00139259 -.00000001 .00116892 -.00000001 21.04 .85 R47 24 2.75000000 3.00000000 .02330743 -.00014602 .01989910 -.00014788 20. 74 .83 R48 24 1.66666670 1. 57407410 .00448605 .00000120 .00406692 .00000120 19.57 . 85 R49 24 .50000000 .53968254 .10454512 -.00000002 .11044128 -.00000001 16.82 .88 R50 24 4.83333330 5.00000000 .00286382 .00000123 .00174614 .00000127 29.18 .81
Table III: (continued)
Lower Upper bound bound Error of Error of
s v of ¢2/<Pl of ¢2/<Pl Gl Gl G2 G2 B D
R51 25 .50000000 .51891892 .01097727 .00000000 .01216185 .00000000 16.86 .88 R52 25 .33333333 .35813953 .01135905 .00000000 .01311656 .00000000 16.22 . 87 R53 26 1. 30769230 1. 33333333 .00030970 -.00000000 .00027208 .00000000 22.42 .87 R54 27 1.55555556 1.60000000 .00051831 .00000000 .00044830 .00000000 23.93 .88 R55 28 1.92857140 2.00000000 .00124101 -.00000000 .00104955 -.00000000 25.65 .88 R56 28 .66666667 .68934240 .00163455 -.00000000 .00178741 -.00000000 19.90 • 89 R57 30 .80000000 .81269841 .00009192 -.00000000 .00010303 -.00000000 21. 20 .90 R58 33 .31250000 .32679739 .07223852 -.00000009 .07599691 -.00000007 25.36 .92 R59 35 .53333333 .55384615 .00471039 .00000003 .00502271 .00000002 26.98 .92 R60 39 .72727273 .73803120 .00001199 -.00000000 .00001323 -.00000000 29.69 .92 I-'
(j'I
R61 40 1.80000000 1. 76400000 .00003541 .00000000 .00003041 .00000000 39.32 .91 (j'I
R62 45 1.95555560 2.00000000 .00003927 .00000000 .00003385 .00000000 44.96 .93 R63 48 6.87500000 7.00000000 .00004504 -.00000001 .00002640 -.00000001 71. 06 .88 R64 49 .50000000 .50989011 .00021542 -.00000000 .00023106 -.00000000 39.85 .95 R65 49 .33333333 .34637965 .00022058 -.00000000 .00024282 -.00000000 ,·38.85 .96 R66 63 7.88888889 8.00000000 .00000534 .00000000 .00000309 .00000000 97.99 .91 R67 64 .50000000 .50762195 .00002770 -.00000001 .00002943 -.00000001 54.37 .96 R68 80 8.90000000 9.00000000 .00000061 .00000000 .00000035 .00000000 128.51 1.00
Table** IV: Regular G.D., Among Comparisons
s v Gl Error of G1 G2 Error of G2 B D ~v 1 ~v 2
R5 8 .92660598 -.00000000 2.02812200 -.00000001 .75 .35 2 1 R6 8 .92660598 -.00000000 2.02812200 -.00000001 • 87 .40 2 1 R7 RS* 9 .18084145 .04166931 11. 78 .40 3 1/2 R9* 9 .58054100 .37645660 4.24 .46 3 1/2 RlO* 9 .40114170 .21142836 5.31 .40 II II
Rll* 9 1.82060710 3.43970230 1. 50 .51 II II
Rl2* 9 .75044440 .57646628 3.74 .53 II II
Rl3* 9 2.47599780 6.70399280 1.07 .50 II II
Rl4 10 .39800645 -.00013841 .92952501 -.00014245 1.65 .56 5/2 3/2 Rl5 12 .05296100 -.00000000 .03599987 -.00000000 8.17 .65 3 2 I-'
(jl
Rl6 12 .10760015 .00000000 .14907694 .00000000 4.05 .65 II II ....J
Rl7 12 .06449589 -.00000000 .05349805 -.00000000 6.86 .66 II II
Rl8* 12 .50211678 .35168153 8.20 .60 9/2 1/2 Rl9 R20 12 .52964673 -.00000000 1. 31482570 -.00000000 2.34 .62 4 1 R21 12 .10760015 .00000000 .14907694 .00000000 4.21 .68 3 2 R22 12 .33216065 -.00000000 .47197854 -.00000000 4.10 .68 4 1 R23* 12 1.54557910 2.93234110 3.08 . 70 9/2 1/2 R24 14 .05199792 .00000114 .07729396 .00000116 5.03 . 71 7/2 5/2 R25 14 .04482904 .00000114 .05774582 .00000115 5.93 .72 II II
R26 14 .05199792 .00000114 .07729396 .00000116 5.23 .74 II II
*Starred experiments have G2 and G1 as improper integrals; these integrals were
evaluated to the fourth decimal place.
