5‘48 METHOD OF LEAST SQUARES
APPLIED TO
THE ANALYSIS OF VARlANCE

Thasis for the Degrévfif M. A.
ICHIGAN STATE COLLEGE
Miriam M. Geboo
i940

 

 

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THE METHOD OF LEAST SQUARES
APPLIED TO
THE ANALYSIS OF VARIANCE.

by
Miriam Martha Geboo

A THESIS
Submitted to the Graduate School of Michigan
State College of Agriculture and Applied
Science in partial fulfilment of the
requirements for the degree of
MASTER OF ARTS
Department of Mathematics

1940

I wish to express my apprecia-
tion to Dr. William Dowell Baten
whose suggestions and guidance

made this thesis possible.

he
C?
{S
C)
c)
Q:

Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter

Chapter

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

Table of Contents

Introduction . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . .
Randomized Blocks . . . . . . . . . . .
Single Criterion of Classification . . .
Multiple Criteria of Classification . .
Latin Square . . . . . . . . . . . . . .
Analysis of Covariance-Randomized Blocks
Analysis of Covariance-Latin Square . .
Incomplete Blocks . . . . . . . . . . .
Unbalanced Incomplete Blocks . . . . . .
Youden's Square . . . . . . . . . . . .
Lattice Squares . . . . . . . . . . . .
Factorial Design . . . . . . . . . . . .
Confounded (3x2x2) . . . . . . . . . . .
Values . . .

Estimation of Missing Plot

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77

Introduction

One of the newest branches of the science of statistics
is that of the analysis of variance. This was first intro-
duced by Dr. R. A. Fisher‘s-)in 1923, and he and many other
statisticians have been working constantly since that time
to perfect it.

It is a means for segregating from groups of data being
compared the variability arising from known sources, leaving
an estimate of the experimental error. It can be utilized
in testing significance between means of the groups of data.

The analysis of variance is applicable to large or
small samples and to a large number of eXperimental designs.
In an analysis of covariance it is also possible to analyze
two or more associated variables.

It is the purpose of this thesis to illustrate with
various experimental designs how the experimental error used
in the analysis of variance is derived. To do this the
method of least squares, as suggested by F. Yates(r¥)and
illustrated by him<A3 , will be used. A second method,
called the method of fundamental identities, will be illus-
trated in the case of Randomized Blocks.

In conclusion a method for finding values of missing

plots occurring in an eXperiment will be suggested.

2.
Notation

The notation used in this paper is simple; however, a
reader who is unaccustomed to it will find it complicated
unless this explanation is given.

The letter y will be used consistently through the
paper to indicate the yield of a plot in the experiment.
It could, however, depending on the experiment, be used as
a measurement of height, weight, amount of growth, etc.
The letter x will be used in the analysis of covariance
as the "stand" while y will be used, as before, as the
yield.

The subscripts on the y, such as in. Yipou. , will
be eXplained in the individual chapters. -

The general mean of the y's will be denoted by y.

Summing will be denoted by dots in the subscripts.
For example, let us suppose that the data were arranged in
n columns and r rows and that y” indicates the yield
occuring in the i th column and the j th row. The sum of
the yields of the first row would be denoted by I‘l, which
indicated that the yields having a l in the second place
of the subscript--that is, those in the first row--would
be totaled. Similarly, Y3. would be the total of the sec-

ond column. I denotes the grand total.

Throughout the thesis
T; is used to denote treatment 1,

B; is used to denote block 1,

3.

RI is used to denote replication 1,
C; is used to denote column 1,
V; is used to denote "variety" 1.

After the first chapter the sum of squares of the
residual errors will be denoted by SS, and the degrees
of freedom by D. of F.

Each chapter will have equation indexing independent
of any other; that is, the first equation in each chapter
will be numbered (1). If it becomes necessary at any time
to refer to an equation in a previous chapter, for example,
chapter 3, equation (1), this will be done by referring to
equation (3.1).

Other notation will be described in the individual

chapters.

4.
Chapter 1.
Randomized Blocks

One field arrangement that has been found extremely
useful and at the same time Specially suited to the appli-
cation of the analysis of variance, is that of randomized
blocks. In this design blocks are set up, each of which
contains a complete replication of the treatments (or va-
rieties) arranged at random. The number of blocks used
depends upon the desired number of replications of each
treatment.

As has been stated previously, there are two ways in
which the experimental error used in the analysis of var-
iance can be derived algebraically: l. the method of fun-
damental identities, and 2. the method of least squares.
The first method may be explained as follows:

Consider the randomized block layout shown; in Table
1. This is not the arrangement used in the field layout,
but that used for computation purposes. Let n be the

number of treatments and r the number of blocks.

5.

 

 

 

 

Table l.
Treatments

Blocks T, ' T, ... T, Totals

BI yu ya: "° ya: Y"

B1 yll ya: .00 y"; Yet

0 O O O 1 0

Br- ylr yer "’ ynr Y”.
Totals Yh ‘Ya. ... Y,“ Yo.

 

 

 

 

 

The total sum of squares is equal to the sum of

squares of the deviations of the variates from the general

mean. In symbols this is:

g.

" -2.

Total sum of squares = Z; (yii - y)
tn

5-!

where y' is the general mean and yfli indicates that y
appears in the i th treatment and the 3 th block.

Now consider the identity

m; -'§> = (a. -'§)+ (37,, ~37“) + (a).
3 I.

where

(1) d=yu ”I": -yti+ y .

and ‘5‘ is the mean of block 3 and y}. is the mean
J I.

of treatment 1.

Squaring and summing,

6.
zcyi. - if: nzo', - if). r235... 4% 23a)"
«2]. 5 J J c ‘v (j
2 " -" " -" 2 "‘ -‘ d
+ 8y"; YHYtL y)+ ZOE; .v)()
+2267, -3?)(a) .

The fourth term of the right hand side of the equa-

tion can be shown to equal zero thus:
is -'§r’)<‘§ -'y'>-.-. 2:5: (‘5; -§)-'§Z<§r’ rm]
5; t. J 532.: *1 a ‘a
:= O ,

since g6? - y) =- O, and since the mean of the yt,’s
I. l.

is equal to Ԥ .
The fitth and sixth terms equal zero likewise. Therefore

we have
,. b g 0 n T d .
§<yu Y) :12;be y) + rig)!” J) + §( )

The correSponding equation for the degrees of freedom is:
nr-l=(r-l)+(n-l)+(n-l)(r-l).

This has been proven in the literature quoted.
In calculating the sums of squares the following for-

mulae are convenient:

z
I
2 t l 0' - = .. - —
() Toa 25s., y) Ear” nr .
-2, 235.3 Y";

 

(3) Blocks nZGb. - y) = _s_____ __ ,
J J

 

 

(4) Treatments 1:235? ~37) = L _ ,
l- ‘- r nr
1
(5) Error 2(a) = (2) - (3) - (4) .
‘.
The analysis of variance table is shown in Table 2.
Table 2.
Source of Degrees of Sums of Variance
Variation Freedom Squares SS/D.of F.
Total nr - l (2)
Blocks r - l (3)
Treatments n - l (4)
Error (rr1)(n-l) (5)

 

 

 

 

The standard error of the experiment is given by

 

 

 

(6) S=/\/-(r—_—-D(%%z _ 1) =4/error variance.

It is possible to verify the analyses of variance of
other designs in this way; however, it is the purpose of
this paper to illustrate the second way, that of the
method of least squares.

Let ti be a factor concerning treatment 1 which affects
yields,

b5 be a factor concerning block 3 which affects

yields,
and m be a constant.
We will assume that §ti a 0, because, while factors

concerning some treatments tend to increase the mean yield

8.
of a plot above the general mean for the whole experiment,
other factors tend to decrease it. Similarly, élbsta O .
We will assume that the ti's, bi's, and m are coef-

ficients in the following linear equation:

t G '1' ...*tnGn + b'H|+ ooo+err + mK+ euv ,

(7) y“: . .

where G H-, and K are variables which take on the fol-

i’ 3
lowing values for y‘s

GK:O,whenu-,(i;

‘0

Gual,whenu=i

Hv=l,whenv J;HV=O,whenv-,£J;
K = l, for all u and v.
In equation (7) the e‘u, represents the variation
due to chance.
After substituting values for G6, Hj,'and K, equation

(7) reduces to

(8) yes: tL*b3"’m*e£j ,

or
eii:yii‘ti'b3‘mo
The method of least squares is used to find values of
ti' b3, and m that make the sum of the squares of the re-

sidual errors a minimum. This is done by setting the par-
tial derivatives with respect to ti' b5, and m equal to
zero and solving the resulting simultaneous equations.

Let -
F=Zez= {5(3) -ti-b3-m)z.

Taking the partial derivatives,gives:

9.

 

by t‘
W 2 Z(-2)(yu " I... " b5 - III): 0 ,
or
(9) It. - r ti - r m : O ;
and
er - 2 . ..
‘83} '- (”2“in -t‘-b..\ m)- 0 a
or
(10) Y’s-an-nm=0;
and
OF ‘ -‘ - . - ' - -
TIT- "’ §(-2)(y‘,3 t‘ b; H1)- 0 D
or
(11) Y.. - N m = 0, where N = nr .
Directly from equation (ll), it is seen that
YO. '-
1.(12) sz :y 0
Substituting this value of m in (9) and solving
for ti. , gives
Ya. -
(13) ta: r "Y .
Similarly,
2.5 _
(14) b3 2-?- -y 0

Substituting the values from (12), (13), and (14) in
(8), and simplifying,

Y Y '

-—£.— 45—-.. 0"
yii' rIn y+e”,

this becomes, when predicting the value of y on the

10.

average,

... Ya. Y.;

(15) y=r+n-y~
If an equation is of the form given in (7), the stan-

dard error of estimate is given by

‘ _ w
(16) S ' x/gagrees of freedom ’

where

 

 

w: Zy” - t.Zy G, - tLZy G3," - tNZy en
- b'Zy H. - szy Hz- ... - brEy Hr - mZy K ,

and the degrees of freedom in this case are
D.ofF.=N-l-(n-l)-(rel) .
Substituting the values given in (12), (13), and
(14) in (16) and simplifying, gives:‘

I. 3"
..zyz-§(-;—-—-r)r.. ~2;(;. ~37 In. - W-

a z. a
, x ’- [v.1 y Zr. I"
= Z)! "7;?" ' -—--i h '__._” - 3'5 - .
r N n N
Comparing this final value for w with that indi-

cated in (5), we see that

 

- 11.
and is equivalent to that given in (6), as found by the

method of fundamental identities.

