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An Exact Test for Additivity in Two-Way Tables under BiadditiveModellingAmina Barhdadia; Sorana Frodab

a Centre de Recherche, Institut de Cardiologie de Montréal, Montréal, Québec, Canada b Départementde Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada

Online publication date: 28 May 2010

To cite this Article Barhdadi, Amina and Froda, Sorana(2010) 'An Exact Test for Additivity in Two-Way Tables underBiadditive Modelling', Communications in Statistics - Theory and Methods, 39: 11, 1960 — 1978To link to this Article: DOI: 10.1080/03610920902944078URL: http://dx.doi.org/10.1080/03610920902944078

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Communications in Statistics—Theory and Methods, 39: 1960–1978, 2010Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920902944078

An Exact Test for Additivity in Two-Way Tablesunder BiadditiveModelling

AMINA BARHDADI1 AND SORANA FRODA2

1Centre de Recherche, Institut de Cardiologie de Montréal,Montréal, Québec, Canada2Département de Mathématiques, Université du Québec à Montréal,Montréal, Québec, Canada

Interaction in a two-way classification table with one observation per cell has to bemodeled in order to avoid a saturated model. Different models for the interactionterm have been proposed, and multiplicative models are the most common. In thesemultiplicative (bilinear or biadditive) models, the interaction term is written as aproduct of two terms or as a sum of such products (see Denis and Gower, 1994).In particular, the main effects appear as parameters in the interaction term. At thesame time, a graphical tool called biplot has been used to diagnose models for theinteraction. In this article, we propose a polynomial description of the interactionterms, when the biplot suggests that nonlinear functions of the main effects are abetter description of the interaction. Our modelling permits to introduce an exacttest for the presence of the interaction. We illustrate our method on data sets andsimulated data, and these suggest that our test is more powerful than Mandel’s(1961) test.

Keywords Biplot; Exact test for additivity; Mandel’s additivity test; Polynomialbiadditive models; Two way tables with one observation per cell.

Mathematics Subject Classification 62H25; 62J10.

1. Introduction

The general model for a two-way �I × J� table is

yij = � + �i + �j + �ij + �ij� i = 1� � p j = 1� � q�p∑

i=1

�i =q∑

j=1

�j =p∑

i=1

�ij =q∑

j=1

�ij = 0� (1)

Received October 5, 2008; Accepted April 1, 2009Address correspondence to Sorana Froda, Département de Mathématiques, Université

du Québec à Montréal, C.P. 8888, Succursale centre-ville, Montréal, Québec H3C 3P8,Canada; E-mail: froda.sorana@uqam.ca

1960

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where �, �i, �j , �ij represent, respectively, the general mean, the ith row treatmenteffect, and the jth column treatment effect, and the interaction in the �i� j� treatmentcombination. The errors �ij are assumed to be normal i.i.d. variables, � �0� �2�.

For example, in the case of plant breeding and crop production field, yij wouldbe the yield of variety (or genotype) i when grown in the environment j, �i, and �j

are, respectively, the main effects of variety i and the environment j, and �ij is theinteraction between variety i and environment j.

The model is said to be additive if �ij = 0 for all i and j, i = 1� � p j =1� � q. In the plant breeding example the interpretation of an additive modelis that the difference between the mean production field of any two varieties (orgenotypes) is stable (i.e., the same) across environments.

When interaction terms are present, the model is said to be non additive.In the plant breeding example, it means that, given two genotypes, the differencesin mean production field will vary accross environments. That is, one genotype willbe more productive in some environments than in some others. The ability to detectinteraction with a graphic tool (biplots) or with statistical tests is an importantproblem.

In plant breeding, as in many other examples, there is only one observationper cell, which means that the model given by Eq. (1) is saturated and no testcan be performed to estimate the variance �2 (no degrees of freedom are left).The modelization of the interaction term �ij is a way to circumvent this difficulty.The following multiplicative models were the first ones to be proposed in theliterature:

�ij = ��i�j� (Tukey) (2)

�ij = i�j� (Mandel-row) (3)

�ij = �i�j� (Mandel-column) (4)

�ij = i�j� (Multiplicative model with one term) (5)

�ij =r∑

k=1

�k ik�kj� (Multiplicative model with r terms) (6)

Tukey’s model (2) is discussed in Scheffé (1959) and Ghosh and Sharma (1963).Mandel’s models (3), (4) are called the row regression and the column regressionmodel, respectively. Gollob (1968) and Mandel (1969, 1971) independentlyintroduced the general multiplicative model. A basic difference between Gollob’sand Mandel’s method is in the way they assign the number of degrees of freedomto each term. Denis and Gower (1994) introduced the term biadditive for a largeclass of bilinear models (which comprises those listed above).

