ORTHOGONAL CONTRASTS: DEFINITIONS AND CONCEPTS

(1)

Review

ORTHOGONAL CONTRASTS: DEFINITIONS AND CONCEPTS

Maria Cristina Stolf Nogueira

USP/ESALQ - Depto. de Ciências Exatas, C.P. 9 - 13418-900 - Piracicaba, SP - Brasil. e-mail <mcsnogue@esalq.usp.br>

ABSTRACT: The single degree of freedom of orthogonal contrasts is a useful technique for the analysis of experimental data and helpful in obtaining estimates of main, nested and interaction effects, for mean comparisons between groups of data and in obtaining specific residuals. Furthermore, the application of orthogonal contrasts is an alternative way of doing statistical analysis on data from non-conventional experiments, whithout a definite structure. To justify its application, an extensive review is made on the definitions and concepts involving contrasts.

Key words: analysis of variance, partition of sum of squares, experiments with additional treatments

CONTRASTES ORTOGONAIS: DEFINIÇÕES E CONCEITOS

RESUMO: A técnica de contrastes ortogonais com um grau de liberdade é simples e bastante eficiente na análise de dados experimentais, como por exemplo, na obtenção de efeitos principais, de efeito de interação e de efeitos aninhados, nas comparações entre grupos de médias e na obtenção dos resíduos específicos. Além disso, sua aplicação tem revelado ser uma forma alternativa para análise de dados obtidos de um experimento que não segue uma estrutura definida. Com o objetivo de justificar a sua aplicação, foi realizada uma revisão sobre as definições e os conceitos envolvendo contrastes.

Palavras-chave: análise da variância, partição da soma de quadrados, experimentos com tratamentos adicionais

INTRODUCTION

The orthogonal contrast technique is a simple and efficient way of analysing experimental data to obtain, for instance, the main effects, interaction effects and nested effects, for comparisons between groups of means and/ or to obtain specific residuals. Additionally, the applica-tion of orthogonal contrasts is an alternative way of do-ing statistical analysis on data from experiments without a definite structure, like the experiments with additional treatments. The objective of this paper review was to jus-tify the application of the single degree of freedom or-thogonal contrasts in the analysis of experimental data from non-conventional experiments, recently published by Nogueira & Corrente (2000) and Corrente et al. (2001).

Definitions and concepts for mean contrasts with equal number of replications

Scheffé (1959), Winer (1971), Steel & Torrie (1981), Mead (1988) and Hinkelmann & Kempthorne (1994), among others, define a contrast between treatment means, represented by Y, as a linear function that can be estimated, considering an equal number of replications for all treatments, as follows:

i I

1

i i

c

Y

=

_∑

µ

=

where c_iare values of coefficients associated to µi, and

µi the mean attributed to the treatment i, so that

0 c

I

1

i i

=

∑

=

.

Supposing the mathematical model Y_ij = µ + t_i + e_ij, when i = 1,..., I and j = 1, ..., r, and µ being a constant,

t_i the treatment effect i, so that:

∑

=

I 1 i

i 0

t _{, and e}

ij the

ex-perimental error, so that e_ij ~ N(0, σ2_{) and independent}

of each other.

Supposing that µ_i = µ + t_i, then

t c t

c c ) t ( c

Y I _i

1 i i i I

1 i i I

1 i i i

I

1

i i

∑

= =

=

= µ+ =µ + =

=

Two contrasts I _i

1 i hi

h c

Y =

_∑

µ

=

and I _i

1 i hi

h c

Y =

_∑

µ

= ′

′ ,

since h ≠ h’ and h = 1, ..., (I-1), are orthogonal, if Cov(Y_h, Y_h’) = 0. Thus, for the adopted model

Cov (Y_h, Y_h’) = c c

r h'i

I

1 i

hi 2

∑

=

σ _{, occurs when} _c _,_c ₀

i h hi ′ = ,

that is Ic c_h'_i 0

1 i hi

=

∑

= .

