UNIVERSITÀ CATTOLICA DEL SACRO CUORE

ISTITUTO DI STATISTICA

Angelo Zanella – Laura Deldossi

Combined arrays in Taguchi approach: testing the hypotheses on the fixed effects when the

estimated noise factor variances are taken into consideration

Serie E.P. N. 110 - Dicembre 2002

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Combined arrays in Taguchi approach: testing the hypotheses on the fixed effects when the

estimated noise factor variances are taken into consideration(1)

Angelo Zanella(2) - Laura Deldossi Università Cattolica del S. Cuore, Istituto di Statistica, Milano

1. Introduction. A major undeniable original merit of Genichi Taguchi – a celebrated Japanese quality

expert, whose methods began to spread out all over the world in the eighties – has been

that of giving a new impetus to the use of statistical design of experiments in the

technological investigations aimed at improving a field product performance, see for

example Giovagnoli et al. (1994), and Kackar (1985) for a general review of Taguchi’s

approach and methods. Following Shoemaker et al. (1991), for instance, the basic ideas

in Taguchi’s approach to industrial experimentation can be summarized as follows: a)

products and their manufacturing processes are not only influenced by factors

(operating variables, etc.) which are controlled by the designers (controllable factors)

but also by the so-called noise factors, the effects of which happen at random, and

which may express randomly variable process environmental conditions, raw material

properties, factors which depend on the use of the customers, etc.; b) the variability of a

product quality characteristic (response) is obviously also related to that of the noise

factors; however the novelty is that, in general, we have to suppose that there exist also

controllable factors which have an effect on the response variance. Thus, process

(1) This preliminary version was presented and discussed at the Meeting of 6 March 2003 of Istituto Lombardo – Accademia di Scienze e Lettere and accepted for publication on the “Rendiconti dell’Istituto Lombardo”. (2)Angelo Zanella is the author of the methodological part of the paper, especially of sections 1., 2a),b), 3., 4.a); Laura Deldossi developed all aspects of the numerical application presented in the paper and of the

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designers and manufacturing operators can look for levels of controllable factors so that

the latter can not only ensure the aimed at response mean value but also allows reducing

the overall response variance. This gives rise to the so-called “robust product and

process parameter design” which has the goal of specifying operating conditions so that

quality characteristics of interest are little sensitive to variation due to noise factors. To

make the content of the paper easier to understand we shall consider a very simple

example (which can be considered a simplified version of the example developed later

on), that is appropriate to summarize the main traits and the related statistical methods

which are typical in the context under study.

a) We can start with the “product array” approach, from which the “combined array”

approach considered here will follow immediately. Let Y be a random variable

describing a product characteristic subject to random fluctuations (in the following

example, concerning the manufacturing of a rectangular plastic sheet, Y represents the

material strength loss under stress). Suppose that there is only one process controllable

factor x (for instance conditioning temperature) and that an experimental design is

carried out composed of m×n independent trials so that m levels x1, x2,…, xm, of the

controllable variable x are considered and for each condition xi, i=1,2,…,m, the

experiment is repeated n times, without any other known modification of the

experimental asset. Assume that the observable results Yij of the experiment can be

interpreted according to the following models:

Yij = β0 + β1xi + Zij + Eij = ηi + Zij + Eij (1.1)

i=1,2,…,m; j=1,2,…,n, where for simplicity we assume that Zij are stochastically

independent random variables, describing the effects of an unobservable noise factor Z,

with means zero and variances σ2Z(xi), ∀j, Eij are random variables (errors) independent

of each other and of Zij, with means zero and common variance σ2E, β0, β1 are real

unknown parameters.

The variance of Yij is obviously:

σ2Y(xi) = σ2

Z(xi) + σ2E,

corresponding tables and graphs; she is the author of sections 2.c), 5. and contributed to section 4.b). All work was discussed and agreed on by both authors.