Gabriel (1971) and Bradu and Gabriel (1988) introduced a graphical toolnamed biplot where each of models (2)–(4) has a unique corresponding geometricalrepresentation. The main effect �i is represented by a point �i and the main effect�j is represented by a point �j , with �i��j in the plane. Collinearity of points �i

and/or collinearity of points �j may show up. Thus the geometrical configuration ofthese points is indicative of some model, namely: additivity (case of two orthogonallines), Tukey’s model (case of two concurrent lines), Mandel’s row regression model(�i are on a line and �j are scattered), or Mandel’s column regression model (oneline in �j and a scatter in Pi). Gower (1990) showed that for the multiplicative

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1962 Barhdadi and Froda

model with one term in the interaction, the points �i and/or the points �j mustbe coplanar in a three-dimensional space. In general, the multiplicative model (6)has a representation in an �r + 2� dimensional space, and appears as a scatter (i.e.,a nonlinear alignement) when projected on the plane. An important note is thatall these representations apply to the exact model (i.e., no random errors present).Thus, when a scatter is present, one may want to settle whether it is due to a specialform of the interaction or it’s simply noise.

More precisely, for the exact model, collinearity corresponds to rank 2 andcoplanarity to rank 3; if neither is present one may consider higher-order modelsif the points of the biplot seem to have a geometrical configuration. In this workwe propose using the biplot as indicative of models where interaction is present,whenever the points �i are approximately on a polynomial curve (e.g., quadraticor cubic) in a plane orthogonal to the plane containing the scatter of points �j .Further, we consider models for this kind of interaction terms. Finally, we proposea test for these interactions to confront the assessment based on the biplot with astatistical test.

Before introducing our models, we mention that, in the special case of amultiplicative model with one interaction term, Milliken and Graybill (1970)proposed replacing the i’s and �j’s of the interaction terms with unknown functionsf and g, i.e., to write �ij = �f��i�g��j�. Their interest is solely on testing the presenceof the interaction and therefore their null hypothesis is � = 0. On the other hand,they do not give an explicit expression for f and g and thus their parametrizationdoes not allow for an exact test.

If the points �i lie on a quadratic curve and the points �j are scatteredin another plane, then the most simple functional model is �ij = �i�1j + �2i �2j .We notice that this interaction is a sum of two terms so these models are alreadyin the general multiplicative form with two terms of interaction. Statistical tests aredifficult to elaborate with these models.

As an example, consider the data analyzed in van Eeuwijk and Kroonenberg(1998) on the maize data from the official Dutch Maize Variety Trials. These dataare classified as a three-way table with variety by location by year structure: 6varieties, 7 years, and 4 sites. From these data we constructed a two-way table bycombining the location and year to obtain a 6 by 28 table. Figure 1 shows thebiplot of this 6 by 28 table: the row points �i (maize varieties) lie at the left sideof the plot and the column points �j (year by location) are on the right side ofthe plot. We could say that the �i’s seem to describe some curve while the �j’s arescattered over the plane. The number of points �i is too small to conclude visuallythat the points lie on a quadratic or cubic curve. As shown in the sequel, the testwith our proposed model confirms interaction, while Mandel’s test concludes anadditive model (see Sec. 4). In other words, this graphical configuration may indicatethat the interaction has a special structure that cannot be described by Tukey’s orMandel’s models, but can be best described by a general multiplicative model (6)where, e.g., �ij = �i�1j + �2i �2j .

This article is organized as follows. In Sec. 2, we present the polynomial modeland the estimates of its parameters. A test for the presence of the interaction whichis modeled by these models is also discussed in this section. In Sec. 3, we carry out asimulation study to assess the performance of the proposed test. In Sec. 4, we applyour model to three real data sets and we present the results of our analyses, while ashort discussion is given in Sec. 5.

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Exact Test for Additivity in Biadditive Modelling 1963

Figure 1. Biplot of Dutch maize variety trials (van Eeuwijk and Kroonenberg, 1998).

2. Polynomial Model

2.1. The Model and Its Equations

With respect to the investigation of non additivity, we can view the proposed modelsof interaction from two different perspectives. Models (2)–(4) use regression on themain effects, whereas in model (6) the interaction is partitioned in r multiplicativeterms.

In this article, we propose a model that combines these two approaches. Ratherthan considering the interaction as linearly related to the row main effect �i as inMandel’s model (4), �i is replaced by a polynomial in �i of any desired degree lessthan p. Further, we partition the interaction in a sum of r multiplicative terms as inmodel (6), but with each of the r terms a product of a polynomial in the row effect�i with a parameter �kj which is not necessarily a function of the column effect �j .