(2)

y c

Yˆ I _.i

1 i hi

h

∑

=

= , with

r y y

r

1 j

ij

. i

∑

₌

= ,

considering e_ij~ N ( 0, _σ2_{) and independent of each other,}

then Yˆ is an unbiased estimator for Y_h _h and

∑

=

= = = σ

σ =

= I

1 i

2 2 hi I

1 i

I

1 i

2 2 hi i.

2 hi

h) c V(y ) c _r c_r Yˆ

(

V .

A special characteristic of orthogonal contrasts is that they may easily be included in the analysis of vari-ance, in such way that they originate sums of squares with one degree of freedom which correspond, each of them, to the (I-1) subdivisions of the sum of squares due to the treatments with (I-1) degrees of freedom. That is, the sum of squares due to the treatments can be decomposed in (I-1) sums of squares due to the contrasts with one de-gree of freedom.

Mead (1988) demonstrated this characteristic of the orthogonal contrasts and defined that the sum of squares due to Y_h is given by:

∑

= =

= ₌

= _I

1 i

2 hi 2 h I

1 i

2 hi I

1 i

2 .i hi

h

c Yˆ r c

] y c [ r

SSY with one degree of freedom

and consequently, the sum of squares due to treatments is given by:

SST =

∑

−

=

1 I

1 h h

SSY with (I-1) degrees of freedom.

Hence, supposing

∑

= = I

1 i

i hi h c t

Y is a contrast of treatment effects, so that

∑

= =

=

= I

1 i

i' h hi I

1 i

hi 0 and c c 0

c _{, with h}_≠_{h’ , for h = 1, ... ,}

(I-1) , and g_hiis a coefficient defined as:

∑

=

I

1 i

2 hi hi hi

c c

g ,

so that

∑

= =

=

= I

1 i

i' h hi I

1 i

hi 0 , g g 0

g , with h ≠ h’ , h = 1,

..., (I-1) and

∑

= =

I

1 i

2 hi 1

g , originating orthonormal

con-trasts. Consider also a group of (I-1) orthogonal linear functions of the treatment effects expressed in the

follow-ing matrix form:

τ ′ = τ    

 

   

 

′ ′ ′ =

−

ˆ G ˆ

g g g

z

1 I 2 1

M ,

where

g

′

_h

=

(

g

_h1

,

g

_h2

,

...

,

g

_hI

)

, for h =1, ... , (I-1);

τ

'

=

(

t

1

,

t

2

,

...

,

t

I

)

is the vector of the estimates of the treatment effects. Thus, the vector z will be a group of (I-1) estimates of the considered orthogonal contrasts, and that, due to the contrast orthogonality, the contrast estimates are independent, that is, the z_hare independent of each other. Thus,

z

_h

=

g

′

_h

ˆ

τ

, for h =1, 2, ... , (I-1). If the variable observations follow a normal distribution, then the vector z is a normal aleatory variable vector and independently distributed, and each element has mean = zero and variance = 1.

Adding to the matrix G’ a line vector g’₀ , so that

Ε′ = ′

I 1

g₀ _,_{where E is a vector of 1’s, the data matrix} will take the form:

τ ′ = τ    

′ ′ = τ    

 

   

 

′ ′ ′ =

−

ˆ G ˆ G g ˆ

g g g

*

z 0 *

1 I 1 0

M ,

where the data matrix G*’ is an orthogonal matrix, hav-ing G*’G* = I, and for this reason G* = (G*')-1_,

result-ing G* G*’ = I . Supposing

∑

= =

τ ′

= I

1 i

2 i. hi 2

I

1 i

i hi 2

h 2

h [ g ˆ] [ g tˆ ] [ g y ]

z

where

∑

= =

I

1 i

2 hi hi hi

c c

g , for h = 1, ... , (I-1), thus

∑

= =

= I

1 i

2 hi I

1 i

2 . i hi 2

h

c ] y c [

z (1)

considering , y

r 1 r y

y r

1 j

ij .i

i.= =

∑

=

and y_ij = _µ + t_i + e_ij ,

the expression can be written:

r e t y ) e r t r ( r 1 ) e t ( r 1

y i.