The standard error of estimate used in this and the
following problems is the experimental error which gives ria

rise to the analysis of variance.

12.
Chapter 2.
Single Criterion of Classification
The simplest set-up to which the analysis of variance
can be applied is that of a single criterion of classifi-
cation with either equal or unequal frequencies in the
classes. Such a set-up is illustrated in Table 3.
Table 3.

 

_Treatments

T T 0.. T

I 2 H

 

 

 

y“ y“ '” ym

 

yu. yu. "’ ynz.

 

 

 

y'r‘v yzr, ... ym,"
Y Y ... Y

I. z. n.

 

 

 

 

 

 

In the general case we will consider n treatments,
and the frequencies in Treatment 1, 2, ... , n will be
respectively r., r1, ... , r?! where the r 's may be
equal or unequal. V

The equation from the fundamental identity is

Y " it." I
(1) Zyz- N ='(Z—-‘—‘—'-§'-:)+ Z.(d)z .

I‘

 

where d is an eXpression similar to that in (l.l), and
N 9- Zr}.
4; i.
Let ti be a factor concerning treatment 1 which
affects yields,

and m be a constant.

13.
We will assume that the ti's and m are coefficients

in the following linear equation:
(2) tu '-'-' tiEi + tzEZ. ‘l' 000 ‘* tNEH +m‘ 4' e‘v,

where z;t£ a O , and where the variables E and O»take
in

on the following values for y :

Eu: 1, when u = 13E“: 0, when u i i;
0 =1, for all u's and v's.

Then (2) reduces to

(3) yij. = ti+ m + eij .
or

6"” = y‘oj “ti‘mo
Then let

P = 26": Pm. - ti ~m)z .

x
(

Taking the.partial derivatives and setting them equal

to zero, gives

Ti
3-1.- -Z(-2)(yij " t; " m) ’- 0 a

ati "
or
(4) Yip - riti " rim =0 3
and
B: =Z(-2)(yi5 " t, " m) = 0 o
‘3
0r
(5') Y..-Nm=O o

Directly from (5) we get

l4.
(6) m =5; .
- Substituting this value of m in (4) and solving for
ti gives
(7) ti:

Ya.

ri

- y .

 

Substituting the values in (6) and (7) in (3). and

simplifying, gives

Ya.
(8) yii:;—.—+ei30

I.

If an equation is of the form (2), the standard error

of estimate is given by

 

 

(9) S " Degrees of Freedom

where

w = Zyz' - t.ZyEl - tzZyEz - ... - chyE,‘ - mZyD ,
and

Degrees of Freedom =jN - l - (n - 1)
Substituting the values of ti and m given in (6)
and (7) in the equation for w yields

1- 7'
2 o — —
W =Zy .Z(; .y)ri. -y Yo.

t

a. - In a: -25) .

t

 

 

 

 

which is essentially the same as that given in (1) if 2d,.

is replaced by w .

The analysis of variance is given in Table 4.

15.

 

 

 

 

 

 

Table 4.
Source of Degrees of Sums of
Variation Freedom Squares
Total N - l (a) Zyz - i;
2. 2.
Y- Y
Treatments n - l (b)2:_ " - _LL_
Error N-l ~(n-l) (a) - (b)

 

 

 

16.
Chapter 3.
Multiple Criteria of Classification

Some times an experiment is carried out in which
there are more than two criteria of classification, say
three; in such an eXperiment, the problem of interactions
occurs. For instance, if we have an arrangement of four
varieties and three treatments, replicated three times, we
have an interaction appearing between varieties and treat-
ments because of the different reSponses of the different
varieties to the same treatment; similarly, we have inter—
actions between varieties and replication, and treatments
and replications.

Suppose in the general case we have n treatments,
r varieties, and s blocks or replications. The arrange-

ment for computation purposes is illustrated in Table 5.

17.

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 50
Treatments
Varieties Tl TQ' .... Tn “Totals
I ym; ya,” ynu
BS yns yzts "H Y?!”
Y“. Yale .... Ynio YO‘O
Bl ylzl yzz; "" ynz.
Va : : : :
BB y‘s: yhzs °°°' ynzs
le0 Ya. In}. Yet.
Bl y‘r; yirl ’°°' ynrl
v C O O .
r O O . 0
53 yin yer: "" ynrs
er. .... th. Y.r.
9...

 

 

 

 

 

   

 

By yijk it is indicated that the yield is in the
i th treatment, 3 th variety, and the k th block. The

total for Block I: :if”k .

Let t; be a factor concerning treatment 1 which
affects yields,
Let vj be a factor concerning variety 3 which affects

yields,

18.

bK be a factor concerning block k which affects

yields,

m be a constant,

(tv)i3 be a factor concerning the interaction of
treatment 1 and variety 3 which affects
yields,

(tb)U( be a factor concerning the interaction of
treatment 1 and block k which affects
yields,

and (vb)j,t be a factor concerning the interaction of va-

riety j and block k which affects yields,

where

it; = o, 2,sz

i 0, gb“ = o, g our)”. 3 0,

Z (tithe) a 0, z;(tb)ik = 0, £033)“ 3 0,
i 4. k
{EWb’ik = o, 29%).“ .-. 0.

We will assume that these factors are coefficients in

the following linear equation:

(1) yum t'El+...+t"En + v'F‘ +... err + blG. +...

+ b‘G‘ + m H + (tv)H J“«b...1-(tv)nrJ

"P

+ (131))” K '1"... +(tb) + (Vb)|l L” 1'...

u nsKhs
+ (vb)”L,s 1- euvw ,

where the variables take on the following values for yisk:

19.

whenu =13Eu= 0, when uii;

‘
ll
9

whenvzj ;F,=O,whenvfi;

<
II
HHH
U

whenw=k3Gw=O, whenwfk;

= l, for all u, v, and w ;

qmmmm
II

al,whenu=iandv=3;
J." =0, in all othercases ;
MI,al, when u-a i andw =k;

= O, in all other cases ;

.0

K
K
L,,,,=l, whenv a.) andw: k
L

:0, in all other cases .
Then (1) reduces to
(2) yixk a t3. 4- ys + b" + (tv)u~ +- (tb);k + (vb):H< + eijk*”‘-

Then

2
F = Zez= ZCYijk " ti. ' V's - bk " m ' (tv);3 - (“flin- (Vb)3k] .

. The partial derivatives result in the following equations:

(3)‘%FE'.:3Y;”~rst-‘-rsm=o,
3
(4)7%;Y,3,-nsv3-nsm=0,

(5)-%%; Y..k-nrb,‘-nrm=0,

 

 

(6)-%—E—;Y,,,-nrsm=0,

(7) 3:05;“ , Yij. -st; -svj -sm-s (tv)i's -.-. O,
(8) 353;)“ ; Yuk -rt£ -rbK-rm-r (um-k; o,
Q9)-5%-§—S——;de-nv3-nbk-nm-n(vb)jk=0.

20.
From (6) we see immediately that
(10) mg? .
Substituting this value of m in equations (3), (4),

and (5) and solving for to , vi , and b,‘ reapective-

 

 

1y, we get
Yioo '-
(ll) ti, 3 rs " y 2
Y4. —
(12) V5 3 ns "' y a
Y K _ .
(13) bit 2'. .. " y o
nr

 

Utilizing the values of (10), (ll), (12), and (13) in
equations (7), (8), and (9), we get

 

 

 

 

 

 

4 Yes. Yil. Y's. '-
1'. ~- : - r
(l ) ( v)‘J 8 er ns +~y '
- Y- Y,
1 tb = (OK - ‘0. - ..k - ’
( 5) ( )ik r Br nr 4' .Y
YOIK Yojo - Yook

 

 

 

u
I
“H

C

n ns nr

(16) (Vb)jk

After substituting values in (10) through (16) in
equation (2), we get the following equation for predicting
yijk. on the average:

Y. Y Y

Y.. . . Y
1 N‘. = t]. ‘Ik 03K - to. - 030 - 00K.
( 7) y‘s“ s I n I r sr ns ‘ nr

 

 

 

 

 

 

The standard error of estimate is given by

- ....JL...
(18) S " )/D. of F. '

where

 

 

 

2. Y”1 Zr; .2, x”‘
w - izyijk - nrs - rs - nrs

ns nrs nr nrs

“22...”; 2..) (:2...‘ ...2) (:2...‘ 2...)
[( ) ( > >1
- [(21.33 _. Inf) _ (2:14." ‘ x. ”2) -(ZYHK" - Y...‘)]

= (Total Sum of Squares) - (8.8. for Treatments)

 

K

 

 

 

 

 

 

 

 

 

- (8.8. for Varieties) - (8.8. for Blocks)

- [8.8. for Interaction TxV]

«- [8.8. for Interaction TxBJ

- [3.8. for Interaction VxB]

=(1) - (11) - (111) - (iV) - (V) - (v1) - (v11) .
and
D. ofF.=(nrs-1)- (n-l)- (r-l)- (s-l)
- (n - l)(r - l) - (n - l)(s - l) - (r - l)(s - l) .