**Lower and upper bounds of ¢2/¢1 are given in Table III.
Table IV: (continued)
s v Gl Error of G1 G2 Error of G2 B D :k:v 2 1 kv
2 2
R27 15 .18129859 -.00014162 .50217754 -.00014551 3.05 .68 5 3/2 R28 15 .11611093 -.00013932 .20327436 -.00014065 4.93 .70 II II
R29 15 .05347859 -.00013793 .04282467 -.00013764 10.91 .72 II II
R30 15 .18129859 -.00014162 .50217754 -.00014551 3.17 . 71 II II
R31* 15 2.16696860 6.96629570 2.73 .73 6 1/2 R32* 15 .44699438 .32561219 12.04 .66 II II
R33 15 .04206644 -.00013774 .02922470 -.00013730 12.62 .65 5 3/2 R34 15 .10340764 -.00013898 .14842798 -.00013971 6.15 .78 II II
R35 16 .37627084 -.00000001 .99309690 -.00000000 3.64 .68 6 1 R36 16 .09018610 -.00000000 .05430287 .00000000 16.04 .72 II II
R37 16 .06200081 -.00000000 .02843526 .00000000 21.19 .66 II II
R38 16 .11214239 -.00000000 .08637660 -.00000000 12.67 . 71 II II I-'
°' R39 16 .28767554 -.00000000 .52623532 -.00000001 5.35 .77 II II Q)
R40 18 .00877483 -.00000000 .01045750 -.00000000 9.82 .80 9/2 7/2 R41 18 .00661897 -.00000000 .00595953 -.00000000 13.04 .80 II II
R42 20 .00401108 .00000000 .00467043 .00000000 11. 79 .83 5 4 R43 20 .05062569 .02532839 -.00000000 27.58 . 70 8 1 R44 21 .01032799 .00000118 .01424629 .00000118 10.67 .Bl 7 5/2 R45 24 .03353262 .00000002 .11118008 .00000005 5.28 .80 9 2 R46 24 .00398032 -.00000000 .00597248 -.00000000 11. 77 .84 8 3 R47 24 .24051216 -.00000006 .67638281 .00000012 6.30 .76 10 1 R48 24 .01588761 .00000001 .02485663 .00000001 11.34 .81 9 2 R49* 24 .28425893 .16907593 29.88 .77 21/2 1/2 R50 24 .03353262 .00000002 .11118008 .00000005 5.37 .81 9 2
Table IV: (continued)
s v Gl Error of G1 G2 Error of G2 B D ~v 1 :J.::v 2 2
R51 25 .01445477 -.00014358 .00832752 -.00014303 32.42 .82 10 3/2 R52 25 .00989200 -.00014338 .00418149 -.00014265 44.30 .77 " II
R53 26 .00043324 -.00000000 .00050744 -.00000000 16.82 .88 6.5 5.5 R54 27 .00132261 -.00000001 .00185870 -.00000001 14.73 .86 9 7/2 R55 28 .00518073 .00000122 .00904618 .00000123 12.42 .85 10.5 2.5 R56 28 .00224372 .00000121 .00169697 .00000120 28.77 .86 II II
R57 30 .00007635 -.00000000 .00006869 -.00000000 26.41 .90 7.5 6.5 R58* 33 .14811017 .05181450 76:. 26 .84 15 1/2 R59 35 .00825644 -.00014972 .00495119 -.00014919 47.97 .86 15 1. 5 R60 39 .00001325 ,.-.00000000 .00001075 -.00000000 40.38 .91 13 5.5 R61 40 .00012458 .00000000 .00019609 .00000000 21.45 .91 15 4 R62 45 .00018122 -.00000001 .00031959 -.00000001 21.98 .91 18 3.5 I-'
°' R63 48 .00088430 -.00000000 .00386011 -.00000001 9.54 .89 20 3 l.D
R64 49 .00031901 .00000132 .00017518 .00000132 77.83 .91 21 2.5 R65 49 .00021694 .00000132 .00008413 .00000131 110.29 .88 II II