i i. i. i r

1 j

ij i

.i=

∑

µ+ + = µ+ + ⇒ =µ+ +

=

(3)

Substituting equation (2) into equation (1) the fol-lowing expression is obtained:

∑

=

= µ+ +

= _I 1 i 2 hi 2 I 1 i i. i hi 2 h c )] r e t ( c [

z , (3)

Considering that I

c

0

1 i hi

=

∑

=

, ever h = 1, ... , (I-1), and applying to (3), the obtained form is:

∑

= = + = _I 1 i 2 hi 2 I 1 i i. i hi 2 h c )] r e t ( c [ z

Considering that E(t_i) = t_i , E (e_ij) = 0 and

2 2 ij )

E(e =σ and also, E( t_i e_ij ) = 0 , thus

∑

= = = = = = σ +     = ⇒ + = _I 1 i 2 hi I 1 i 2 2 hi I 1 i 2 hi 2 i I 1 i hi 2 h I 1 i 2 hi 2 I 1 i i. i hi 2 h c r c c t c ) E(z ] c )] r e t ( c [ [ E ) z ( E r c t c ) z ( E 2 I 1 i 2 hi 2 i I 1 i hi 2

h +σ

      =

∑

=

= _⇒ 2 2

h 2

h ) r Y

z ( E

r = +σ

.

If Y_h = 0, E(z _h ) = 0, and since z_h follow a nor-mal distribution, and supposing the null hypothesis H₀ : Y_h = 0, thus

2 (1) 2 2 h ~ z

r σ χ .

Now supposing ˆ G G ˆ z

z′ =τ′ ′τ , (4) and that G G I 1 E E I 1 G G g g * G *

G ′= ₀ ′₀+ ′= ′ + ′

where EE’ is a matrix of elements 1, thus

τ ′ τ′ + τ ′ τ′ = τ ′

τ′ ˆ EE ˆ ˆ G G ˆ I 1 ˆ * G * G ˆ

and like matrix G*’ is also a orthogonal matrix and

0 ˆ E

E ′τ= . Due to the restriction relative to

∑

= = I 1 i i 0 t _, thus ˆ G G ˆ ˆ * G * G

ˆ ′τ= τ′ ′τ

τ′ .

Considering that G* G*’ = I , due to the orthogonality of matrix G*’, which means

τ ′ τ′ = τ

τ′ˆ ˆ ˆ G G ˆ (5) The comparison between (5) with (4) evidences that

τ

τ′

=

′

z

ˆ

z

and consequently,

(

)

∑

= = τ τ′ = ′ I 1 i 2 .. i. -y

y ˆ ˆ z

z . (6)

The one by one element multiplication of (6) by r, results in:

(

y -y

)

SST r z) z ( r I 1 i 2 .. i. = = ′

∑

=

In this way, the sum of squares due to Y_his ob-tained by

∑

= = = = ₌ = = _I 1 i 2 hi 2 .i I 1 i hi I 1 i 2 hi 2 .i I 1 i hi 2 h h c r ) y c ( c ) y c ( r z r

SSY , with one

degree of freedom, having the following properties:

(i) If Y_h= 0 , the SSY_h~ 2 (1) 2 _χ

σ ;

(ii) The SST

∑

−

= =I1

1 h h

SSY , with (I-1) degrees of freedom.