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i=1,2,…,m, say, and if we indicate by S2i = ∑

j=1

n (Yij-Y

_i)2/(n-1), the usual unbiased

estimate of σ2Y(xi) obtained through the n replications, model (1.1) can be completed, for

instance, by a relationship of the type:

loge S2i = γ0 + γ1xi +γ11x

2i + E*

i (1.2)

i=1,2,…,m, where E*i still can be assumed to be zero mean independent random

variables, γ0, γ1, γ11 are unknown real parameters which allow us to connect the

controllable factor x with the variance of Y. Models (1.1), (1.2) give a simplified

example of the dual approach proposed by Vining et al. (1990), see also Magagnoli et

al. (1990) and Zanella (1992). In theory after fitting models (1.1), (1.2) to the

observations, typically by having recourse to the least square criterion, possibly

weighted, we can try to determine the level of the controlled factor x which minimizes

σ2Y(⋅) subject to an appropriate target constraint on the mean value η.

b) The original “array product” of Taguchi raised some criticisms followed by

corresponding improvements which led to the “combined array approach”:

1) a major criticism is that the product array approach requires a full factorial

experimental design for combining the controllable factors levels with the noise factors,

since it requires for each of the m sets of experimental conditions on the controllable

factors – that is for each “row” of the so-called “inner array” of the experiment – to

carry on a complete replication of an “outer array” with n trials, in which random

variation of the noise factors effects can be assumed. Thus the required number of trials

is n×m and it can become very big (e.g. if the inner array corresponds to a fractional 25-1

experiment in 5 controllable factors at 2 levels with 16 trials and an outer array also

with 16 trials is considered, to study the effects of say 5 additional noise factors, the

experiment would include 256 runs, etc.).

2) The product array approach allows what in my view is the most direct and coherent

use of the method, since the outer array may consist of n random independent replicates

for each set of conditions chosen for the controllable factors. It follows that the

statistical analysis of the observations can be based on two neatly distinct models –

named “dual models” in Vining et al. (1990) – a first model, like (1.1), for the study of

the effects of controllable factors on the mean value η of the response; a second model,

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like (1.2), for the study of the effects of the controllable factors on the response

variability. However a drawback of a model like (1.2) is that it does not allow us to

describe the origin of the latest mentioned effects, which seem naturally related to

interactions between controllable and noise factors.

3) There is a surprising ambiguity in defining the nature of the noise factors: on one

hand they are presented as factors whose behaviour is “by definition unsystematic and

possibly random” see for instance Alt (1988), p.166; on the other hand it is supposed

that the experimenter is capable of setting up conditions so that a noise factor can either

produce a negative or an intermediate or a positive effect, see for instance Kackar

(1985), p. 185 and also Park (1996), p. 281: this latter is speaking about a noise factor of

“three levels or conditions: good, normal or bad”.

The “combined array approach” unifies both the controllable and the noise factors in a

single “design matrix”, which can be possible under the assumption that the noise factors

can be reproduced as controllable factors in a laboratory or on the pilot plant level, see

Myers et al. (1997), p. 430, and also Khattree (1996), p. 189. In other words it is assumed

that for each guessed noise factor the experimenter can identify a controllable factor that

will reproduce the noise factor effect when its levels appear at random.

From the former point of view the study of both types of factors, controllable and noise

factors, can be conducted on the basis of a unique factorial design, typically fractional,

with a corresponding often drastic reduction of the number of trials and the possibility

of describing controllable by noise factor interactions explicitly (as a very recent, from a

certain point of view, problematic on a large scale application of the “combined array

approach” we mention Dasgupta et al. (2002)).

