Reproduced by Sabinet Gateway under licence granted by the Publisher (dated 2011)
17
South African Statist. J. (1994) 28, 17":'21
MULTIPLE COMPARISON,S IN A MIXED
TWO-WAY'MODEL
Andres A. ManiJJ:
Catedra de' Bioestadistica, . Facultad de Medicina::
Universidad de Granada, 18071 Granada,SPAIN. . .
Keywords:
Dunnett's method;
intraclass' correlation model;
mixed two-way .. model; ' ..• multiple . coDiparison;
Scheffe's method; Tukey's method.
Srunmacy:
This article presents an alteriui.tive andjlinipler proof
of the results of Bhargava and Srivastava (1973) and of
Hochberg and Tamhane (1983) on the validity of the
methods of multiple comparisons (T~ey, Dunnett and
Seheffe) in the mixed two-way model. This paper
gives a shorter and easier alternative proof of those
results, which may also be applied to more general
designs. .
.
1.
INTRODUCTION
! .
Graybill (1961, chapter 18) sets out a mixed two-wa.y model in which ea.ch·
observation ~jh = 1£ + ~ + bj + eij + eijh is. the sum ofa general- m~an (1'),
plus the fixed effect (a.)
of each row, plus the random effect (b:)
of each
1
J
.
column, plus the random interaction (c.. ), and of the random error (e..h),
~
~
with i = 1,2, ... , ri j = 1, 2, ... , t; h:: 1; 2, ... , n. Iii this model the variables
e"h '" N(O, i), b. to N(Oi (1B2) . and c... to N(O, . (r-l)(1A2B/r) are all
IJ
J
IJ
independen t from each other; for .any .index 'wi th the exception of Cov(c. .,
2
.
-
: ' ..
IJ
c.,.)
::: -aAuir. Then it follows that the
mean vector
of fixed treatments
1J
.
_.
.
(x!.., x 2..' ... ,
xr)
with means E(iJ
is distributed as an r-dimensionalnormal distribution
= 1-£ + ~,
variances V(xJ::: {ra 2 + nrcl~ +li(r-l)a1B}
i
rtn, and covariances Cov(xi.. , ii') =(ra;- (11B)/rt =C. This is eqUivalent
to what Scheffe (1956) caJled the symmetric model, and it is known in the
STMA 1994 SUBJECT CLASSIFICATION: 08:090(08:030)
Reproduced by Sabinet Gateway under licence granted by the Publisher (dated 2011)
18
literature as the intraclass correlation model. Bhargava a,nd Srivastava
(1973) proved that the classical methods of multiple comparisons apply to
thismodel-for- the case n = Ii Hochberg and Tamhane (1983) extended the
result to the casen·> 1. This paper gives a shorter and easier alternative
proof of those results, which may also be applied to more general designs.
2.
THE BASIC THEOREM
'.!-'BEOIlEII: Let (XI'~' ""~) be a random vector normally distributed
With covariance matrix R given
by
R
.. = V and. R..IJ = C (i # j). Then,
it
.
_
.u .
.
is always possible to define a new vector ,(Yl' Y2' .... , y,,) which is normally
distributed with E(y.)
= E(x.),V(y.)· == V - C and Cov(y.,y.) = 0 (i # j).
1
1
1.
1
J
PROOF: Let us consider
y. = x. + x,
is a normal random variable
with E(x) = 0 and Vex) apd Cov(x,~) have yet to be' fixed .. Then
E(y.)
= E(x.),
and:. .
1
I
-, "
I
1
where
X
(a), If C < 0, by taking V(x) = ;"C and Cov(x,-x.)::: 0, it follows that
"
I
• V(y.) = V - C and Cov(y., y.) = C - C = 0 (i# j). I
""
I
J,
, -
(b) If C> 0, by taking Vex) = C and Cov(x,~) = -C, it follows thatV(Yi) .=, V, - 2C + C = V - C and Cov(Yi' Yj) =-C:- 2C t C': 0.Note ,t~at to choose Vex) = C and Cov(x, X;,)= -C is possible
provided that the covariance matrix,~ +, of the vector, .,exl , x2 ' .'"
~, x) is positive defini,te. Indeed, since R is positive definite, we
, need only' to verify' that the determinant -of the matrix R + ' is','
, pO~itive. But
V
C
det
:c:~,~.,
v-C
C ',,-
V_ •• ·., C
. ....
.-c
0
...
o v- C···
0
.0
0,' 0
. = det
C c .. · V-C
I
-C -C .. · ~CC
I
and the proof follows.