Thus we have derived the eXperimental error used in
the analysis of variance of an experiment with three
classifications.

The resulting analysis of variance is given in Table 6.

22.

 

 

 

Table 6.
Source of Degrees of Sums of Squares
Variation Freedom
Total N - 1' (1)
Treatments n - 1 (ii)
Varieties r - 1 (iii)
Blocks s - 1 (iv) I
Interactions .
TxV (n - l)(r - l) (v)
TxB (n - l)(s - 1) (vi)
VxB, (r - l)(s - 1) (vii)
Error(TxVxB) (N-D(n-l)-(r-l) (i)-(ii)-(iii)
-(s-l)*(n-1)(r-1) -(1V)-(V)-(v1)
~(n—l)(s-1) -(vii)

 

~(r-1)(8-1)

 

 

*Nanrs.

23.
Chapter 4.
Latin Square
An experimental design which is frequently used is
the Latin Square. If n varieties (or treatments) are to
be tested, a plot of land is divided into a checkerboard
arrangement of n rows and n columns, and the n vari-
eties are distributed at random in the plots, but with the
double restriction that each variety appear once and only
once in each column and each row.“ If V,, V1, V3, V¥,‘%r
are five Varieties, we can form a Latin Square as shown in
Table 7.
Table 7.

 

Columns

 

 

 

 

 

 

‘< ‘4 '<
~ “<12“.

 

 

 

 

 

 

R fill V V
3 a

If n is the number of rows, columns, and varieties,

 

the fundamental identity for the Latin Square, written in

terms of summations used in calculations, is

 

* Rider, P. R., g3 Introduction to Modern Statistical
' Methods, pp. 166- -9.

 

 

24.

 

 

 

2.

(1) Z 1 y ‘ Y; Y 1 Y4." x. f
y" - ... = . .. - 0.. + - a
85“ ”k n" n n1 n 7
a. a
Y. Y... 1
+ 'k - 2' +-Z:d. ,
n n

where d is an eXpression similar to that for randomized
blocks and yijk is the yield of the variety k which
appears in the i th row and the 3 th column.
Let vk be a factor concerning variety k which affects
yields ,
ri be a factor concerning row i which affects
yields ,
c5 be a factor concerning column 3 which affects
yields ,
and m be a constant,

where
gvkzo,§ri=o,§:ci=o.
We will assume that the v's, r's, and c's and m are
coefficients in the following linear equation:
(2) yuvw=V|G'+ooo+van+r'H.+ooo ran+ 0"].
+ ...-renJ,‘ + m K + e“,W ,
where G... H“, JW and K are variables which take on the
following values for yijk:
G» :1, whenw: k; G“: O, whenw fk;

“=1,whenu=i;Hu=-.O,whenufi;
V V

H
J = 1, when v =.J; J = 0, when V i J;
K

=.1 , for all u, v, and w.

25.

Then (2) reduces to

(3)

= Vk+r£+ 03+ m *e‘ssk o

yijk
The expression of the sum of squares of the residual

errors is

3:281 = {(3,in ' Vk " CS "' rt --m) o

ijk
The partial derivatives result in the following

equations:

3F 0 - - - O
(4') W a Y"" n Vk n m - O ,
(5) 3F ' Y - - n c- - n m = o ,

3c) ' '1' J

bF O . C o - - .
(6) .5;- : Y‘H‘ nr. nm-O,

a:

I
”P
s
u
o
O

(7) g;—

Directly from equation (7) it is seen that

(8) YO... '-
m = 3 y o
n.-

 

Substituting this value for m into (4), (5), and

(6) and solving for vk, c5, and r; respectively, we get

 

 

Yank -
(9) vk =- n - y D

Y‘s. _
(10) cj = n " Y o
and

I. _
(11) r- 7- u. " Y o

 

l n

26.
Now substituting these values into (3) and simpli-
fying we have the following equation for predicting y

on the average:

. Y. _
(12’5’:1~=‘§"'*7{T’L*‘fi*""2y ‘

The standard error of estimate is given by

_, W
(13) S - fl. Of F. '

where

 

 

2.

' (22’ - ‘11: )'(>;‘ - 1'5)

n.
Y. 3 x " 2,1 x 1
- Z ... - no. 2 0!. . Doc ’
i n n1 3 n n1

which is essentially the same as (l) is we let
a
W .1: Ed 0

Thus we have derived the eXperimental error used in

 

 

 

 

 

 

the analysis of variance for a Latin Square design.

27.
Chapter 5.
Analysis of Covariance

It is the purpose of the analysis of variance to re-
move from the experimental error all variability except
that due to the chance variation within the factor being
tested. This is done sometimes through an attempt to hold
other factors constant, or through replications. Sometimes,
however, a factor enters into the experiment which can not
be held constant. For instance, in an experiment which
deals with the yields of a certain variety under various
treatments, there may be a variation in the yields caused
by unequal stands in addition to the variation caused by
the treatments. Also, in an experiment regarding the
weights of animals fed different diets, original weight
may cause variation which should be considered.

Fisher“y and Goulden** have discussed problems of this
nature and analyzed them by the analysis of covariance.

When an experiment is analyzed by the analysis of
covariance, two separate measurements x and y are
made on the items. Now, in the first case above, x
would refer to stand and y would refer to the yields,

while in the second case x would refer to initial weight

 

* Fisher, R. A., Statistical Methods for Research Workers,
Sixth Edition, pp. 275-90.

**'Goulden, C. H., Methods 9; Statistical Analysis, pp.
247’60 .

28.

and y the final weight.

Randomized Blocks

Let us consider a randomized block experiment in

which measurements were made both on stand and yield.

n be the number of treatments applied and

her of replications.

Table 8 illustrates such an experiment.

L

be the num-

et x : stand and y.= yield.

 

 

 

 

 

 

 

 

 

Table 8.
Treatments
Repli-
cation T . T1 . . T n Totals
R. H x“ o 0 HI X”
y” y.“ y,” You
R2. |L x21 n1. x.1
ygl yzz yhl— Y0;
R? W‘ x1? '5? Xor
yd? ylr
Totals " "
Y‘O Y2.-

 

 

 

 

 

 

 

 

 

 

 

Let ti be a factor concerning treatment 1 which

.r.
J

m

be a constant;

affects yields;

be a factor concerning replication 3

affects yields;

which

29.
and b be the regression coefficient between stand and
yield;

where

Zt"=0 , 21‘. :0 O
t 3 ’
Then the equation similar to (1.8) is

(l) yijatt+r3+m+bxij

+ei".

The eXpression for the sum of squares of the residual
errors is
z

)2.
Setting the partial derivatives equal to zero and

simplifying, we have,

(2) .13.; Y- -rti-rm-bX-‘.=O;

at; "
3F . - - - . .
(3) 3-5;.- 9 Y.) n I“ n m b X‘5 I O ,

(4) 3.3.; x -Nm-bX. :O,whereN=nr;

‘Bm. °° -
(5) .921. - Zx‘..y.. - Zt-x- - Zr-x. - mX" - be --" = o.
Ob ’ is ‘5 ‘5 a ‘ L' 3 i a; ij «.1

By straight forward algebraic manipulation on equa-
tions (2), (3), (4), and (5), one can arrive at a solution

for b ; it is

(5) b .3 Sum of products due to error ,
Sum of x squares due to error

where

30.

(7) Sum of products due to error =

———><2 —-——-—>( —--—-—=>

> -( 31;") -

This b is used for adjusting y for stand and has the

 

 

and

(8) Sum of x squares due to error =

(21* - 3-‘-‘-'7-) (L ~12

 

 

N

covariance due to treatment and replication removed from

it.

It is possible to solve for t , r , and m in terms

of b ; by doing so we get

and
Y ‘_ X _
(io)r=__'§_-y - __'_5_-x ,
n n
and

(11) m = ‘Y - b X ;
where i: is the general mean of the y's and ‘2 is the
general mean of the x's.
Substituting (9), (10), and (11) in (l), we get
(If. Yo' -) Xi. X4 —
(12) Y: ——+—--Y -b(—§—+T-X)+bxij+eii .

The standard error of estimate is given by

w a
(13) S ’ VD. of F. ’

 

31.

w = Sum of y squares due to error -
b (sum of products due to error)
where

(14) Sum of y squares due to error ::
1 L
:3." Zr. Y- 1
- .’ II - ..

D.OfF.=N-l-(n-l)-(r-l)-l.

 

 

 

and

Hence we have derived the experimental error which
gives rise to the analysis of covariance suggested by

*
Fisher.

 

*’ Fisher, R. A., Statistical M_§hggg for Research Workers,
Sixth Edition, pp. 275-90.

32.
Chapter 6.
Analysis of Covariance
Latin Square
Let us now consider an experiment to be analyzed by
the analysis of covariance which is set up in an n by n
Latin Square. Let x 2 stand and y a yield. Table 9

illustrates such an experiment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 9.
Columns I
ROW C, C" o o C“ P T013818
R T, Tz . . Th
e x,” x“, . . ng X”.
y,” yuz ' ' yam Yd.
R T" T, . . TM
2 x12» xu/ ° - Kan...) X4.
5'12» ya: I ‘ ' yum-z) Y. a .
T1 T3 0 e
" xlni xina - .
ylhi yawn ' '
' x x . . '
Totals "‘ 1"
Yd. Yt“ 0 o

 

 

 

The sum of the stand for plots with treatment k

will be denoted.tw' Iflk. denotes the yield of the

y‘jk
plot in the i'th column and the 3 th row, where treatment
k was applied.

Let c; be a factor concerning column 1 which affects

33.

yields;

rj be a factor concerning row 3 which affects
yields;

t“ be a factor concerning treatment k which

affects yields;
m be a constant;
and b be the regression coefficient between stand and

yield;

where Zci , Zr.