Hinkelmann & Kempthorne (1994) considered that,

SST

∑

−

=

=I 1

1 h h

SQY , with (I-1) degrees of freedom in

order to demonstrate that

y G y g g g z 1 I 1 0 ′ =             ′ ′ ′ = − M

where G’ is an orthogonal matrix, and thus

∑

= = ′ = ′ = ′ ′ = ′ I 1 i 2 i. y y y y I y y G G y z

z (7)

These authors observed also that

[ ]

∑

− = = = = = ′ +     = ′ + ′ = ′ = =

′ I1

1 h 2 h 2 I 1 i i. 1 -I 1 h 2 h 2 0 2 1 -I 0 h h 1 -I 0 h 2

h y g y

(4)

2 .. 2 .. 2

.. 2 r

1 j

ij I

1 i 2 I

1 i

i. Iy

r I y I I 1 r y I 1 r y I

1 y I 1

=     =     =     

   =      

∑

= = =

resulting

[ ]

∑

=

′ + =

′ I-1

1 h

2 h 2

.. g y

y I z

z (8)

Using equation (7) in (8), results that

[ ]

∑

= =

′ +

= I-1

1 h

2 h 2

.. I

1 i

2

.i Iy g y

y

[ ]

∑

= =

′ = I-1

1 h

2 h 2

.. I

1 i

2

.i- Iy g y

y

(

)

_∑

[ ]

∑

−

= =

′ =I 1

1 h

2 h I

1 i

2 ..

.i- y g y

y (9)

The one by one element multiplication of (9) by r, results in:

(

y - y

)

r

[ ]

g y

r I 1

1 h

2 h I

1 i

2 ..

.i =

∑

′ =

∑

−

= =

SST

with (I-1) degrees of freedom.

This result can be included into the analysis of variance table, originating a more detailed table, as the example in table 1.

In order to test the hypothesis H₀ : Y_h = 0, which may also be written as H₀ : c' τ = 0, test F is applied, so that

where MSR is the mean square residual and v corresponds to the degrees of freedom related to the F denominator, that is, in this case, v = I(r-1).

Alternatively, a confidence interval for Y_h , with (1-α) 100 % , is given:

) Yˆ ( Vˆ t

Yˆ h

v) , 2 ( h ± α

where v refers to the degrees of freedom related toVˆ(Yˆ_h), so that

and where σˆ2 refers to SSR, with I(r-1) degrees of free-dom, hence v = I(r-1).

According Scheffé (1959), the hypothesis H₀ : t₁ = t₂ = ... = t_I = 0, initially tested by the test F, de-scribed in table 1, is equivalent to the hypothesis

,

(an alternative hypothesis H_a which consists of at least one contrast Y_h diferent from zero),and where {Y₁ , Y₂, ... , Y_I-1} is a group of predictable independent linear func-tions.

Supposing Yˆ is an unbiased estimator of Y_h _h, given by

y c t c

Yˆ I _.i

1 i

hi I

1 i

i hi h

∑

= =

=

= , where

r y y

r

1

j ij

.i

∑

=

= ~ N (µi ,

r

2

σ

) and independents and

the author demonstrates that the probability of all con-trast values to fulfill simultaneously the unequal expres-sion written below, is (1- _αααα_α):

Yˆ h h Yˆ

h S ˆ Y Yˆ S ˆ

Yˆ − σ ≤ ≤ + σ (10)

or else,

Yˆ

h Sˆ

Yˆ ≥ σ

where the constant S is obtained by the following expres-sion

F 1) -(I

S= ₍_α_,_I_-_1,_I(r_-₁₎₎ and

σ

ˆ

_Yˆ

=

Vˆ

ar(

Yˆ

_h

)

.

The method described is the Scheffé method ap-plied to multiple comparisons using intervals for the con-trasts. This method is related to the F test, when testing the hypothesis H₀ : t₁ = t₂ = ...=t_I = 0, as follows: if Yˆ_h is considered significantly different from zero, which implies that Y_h = 0 is excluded of the interval (10). Now, if 0 Yˆ_h Table 1 - Analysis of variance with decomposion in

orthogonal contrasts. f

o s e c r u o S

n o i t a i r a

V DF SS E(MS)

t n e m t a e r

T (I-1) SST

Y₁ 1 SSY₁ or Y₂ 1 SSY₂ or

Y_I_-₁ 1 SSY_I_-₁ or l

a u d i s e

R I(r-1) SSR

∑

=

+

σ I

1 i

2 i

2 _t

1 -I

r

2 1 2₊_r_Y

σ σ2 +r (c1′τ) 2 2

2 2₊_r_Y

σ σ2 +r (c′2τ)2

2 1 -I 2 ₊_r₍_c_′ _τ₎

σ

2 1 -I 2₊_r_Y

σ

2

σ

M M

(5)