If we adopt the combined array approach in the example examined above, for the

statistical interpretation and analysis of the experimental results obtained in laboratory

or on pilot plant level, model (1.1) has to be replaced by another one, for instance, of the

following type:

Yi = β0 + β1xi + γξi + δξixi + Ei (1.3)

i=1,2,…,n, where n is now the total number of trials, ξi the chosen level of a

controllable variable ξ, which is the systematic counterpart of the random noise Z, Ei

still are stochastically independent random errors, with zero means and variance σ2E, β0,

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β1, γ, δ are unknown real parameters. Model (1.3) describes the results obtained in

“laboratory” through an appropriate designed experiment and it is instrumental in

pointing out the relevant factor effects. However, on the other hand, the former

experiment is conceptually related to a “second one” in which the levels of ξ happen at

random as values of an observable noise variable Z and which would produce some

results actually similar to those experienced for instance by a customer in the real life of

a product. As regards the simple example considered above a natural model for this

second experiment would be the following:

Yi = β0 + β1xi + γZi + δZixi + Ei (1.4)

where we preserve the assumptions on the stochastic independence of the random

variables Zi, Ei, etc. made in connection with (1.1), see Box e Jones (1992) for a general

presentation of this type of model. It is immediate to show that for model (1.3) the

response variance is

Var(Yi ) = σ2Y = σ2

E ,

while for model (1.4) we obtain:

Var(Yi ) = σ2Y(xi) = σ2

E + (γ + δxi)2v2, (1.5)

if we further assume that the sample noise variables Zi have constant variance v2,

besides having zero means. In the example presented in the paper one of the noise

variables is associated with irregularities in the plastic sheet from which the finished

product – a photographic film strip – is obtained. One of the sources of this irregularities

is related to an axial trend along the longer sheet dimension, that is to a spatial

coordinate ξ. When the product is retailed one can assume that the product unit received

by a costumer corresponds to a strip for which the level of ξ has been chosen at random

and that it actually is the value of a random variable Z. With regard to model (1.3) it is

of primary importance also to assess whether the effect β1 of the controllable factor x is

clearly distinguishable from the random fluctuations which affect the response Y. In the

framework of the laboratory experiment if β̂1 is the usual least squares estimate of β1

this can be carried out, under suitable assumptions (in particular that the errors Ei have

also a normal distribution) by having recourse to the well-known T-test. However this

standard procedure would only point out whether β1 is sufficiently large with regard to

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random fluctuations due to the errors Ei of variance σ2E. In the context of the combined

array approach what really interests whether the effect β1 too is really distinguishable

from the overall random errors including the noise effects, as it is shown by relationship

(1.4), whose corresponding possible variance increment is given by (1.5). This paper

has the purpose of going into the last subject. In section 2. the reference experimental

framework, based on a completely orthogonal design, and the statistical model used for

the interpretation of the corresponding observations are presented together with an

example which later on, in section 5. will allow us to illustrate the application of the

obtained theoretical results.

In section 3. a generic controllable factor effect, regardless of its being either a direct or

an interaction effect, say βh, is considered with the corresponding estimate β̂h obtained

by the least squares criterion applied to the results of the first laboratory experiment.

Then the true overall variance hσ2T of β̂h is considered, which takes the noise factor

effects into consideration and would arise when the observations were obtained

according to the second conceptual experiment. After some preparatory Lemmas an

unbiased estimate hσ̂2T of hσ

2T is established by having recourse to the sole first

experiment observations; adding the normality assumption for the random error

probability distribution also a condition is pointed out so that the probability distribution

of hσ̂2T becomes a noncentral chi-square distribution.

In section 4. two procedures, that have recourse to the observations of the first

laboratory experiment only, are proposed, aimed at assessing the distinguishable value

of an effect βh with respect to the real overall variability resumed by the standard

deviation hσT of the estimate β̂h. The first proposed method is a statistical test, whose

percent point, defining the acceptance/rejection regions of the hypothesis under study, is

the appropriate (1-α) percentile of a doubly noncentral F distribution. The second

method considers the least squares estimate β̂^

h of βh which would be obtained if the

second real experiment should also be carried out. The method suggests to have

recourse to a conditional indicator whose value approximates the probability of

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accepting the hypothesis βh=0 when applying to β̂^

h the usual T-test, conditionally to the

results of the first experiment and, thus, in particular, to the estimate β̂h obtained from

the same.