More precisely, the proposed orthogonal polynomial model in the main roweffect is given by:

yij = � + �i + �j +r∑

k=1

Pk��i��kj + �ij� i = 1� � p j = 1� � q� (7)

where Pk��i� has degree k in �i. We assume that

∑�i =

∑�j = 0� E��ij� = 0� var��ij� = �2

Because of their orthogonality, the polynomials Pk do verify

p∑i=1

Pk1��i�Pk2

��i� = 0� k1 �= k2 (8)

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1964 Barhdadi and Froda

For example, with r = 2 we have

P0�x� = 1� P1�x� = x� P2�x� = x2 −(∑p

m=1 �3m∑p

m=1 �2m

)x −

(∑pm=1 �

2m

p

)

In the special case r = 1, our model coincides with Mandel’s row regression model.To ease some calculations and without loss of generality, we require that the normsof the orthogonal polynomials be 1, that is,

∑pi=1 P

2k��i� = 1. In matrix form, the

polynomial model (7) can be written as

Y = ��1p + ��1′q + 1q�′ +�����+ �� (9)

where

Y =

y11 · · · y1qy21 · · · y2q

yp1 · · · ypq

�

���� =

P1��1� · · · Pr��1�

P1��2� · · · Pr��2�

P1��p� · · · Pr��p�

�

� = ��1� � �r �′

=

�11 · · · �1q�21 · · · �2q

�r1 · · · �rq

�

with 1p (respectively, 1q) a column vector of ones only, and of length p (respectively,length q).

In a similar way, we could consider orthogonal polynomial models in the maincolumn effect. Such polynomial models are written as:

yij = � + �i + �j +r∑

k=1

kiPk��j�+ �ij� i = 1� � p j = 1� � q (10)

for simplicity, in the sequel we consider (7) only.

2.2. Biplots of Polynomial Models

In short, the biplot is a convenient graphical planar display of the elements, as wellas rows and columns, of a matrix of rank ��� (for more details see, e.g., Krzanowski,1988, Ch. 4). It was introduced in Gabriel (1971) and is based on the singularvalue decomposition of the original matrix as a sum of orthogonal matrices ofrank 1; in the case of a matrix of rank 2 the biplot is exact, while for higher

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Exact Test for Additivity in Biadditive Modelling 1965

rank matrices one retains the first two terms of the singular value decomposition,which is the 2-rank approximation to the original matrix. Then one can considerthe approximation Y ≈ GH ′, where G is an p× 2 (usually) orthogonal matrix andH is an 2× q (usually) orthogonal matrix (thus their product is of rank 2); G is therow matrix (matrix of row points), and H is the column matrix (matrix of columnpoints). So, for fixed �i� j� each element of the original matrix is approximated bythe inner product gih

′j of two 2-dimensional vectors, gi� hj ; thus, the mn entries of

Y are represented by only m+ n vectors in the plane, and various relationshipsamong them. Of course, for Y of higher rank than two, one can consider higher rankmatrices, G (of dimension s × p) and H (of dimension s × q), but then the graphicalrepresentation is not necessarily planar (see Gower, 1990). Bradu and Gabriel (1988)introduced the idea of using the biplot for diagnosing special interaction models.For the intuition behind this approach, consider the model with no interaction,of model matrix Y , with elements yij =

{� + �i + �j

}�i=1��p�j=1��q�

. This matrix can

be written as Y = GH ′ with row points G = ��� + �i� 1���i=1��p� and column pointsH = {

�1� �j�}�j=1��q�

; thus, the row points are aligned on a horizontal line, while thecolumn points are aligned on a vertical line. The orthogonality of these two lines iswhat characterizes additivity (no interaction). The decomposition GH ′ is not unique,but this does not affect the final diagnostics (see Bradu and Gabriel, 1988).

In our case, we consider matrices of higher dimensions p× �r + 2� andq× �r + 2�, where r is the number of terms in the interaction of model (7);the rank of this model (in the singular value decomposition sense) is � = r + 1.Indeed, the row points �i, i = 1� � p, are the p lines of the row matrix �� +�i� 1� P1��i�� P2��i�� � Pr��i��

′�i=1��p�; the column points �j , j = 1� � q, are the q

lines of the column matrix �1� �j� �1j� �2j� � �rj�′�j=1��q�. For r = 2, the rank of our

model is 3, and the coordinates of �i can be taken �1� �i� P2��i��. We notice that thefirst coordinate is constant and equal to 1 and thus the points �i lie in the sameplane, on a quadratic curve. The column points �j have coordinates �� + �j� �1j� �2j�and, therefore, they are scattered in a three-dimensional space. If r = 1, we recognizethe usual Mandel column regression model (i.e., the points �i, i = 1� � p, lie onthe same line). For a detailed presentation of the biplot in the polynomial case werefer to Barhdadi (2003, Ch. 2).