I
Then we have the following linear equation

, EZtk' are all equal to zero.

(1) yo." :3 c, + rj + t“ +m +b xi,“ +eijk .
The expression for the sum of the squares of the
residual errors is
F =Zea': 3’2"]qu - c,- 8- rJ- - tk- m - b x5501.
Setting the partial derivatives equal to zero and
simplifying, we have

(2)_%.E._ Yi. -nzci-nm-bx,‘“=0
‘ 0

H
O

‘0

BF .. 9-. .. .. .
(3) T;— Y°S~ nr“ nm bx.”

3F . ‘“ - 12 - .“ .
(4) btk , T's n tic n m 5'1““: 0 .
(5) 33-: Y... ~n‘m-bX... =- 0:
am ‘
(6) ..gg— ; té‘xy - nizcixi” - ninixg‘. - ngtkrf,‘k

34.
Again as in the preceding chapter, by a straight for-
ward algebraic manipulation on equations (2), (3), (4),

(5), and (6), we can arrive at a solution for b-; it is

I

(7) b ==.Sum of_products duerto error ,
Sum of squares of x s due to error

where

(8) Sum of products due to error 3

(Zn - 1.1;...) - (gxir-lfliu _ L9)

-(§x.3,x.3; - xmim) (Exit, 53-3., - xmx...)
"“1fi“"‘ "'THS"‘ "fi““'

 

 

n
and

(9) Sum of squares of x's for error =

(23*- x'tt)? (5—— - X5)

n N n N

 

 

 

 

We can solve equations (2), (3), (4), and (5) for
CI' r3 tK, and m respectively in terms of b as fol-

lows:

. X.
x. _ x . _
(11) r; =( 'i' - I) - b( 'i' - X)
n

(12) tk:(

 

 

.0

 

 

 

35.
and

(13) m=Y-b3'<;.
- 4 Equation (1) becomes, after substitution of values

given in (7): (10’s (11): (12), 3nd (13) 3

 

 

Yi' Y'S' E‘ _
(14) yak: n' + n +—-5n - ‘21

 

 

Xi.“ X.3_ if, _
-bn +11 ‘,_fi.5,u2x +bxisk+eilk.

The standard error of estimate is given by

(15) 3 “VD. of, F.

where
w .1: Sum of squares of y's for error

e b (Sum of products for error) ,

where Sum of products for error is given by (8) and the
Sum of squares of y's for error is the same as (9) except
that x is replaced by y ;
and
D.‘ofF. =N-l-3(n-l)-l.

Hence we have derived the experimental error which
gives rise to the analysis of variance for a Latin Square

eXperiment.

36.
Chapter 7.
Incomplete Blocks
If the number of varieties being tested is very large,
it often becomes impossible to use the complete randomized
block layout because of the amount of field space needed.

(I4) (10)
Yates and Weiss and Cox

have develOped and extended
methods by which a large number of varieties can be tested
economically. Two designs which are useful in this respect
are: l. Incomplete blocks, balanced or unbalanced, and 2.
Lattice Squares or Quasi-Latin Squares.

The requirement of the balanced incomplete block de-
sign is that every variety occur with every other variety
in the same number of blocks._ Since the number of replica-
tions and block size must be kept within practical limits,
it is possible to arrange such designs for only specific
numbers of varieties. If n is the number of varieties,

s is the number of varieties in each block, r is the
number of replications of each variety, N or nr or sp
is the total number of items, and w is the number of
times two varieties appear together, it is seen that certain
relationships must exist between these numbers, since they
must all be integers. Two of these relationships are

l. (n - l) w = r (s - l) ,

2. p=uo

For any value of n and s , there do exist values of

w and r which do satisfy the first relation, but it is

desirable to keep the number of blocks (p) to a small

37.
number or all benefit of such a design is lost.
There are simple devices for deriving the combinations

(’4)

which are possible, yet which retain balance. Yates

w‘ ) gives some of these arrangements.

Structure of balanced incomplete block arrangements is dis-

(ߴ), illlllllllllllllll|( ),

cussed thoroughly by Yates
and binoulden..

Let us consider in connection with the derivation of
the experimental error used in the analysis of variande,
the special case in which n 2 7 and s = 3 . By carrying
through this problem, yet keeping in mind the general case,
we will arrive at a result which will be applicable to any
eXperiment set up in balanced incomplete blocks.

Let the blocks be set up as indicated in Table 10.

 

 

 

 

 

 

 

 

 

 

 

 

Table 10.

Blocks ‘Total
B I Y” Y“ 3,, E,
B 2. ya. yea-z 5'52. E).
Ba 3713 3'53 37.; 73-3
34- y1¥ y4+ ye$ Eu
5: er ya" 3’7: gr
Ba 5'36 3751. y“ E‘
B"! 5’37 374-7 5'77 E7

 

 

 

 

In this figure, is the yield of the i th variety in

5'55

 

* Goulden, C. H., Methods 9; Statistical Analysis, pp.
175-202.

38.
3 th block. '8; denotes the sum of the yields of the

plots in block 3 . ’7; denotes the sum of the yields of

variety i .
This is a balanced incomplete block design because
variety one occurs in a block with each other variety once

and only once. Similarly for each other variety.

Let v; be a factor concerning variety 1 which affects
yields ,-
b' be a factor concerning block 3 which affects

I
yields ,

and m be a constant ,

Xvi-=0; zbj-so.
l. J.

where

We will assume that the v's, b's, and m are coeffi-

cients in the following linear equation:

(1) ,yUV = VIE, ‘i’ 000 +VnEn * b'F‘ + 000 1’ bPFP + m G 4- 80V,

where E", F,, and G are variables which take on the fol-
lowing values for ya} :

Eval, whenu=i;E‘=0, whenufi;

F's-.1, whenVI-J ;Fv=0,whenv#i;

G :1, for all u and v.

Then (1) reduces to

(2) y£j=vi+bj+m+e£3.

1 2..
Let F: Ze =29)“ -v-L-b.)-m) .

39.
Setting the partial derivatives equal to zero and

simplifying, we have :

Va
(3)_3.F.'_.° V2-rv.-Zb3-rm=0;

 

av,’ '2 t
B.‘
(40%; EJ'ZV1'5b3'5m=O3
i
(5) 35; YH-Nmzo.

Directly from equation (5) we see that
(6) m=§o

Now considering the special case indicated above,
where n = 7 and s = 3 , and writing out the equations

indicated by (3) and (4) , we have

(a) V,-rv,-b‘-bz-b3-y=0,
(b) Va-rvz-b,-b,,-b5.-§'=o,
(c) Vz-rva-b,-b6-bq-y7:0,
(d) V,-rv,-~bz-b,,-b.,-§'=o,
(e) V5-rv5-b2-by-b‘-y=o,
(f) V6‘rv5’b3'b4'bc'§=0.
(s) Yq-rV1-b3-bg-b7-yzo,
(h) -B',-v,-va-v:-sb,-y=0,
(i) I5,-v,-v4-v;-sb,_-y=0,
(3)33-V,-v‘-v1-sb-§=o,
(k) Sa-vz-v -v6-sb4_-y‘=0,
(l) 'B'r-va-vf-vq-sbs-yso,
(m) §‘-V3-v5.-v6-sb-y'=o,
(n) S7-v -v*-v1-sb1-§=O.

 

40.
These, together with (5) are called the normal equations.
Subtracting [(h) + (i) + (3)] / s from (a) and

simplifying (making use of the fact that 2"; = O) we
6

have VI
Z'B“
or

I
-Z§ -v'hs-r+1)=o.

Solving this equation for v, we have

 

I
(7) v 1: B V, 'JZ:B3 3
' rs - r +'l

in general this becomes

(8) V'=Q;NE::%)).‘

where
(9) Qt 3 8 Vi “:E- o I

Substituting the value for vc given in (8) into

(4) and solving for bj , we have

C”

 

' i
10 be 3 3 . ( " 1 - ‘I' Y“ .
( ) J E— Ns?s - I) ZQ" N

Equation (2) now becomes

i5 ( L _( )Bj
..- o n‘1 - 11-1 . ,. .
(11) yu" s +Q‘N (ed-l) Ns(s-l Qt+e"5

If an equation is of the form (1) , the standard

error of estimate is given by

41.

 

 

‘12) s =VBT£T
where '
Y "' Z'BVL y 1
= 1-- 00 _ l _ oo _ ( -1) 1
W (Z)! T) (CL—5— T) (Ns?s -1 Q‘)’
and

D.0fF.=N-l-(p-l)-,-(n-l).
Thus we have derived the experimental error used in
the analysis of variance of an incomplete block design.

This gives rise to the analysis of variance given by

 

 

 

 

 

 

Yates(’4). The analysis of variance is illustrated in
Table 110
Table 11
Source of Variation Degrees of ‘ Sums of Squares
Freedom
Total N 1 Zy" Y"- (1)
‘ t
{Sf Y0”
Between Blocks p - l s - 'fi” (11)
Between Varieties n - 1 n " 1 ZQ-zuii)
Nels-l) L
Error ° N-p-n+1 (1) - (11) - (111)

 

 

 

This method of balanced incomplete blocks works very

well in the case where the number of varieties is equal to

at or sz'- 5 r 1 where s is the number of items in a

block. Goulden' discusses in great detail these two Spe-

cial cases.
a Goulden, C.H., Method 2; Statistical Analysis, 3» 175*
202.