In section 5. the theoretical results are applied to the example already presented in

section 2..

2. A completely designed experiment and its conceptual counterpart. The

corresponding models.

a) First experiment: completely designed.

Consider an experimental design with n trials in p+q quantitative continuous

controllable factors aiming at the study of their direct and first order interaction effects

on a quantitative response Y. Denote by x1, x2, …, xp the variables describing the levels

of the first p factors and by ξ1, ξ2, …, ξq similar variables related to the other ones,

which in ordinary cases, that is outside the programmed experiment, would assume

random values and, thus, correspond to the so called noise factors (e.g. environment,

temperature, humidity, light intensity, etc.). Assume that the response value in a generic

trial can be expressed as follows:

Yi = β0 + ∑j=1

pβjxji + ∑

j=1

p-1 ∑s=j+1

p βjsxjixsi + ∑

j=1

qγjξji + ∑

j=1

q ∑s=1

p δjsξjixsi + Ei, (2.1)

where β0, βjs, γj, δjs denote sets of unknown real parameters, xji, xsi, ξji are the factor

levels in the i-th trial, i=1,2,…,n, n≥1+p⋅(p+1)/2+ q⋅(p+1), Ei are independent normal

variates with mean zero and constant variance σ2E , Ei∼N(0,σ2

E), i=1,2,…,n.

Now define the column vectors hx = (xh1,…,xhn)’, for h=1,2,…,p, and put hx = jx * sx for

h=p+s+(j-1)(j-2)/2, s=1,…,j-1, j=2,..,p, with * denoting Hadamard’s product of two

vectors (product component by component) and likewise hξ = (ξh1, …, ξhn)’, h=1,2,…,q,

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and define hξ = jξ * sx for h=q+(j-1)p+s, s=1,2..,p; j=1,2,..,q. In matrix notations (2.1)

becomes:

Y = Xβ + Ξ0γ + Ξ1δ1 + … + Ξqδq + E (2.2)

where the vectors and matrices are defined as follows: Y = (Y1,…,Yn)’, E = (E1,…,En)’;

if m=p⋅(p+1)/2, β=(β0,β1,…,βm)’ is a column vector (m+1)×1 summarizing the β’s

parameters: β=(β0,β1,…,βh,…,βm)’, for h≠0 with βh≡βjs defined and ordered according

to the above rule for h=p+1,…,m; γ=(γ1,…,γq)’; δh=(δh1,…,δhp)’, h=1,2,…,q; X is a

n×(1+m) matrix with the first column of unitary elements, say 0x, and the other columns

given by hx, h=1,2, …,m; Ξ0 is a n×q matrix with columns hξ, h=1,2,…,q, Ξj are n×p

matrices whose columns are the vectors hξ, h=q+(j-1)p+s, s=1,2,…,p, j=1,2,…,q,

respectively.

We state an assumption which will result essential later.

Orthogonality assumption. The “design matrix”:

(XΞ0 Ξ1…..Ξq)

has columns which are all mutually orthogonal.

As well known the linear efficient estimators of the unknown parameters of model (2.1)

are obtained by applying the least squares criterion which leads to the expressions listed

below.