is not considered significantly different from zero, this implies that Y_h = 0 is included in the interval (10). Thus, when H₀ is rejected by F test, the author infers that at least one contrast should be significantly different from zero. In other words, if (and only if) under the significance level a it is concluded by the F test that the contrasts are not all nulls, then the Scheffé method will show contrast esti-mates that are significantly different from zero.

The level of significance α is the probability of the global type I error or experimentwise, that is, the prob-ability of at least one contrast be significantly different from zero, from a group of (I-1) contrasts, which means:

α = 1 − (1−α')(I−1)

where α’ is the probability of type I error for a particu-lar contrast (or comparisonwise) applied when the null hypothesis is rejected H₀ : Y_h= 0, for h = 1, ..., (I-1) .

According to Kuehl (1994), the formula

1 I

1

) 1 ( 1₋ ₋_α − =

α′

expresses the probability of type I error for a particular contrast (comparisonwise) as a function of the global type I error (experimentwise). For ordinary calculations a’ is considered closely _α/(I-1).

However, it may occur that none of these contrasts significantly different from zero are of practical interest. An example of a non-practical contrast is the normalized maximal contrast Yˆ_max estimate.

Definition and estimation of Yˆ_max

Scheffé (1959) demonstrated that SST = SSY_max = 2

max

Yˆ , where Yˆ_max is the normalized contrast estimate for the effect {t_i } maximal, defined as: suppose L is the sampling space for all the contrasts in {t_i}, and L” is the group of all normalized contrasts in L, that is, the group of all Y in L, so that the Var(Yˆ )=Cσ2, and that C =1. Thus, Yˆ_max is maximal Yˆ with Y in L”.

If

∑

= = = 

   

   = = I

1 i

r

1 j

ij j I

1 i

i i.

i y c w y

c

Yˆ ,

where w_j = 1/r , so that w_j≥ 0 and

∑

=

r

1 j

j 1

w , and

thus

∑∑

= = = I

1 i

r

1 j

ij j i w y

c

Yˆ ,

with

∑∑

= =

σ = I

1 i

r

1

j ij 2 2 j 2 i

K w c )

Yˆ (

Var , considering in this case

K_ij = 1, for all i = 1, ..., I and j = 1, ..., r , which repre-sents the number of times in which i and j appear to-gether, that is, for the present case in study the

consid-ered structure is a treatment factor with I levels and com-pletely random r replications, each level i of the consid-ered treatment factor occurring only once with the j rep-lication. Supposing the above considerations the follow-ing expression may be written:

∑∑

= = σ

= I 1 i

2 r

1 j

2 j 2 i w

c )

Yˆ (

Var ,

The restrictions for Yˆ being in L”, are the fol-lowing:

∑

=

I

1 i

i

0 and

c

_{∑ ∑}

= = =

I

1 i

r

1 j

2 j 2 i w 1

c

Hence, to maximize

∑

= = I

1 i

i. i y

c

Yˆ , variations on

{c_i} are observed while {y_.i} are remained fixed and thus, it is convenient to consider that

r

r 1 1

w 1

w 1 c 1 ] w [ c 1 w

c _r

1 j 2 r

1 j

2 j I

1 i r

1 j

2 j 2 i I

1 i

r 1 j

2 j 2 i I

1 i

r 1 j

2 j 2

i = ⇒ = ⇒ = = = =

∑ ∑ ∑

∑∑

= = =

=

= =

,

and thereafter, this leads to

∑

= =

I

1 i

2 i ₁

r

c _.

In order to maximize Yˆ, the following conditions are imposed:

∑

=

I

1 i

i

0 and

c

_∑

= =

= ⇒

=

I

1 i

I

1 i

2 i 2

i _-₁ ₀

r c 1 r c

.