 

42.
Chapter 8.
Unbalanced Incomplete Blocks
The problem of incomplete blocks becomes much more
complicated if the blocks are not balanced; that is, if
each variety does not appear with every other one the same
number of times. Goulden“ discusses a simple casetof this
kind in which the number of varieties is equal to pz'.
If the varieties are arranged in the form of a square,
the blocks may be set up by first considering the rows as.
blocks and then the columns as blocks. For instance, con-
sider the following nine varieties arranged in the form
of a square:
ll l2 13
21 22 23
31 32 33

The blocks which may be set up as indicated above are:

 

 

 

 

 

 

Group X Group I
11 12 13 11 21 31 '
21 22 23 12 22 32
31 32 33 13 23 33

 

 

These blocks are unbalanced because variety 21 ap-
pears only with 11 , 31 , 22 , and 23 . ~It does not ap-
pear with the other four varieties. Similarly for each
other variety.

The groups may be replicated as many times as needed

to make the desired number of replications.
* Goulden, C. H., Methods of Statistical Analysis,pp.l79-
185.

 

43.

Let p be the number of items in each block, then
pa equals the number of varieties, 2 np equals the num-
ber of blocks, 2 equals the number of groups, and n is
the number of times each group is replicated.

Then we can set up summary tables in which each xcf
and yis is the sum of n yields of variety 13 in the

n replications of group X and group Y reSpectively, and

 

 

 

 

 

 

 

 

 

 

 

 

 

Group X Total Group Y Total

B'K X” x|1 XI} XI. B“! y” Y1‘ Ya. Yo'
BM x9.) x 11 xat xl- B2. yn. 3'12. 3(3ij Y1?-
Bsx 1:" 3‘12. x." X8- B3 y|3 ya: 3’12. Y-s
Group Total X.. Group Total ‘Y..

Variety Summary

 

Tu Tu. Tu

 

T2» T 12. T13

 

Tu T 32. T33

 

The grand total X . + Y” = T

It is to be noted in this case that the first sub-
script of the variety indicates the block in the X-group
to which it belongs, and the second subscript indicates
the block in the Y-group.

Let v€j be the factor concerning variety 13 which
affects yields;

b. and b1 be factors concerning blocks ix and

1x 5%

3y respectively which affect yields;

44.
s and s be factors concerning group X and group
Y reSpectively which affect yields;
m be a constant ;

Where {3&5 = o ; gb‘o‘+iZ'bJ%-.c; sx+ 8}: o .
Instead of one equation as in previous chapters, we

will use the following two for the yields of the X group

and Y group reapectively:

x“ =yi-‘ *bix +s‘+m+ eijx ,

ya}. = V13 +b3'1 +s3+m *6“: .

(1)

For this experiment the sum of the squares of the
residual errors is:

2. ‘2.
(2) F=Ze =§(x;£ -v€i -bix-8,"m)

, 1
+2 ,. - .. a b. n a .
aw“ v” ‘3 a} m)

Setting the partial derivatives equal to zero and

simplifying, we have:

 

(3).-o-?,—f}i-; T‘s-2nv‘5~nch-nb’3-2nmzo;

31“ Bin
(4) 5«6.5:»; XhunZvi‘i-npr‘---npsa npmzo;

B.

3F 0 . .. ,. .- b. c- .- = '
(5) .55.;- a Y.) n V‘s hp ’3 np 83 Hp :11 O ,
(6) Z: ; X..-np"s,-np”m=0;

I

(7) ‘BF ; Y,. - n p1 s} - n p1m =.O ;

 

45.

(8)—1L3T ~2np"m=0.

am .0

Solving (8) for m , recalling that N = 2 n pl,
we get
(9) m = M .
where M is the general mean.

Substituting this value for m in (6) and (7)

 

and solving for s, and 3‘ reapectively, we get
(10) 3‘: 1‘15;

n P
and

(11) B} fi‘m o

By straight forward algebraic manipulation the values
of the other factors are determined to be:

X‘.o-Yio-2np8

 

 

 

(12) bus __ ;
n p '
Y . - X . u 2 n p B
(13) b5 8 .5 .5 1 3
. 1 n p
(14) v” = p T‘s - (Xi..- - Xi.) O (Y's - X03) 6 2 n p M .
‘1 2 n p

By substituting the values of equations (9), (10),
.(ll), (12), (13), and (14) in equations (1), we get the

following equations for predicting the yields x--

‘1 and yij

on the average:

46.

 

 

pT“ +(X-,-Y',)-(Y.- ~X.°) X.
(15) xi. = ‘3 " ‘ ‘ 3 - -:-i +M ’
_ , J 2 n p up
and
(16) 3"“ = p T43 - (X... - Yi') + (Y '3 - X4.) - Y’; +M o
‘3 2 n p up

The standard error of estimate is given by

<17) 8 2 W
D. 0f Fe ,

where

 

w - Total Sum of Squares - Sum of Squares due to varieties

- Sum of Squares due to blocks ,

 

 

 

 

 

where ( 1 )1
a. z. X..'.+ Y”
(18) Total Sum of Squares 32x” 4- Eyes . ;
N
and
(19) Sum of Squares due to Varieties =
I. 2. 1.
To. (X' - Y. ) (X ’ - Y 0)
z t! * Z 0 " + z 0’ ..
2n 2 hp 2 up
1. 1. 1.
(X.. C Y..) (in. +ZY°S )
2 n p" np

and

(20) Sum of Squares due to Blocks ;_-

1
2": *ZYNL T..

1

 

np I;
and

1 2
D.ofF.=2np-(2np-l)-(p -l)-l.

Thus we have derived the eXperimental error used in

47.
the analysis of variance of an unbalanced incomplete block
design. Equation (17) gives rise to the analysis of var-

a
iance which Goulden discusses.

 

* Goulden, C. H., Methods 2; Statistical Analysis, p. 180.

48.
Chapter 9.
Youden's Square
W. J. Youden(’7) has modified the method of incom-
plete blocks in order to eliminate variations due to
replications from the error term. Consider the arrange-
ment in Table 12 in which there are seven treatments (A
through G) and three plots in each block, the blocks being
the vertical rows:
Table 12.

A B C D E F G

D E F G A B C

B C D E F G A
One will notice that there are three replications of each
treatment. The modification which Youden introduced was
that of placing the seven blocks side by side and arrang-
ing the treatments within the blocks in such a way that
each of the horizontal rows contains a complete replication
of the treatments. Table 12 shows this. Youden also sug-
gests that this can be done for various combinations of
treatments and blocks, when certain restrictions are
placed on the number of treatments and replications. It
will be noted that the number of incompleter(vertical)
blocks is equal to the number of treatments (n); that the
number of replications (s) of each treatment is equal to
the number of items in each block; that n and s are
connected by the relation

n-1=S(S-l);

49.
that ns =N is the total number of items .
Let yijk. be the yield of a certain plot, where the
plot appears in row i , block 3 , and treatment k .
Let r; be the factor concerning row i which affects
yields ,

b.

’ be the factor concerning block 3 which affects

yields ,
tk be the factor concerning treatment k which af-
fects yields ,

and m be a constant ,

where Zri20, ij=0,and it‘so.
i 'J- n

We will assume that the r's , b's , t's , and m

are coefficients in the following linear equation:
(1) vasgnirc 4- {$13,125 +¥Fkth +G m «mum,

where D“, EV, E, and G are variables which take on the

following values for Ytju‘

Du=1, whenu:i;D,,-O,whenu¢i

E,:1,whenvcj;E,=0,whenv¢1

‘0

F~=1,whenw=k;F.sO, whenwfk

G =1, for all u, v, and w .
Then (1) reduces to

(2)

yiik 3 r‘- + bi fit“?e£i|‘e
The sum of the squares of the residual errors is

Elven by

‘ 1
F329 ‘Z‘ngg ’ri 'bE-tk) °

50.
Setting the partial derivatives with reSpect to the

various constants equal to zero, we have

OF .
(3) W L"

+<
1
u
H
u
s
B
u
C)

I
m
0'

b

I

¢+
K

I
m
E
ll
0

éF .
“Wwfi'_y

F
(5) "Pa-f: 00K

H
I
m
d-
x
I
O’
h-
I
m
E
II
C)

.0

Y “NIH-‘30.

<5> €1-
B.
In (4’) Zit denotes the sum of the factors con-

cerning the treatments appearing in block 3 . In (5)

T
‘ifbi denotes the sum of the factors concerning the blocks
in which treatment k appears.
Directly from (6) we see that
(7) m =.§ 0
Substituting this value of m in (3) and solving for

r, , we get

 

(8) I": :1” -§ 0

Using a method similar to that used to find v; and

 

 

b3 in Chapter 7, we find
B.
-Yd.-:Q --
(9) b, - ...7?______ y ,
‘3 s(s -s+1)
and
(10) th" k ,

51.

TX
Qk=8 YOOK " 21.3. o

If we substitute the values of equations (7) through
(10) into equation (2), we get the following equations for

predicting y.

Lik on the average..

 

 

B:
~ Y. _ Y. Q Q
(11) Fisk-r. ‘" -y+_'_3;.- Z1); +__{__F_ -
8 s(s -s +1) 8 -s+1

The standard error of estimate is

(12’ S =Vsrl§fi7

where

‘1
1 90. 1 7-
' = O. - —_ - Q
zy‘l“ N s(sl- add); k

 

 

s N
and
D.ofF. =N-l-(s-l)-2(n-l).
Thus we have derived the experimental error used in
the analysis of variance of an experiment set up in a
Youden's Square. The analysis of variance which results
from this value of the experimental error is given in

Table 13.