In force of the assumed orthogonality we have precisely:

β̂h = ∑i=1

n

b

xhi /∑

i=1

nx2

hi Yi = hx′(Xβ + Ξ0γ + Ξ1δ1 + …… + Ξqδq + E)/ hx2 =

βh +∑i=1

n ( )xhi / hx2 Ei = βh + ∑

i=1

nahiEi (2.3)

h=0,1,…,m, where hx denotes the modulus of vector hx, n×1, and we put (xhi/ hx2)=ahi;

likewise:

γ̂j = ∑i=1

n

b

ξji /∑i=1

n

ξ2ji Yi = γj + ∑

i=1

n

( )ξji /jξ2 Ei = γj + ∑i=1

nbjiEi , (2.4)

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j=1,2,…,q, where jξ is the modulus of jξ and we put (ξji /jξ2) = bji;

δ̂js = ∑i=1

n

b

ξhi /∑i=1

n

ξ2hi Yi = δjs + ∑

i=1

n

( )ξhi /hξ2 Ei = δjs + ∑i=1

ncjsiEi (2.5)

where h=q+(j-1)p+s, j=1,2,…,q; s=1,2,…,p, hξ is the modulus of hξ and we put,

resuming the original definitions, (ξji xsi/∑i=1

n

ξ2jix2

si ) = cjsi.

Thus any estimator equals the value of the corresponding parameter plus an “error”,

which is a zero mean normal random variable, since it is a linear combination of the

zero mean normal random variables Ei, i= 1,2,…,n which have constant variance σ2E; it

ensues from the orthogonality assumption that all these errors are stochastically

independent. Obviously for a given realization of the experimental design the

parameters’ estimates are obtained by substituting the obtained observations yi for Yi,

i=1,2,…,n, in (2.3), (2.4), (2.5).

b) Second experiment: a conceptual counterpart of the first experiment.

In the real world the factors ξ1, ξ2, …, ξq are noise factors the levels of which take on

values at random. Therefore we can conceptually associate the former experiment with

another one which is exactly the same as far as the levels of the controllable factors are

concerned, while the levels of the noise factors are realizations of some random

variables Z1, Z2, …, Zq. Let us assume that the latter are zero mean normal variables

with constant variance v2, also stochastically independent of each other and of the errors

Ei, i=1,2,…,n, and that the n trials originate a simple random sampling from them which

will be described by the variables Z1i, Z2i, …, Zqi, i=1,2,…,n. Therefore Zji∼N(0,v2), ∀i,j

and all the Zji are stochastically independent of each other and of the errors. We note

that assuming Var(Zj) = v2j = v2, j=1,2,…,q, doesn’t represent a major restriction. In the

designed experiment ∑i=1

nξji = 0 ∀j, because of the orthogonality assumption. Thus the

levels ξji have to be thought as differences from a mean value ξ-

j . For an appropriate

choice of these differences one has to guess the possible ranges 2Rj , say, so that ξj ∈

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{-Rj, Rj}, j=1,2,…,q, by the symmetry of the noise factors distributions and 2Rj will also

be the ranges of the latter. Then we can put, e.g., vj≅Rj/3 and if we re-scale the ξj

variables as 3ξj/Rj, we can assume that for the corresponding random variables Zj we

have Var(3Zj/Rj) ≅ 1 ∀j, when they are expressed on a same conventional scale by

putting 3Zj/Rj . We shall actually introduce a parameter v2, which can also be different

from 1, in order to allow one to study the effects of possible adjustment of the scales

definition, in particular see later Remark 3.1. We can now summarize the preceding

point by the following assumption.

Scale assumption. The scales of the variables ξj related to the noise factors are chosen

in such a way that Var(Zj)=v2, ∀j = 1,2,…,q, with v2 a known constant.

For the second experiment we assume the following model:

Yi = β0 + ∑j=1

pβjxji + ∑

j=1

p-1βh ∑

s=j+1

pβjsxjixsi +

∑j=1

qγjZji + ∑

j=1

pβh ∑

s=1

qδjsZjixsi + Ε

∼i, (2.6)

i=1,2,…,n, where the parameters βj, γj and δjs are the same as in model (2.1);

furthermore the errors Ε∼

i have the same stochastic structure as in (2.1) with Var(Ε∼

i) = σ2E

∀i, and are assumed to be stochastically independent of Ei; furthermore, as stated above,

Zji, j=1,2,…,q, i=1,2,…,n, are zero mean independent normal variables with constant

variance v2 independent of Ε∼

i as well as of Ei, pertaining to the first experiment,

i=1,2,…,n.