Examples of biplots of polynomial models are shown in Figs. 2(i) and (ii). As wecan see, the structure of the interaction is well visualized in these figures. Figure 2(i)shows the biplot of the polynomial model with r = 2. The row points are on a

Figure 2. (i) Quadratic model and (ii) Cubic model.

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1966 Barhdadi and Froda

parabola while the column points are scattered. Figure 2(ii) displays the biplot of apolynomial model with r = 3, where the curve described by the row points can beparametrized as a polynomial of degree 3.

2.3. Analysis of Variance Table

We start by estimating the main effects � and � as well as the matrix ����. Theseestimators of the main effects � and � are

� =[1qJ1×q ⊗

(Ip −

1pJp×p

)]y

� =[1pJ1×p ⊗

(Iq −

1qJq×q

)]y�

where y = �y11� � yp1� � y1q� � ypq�′, Jp×p is the matrix 1p1

′p, and J1×p = 1′p

(respectively, J1×q = 1′q) are row vectors of ones only and of length p (respectively,length q).

Then a natural estimate of the matrix ���� is

���� =

P1��1� · · · Pr��1�

P1��2� · · · Pr��2�

P1��p� · · · Pr��p�

� (11)

where the notation Pk�·� stands for the fact that Pk is the polynomial Pk with �ireplaced by �i in the sums

∑pi=1 �

2i �∑p

i=1 �3i , etc.

To estimate the vector � we use the vector of residuals z which can be written as

z =[(

Iq −1qJq×q

)⊗(Ip −

1pJp×p

)]y

This gives

� = (Iq ⊗ ����

)z (12)

Further, we look for a test for interaction under our polynomial model. Westart by giving the corresponding analysis of variance table and a theorem on theconditional distribution of the sums of squares in this analysis. In the next section,we give the unconditional distributions and generalize Mandel’s test for additivity.

In Table 1, we give the analysis of variance table of our polynomial model,where

SSA = qp∑

i=1

�yi − y�2�

SSB = pq∑

j=1

�yj − y�2� (13)

SSk =p∑

i=1

�Pk��i��2

q∑j=1

�kj2 =

q∑j=1

�kj2�

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Exact Test for Additivity in Biadditive Modelling 1967

Table 1Analysis of variance of polynomial model

Source df Sum of squares

Mean 1 pqy2Row effect p− 1 SSAColumn effect q − 1 SSBNon additivity r�q − 1�

∑rk=1 SSk

Residuals �p− r − 1��q − 1� SSETotal pq

∑pi=1

∑qj=1 y

2ij

and

SSE =p∑

i=1

q∑j=1

(yij − y − �yi − y�− �yj − y�−

r∑k=1

(Pk��i��kj

))2

=p∑

i=1

q∑j=1

(yij − y − �yi − y�− �yj − y�

)2 − SSk (14)

All our tests are based on the SSk and SSE sums of squares. These sums arefunctions of the vector z and also functions of the vector �. Note that z is not avector of independent observations. Moreover, it is a function of the vector �. Theconditional distributions of SSk and SSE are given in the following theorem; its proofis sketched in the Appendix.

Theorem 2.1. Consider the polynomial model given in (7), and the sums of squares∑rk=1 SSk and SSE , defined in (13) and (14), respectively. Then, conditionally on �,

we have:

(1) 1�2�∑r

k=1 SSk� and1�2�SSE� are statistically independent.

(2) These two sums are distributed as �2�r�q − 1���1� and �2��p− r − 1��q − 1���2�,respectively, where

�1 =12�2

r∑k=1

q∑j=1

[p∑

i=1

Pk��i�r∑

l=1

Pl��i��lj

]2

�

�3 =12�2

p∑i=1

q∑j=1

( r∑l=1

Pl��i��lj

)2

�

�2 = �3 −�1

2.4. Test for Additivity

Various models and tests for non additivity were proposed in the literature byimposing an additional structure on the interaction �ij in model (1). The tests fornon additivity proposed by Tukey (1949) and Mandel (1961) are based on statisticsthat follow a central Fisher distribution under H0. It is important to note that thesetests for non additivity were obtained by approximating a nonlinear model with alinear model, and this allowed to perform an exact Fisher test.

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1968 Barhdadi and Froda

In the approaches of Tukey and Mandel it is assumed that, for the interactionto be present, the model has main effects, i.e., �i �= 0 and �j �= 0, i = 1� � p, j =1� � q. Therefore, we work under the same assumption.

Consider the polynomial model given in (7), i.e., let

yij = � + �i + �j +r∑

k=1

Pk��i��kj + �ij� i = 1� � p j = 1� � q

In this model, additivity is present when the null hypothesis H0 is satisfied, with

H0 � �1 = �2 = · · · = �r = 0

If H0 is accepted, then the model is additive; otherwise, the model is non additiveand interaction is present.