52.
Table 13.

 

 

 

 

 

 

 

 

Source of Degrees of Sums of Squares
Variation ,Freedom 1
Total sn - 1 a. m1
zyc‘ik T (1)
2.
Zr; 1 '-
Between 3 - 1 .1____'.'._ - _.'__'..1_ (ii)
Rows n N
Between n - 1 Y};.L Y...L
Blocks 3 - N. (iii)
Between. _ n - l 2..
Treatments 1 23;“ (iv)
s(s -s l) “
Error sn-l-(s-l)-2(n-l) (i)-(ii)-(iii)-(iv)‘

 

 

 

53.
Chapter 10.
Lattice Squares
The second design for a large number of varieties
that was suggested previously is that of Lattice Squares.
For this design to be used it is necessary that the number
of varieties being tested be a perfect square. In Table

14 an example of such a design is given for 25 varieties.

Table 14.
12345 110141822 18151724
6 7 8 9 10 2O 24 3 ”7 11 19 21 3 10 12>
1112131415 23261519 25291118
1617181920 12162548 13202246
21 22 23 24 25 9 13 17 21 5 7 14 16 23 5

In this lattice square arrangement every pair of
varieties occurs together once only in either a row or a
column of any one of the squares. Also, every variety
occurs with every other variety once in one column and one

row from each square. Complete discussion of this example
(IO)

(

has been presented by Weiss and Cox Other examples

(’2)

have been discussed by Fisher and Yates ), and Yates i.

Let yiikf' be the yield of variety 3 in square i,
column k and row p . Let nz' be the number of varie-
ties and t the number of squares.

Let s; be a factor concerning square 1 which affects
yields,

v3 be a factor concerning variety 3 which affects

yields,

\—

54.
c be a factor concerning column k which affects
yields;
r be a factor concerning row p which affects
yields;

m be a constant;

£2"
ll

0 ; 2:v5 = O ;‘Z:ch_=.0 szizr,,=.0 .
J Si.

The yield y.

ijp is given by

(l) y s11- v3+ cx+rp+m+e£3KP,

0. 3
«pap
and the eXpression for the sum of the squares of the re-

sidual errors is

z
(2) F=Ze =Z(yi'3kp'9i."’3 -ck-r’-m) .

Setting the partial derivatives of F with reapect
to the various factors equal to zero, and simplifying,

gives the following equations:

”"2527331... ”“15; “n”!!! =0 3
V3 V3
OF ' . d . C n - ‘ '
(4) 79? ' Y.1°' t V) ch if? t: m — 0 ,
Ck
aF‘ ’ . Q a u —
(51-36:. Y...“ mstk 2v3 nck nm_O ,
R
.F . ff
(6) 75;, IN? as“ v -an-nm-0 ,
(7) .1flE. Y - t n1 m :10 ;

0’
B

55-
where 55,, is the factor concerning the square in which

column k occurs; similarly, s is the factor concerning

i
P
the square in which row p occurs. In equation (4)

Z:cK denotes the sum of the columns in which variety 3
V;

occurs, ander denotes the sum of the rows in which

variety J occurs. Inequation (5) :v’. denotes the

sum of the varieties occurring in column k . In equation

(6) E'vi denotes the sum of the varieties occurring in
row

Solving equation (7) for m results in
(8) m =:§' .

Substituting this value in (3) and solving for so :

 

< ) Y" ‘
9 s-:: " - y .
L n‘

We can now solve for the other factors in a manner

very similar to that in Chapter 7; when we do so, we get

Q; +tn t);
(10) viant-2t4-l

1 Q5+nt§. _]
a Y..k. ’ n Bik_- nt-2t-r1 n y '

 

 

 

(11) OK:

310, ..nt'y' ~..
12 ,1 .. . - _1 - "‘ .
( ) r9 n[Y'°'P n81. \nt-2t+l) ny]
where

56.

-The standard error of estimate is given by

_. W
(13) s - M

where

 

w 8 Total Sum of Squares - Sum of Squares due to squares
- SS due to columns in squares
- SS due to rows in squares - SS due to varieties ,5

where

9.
Y
(14) Total SS uzy1_ .N... ,
Y I Y ‘
(15) SS for squares 2 c h" .- .. .. ’
11‘ N

(16) 83 for Columns in Squares —

22.1.” I...) -(ELL: :___)

K - n
n N 11%) N

(17) SS for Rows in Squares =

I.
ZY ‘ 'Yoeoo‘ (ZYI.°” Yea-0‘)
...—2.1!... - ...—....— “ ..L...___.__. -___.__.._ 9

 

n N n‘ N
and (I: )‘
1 HQ-
'r - 1 —- o - —————’-— .
(18) SS for Varieties ._ n(nt-2t+l Z(nQ,’) n" 1,
and

D.ofF.=n"t-(t-l)-2(nt-t)-(nz-l)-l.
Thus we have derived the eXperimental error used in

the analysis of variance. This gives rise to the analysis

of variance for a Lattice Square which has been discussed

by Weiss and Cox(’o).

M 57.
Chapter 11.
Factorial Design

Let us consider an eXperiment in which we have three
kinds of fertilizer, a', b', c'. We may apply the ferti-
lizers one at a time, two at a time, or all together, so
that instead of three treatments we have eight, which may
be designated by

a'b'c', a'b', a'c', b'c', a', b', c', (l)
where (1) denotes the absence of each fertilizer and is
used as a control "treatment". In the field set-up we
would have blocks of land with eight plots in each, the
treatments scattered at random in each block, with the
one condition that each block contain all the treatments.

The analysis of variance could be considered as in
a randomized block experiment with eight treatments; how-
ever, the effect A of treatment a' can be found by compar-
ing the yields of all plots containing a', with or without
any other ingredient, with the yields of plots not con-
taining a' at all. A comparison may also be made of the
effect of a' in the presence of b' with that of a‘ in the
absence of b'. This effect we call AB, the simple inter-
action between a' and b'. Thus we can make seven differ-
ent comparisons:

A, B, C, AC, AB, BC, ABC.
The first three of these we shall call the main effects

and the remainder we shall call the interactions.

58.

We might also consider that the three fertilizers
are applied in two ways -- some fertilizer and no fertili-
zer; we will call these a", b,‘, c,’ and a0}, bo', co‘
respectively where the subscript 1 indicates the pre-
sence of the fertilizer and the subscript 0 indicates
its absence.

For simplicity we will let yijk: be the yield of the
plot in which a;, bj, c,‘ were applied; the subscripts of
the y indicate the levels on which the a, b, c, were
applied respectively. We will let the number of replica-
tions be four, and the fourth subscript on the y will
indicate the number of the replication.

Let r ,be the factor concerning block p which affects

P
yields ;
a6, a, be factors concerning main effect A which
affect yields ;
b‘, b, be factors concerning main effect B which
affect yields ;
co, c, be factors concerning main effect C which

affects yields ;
(ab) be the factor concerning simple interaction AB
which affects yields ;
(ac) be the factor concerning simple interaction AC
which affects yields ;
(be) be the factor concerning simple interaction BC

which affects yields ;

 

 

59. 1
(abc) be the factor concerning double interaction
ABC which affects yields ; J
and m be a constant.
We will assume that these factors are coefficients

in the following linear equation:

(1) yww“: aaDo + a'D' 4- boEo + b,El + °oFa + c,F,

-. ~4-—. ~1-q-———. vv—w- .
. ..a' . ..ana - .“ "
-
". w" a.
‘ a 1 __

+(ab)G + (ac)H + (bc)I + (abc)J + Zb,L,.
P

+mK +euvw1’

 

where D“, Ev, F”, G, H, I, J, L, and K are variables which
take on the following values for yt'fkf:
D, a 1, when u :1 ; D u 0 elsewhere ;

k

= 1, when v =3 ; E, = O elsewhere ;
= 1, when w = k ; F... = 0 elsewhere ;
:1, when uv a 11 or 00;

:e-l, when uv = 10 or 01 3

EV
F”
G
G
H 8 1, when uw :11 or 00;
H =-1, when uw =10 or 01 ;
I :1, when vw s 11 or 00 ;
I -'-'-1, when vw = 10 Or 01 ;
J = 1, when uvw slll, 100, 010, 001 ;
J =-1, when uvw =-llO, 101, 011, 000 3

K = l, for all values of u , v , and w 3

L} w 1, when 2 s p ; Lia. O elsewhere .
The values of the variables G, H, I, and J were determined

as follows:

- 60.
Goulden* states that "algebraically, all the treat-
ments can be represented as follows:
N =(N' - Na)(K, +- K,)(P, + 1;) .
p =(N,+N,)(x, . mm», -r,) ,
K,)(P, +Po) .
K,)(P, -P,) .
KO)(P, * Po) 9
K,)(P, ‘ Pa) .
N x? x K =(N, - N°)(K, - K0)(P' -Po) " .

K ::(N, + NO)(K,

...

N X P =(N. - NO)(KI
No)“:
P x K = (N, + N°)(K,

N x K =:.(N|

41v""ori._ f. '.

where N, P, and K are the three treatments correSpond-
ing to the A, B, and C in our experiment. 1 in the
subscript of the N, P, and K denotes presence of the
fertilizer, while 0 denotes absence.

We shall make use of N x P to determine the values
for G . This is the interaction which corresponds to A3,.
the factor concerning which is the coefficient in (1) .

In the notation of our experiment N x P is written:

A B =(a' - a°)(b, - b,)(c‘¢- °o) .

Expanding this we have :

A B : a,b‘c,-+ a'b'c - a,boc, - a’boc - aoblc,

O O

- aob,c° +-a.b°c,i+ a‘boco .
The coefficients of‘the terms containing a,b, and aoho
are ~tl while the coefficients of the terms containing
a,b° and aab, are - 1 . These are the two values of
G. H and I are determined similarly.

We shall make use of N x P x K to determine the

values of J . In our eXperiment this relation is written:
* Goulden, C. H., Methods of Statistical Analysis, p.161.

 

ov-. w"!

INI

61.
ABC :(a‘ - ao)(b' - b°)(c‘- co) .
Expanding this we have :

A B C = a'b'c‘ - a'blco - a'boc‘ + a.b.c° - aob‘c‘

+ aoblco+ aob¢c, - a°b°c° .