The expression in square brackets of (2.6) shows the “increment” of random error which

arises in the second experiment.

According to model (2.6) the ordinary least squares estimator of a parameter βh

becomes, in view of (2.3) but with reference to (2.6):

β̂^

h = ∑i=1

n

b

xhi /∑

i=1

nx2

hi Yi = βh +∑i=1

n ahi Ε

∼i + ∑

i=1

n ahi

∑j=1

qZji (γj+∑

s=1

pδjsxsi) =

βh +∑i=1

n ahi Ε

∼i + ∑

i=1

n ahi Z′i (γ+Dxi) , (2.7)

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h=0,1,…,m, where in the last expression we introduced the vectors Zi = (Z1i,Z2i,…,Zqi)′,

xi = (x1i,x2i,…,xpi)′, γ and the matrix D with rows δ′h, h=1,…,q, see above (2.2). Now,

under the above assumptions and definitions we have:

Var(∑i=1

n ahi Ε

∼i) = σ2

E (∑i=1

n

a2hi ) = σ2

E /hx2 = hσ2E* ; (2.7’)

Var

∑i=1

n ahi Z′i (γ+Dxi) = ∑

i=1

n

a2hi (γ+Dxi)′E(ZiZ′i) (γ+Dxi) =

= v2∑i=1

n

(x2hi/hx4)(γ+Dxi)′(γ+Dxi) = hσ

2D, (2.7’’)

when E(⋅) indicates the expected value and the scale assumption is taken into

consideration. It follows from (2.7’) (2.7’’) that (2.7) can be given the following forms:

β̂^

h = β̂h +β̂*h = βh + hσ

E*U1+ hσ

.DU2 = βh + hσ

2E*+ hσ

2D U (2.8)

where in force of the normality and independence assumptions U1, U2 are standardized

normal random variables, independent of each other and, thus, also U is a standardized

normal random variable. Finally we also have:

Var(β̂^

h) = hσ2E* + hσ

2D = hσ

2T , (2.9)

h=0,1,…,m.

Remark 2.1. With reference to (2.6) note that in the second experiment the observed

variables Yi are heteroscedastic. In fact it is:

Var(Yi) = [σ2E + v2 (γ+Dxi)’ (γ+Dxi)] = 1/w2

i , (2.10)

say, an expression which depends on the experiment conditions xi in the i-th run,

i=1,2,…,n.

As well-known, should Var(Yi) be known, efficient estimates of βh, h=0,1,…,m, could

be obtained by having recourse to weighted least squares instead of ordinary ones, that

is by minimizing (Y-Xβ)’W (Y-Xβ), where W is the diagonal matrix with non null

elements w2i , i=1,2,…,n, equal to variance reciprocals, instead of (Y-Xβ)’(Y-Xβ). The

corresponding estimators would be:

β̂ = (X’WX)-1X’WY;

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however the variances (2.10) are unknown and should be estimated, for instance by

using the results of the first experiment, so that the weights w2i and the resulting

estimates would be only approximated. Also the observations of the second experiment

should be known, which to avoid is the purpose of the paper and it turns out to be much

more easily done if we refer to ordinary least squares.

c) An example

To illustrate the techniques here presented we consider the experiment reported in

Zanella and Cascini (1997) concerning the optimization of a mechanical characteristics

of a plastic film produced on industrial scale.

In particular the goal of the experiment was to assess the effects on the mean film

strength loss under stress η, measured in laboratory on a test strip as y, of two process

controllable factors: x1 (stretch ratio), x2 (conditioning temperature, oC ) also having

regard to two between product noise factors z1 and z2, related to manufacturing

imperfections. More precisely we have to mention that the finished product is obtained

by cutting an appropriate strip from a many meters long film roll, “whole roll”, which

after unwinding can be represented by a rectangle as it is shown in Figure 1 where also

a corresponding coordinate system (ξ1,ξ2) is indicated.