Applying the technique Lagrange’s multipliers, we have:

∑

= =

= +λ +λ

=

λ + λ + = λ λ

I

1 i

2 i 2 I

1

i i

1 I

1 i i i.

2 1

1] -r c [ c

y c

h(c) g(c)

f(c) ) , , c ( L

calculating

r c 2 y

c

L i

2 1 i. i

λ + λ + = ∂

∂ _{, and considering}

0 c

L

i

= ∂

∂ _{, for all i, leads to the form}

0 r c 2

y_i.+λ₁+ λ₂ i = .

Isolating the c_ielement we obtain:

2 1 i. i

1 i. i

2 c_r - (y ) c -r (y₂ ) 2

λ λ + =

⇒ λ + =

λ . (11)

Adding (11) in

∑

= =

I

1

i i

(6)

∑

= = λ =

λ + =

I

1 i

I

1

i 2

1 .i

i - r(y₂ ) 0 c

resulting

.. 1 .. I

1 i

i. 1 I

1 i

.i I

1 i 1 1

I 1 i i.

y Ir y - Ir

r y r r y r 0 ) y (

r +λ = ⇒ λ = ⇒λ = ∑ = ⇒ λ =

∑ ∑

∑ =

= =

= (12)

Hence, placing (12) in (11) , leads to

2 ) y -y ( r -c

2 .. i. i

λ

= , (13)

and placing (13) in

∑

= =

I

1 i

2 i ₁

r

c _{, leads to}

2 .. I

1 i i. 2 2 I

1

i 22

2 .. i. I

1 i

2

2 ..

.i ₁ ₄ _r ₍_y _y ₎

4 ) y - y ( r 1 r 1 2

) y -y ( r

− = λ ⇒ = λ ⇒ =    

 

λ

− ∑ ∑

∑

= =

= (14)

Finally, placing the result obtained in (13) in the formula

4

] ) y -y ( r [

] y 2

) y -y ( r [

] y c [ Yˆ

2 2 I

1 i

2 2 .. .i I

1 i

2 i. 2

.. i.

2 .i I

1 i

i max

λ =

=

∑

= = =

(15)

substituting into (15) the result of (14) ,

2 .. I

1

i i.

2

2 r (y y )

4λ =

_∑

−

=

,

) y y ( r

Yˆ 2

.. 1

i i.

2

max=

∑

− =

=

I

SST .

Thus, it is concluded that Yˆ2 SQY_max

max= = SST.

If the interest is to test the null hypothesis H₀: Y_max = 0, the rejection area for this test is given by

,

or by

where α is the significance level that gives the quantity of the F distribution with (I-1) and I(r-1) degrees of free-dom.

It is noticed that the way the test was applied to check the null hypothesis H₀: Y_max = 0 is identical to the test applied to verify the hypothesis of no variation due to treatments. If the tested hypothesis in terms of com-parisons is not rejected, the resulted implication is in the correspondent comparison in populational terms that it is not significantly different from zero. Then, if the null hy-pothesis, H₀: Y_max = 0, is not rejected, that is, if Y_max is not significantly different from zero, then no other com-parison to be tested will be significantly different from zero, when the Scheffé method is used. If the null hypoth-esis H₀: Y_max = 0 is rejected, this impplies that the corre-spondent comparison is significantly different from zero. For the group of tests used in the considered compari-sons, the Scheffé method is referred to the a level of sig-nificance, which means that the a level of significance is the probability of the global type I error (or experimentwise), correspondent to the totality of tests used.

Definitions and Concepts for mean contrasts with un-equal number of replications

In the case of treatments with unequal number of replications, that is, for j = 1, ..., r_i, (Steel & Torrie, 1981, cited in Nogueira, 1997), consider that for a given con-trast

∑

= =

=

= I

1 i

i. hi i I

1 i

i. hi

h c y r c y

Yˆ _,

where

i i. i r

1 j

ij .i

r y r

y y

i

=

∑

= ~ N (µi ,

i 2

r σ

) and independent

of each other, and, r_i is the number of replications of treat-ment i; c_hi is the coefficient to be attributed to

y

_.i ; and

.i

y

is the mean estimate of treatment i, so that

0 c

r

_hi I

1

i i

=

∑

=

.