We propose the test statistic T0, with

T0 =�p− r − 1��q − 1�

r�q − 1�

∑rk=1 SSk/�

2

SSE/�2

(15)

As a consequence to Theorem 2.1 (see also Barhdadi, 2003, Ch. 4) we have thefollowing result.

Corollary 2.1. Under H0, the statistic T0 follows a central Fisher distribution,

T0 ∼ � �r�q − 1�� �p− r − 1��q − 1��

Proof. For fixed �, in Theorem 2.1 the centrality parameters �1� �2 are null underH0, and the independent sums of squares

∑rk=1 SSk/�

2 and SSE/�2 follow central

chi-square distributions. In order to show that the result holds unconditionallyas well, one can repeat the argument given in the proof of Theorem 4.8 ofScheffé (1959), which is based on the following fact: under H0, the three sets oflinear forms, ��i�i, ��j�j , and � ij�ij = �yij − yi − yj + y�ij span orthogonal spaces,and the conditional distribution of � ij�ij given ��i�i and ��j�j is the same asthe unconditional one. Then, the unconditional joint distribution of the sums∑r

k=1 SSk/�2 and SSE/�

2 is the same as the conditional one, of two independentcentral chi-square variables. Hence, T0 follows a central Fisher distribution withr�q − 1� and �p− r − 1��q − 1� degrees of freedom, as stated.

If H0 is rejected, then we have at least one vector �s �= 0, which means that theinteraction is at least a polynomial of degree s in �i and the model is non additive.

In practice, one can use r = 2 or 3, as typical cases of even and odd functions,respectively. The biplot can be taken as indicative starting point.

3. Simulation Study

In this section, we study the performance of the proposed method by a simulationstudy. Our objective is the detection of the presence or the absence of the interactionusing the global test of additivity. The simulation results reveal the differences inthe performance of our global test for additivity and Mandel’s test. The size of thevariance �2 and the relative contribution of interaction’s terms play an importantpart in these results.

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3.1. Parameter Values

We generated data sets with p = 10 rows and q = 6 columns according to apolynomial model of degree r, with r = 2 or r = 3. Such a model can be written as

yij = � + �i + �j +r∑

k=1

Pk��i��kj + �ij� i = 1� � p j = 1� � q

The row effect �i and the column effect �j were chosen equidistant and were scaledso that the row effect sum of squares SA =∑p

i=1 �2i be 200 and the column effect

sum of squares SB =∑qj=1 �

2j be 100. The random errors �ij were drawn from a

normal distribution with mean 0 and variance �2. We considered different sizes for�2: low, average, and high. Different values were assigned to the interaction sumof squares SI =∑

i�j �2ij and for each value of �2 we considered the following four

cases:

SI = �SA+ SB�

2� SI = �SA+ SB�� SI = 2�SA+ SB�� SI = 3�SA+ SB� (16)

In other words, we took the sums of squares of the main effects such that their sumwas twice the one of the interaction, equal to the sum of squares of the interaction,etc.

3.2. Experiments and Simulation Results for r = 2

For r = 2, we considered the following three values for the variance �2: 2.5, 10, and25. With these values of �2 the error sum of squares, SE, is set around 100, 400, and1,000, respectively. As noted above, the values of SA and SB were fixed, respectively,at 200 and 100 in all tables. Then the SI value varied between 150 and 900. Next,consider the three values of �2. When �2 = 25, the sum of squares SE is around 100and SA� SB, and SI are all greater than SE. When �2 = 10, the error sum of squaresSE is around 400. In this case, when SI is set equal to �SA+ SB�/2 or SI is set equalto SA+ SB, SI is lower than SE, but SI is greater than SE for the two other valuesof SI . Finally, when �2 = 25, the error sum of squares SE is around 1,000 and isalways larger than SI .

Further, consider the interaction, and recall that we want to take into accountthe quadratic structure. In order to do this, we have to vary the ratio SD1/SD2,where

SDk = ��k�2 =q∑

j=1

�2kj� k = 1� 2

Thus, for each value of �2 and for each value of SI we took seven values of theratio SD1/SD2 and we did 100 simulations for each one of these values; in all,we performed 21 experiments of 100 simulations each.

To summarize, our goals in setting these parameter values were as follows. First,the purpose was to assess the behavior of our global test when the total sum ofsquares of the interaction is partitioned into various combinations of individualsums of squares inside the interaction (linear and quadratic). Second, the interaction

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1970 Barhdadi and Froda

had to be contrasted with the contribution of the main effects. Finally, varying thevariance �2 allowed to assess how well the test performs with errors of differentmagnitudes.