Fig. 1 Geometric representation of a “whole roll” of plastic material

Now technical experience suggested as unrealistic to consider a whole roll, when

unwound, as equivalent to a uniform plastic sheet and it seems more appropriate to

ξ2

0

Test strip

ξ1

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assume, with obvious simplification, that the local film strength loss under stress of a

small rectangle film strip depends linearly on the coordinates (ξ1,ξ2) of its centre. For

simplicity the test strip centre will be supposed to resume the strip location within the

whole roll and to define a uniform modification of the film strength loss under stress

within a given strip, also when this corresponds to a finished product unit. When such

units are retailed we can assume that a customer receives a finished product unit

corresponding to a strip the location of which in the original whole roll was chosen at

random. Thus while the values of the coordinates (ξ1,ξ2) can be chosen as desired by the

experimenter in the first part of the investigation and thus correspond to two

controllable factors ξ1, ξ2 the values of ξ1, ξ2 will appear at random in the successive

real life of the product as values of two random variables Z1, Z2.

In conclusion the experimental setting we have just illustrated complies with the

conceptual framework assumed in §2.a) above, with x1, x2 as fixed effects controllable

factors and ξ1, ξ2 as controllable counterpart of the noise factors, which are actually

described by the random variables Z1, Z2. Correspondingly a 24 full factorial design

with 16 runs in the four variables x1, x2, ξ1, ξ2 could be carried out, with conventional

units x1= (xA-1.425)/0.025; x2= (xB-80)/10 for the first two variables, while (-1,-1),

(-1,+1), (+1,-1), (+1,+1) express, in the conventional units chosen for ξ1, ξ2, the

coordinates of the vertices of the rectangle describing a whole roll, see Fig. 1.

According to Zanella and Cascini (1997) the completely designed experiment, with

conditions given in conventional units, together with the ensuing experimental results

are summarized in the following table.

Table 1: Experimental design and observed film strength loss under stress yi as reported in Zanella and Cascini (1997).

Design Factors Noise Factors (ξ1, ξ2)

x1 x2 (-1,-1) (+1,-1) (-1,+1) (+1,+1)

-1 -1 3.20 3.21 3.19 3.22

+1 -1 3.10 3.08 3.11 3.12

-1 +1 2.10 2.05 3.22 3.05

+1 +1 2.66 2.65 2.64 2.65

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1) First experiment: completely designed.

Model (2.1) for the example we are considering becomes

Yi=β0+β1x1i+β2x2i+β12x1ix2i+γ1ξ1i+ γ2ξ2i+δ11ξ1ix1i+δ12ξ1ix2i+δ21ξ2ix1i+δ22ξ2ix2i+Ei

where i=1,2,…,n=16 enumerate the trials. Later on we shall suppose that all

assumptions considered in §2.1, a) and b) hold true. The total number of unknown real

parameters in this case is 1+p⋅(p+1)/2+ q⋅(p+1)=10 that is less than the number of the

trials (n=16) so that all the parameters in the model are estimable; in particular σ2E with 6

degree of freedom.

The matrix X is

X =

+1 -1

+1 -1+1 -1+1 -1+1 +1+1 +1+1 +1+1 +1+1 -1+1 -1+1 -1+1 -1+1 +1+1 +1+1 +1+1 +1

-1 +1

-1 +1-1 +1-1 +1-1 -1-1 -1-1 -1-1 -1+1 -1+1 -1+1 -1+1 -1+1 +1+1 +1+1 +1+1 +1

where the first column of unitary element corresponds to 0x, while the other columns are

respectively 1x, 2x, 3x = 1x * 2x; the matrix Ξ0 = [1ξ , 2ξ ], Ξ1 = [3ξ , 4ξ ] and Ξ2 = [5ξ , 6ξ ]

with 3ξ = 1ξ * 1x; 4ξ = 1ξ * 2x; 5ξ = 2ξ * 1x; 6ξ = 2ξ * 2x.