Moreover, I _h

1

i i hi i

h ) r c Y

Yˆ (

E =

_∑

µ =

=

and

∑

= σ

= I

1 i

2 2 hi i

h) r c

Yˆ (

Var _{. Thus,}

h

Yˆ is defined as the con-trast between treatment means obtained from data with unequal number of replications.

Supposing that µi = µ + ti, where µ is a

con-stant and T_i is the effect of treatment i, the following ex-pression is written:

t c r t c r c

r ) t ( c r

Y I _hi _i

1 i

i i hi I

1 i

i hi I

1 i

i i

hi I

1 i

i

h

∑

= =

= +

(7)

Two contrasts i I

1 i i hi h

r

c

Y

=

∑

µ

= and

i I

1 i

i h i

h r c

Y =

∑

µ

= ′

′ where h ≠ h’ and h = 1, ..., (I-1), are

orthogonal when r_ic_hi ,r_ic_h′_i =0, that is,

0 c c r _hi _h'_i

I

1 i

i =

∑

=

.

Hence, the null hypothesis

0 c

r Y :

H I

1

i i hi i h

0 =

∑

µ =

=

can be tested by means of

where

∑

= = _I

1 i

2 hi i

2 h h

c r Yˆ

SSY and v =

∑

= −

I

1 i

i 1)

r

( _{= n-I, where}

n is the total number of observations, that is, n =

∑

=

I

1 i i

r _.

Kirk (1968), for average data from unequal num-ber of replications, considers that

∑

= = I

1 i

i. hi h c y

Yˆ _,

so that

∑

= =

I

1 i

hi 0

c _{and the orthogonal condition is as} fol-lows:

∑

=

′ ₌

I

1 i i

i h hi ₀

r c c

Thus, the null hypothesis

0 c

Y :

H I

1 i hi i h

0 =

∑

µ =

=

can be tested by means of

where

∑

= = _I

1 i i

2 hi 2 h h

r c Yˆ

SSY and v =

∑

=

−

I

1 i

i 1)

r

( = n-I.

Winer (1971) and Kirk (1968) showed that the sum of squares due to Y_h , for the case of unequal num-ber of replications, is given by the following expression:

∑

= = _I

1 i i

2 hi 2 h h

r c Yˆ

SSY when

∑

=

I

1 i

2 .. i. i

.. i. i i

) y -y ( r

) y -y ( r c

and, suggested that the SSY_max is obtained when and this leads

SSY_max=

∑

=

− = I

1 i

2 .. .i i 2

max r(y y )

Y _=SST

to according to Scheffé (1959).

REFERENCES

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experiments. New York: Wiley-Interscience, 1994. v.1, 495p. KUEHL, R.O. Statistical principles of research design and analysis.

Belmont: Duxbury Press. 1994.681p.

MEAD, R. The design of experiment: Statistical principles for practical application.Cambridge: Cambridge University Press, 1988. 618p. NOGUEIRA, M.C.S. Estatística experimental aplicada à experimentação

agronômica. Piracicaba: ESALQ, DME, 1997. 250p.

NOGUEIRA, M.C.S.; CORRENTE, J.E. Decomposição da interação tripla significativa utilizando o comando contrast do PROC GLM do SAS, aplicado ao modelo de classificação tripla para dados balanceados.

Bragantia. v.59, p.109-115, 2000.

SCHEFFÉ, H. Analysis of variance.London: John Wiley & Sons, 1959. 467p.

STEEL, R.G.D.; TORRIE, J.H. Principles and procedures of statistics: A biometrical approach. 2.ed. New York: McGraw-Hill International Book Company, 1981. 625p.

WINER, B.J. Statistical principles in experimental design.2.ed. New York: MacGraw-Hill Book Company, 1971. 897p.