The results are given in Table 2, where, for each value of �2, there are four othersubtables. Table 2 describes the comparative results of the additivity tests: namelyglobal test, i.e., our test, and Mandel’s test. We list the results according to eachvalue of the sum of squares of the interaction (i.e., SI) as compared with �SA+ SB�as in (16). Further, each subtable has three columns. In the first column we have thevalues of the ratio SD1/SD2, in the second column there are the results of our testand in the third column the results of Mandel’s test. We are reporting the numberof times where the two tests rejected the null-hypothesis of additivity for each valueof SD1/SD2. In order to read our results, recall that SI is divided between SD1 andSD2, and that the sums SD1 and SD2 show the respective importance of the linearand the quadratic term in the interaction.

In the low variance case, i.e., �2 = 25, for each value of SI and each valueof the ratio SD1/SD2 our global test of additivity detected the presence of theinteraction 100 times in the 100 simulations. Mandel’s test was also able to detectthe interaction when SD1/SD2 was high but was less able to do so when SD1/SD2

was low. Note also that the number of times Mandel’s test was able to detect thepresence of the interaction decreased dramatically when the ratio SD1/SD2 wentfrom 1 to 1/4, i.e., when the importance of the linear term decreased. When �2 = 10(moderate variance), it can be seen that our global test for additivity performedbetter than Mandel’s test. As had to be expected, the number of times when thetwo tests detected the presence of the interaction increased with the interactionsum of squares SI . These numbers were generally lower than those for �2 = 25.We concluded that with a medium size of the variance, our global test performedwell for each value of SI , whereas Mandel’s test was good only when SD1/SD2 wasgreater than one and SI was high (i.e., in the case of an important linear effectand strong interaction). When �2 = 25 (large variance), our global test of additivitywas good for the last values of SI , whereas Mandel’s test results were very poorall the way. Our global test performed better when the interaction sum of squareswas high.

To conclude, we can say that applying our global test for additivity gave goodresults with different sizes of the variance �2. Moreover, our test detected more oftenthe presence of the interaction and performed better than Mandel’s test even whenthe sum of squares of the linear term, SD1� was higher than SD2.

3.3. Experiments and Simulation Results for r = 3

The simulation results for the comparison between the two additivity tests whenr = 3 are given in Table 3. In this case, we have three components: SD1� SD2, andSD3 where SD3 =

∑6j=1 �

23j . We considered the sum SD, defined as, either (i) SD =

SD1 = SD2, or (ii) SD = SD1 + SD2, and we set the ratio SD/SD3 at a few valuesbetween 4 and 1/4. We report the results for a medium variance, �2 = 5, for whichthe error sum of squares SE was set around 200.

In all, it appears that our additivity test performs definitely better than Mandel’stest. Moreover, we can see that the rejection number is not greatly affected by theSD/SD3 ratio.

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Exact Test for Additivity in Biadditive Modelling 1971

Table 2Additivity test for r = 2 and given �2: comparison of our

method (global test) and Mandel’s test

�2 = 25

SI = �SA+ SB�/2 SI = �SA+ SB�

SD1/SD2 Global Mandel SD1/SD2 Global Mandel

4 100 99 4 100 1003 100 98 3 100 1002 100 94 2 100 991 100 79 1 100 931/2 100 49 1/2 100 621/3 100 27 1/3 100 381/4 100 17 1/4 100 21

SI = 2�SA+ SB� SI = 3�SA+ SB�

SD1/SD2 Global Mandel SD1/SD2 Global Mandel

4 100 100 4 100 1003 100 100 3 100 1002 100 100 2 100 1001 100 98 1 100 981/2 100 71 1/2 100 771/3 100 44 1/3 100 461/4 100 26 1/4 100 25

�2 = 10

SI = �SA+ SB�/2 SI = �SA+ SB�

SD1/SD2 Global Mandel SD1/SD2 Global Mandel

4 77 55 4 80 743 76 49 3 89 692 74 40 2 91 631 69 25 1 91 441/2 67 17 1/2 77 271/3 65 13 1/3 75 171/4 62 12 1/4 75 15

SI = 2�SA+ SB� SI = 3�SA+ SB�

SD1/SD2 Global Mandel SD1/SD2 Global Mandel

4 91 74 4 99 963 91 69 3 98 942 89 63 2 96 901 80 44 1 89 671/2 77 27 1/2 85 421/3 75 17 1/3 84 281/4 75 15 1/4 83 23

(continued)

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1972 Barhdadi and Froda

Table 2Continued

�2 = 25

SI = �SA+ SB�/2 SI = �SA+ SB�

SD1/SD2 Global Mandel SD1/SD2 Global Mandel

4 50 13 4 62 273 49 13 3 60 242 48 13 2 58 231 44 11 1 56 181/2 39 9 1/2 51 151/3 39 8 1/3 48 111/4 38 8 1/4 45 10