The coefficients of terms a,b,c,, a,b,c°, aob’co, and
aoboc, are +1 while the coefficients of terms a’bico,
a’boc', a‘b'c’, and aoboco are - 1 . These are the
two values of J .

Using the values of these variables (1) reduces to

the following equations, where each equation represents

four equations since p takes on values from one to four:

(2) yam" = a0 +- be «r c, + (ab) 4- (ac) + (be) - (abc)

+m +rP +e°°°P;

(3) yo =. a +b +c + (ab) - (ac) - (be) 1» (abc)

cup 0 o I

+m +r, e..";

(4) y"’P a an + b. «ca - (ab) +|(ac) - (be) + (abc)
+m er, + 90109 ;

(5) your - an «I» b, + c, - (ab) - (ac) «I» (be) - (abc)
+m +ro+ soup ;

(6) ym" - a. + be +00 - (ab) - (ac) + (bc) 4- (abc)
+m+r' +e‘”p;

(7) y'°'P - a. + ba «0- cI - (ab) 4- (ac) - (bc) - (abc)
+m +r', +em-‘P;

(8) y”OP - a' + b, +'c° 4-(ab) - (ac) - (bc) - (abc)

+m 4-r +e
P nap ’

62.

(9) ylup = a'q- b. + c. + (ab) +(ac) +(bc) + (abc)

+m+rP+GHIP 0

fl

_ 2.
Now let F- Ze 1351:?

Taking the partial derivatives and simplifying we

 

have
3F . , _ _ _. .
(10) 7A.: , Yb” 16 a‘. 16 m .. 0 ,
BF . . _ ... m _. .
(11) W , Y.’.. 16 b, 16 I- O 3
OF . .. - ..
(12) .55... , Y...“ 16 cK 16 m .. O ,
K
OF 0 u. .- - °
(13) Or, , Y”)? 8 rP 8 m .. O ,

(14) 31‘" ° (AB) - 32 (ab)= o .

a(ab)'
where (AB) 8 Yoga. + Yool.+ Yuo- +Ym. " You).
' Yon. " Yugo. " 1161-3

(15) {go—7: (AC) - 32 (ac) = o .

where (AC) = Yo,” 4-ng 4' Ynal. ‘* YIN. " You.

-Y9II0 ..Y

(16) 3%; (BC) - 32 (be) = o .

‘1'

I00- Ho. '

where (BC)=Y°°°.+YOH,+Y 4'! mywe].

I00, III.

- YOla. - Yml. " Yno. 3

DP . . ..

where (ABC) = Yool. +1010. 4- Ym.’ + Y”). " Y 00..

" Yon. " YIsl. - YIla. 3

I I :3 L :i'i . 1.1.... .‘

63.
~32m30.

(18) —%—%—3 I
From (18) we have immediately
(19) 131:? .
. Solving the remaining equations for the factors de-

sired, making use of (19), we have :

 

Y‘... "" a;
(20) a; =_l_6_:. y ; ‘.
and
3.3.. .-
(21) bi '3 T " y 3 i
and g
Yo-K- _ L.»
(22) OK: Tg— " y 9
and
YOQOP "
(23) r? = T - y 9
and
(24) (ab) = 93131;
and
(25) (ac): Mil ;
32
and
(26) (be): 1%;
and 2
(27) (abc) c 1%539-1 ';

The standard error of estimate is given by

(28’ S 51/573233? '

D.ofF.=31-(4-1)-(8-1);
and

 

- :6; ""- 5;?)me - 1%?(AB)- lgglhm)

 

O... ’

Y ” Y 1

:(Zy ---'s:e' )(§—e—“" -—--'s‘2‘)
{(Zyi...‘ Inn‘) (ZY.:.." Yooc.‘)
“"‘6‘1 "" ' 32 + J""1'6' '- " "3'2""

1.
is 32 32 32

- 13.0.).(30) - .L__1A30(Asc)- 3' Y
32 32

 

 

,, (ABC) ]
32

Total Sum of Squares - Sum of Squares due to Blocks
- Sum of Squares due to Treatments ,

where

65.
Sum of Squares due to Treatments :
SS due to A + SS due to B + SS due to C
+ SS due to AB +-SS due to AC +-SS due to BC
4 SS due to ABC

:88 due to main effects
4-SS due to simple interactions

4-SS due to double interactions .

Thus we have derived the experimental error which
gives rise to the analysis of variance of a factorial
experiment. This analysis of variance has been discussed

,
by Yates(’5), Fisher , and Rider** .

 

* Fisher, R. A.. The Design 2: Experiments .

** Rider, P. R., An Introduction to Modern statistical
Methods, pp. 1813-1 2).

 

 

66.
Chapter 12.
Confounded (3x2x2)

Confounding is a method whereby the necessity of
including every treatment combination of a factorial ex-
periment in each block is avoided. This has been dis-
cussed in some detail by Yates(A’). By use of this method
block size may be kept small even though the number of
treatment combinations is large.

The combinations of treatments are divided into two

(or more) groups so that the contrasts between the differ- é

 

ent groups represent some interaction of higher order
which is not important in the experiment. In the analysis
of variance, information concerning the interaction which
is "confounded" is lost but this loss is compensated for
in greater accuracy in other comparisons. When an inter-
action is confounded in only several of the replications,
we say that it is partially confounded.

Let us consider a 3 x 2 x 2 emperiment in which
treatment a is applied in three amounts: 2, 1, 0;
treatments b and c in two amounts: 1, O . This ex-
periment is one which has been discussed by Yates“. In-
teractions BC and ABC are partially confounded with
block differences. Table 14 shows the design of this

experiment.

 

* Iates, F., The Desi n and Analxgis g; Factorial EXper-
iments, pp. 55- l.

67.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 14.

'Blocks
. i" '1', j‘fi', n, ”n“ Tfi'gfl'
you ycan yooo yaw yooo y 00/
yoga you you you: you yalo
ywo ylo: yin! yloo y/oo y/O/ “
ym yllo yua y”, y”, 3’1“) i F
yto. y“, y.‘wo 5'10! 5’20] yLOQ
ya." yalo Lyn: yaw yzlo ya”

F

 

Yyyw denotes the yield of a plot on which a was

applied on the u level, b on the v level, c on the
w level. ‘ I

We will let the block totals be [Ia]. [15] , etc.,
and [Lib] - [1,] =3' , {[115] - [11.3} :51 .
{[IIIJ - [IIIJ .2 33. This is the notation which is used
by Yates.

Let the factors which affect yield be
for main effect A: a0, a“ at;

B: b0, b, 3

Czc c

O, I ;

simple interaction
AB: (3b).: (8.13),, (ab): 3
AC: (ac)‘, (8.0),, (so),- 3

BC: (bc) ;

 

68.
double interaction
ABC: (abc)o, (abc)’, (abc)z;
I

replications: I II“, IIb’ III“! IIIb 3

c’, b!

constant: m ;
where the sum of the factors concerning treatments equals
zero and the sum of the factors concerning blocks or
replications equals zero.

In the factors concerning the simple interaction AB,
(ab); denotes a contrast between ai in the presence of
b and a; in the basence of b . Similarly for the
factors concerning AC and ABC.

We have the following linear equation for y :
(l) yum” a ga‘D‘w- gb'E' d- E'CVF‘V + §(ab)mGM

+ Z“(ac)‘Hu+'. '(bc) J + ;(abc)KKu

+ IIIbL” +- m P + em”, .

where the variables take on the following values for

yéjk :-
Du¢1,whenu=i;D“=O,whenufi;
Evsl,whenv=j ;EV=O,whenv¢j;
F.=l,whenWsk;F'«-o,whenw#k;
G. :1 , when uv=il , -l,when uv =10 , Oelsewhere;
H” 3.1 , when uw = 11 , -l,when uw 3.10 , O elsewhere ;
J = l , when vw =»ll and CO , -1 when vw =.Ol and lo ;

 

69.
K“ s l , when uvw s 100 and ill, --1 when uvw = 110 and
101, O elsewhere ;

I. . l , when y appears in I... O elsewhere ;

(a
similarly for L“, La... LIV Lk' Lg”;
P = 1 everywhere.
These values for the variables were determined in a
manner similar to that in the chapter on factorial deSign.
Using these values for the variables we can get an
expression for yijk in general; this is quite complicated

to write down in the general case. However, in a special

case, say for y,” appearing in block II“, we would hage

(2) y = a'+ b' + cl +(ab)t + (ac)I + (bc) +(abc).

«b II‘+ m + smut.)

Again we will let F equal the sum of squares of the

residual errors. Setting the partial derivatives of F

with reapect to the various factors equal to zero and simrifzy

plifying, we have the following equations:

(3) 2;: Yin-12aa-l2mzo;

(4) b5: 1,5.-18b5-18m:0;

(5) 0., : I”: - 18 c‘.-r 18 m = O;

(6) (ab); : Y“. - Y8”, - 12 (ab) = 0 3

(7) (ac); : Y,” - Y,',, - 12 (ac) ‘5 O;

(8) (be) : (BM-36 (be)+26,+251+2s,=0.

"' you: " you " ylol " yua " yaw " 35.10;

 

- ~ m 44-1.1...
I, ,

 

70.
(9) (abc), 3 (ABC)o - 12 (bc)

l2 (abc)° - 2 g.
'0' 2 51+ 2 53 = O ,
where (ABC)O = y”.+y°” - you - You 3

(io) (abc), : (ABC), - 12 (be)

12 (abc), + 2 g,
- 2 2,3 2 2, =0.
where (ABC), nyna’ y,,, “if/,0 'Y/ol 3

(11) (abc)a .: (ABC); - 12 (be)

12 (abc)L-v-2 g,

‘3‘"
.I‘.