R66 63 .00012800 -.00000001 .00062472 -.00000002 11. 62 .91 27 3.5 R67 64 .00004318 -.00000000 .00002338 -.00000000 106.69 .94 28 3 R68 80 .00001735 .00000000 .00009354 .00000000 13.68 .92 35 4
Table V: LS Type, Within Comparisons
Lower bound Upper bound Error of Error of s v of cfJ2/cfJl=l.. of <P2/cfl1=P Gl Gl G2 G2 B D
1 9 1. 96551720 2.00000000 .31913521 .00000056 .26366817 .00000028 3.29 .52 2 9 1. 965517 20 2.00000000 .31913521 .00000056 .26366817 .00000028 3.48 .55 3 9 1. 44827580 1.49999990 .35289453 .00000074 .32071417 .00000053 3.08 .54 4 9 1. 72391020 1. 79999990 .33148280 .00000062 .28723591 .00000037 3.33 .55 5 9 1.25556860 1. 28571420 .37094270 .00000087 .34998933 .00000071 3.06 .57 6 9 7 9 8 9 9 9
10 9 1.98371340 2.00000000 .31913521 .00000056 .26206513 .00000028 3.41 .54 11 9 1. 98371330 2.00000000 .31913521 .00000056 .26206513 .00000028 3.54 .57 I-'
-..J
12 16 .57142860 .59340660 .07081078 -.00013842 .07925708 -.00013790 8.68 .80 0
13 16 .40000000 .42615020 .07412205 -.00013822 .08779910 -.00013751 8.26 .79 14 16 1.89130490 2.00000000 .05368397 -.00014006 .04682753 -.00014098 10.26 .71 15 16 .50000000 .52439030 .07211909 -.00013834 .08256627 -.00013773 8.52 .80 16 25 .50000010 .55555560 .01089548 .00000000 .01216185 .00000000 16.59 .86 17 36 2.30390930 2.33333333 .00123686 .00000127 .00105736 .00000128 34.46 .91 18 49 2.64363930 2.66666740 .00016765 -.00000000 .00014105 -.00000000 50.60 .93 19 64 2.98104600 3.00001610 .00002149 -.00000001 .00001786 -.00000001 69.36 .95 20 100 3.65261530 3.66668620 .00000030 .00000000 .00000025 .00000000 112.48 .97
Table* VI: LS Type, Among Comparisons
s v Gl Error of G1 G2 Error of G2 B D
1 9 1. 87617270 .00028460 3.43970230 .00028460 1.48 .52 2 9 1. 87617270 .00028460 3.43970230 .00028460 1. 57 .55 3 9 1.46043870 .00028460 2.12674590 .00028460 1.92 .52 4 9 1. 68620910 .00028460 2.88645690 .00028460 1. 67 .53 5 9 1. 29601880 .00028460 1.63940220 .00028460 2.28 .55 6 9 7 9 8 9 9 9
10 9 1. 89020710 .00028460 3.43970230 .00028460 1.54 .54 11 9 1. 89020700 .00028460 3.43970230 .00028460 1. 60 • 57 I-' 12 16 .10180204 -.00000000 .06678308 .00000000 14.81 .75 -...]
I-' 13 16 .07346688 -.00000000 .03668556 .00000000 19.60 .72 14 16 .28204712 -.00000000 .52623560 -.00000001 4.80 .68 15 16 .09018610 -.00000000 .05348033 .00000000 16.46 .74 16 25 .01445477 -.00014358 .00943603 -.00014312 28.36 .72 17 36 .00838014 .00000004 .01730837 .00000004 14.26 .90 18 49 .00136756 .00000135 .00316312 -.00000137 18.40 .93 19 64 .00020687 -.00000000 .00052993 -.00000001 22.50 .94 20 100 .00000388 .00000000 .00001188 .00000000 30.61 .96
*The lower and upper bounds of ¢ 2/¢1 are given in Table V.