SI = 2�SA+ SB� SI = 3�SA+ SB�

SD1/SD2 Global Mandel SD1/SD2 Global Mandel

4 70 47 4 74 553 68 45 3 72 532 63 40 2 70 461 61 25 1 60 351/2 55 17 1/2 57 231/3 54 15 1/3 55 161/4 52 13 1/4 53 14

4. Data Examples

To illustrate the method described in the preceding sections, we analyzed threereal data sets. Testing for interaction is of interest in many fields but the biplotis extremely popular in the study of variety by environment interaction. Since wewanted to link the two approaches, our choice was naturally directed towards datasets in this field. The first data set was described in Sec. 1, and represents maize data(variety by year-location) from van Eeuwijk and Kroonenberg (1998). In the seconddata set, we have the number of newly formed cysts on 11 potato genotypes for fivepotato cyst nematode populations belonging to the species Globodera pallida (thisdata set is part of a larger table in Arntzen and van Eeuwijk, 1992). The numbersare means over five replicates. The last data set contains balanced data from alarge table in Piepho (1999) and displays an example from an official cook’s foot(Dactylis glomerata) variety trial in Bundessortenamt, Hannover, Germany. We willconcentrate mainly to illustrate how one can model and visualize the interactionas an appropriate polynomial model. We present the results obtained by applyingour global test of additivity and Mandel’s test of additivity to the three data sets.Moreover, we provide the three biplots (the one of the first data set was given inSec. 1).

Consider first the data described in Sec. 1. Our global test of additivity rejectsthe additivity, and we conclude that there is an interaction between genotypes andenvironments. On the other hand, Mandel’s test cannot reject the null hypothesis of

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Exact Test for Additivity in Biadditive Modelling 1973

Table 3Additivity test for r = 3, �2 = 5: comparison of our method

(global test) and Mandel’s test

SD = SD1 = SD2

SI = �SA+ SB�/2 SI = �SA+ SB�

SD/SD3 Global Mandel SD/SD3 Global Mandel

4 93 48 4 94 663 91 47 3 93 612 88 44 2 89 581 85 33 1 85 501/2 83 23 1/2 82 341/3 79 22 1/3 81 241/4 77 19 1/4 78 20

SI = 2�SA+ SB� SI = 3�SA+ SB�

SD/SD3 Global Mandel SD/SD3 Global Mandel

4 94 75 4 95 823 93 72 3 94 802 89 69 2 89 721 85 56 1 84 621/2 82 44 1/2 81 471/3 80 29 1/3 77 331/4 78 24 1/4 76 27

SD = SD1 + SD2

SI = �SA+ SB�/2 SI = �SA+ SB�

SD/SD3 Global Mandel SD/SD3 Global Mandel

4 88 44 4 89 583 86 43 3 87 552 85 33 2 85 501 85 33 1 85 501/2 77 19 1/2 78 201/3 72 8 1/3 74 121/4 73 7 1/4 74 9

SI = 2�SA+ SB� SI = 3�SA+ SB�

SD/SD3 Global Mandel SD/SD3 Global Mandel

4 89 69 4 89 723 87 64 3 87 702 85 56 2 84 621 85 56 1 84 621/2 78 24 1/2 76 271/3 72 15 1/3 74 151/4 73 11 1/4 73 11

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1974 Barhdadi and Froda

Figure 3. Biplot of potato cyst nematodes on potatoes data (Arntzen and van Eeuwijk,1992).

an additive model for these data (i.e., no interaction). Note that the biplot in Fig. 1shows that the row markers are roughly describing a parabola, and so it seems thatour global test is able to detect this interaction which is of higher degree.

Consider now the second data set. We decided to work in the logarithmic scaleand use yij = log�yij�; this makes sense because yij are integer counts. The biplot ofthis data set is presented in Fig. 3 and seems to suggest that the row markers aredescribing a curve that has an S shape and thus resembles a polynomial of degree 3.

Figure 4. Biplot of data on 25 cook’s foot varieties in trials at Bundessortenamt, Hannover,Germany (Piepho, 1999).

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Exact Test for Additivity in Biadditive Modelling 1975

Therefore, we fitted the following model:

yij = � + �i + �j + P1��i��1j + P2��i��2j + P3��i��3j + �ij�

i = 1� � p j = 1� � q

Both our global test and Mandel’s test for additivity reject additivity.For the last data set both our global test and Mandel’s test of additivity

conclude the presence of the interaction. These rejections are in agreement withthe fact that the biplot in Fig. 4 suggests an approximate linear structure of theinteraction as a function of the variety effect; Piepho (1999) arrives at the sameconclusion.