+2gz-2

P
0‘2

0

I
0

where (ABC)a.'-’yaao+ygu ’Y‘,."yaol 5

 

r".- =:..

(12) I‘: 2 (be) - 2 (abc)° + 2 (abc). +2 (abc)z
* 6 In.“ [101 :03
(13) It : - 2 (be) +2 (abc)o - 2 (abc)I - 2 (abc);

... 5'11; - [1.] =0 3

(14) II : 2 (be) + 2 (abc). - 2 (abc)‘ «l- 2 (abc):
+6 IIa~[IIa_]=O;

(15) 115: - 2 (be) - 2 (abc)o 4-2 (abc), - 2 (abc):
+ 6 11b - [11.] :0;

(16) III‘: 2 (bc)+ 2 (abc)o +2 (abc), - 2 (abc)1
+6 III,— [IIIJ = o;

h)

(17) III : ~ 2 (be) -

5 (abc)° - 2 (abc), 4-2 (abc):

+6 IIIb- [1115] =0;
(18) m :Y.”-35m::0,
from which we see directly that
(19) m='y' .
Knowing this value of m we can solve for the un-

known factors in equations (3) to (8); the results are

71.

Y‘... _
yo. 0 ‘-
b g ‘ ' y 3

(ab);

 

12

i./ ' Yiko
12

(&C)£ 9

 

we may solve for (be) by making the following
combination of equations:

(12) - (13) * (14) - (15) + (16) - (l7) 1'3 (8) .
making use of the values of g., 52’ g’.. Hence we get:
(21)

_ BC - 1
(be) .1—1 56(8'+61+86) .

By a similar process we get:

(22) (abc), 5321305 .52. + 6,323,) - gen).

(2}) (abc),

Egums), +552, - a, +23)-5 gum) .

(24) (abc)2

gem), .2134 s, + e, - s3) - gum) .

We will solve for the factors concerning replications

in terms of (be), (abc)o, (abc)“ and (abc)7_ :

I .
(23). 142%:1-é6m)- (abc)o+ (abc), 4- (abc),] ,

 

V .“-._ ' ‘ ." ‘_'__A__.IL l'u"! '_‘__-_._z'u 14.-5.:

__ :_

72.

LI.]

15 "-E__
II‘:.C_6_I- jibe) 4- (abc)° - (abc)| i- (abc)z],

=[IIB__6_1-%

IIIt =T - 3 be) +(abe)o + (abc) l - (abc)1] ,

[IIIJ I
IIIb = T- -’-(bc) - (abc)o - (abc). +(abe)1 .

The standard error of estimate is given by

w
(24) S g W/ID: of F. '

where ‘

-%Ebc) + (abe)° - (abc). - (abc)1]

IIb be) -(abc)o +-(abc)| - (abc)1],

 

EY; y ‘
= T013 1 SS - Block SS - H _ no
a 12 T3

235.3} Y...‘ {Ynez Y”)
”FTP’T) '(J'Ia— " 3 )

_(Z(Y‘-h ' Yio-f') - (Zach - 15.0 )1)
l2 12

D
- bC t . 2(8130 )t. o
28% 60
and

Do or F0 :36 " 5 " 11".

 

This is essentially the analysis of variance described
a

by Yates, and thus we have derived the experimental error
for a confounded eXperiment.

* 'Yates, F., The Design and Analysis 9: Factorial EXperi-
ments, pp. 58-60.

 

 

"T§vfi.7fi... l.‘~' ..." 1*
I

73.
Chapter 1}.
Estimation of Missing Plot Values
Sometimes when an experiment is completed, it is
seen that a value of the yield in one of the plots is
missing. As it would be quite wasteful to discard the
entire experiment for this reason, it is better to make
an estimation of the value and carry out the analysis of
variance using this new value. However, when such an
estimation is made, the degrees of freedom for the total
must be decreased by the number of estimated values.
Allen and‘Wishart ( I) and Baten (1’) have derived
formulas for estimating plot values. I am going to show
a way, different from theirs, to arrive at the same for-
mulas, and to suggest how it can be extended to any rea-
sonable number of missing plots.
Let us considers randomized block eXperiment (see
Table l) in which the yield in plot uv is missing.
We wish to make an estimation of this value.

From equation (1.15) we know

Y' Yo
".....f; .19..."
(1) y‘s-rI-n 3!.

which is on the average the "best" value of the yield

Y

(J '
In the case when a missing plot occurs in treatment
u and block v , 1,“. is the sum of the value of the y
yield of the missing plot and the sum of the known yields
in that treatment; we will denote this sum of known yields

by T9 . Similarly, Y.v is the sum of the value of the

‘n “‘3

 

r fiz‘xw.
j

74.
yield of the missing plot and the sum of the known yields
in block v which we shall denote by Ev). Y,. is the
sum of the value of the yield of the missing plot and the
sum of the known yields, which we shall denote by T (no
subscript).

Substituting these sums in (l) we have
+ B

yvv +'TU yov v yvv
_____.____+_________-
r n N

4-T

 

(2) y.v =

Solving this equation for’yvv, we have
(3) y g n.Tu,-h r Bv - T .
0V ’Tn-l)(r-l)

This is essentially the same as that given in the liter-

 

ature cited.

If there are two missing values occuring, we shall
assume in two different treatments and two different
blocks, the problem becomes one of solving the two fol-

lowing simultaneous equations for yuv and y3":

 

 

 

 

(4) y -— -——————y"" *T" +____y”"*B" - y“ ”7" ”l .
UV ' r n N
'yfl-O-T‘ y51+Bt yuvq-y‘cd'l‘ .
This gives:
(6) y" s T“ 1)? ”if; [(r-l)(n-l)(n TV .1» Bv- T)
O r- O

- (n Tsi-r B‘s-'13)] ,

 

 

7S.
and ySt is a similar expression.

The reader will readily see that it is possible to
solve for any reasonable number of missing plots, no
matter where they appear in the layout.

In the case of a Latin Square, equation (4.12) is
used. To illustrate this problem let us assume that the
values of ym,W and y“:W are missing. The values are
in different rows and columns, but they are in the same

treatment. The resulting simultaneous equations are:

 

 

 

 

_ yvvw+ystw*Tw+ yuvw*Ru ... yuvw"0v
yWW” n n n

(7)

 

_ 2 (yovw+ysew+T
N

and

_yuvw*y:tw +Tw+ yet-w *RJ' yttw"'ct
sGtmr' :1 n n

(8)

 

 

 

_ 2 (yuvw+:8tw*‘r) ,

\Nhere TV." is the sum of the known yields in treatment w ;
130 is the sum of the known yields in row u ; CV, is the
sum of the known yields in column v 3 and T is the grand
‘total of the known yields.

YIhe solution for yUVVV is :

76.

 

(9) yuvw = (n-2)[:(n-1)1-41T [(n-1)(n Tw-vn C's-n R, - 2 T)

+ (n Tad-n Ct-I-n R5 - 2 T)] ,

and yStw is a similar eXpression.

Equation (5.12) would be used to estimate the value of
missing plots in the analysis of covariance for randomized
blocks. Similarly, equations (3.17), (6.14), (7.11), (0.15),

and (9.11) may be used in estimating missing plot values for

the various experimental designs.

 

‘ 11.. u 1.3.35 ‘9-

'Im - a

1.

2.

3.

5.

7.

8.

9.

10.

77.
Bibliography
Allen, F. E. and Wishart, J. A method 2: estimating

the yield 9: a missing plot in field experiment-

 

2; mg. Jour. Agr. Sci., 20:399-406. 1930.
Baten, W. D. Formulas £23 finding estimates for two
and three missing_plots in randomized block lay-
ggtg. Mich. State College, Tech. Bul. 165. Apr.
1939. '
Fisher, R. A. Statistical methods for research work-
222° Oliver and Boyd, Londen. 6th Ed. 1936.
Fisher, R. A. The design 2; experiments. Oliver and
Boyd, Edinburgh. 2nd Ed. 1937.
Fisher, R. A. and Mackenzie, W. A. Studies in 3322

variation, I}. The manurial response 9: differ-

 

ent potato varieties. Jour. Agr. Sei., 13:311.
1923.

Goulden, C. H. Methods 2; Statistical analysis.
John Wiley and Sons, New York. 1939.

Goulden, C. H. Modern methods for testing a large
number 92 varieties. Dominion of Canada, Dept.
of Agr., Tech. Bul. 9. 1937.

Rider, R. R. An introduction 29 modern statistical
mgthgdg. John Wiley and Sons, New York. 1939.

Snedecor, G. W. Statistical methods. Collegiate
Press, Ames, Iowa. 1938.

Weiss, M. G. and Cox, G. M. Balanced incomplete

I

11.

12.

13.

14.

15.

16.

17.

78.
block and lattice square designs for testing
yield differences among large numbers 2: 221;
bean varieties. Iowa State College, Research
Bul. 257. Apr. 1939.

Yates, F. A further note 23 the arrangement 2:

variety trials: Qpasi-Latin squares. Ann. of

Eug., 73319-332- 1935-7.

Yates, F. A new method 2: arranging variety trials

involving_a large number 2; varieties. Jour.
Agr. Sei., 26:424-55. 1936.

Yates, F. lggompletg Latin squares. Jour. Agr.
Sci,, 26:301-15. 1936.

Yates, F. Igcomplete randomized blocks. Ann. of
Eug., 7:121-140. 1936-7.

Yates, F. The design and analysis 2; factorial

 

experiments.

Yates, F. The recovery 9: interblock information
in incomplete blocks and quasi-factorial g3-
signs. Ann. of Bug. 1939.

Youden, W. J. Qg3_g§ incomplete block replications
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search, 9:41-48. 1937-8.

0.
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