Table* VII: Triangular Designs, Within Comparisons
Noo Lower Upper Lower Upper of bound bound bound bound Error of Error of
design v of 1/131'1/! 1 of 1/1311/! 1 of 1/1211/! 1 of 1/1211/! 1 G2 or E2 G2 G1 or E1 Gl B D
T6 10 .42857120 .46938750 .57142840 .60204070 .48272526 -.00000000 .37314616 .00011290 2.62 .49
T7 10 1. 00137230 1. 02147270 1. 00102920 1.01610450 .33310462 .00000000 .33199972 .00011336 3.75 .62
T8 10 .42857150 .46938780 .57142860 .60204090 .48272516 -.00000000 .37314613 .00011290 2.98 .56
Tl2 10 1.62068960 1. 64285710 1. 82758620 1. 85714280 .25731060 -.00000000 .27927481 .00011432 3.96 .55
Tl3 10 1.62068960 1. 64285710 1. 827 58610 1.85714280 .25731060 -.00000000 .27927481 .00011432 4.18 .58
Tl4 10 1. 28817730 1. 32142850 1.38423640 1. 42857140 .29228316 .00000000 .30268360 .00011383 3.89 .59
Tl8 10 1. 63700830 1. 64286130 1. 84934440 1.85714840 .25583841 .00000000 .27927454 .00011432 4.16 .58
Tl9 10 .42857450 .43072750 .57143090 .57304560 .48272390 -.00000000 .37672160 .00011286 3.00 .57
T23 15 .46666660 .52647970 .80000000 .82242990 1. 34026490 1. 04359710 6.51 .62
T24 15 .46666660 .52647970 .80000000 .82242990 1. 34026490 1. 04359710 6.80 .65
T30 15 1.12550610 1.14285720 1. 33468310 1. 38095270 .94547610 .92176423 8.57 .72
*The accuracy of E1 and E2 is to the sixth decimal place (see Section 8.11}.
Table* VIII: Traingular Designs, Among Comparisons
No. of E2 or G2 Error of G2 E1 or G1 Error of G1 D design v B
T6 10 1.19818240 1. 05692300 4.96 .37
T7 10 .99986668 .99799656 3.69 .61
TS 10 1.19818080 1.05692300 5.64 .43
Tl2 10 .87732533 .91344069 2.05 .51
Tl3 10 .87732533 .91344069 2.16 .53
Tl4 10 .93602815 .95229941 2.68 .55
Tl8 10 .87476619 .91344014 2.14 .53
Tl9 10 1.19817970 1. 06176610 5.95 .45 ...... -....) w
T23 15 .22027083 .00000000 .18703621 .00011599 8.63 .38
T24 15 .22027083 .00000000 .18703621 .00011599 9.02 .39
T30 15 .17512971 -.00000000 .17211884 .00011664 6.26 .61
*The accuracy of E1 or E2 is to the sixth decimal place (see Section 8.11).
Table* IX: Cyclic Designs, Within Comparisons
Lower Upper Lower Upper No. of bound bound bound bound design v Of 1/13/1/J l of 1/13/1/1 l of 1/12/1/11 of 1/12/1/11 El E 2 B D
cl 13 .33333333 .37037010 .83333333 .84259260 1.06178250 1.92264240 3.38 .40
c2 13 1.19143980 1.20000000 1.76575940 1.80000000 .78509506 .87702937 6.08 .53
c3 13 1.06262920 1.07142860 1.25051680 1.28571460 .90801427 .95631297 6.45 .65
c4 13 .33333340 .33397050 .83333340 .83349270 1.06565970 1.92264370 3.81 .45
cs 17 1.13059700 1.13888880 1.36567160 1.38888880 .88794416 .91941955 10.38 .71
c6 17 .53333340 .53631880 .83333340 .83439960 1.05928760 1.43911020 7.94 .65
c7 17 .53333270 .53539250 .83333310 . 83406880 1.05941500 1.43910700 7. 97 • 65 ..... -..J oj::>
c8 29 .57639040 .58160980 .91666700 .91769380 1.02510200 1.32158740 17.60 .72
c9 29 .27380540 .27535080 .85714210 .85744610 1.04443640 1.69198110 14.04 .59
clO 37 .70634920 .74441510 .88517270 .90005770 1.02648960 1.17377540 26.82 .83
*The accuracy of E1 or E2 is to the sixth decimal place (see Section 8.11}.