Finally, let’s note that Barhdadi (2003, Ch. 4) is proposing an algorithm bywhich one can assess if a higher degree polynomial model can be reduced to one oflower degree. For example, in the last data set, if one starts with a polynomial ofhigher degree, eventually it would be reduced to a linear one. This is consistent withthe fact that Mandel’s row regression model and our polynomial model of degree 1coincide. In particular, based on this algorithm, one could confirm that in the aboveexamples the polynomials are of degree 2, 3, and 1, respectively. On the other hand,the polynomial model is not a goal in itself but only a tool for proposing a test ofincreased power. Therefore, to keep the presentation simple, we did not introducethe algorithm in this article.

5. Discussion

In this article, we propose a polynomial model for the interaction in a two-waytable with one observation per cell. This polynomial model is a generalization ofMandel’s (1961) model. Two ideas are behind our approach. First, a polynomialis a natural way to approximate any functional relationship. Second, there is anextensive literature on the biplot technique, which is also used as a diagnostictool for the presence of the interaction. On the other hand, the biplot is aplanar representation, and in the usual classification (according to Bradu andGabriel, 1988) any relationship that goes beyond linearity corresponds to a scatterin a two-dimensional representation. Still, such a scatter can take the shape ofa parabola, an S-shaped function, etc, and be indicative of a specific type ofinteraction. Thus, we propose a model that allows to detect an interaction when thebiplot is not linear but still has a structured appearance. Moreover, as pointed outin the Introduction, the biplot applies to the exact model, and one may be interestedin separating model from noise.

Thus, we propose an additivity test, which, according to our simulation study,seems to be more powerful than Mandel’s test (up to 20% in some cases). It must benoted that, in contrast with other authors, for example Milliken and Graybill (1970)or Mandel (1969), ours is an exact test based on a statistic which follows a Fisherdistribution with easily computable degrees of freedom (see Table 1).

Based on our simulations and applications to real data examples, we can statethat our additivity test can prove extremely useful when interaction is present butthere is no linear term (indeed, see the second data set). To conclude, we feel thatthe present study indicates that simple, lower degree polynomial models for theinteraction can lead to tests and procedures of good use in applications.

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1976 Barhdadi and Froda

Appendix: Proof of Theorem 2.1

Let z = [�Iq − 1

qJq×q�⊗ �Ip − 1

pJp×p�

]y be the vector of residuals. The vector z

follows a normal distribution N��z�V� where

�z =(�z11� �

z21� � �

zp1� �

z12� �

z22� � �

zp2� � �

z1q� �

z2q� � �

zpq

)′�

with

�zij = E�zij� = E

( r∑l=1

Pl��i��lj + �ij − �i − �j + �

)=

r∑l=1

Pl��i��lj�

and the variance-covariance matrix V is

V = �2

[(Iq −

1qJq×q

)⊗(Ip −

1pJp×p

)]

Note that V is a singular matrix.We obtain

1�2

r∑k=1

SSk =1�2

r∑k=1

�′k�k

= 1�2

z′(Iq ⊗

r∑k=1

�k����k���′)z� (17)

1�2

SSE = 1�2

z′(Ipq×pq − Iq ⊗

r∑k=1

�k����k���′)z

Let �k = 1�2�Iq ⊗ �����k���

′�, � =∑rk=1 �k and � = 1

�2Ipq×pq −�. Then we see

that

1�2

r∑k=1

SSk = z′�z�1�2

SSE = z′�z = 1�2

(z′Ipq×pqz

)− z′�z

Further, note that � and � are functions of �. Moreover, �1/�2�∑r

k=1 SSk and�1/�2�SSE are quadratic forms in zij .

Our proof is based on results in Searle (1971, Sec. 2.7), which come to thefollowing statements in our case:

• the sums �1/�2�∑r

k=1 SSk and �1/�2�SSE are statistically independent if andonly if

(a) V�V�V = 0�

(b) V�V��z = V�V��z = 0�

(c) �′z�V��z = 0

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Exact Test for Additivity in Biadditive Modelling 1977

• these sums are, respectively, distributed as �2�r�q − 1���1� and �2��p− r −1��q − 1���2�, with

�1 =12

(�′z��z

)� �2 =

12

(�′z��z

)�

if and only if

(i) V�V�V = V�V�

(ii) �′z�V = �′

z�V�V�

(iii) �′z��z = �′

z�V��z

Similar formulas apply when � is replaced with �. Finally, note that r�q − 1� =tr��V� and �p− r − 1��q − 1� = tr��V�. The above conditions (a)–(c), (i)–(iii) holdbecause of the following equations:

�2k =

1�2

�k� (18)

V�k = �kV� (19)r∑

k=1

�k

r∑l=1

�l =1�2

r∑k=1

�k (20)

For detailed computations see Barhdadi (2003, Proof of Theorem 13).

Acknowledgements

This research received financial support from NSERC, Canada. We are thankful toP. Rousseau for suggesting this problem and important insights, and to A. Latour,M. Ahmad-Khan, R. Ferland, and an anonymous referee for useful remarks andsuggestions.

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