I
'
o
0
V-C
0
-C
-C
... :'-c
C
Reproduced by Sabinet Gateway under licence granted by the Publisher (dated 2011)
, 19
MULTIPLE COMPARISONS
The theorem has the practical adv~ntage of aJI?wing the application of the
xi
classical methods of multiple Comparisons to the
(i ~ 1) values, by means
of the Yi (i ~ 1) ones, which are already independent. provide~. that one has
'v: - C,
an appropriate estimato! for
whi.ch, is,
i
dist.ributed. ,rhe.t_h~r~m
also applies to non-normal variables.
3.
MULTIPLE COMPARISONS IN A MIXED TWO-WAY MODEL
_
' . a.
. . . . '
_
:.
\'
'
By applying the theorem to the variables, ,xi ~ ~ .. ' ,we can conclude, that
there exist normal independent variables y.1 with means E(y.)
= Jl. +
'
1
variances V(Yi)
error
fs1B'"
sis
(i
a.
1
and
= V -:- C =:( u2+nu1s)/nt. ' Moreover the mean squared
of interaction of, the mixed two-way model is such that
+ DO'1s)
X~'
(~-I)(t-l),so
where f=
that siB is
an unbiased
estimator of the unknown variance {u2 + DO'1B} , It is also ind~endent of
•
~
•
-
T
_-.'
' : . '
••
-
-
•
the random variables y.,'
wruch follows from' the fa.ct'
that
it. is ,independent
I
'
'"
.'
.-
,.,
.'
~
of the x.. Multiple comparisons results for the mixed two-way model are
1
.
' .. " -
easily derived from our theorem:
1
COROLLARY 1: The
,..
"J
";
, .. ~... ~
. '
... · . . •• .. ~,·i~~:.
,,'
mdiffer~nces" ~
probability 1 - a , that
~ _.:'
.i.-
.
'-:-
~,
. . ~. '.~.
. ', . "
,.~~~ ..
verify simultaneOusly,
~:
~ith
~','.
2
1
a.1 - a.,1 E (x.x.,·)
± qOf'{sA'
B/nU2
1..
1..
r,
(1)
where q r is the studentized range,
r,
.
PROOF: One need only apply the result in sectiqn 3:6, of S~,b.¢fe (1959) to
~~~~~~.
The (r-l) differences
simultaneously, with probability 1 - a, that
COROLLARY 2:
~
- al
(i f 1)
verify
Reproduced by Sabinet Gateway under licence granted by the Publisher (dated 2011)
MARTiN
20
(2)
where d
'f is the Duonet distribution.
r-l,
PROOF: One need only apply the result in section 2.5 of Miller (1966) to the
-
y. random variables.
1
COR.OLLARY 3:
The set of all the possible contrasts for the form
E-\~
(E-\ = 0) verify simultaneously, with probability of 1 - a, that
4Ai~ E E-\ xi.. ± J~,f{S!B(EA~)/nt}!
(3)
1
where J~,f = {(r-1)F~_1.rJ2 and, Fr,-l,f is the Snedecor F distribution
with" the same degrees of freedom.
.", .
PROOF: One ~eed only apply the result of section. 3.4 9f Scheffe (1959) to
the y. 'random variables.
1
_
"
, 'From expressions (1), (2) and (3), and fr~m the fact that
"
= 2(i + nO'!B)/nt, the conclusion can be
draw~ ~h~t
'
independent with variance
random variables
{u'l + nO'!B}'
xijb '
(0'2 + nO'!B)/nt,
v(i.- i,)
,I..
"
or
I",
the random variables
i., in the case of multiple comparisons, maybe treated
I..
'" ,
as if they ,'were
•
eq.riv~ently, that 'th~
in the same case, are iJ!.dependent with variance
which may be estimated by means of the mean squared,error
s!B of the row of interaction with (r-l)( t-1) degrees of freedom .
. These results are valid in general for every' K-dimensional vect~~
~Imally distributed with a matrix of covariances of type R indica.ted
I
above.
Reproduced by Sabinet Gateway under licence granted by the Publisher (dated 2011)
MULTIPLE COMPARISONS
21
A9KNOWLEDGMENTS
I am grateful to the referee forbis useful comments.
REFERENCES
BHARGAVA, R.P. and SRIVASTAVA, M.S. (1973). On Tukey's confidence intervals for the contrasts in the means of the intraclass
'
correlation model. J:R.S.S. B 35, 147-152.
GRAYBILL, F.A. (1961). An introduction to linear statistical models
(1lO11I.me I). New York: McGt8.W- Hill.
HOCHBERG, Y. and TAMHANE, A.C. (1983). Multiple comparisons in a
mixed model. Amer. Statist. 37(4), 305-307.
SCHEFFE, H. (1959). The analysis of 1Jariance.New York: John Wiley
and Sons.
Manu8cript received, 1992.10.19, revised, 1993.09.24.