- 175 -
Table* X: Cyclic Designs, Among Comparisons
No. of El design v E2 B
cl 13 1.04573670 1.61058820 4. 72
c2 13 .83119494 .90590690 3.46
c3 13 .92964747 .96708089 5.08
c4 13 1. 04856730 1. 61059310 5.37
c5 17 .90855906 .93483468 7.52
c6 17 1. 04693670 1. 33285940 10.16
c7 17 1.04703650 1. 33286350 10.19
c8 29 1. 02190010 1.27396560 19.83
c9 29 1. 03868120 1. 57294470 17.51
clO 37 1.02348070 1.15243270 30.26
*The accuracy of E1 or E2 is to the sixth decimal place
(see Section 8.11}.
D
.18
.38
.56
.21
.59
.44
.44
.47
.20
.66
- 176 -
Table XI: The p. 's for Triangular Designs l.
No. of design
T6 o, i 1,2, ... ,7; 12 9 pi = = Pa -- Pg = 7 , 7
T7 fl " " .. TS .. II .. ..
Tl2 fl " fl fl
Tl3 fl .. .. II
Tl4 II fl " fl
Tl8 fl " " "
Tl9 II II II II
o, i 1,2, ••• ,11; .§. 1 7 T23 pi = = P12 = P13 = P14 = 3 , , 3
T24 " II fl fl .. T30 .. If .. fl ..
No. of design
Table XII: The p. 's for Cyclic Designs 1
3 1 cl pi=O, i=l,2, ... ,8; P9=2, P10=2, P11=2, P12=2
c2 II II II II II "
c3 II II II ti II "
c4 II II II II II II
7 7 5 43 16 cs p.=O, i=l,2, ••• ,11; P12=3, P13=3, P14 = G' P15 = 30' P16 = 15 1
c6 II II II II " " "
c7 II II II II II " II
5 11 61 97 11 P26 = 1, P27 = 1, c8 p.=O, i=l,2, ••• ,20; P21 = °2' P22 =6, P23 = 24' P24 = 40' P25 =s, 1
1 P2s=2
c9 II II II II II II II II II II
clO 8. =O, i=l,2, ••• ,27; 5 5 47 91 11 37 P2a=2, P29=3, P30=23' P31=40' P32=s, P33=14'
243 8 3059 P34 = 112' P35 = S' P36 = 2960
I-' .....i .....i
ABSTRACT
We know that the best linear combination of the intra-
and inter-block estimates is
Intra-estimatexinter-variance+Inter-estimatexintra-variance Intra-variance + Inter-variance
however, this combined estimate is merely theoretical, since
we do not know in practice the exact inter- and intra-
variances. A reasonable solution is to use a random weight
which can be computed from the data of our experiment, but
so far there has been no practical solution without severe
restrictions on the size of the experiment, and no solution
at all for a clear answer to the question of how much we
i
recovered. In fact, the experimenter applying the methodology
available to him now, cannot be sure that he is really
improving the accuracy of his estimation.
This research has achieved the following:
1. A new method of combining two independent estimates
has been developed. This method has its use in incomplete
block designs, in similar experiments, and in randomized
block designs with heterogeneous variances. The improvement
introduced by this method is very satisfactory, compared with
the utmost possible theoretical improvement.
2. A procedure for recovering the inter-block informa-
tion in B. I. B. designs was given, which is applic'able in
experiments of as small as t = 4.
3. It has been proven that the practical utilization
of inter-block information is possible in any P.B.I.B. with
seven treatments or more.
4. A general procedure for recovering the inter-block
information in P.B.I.B. 's with two associate classes was
given.
5. An inter-block analysis of singular and semi-regular
group divisible designs was discussed_, which makes a partial
utilization of the inter-information possible.
In general, this work has two merits:
1. It makes possible the utilization of the inter-
block information in small and moderate size experiments.
2. As a ratio of the utmost possible theoretical
recovery (by combining linearly), either exactly or a lower
bound of the ratio of recovery is always computable.
Tables which enable the experimenter to use the procedures
described in this dissertation were given. The ratios of
recovery listed in these tables show that the new method
gives good results where the old method is not applicable,
and when the old method starts, hopefully, to be valid, the
ratio of recovery achieved by the new method starts to
approach the theoretical value that can be achieved, assum-
ing the intra- and inter-variance are known.