Explicitly we have:

15

Ξ0 =

-1 -1

+1 -1-1 +1+1 +1-1 -1+1 -1-1 +1+1 +1-1 -1+1 -1-1 +1+1 +1-1 -1+1 -1-1 +1+1 +1

Ξ1 =

+1 +1

-1 -1+1 +1-1 -1-1 +1+1 -1-1 +1+1 -1+1 -1-1 +1+1 -1-1 +1-1 -1+1 +1-1 -1+1 +1

Ξ2 =

+1 +1

+1 +1-1 -1-1 -1-1 +1-1 +1+1 -1+1 -1+1 -1+1 -1-1 +1-1 +1-1 -1-1 -1+1 +1+1 +1

.

Putting β=(β0,β1,β2,β3)’ where β3=β12, γ=(γ1,γ2)’, δ1=(δ11,δ12)’ and δ2=(δ21,δ22)’ the

above model in matrix notations becomes:

Y = η + E = Xβ + Ξ0γ + Ξ1δ1 + Ξ2δ2 + E.

It can easily be verified that the design matrix (16×10)

(XΞ0 Ξ1Ξ2)

has columns which are all mutually orthogonal, so that the “orthogonality assumption”

is satisfied.

Applying the least squares criterion, we obtain the efficient estimators of the unknown

“linear” parameters of model (2.1) listed in Table 2.

From the Table 2 we see that, if we choose α=0.10 as significance level, two main

effects relating to “conditioning temperature” (x2) and to “vertical location of the test

strip in the roll” (ξ2) have an effect on the response mean value η. In addition there are

two large interactions referring to ξ2⋅x1 and ξ2⋅x2. Then, if the goal of the experiment is

to obtain a lower value of η, the best experimental conditions are: a) high values of

conditioning temperature (x2) and choice of the test strip in the lower side of the roll. It

has to be remembered that only x1 and x2 are process factors, always controllable, while

ξ2 is a noise factor which is controllable only in the programmed experiment.

16

Table 2: Parameter estimates and corresponding T-test with regard to the complete model (2.1)

Parameter Parameter estimate Standard Error hσ̂2E* t p-value

β0 2.890625 0.056673177 51.00517 3.81e-09

β1 -0.014375 0.056673177 -0.25364773 0.80823595

β2 -0.263125 0.056673177 -4.642849* 0.0035299595

β3 = β12 0.036875 0.056673177 0.65066054 0.53936760

γ1 -0.011875 0.056673177 -0.20953475 0.84096424

γ2 0.134375 0.056673177 2.3710511* 0.055441083

δ11 0.010625 0.056673177 0.18747846 0.85746479

δ12 -0.015625 0.056673177 -0.27570362 0.79202514

δ21 -0.130625 0.056673177 -2.3048823* 0.060692958

δ22 0.128125 0.056673177 2.2607697* 0.064479931

2) Second experiment: a conceptual counterpart of the first experiment.

In this second experiment we suppose that the factors (ξ1, ξ2) are noise factors the levels

of which take on values at random; for the data of our example it means that the test

strip is taken at random on the roll.

Then, the levels of the noise factors are realizations of two random variables (Z1,Z2) so

that Z1i∼N (0,v2), Z2i∼N (0,v2), ∀i, and Z1i are stochastically independent of Z2i and

errors.

So the matrices Ξ0 = [1ξ , 2ξ ], Ξ1 = [3ξ , 4ξ ] and Ξ2 = [5ξ , 6ξ ] become

17

Ξ0 =

z11 z21

z12 z22z13 z23z14 z24z15 z25z16 z26z17 z27z18 z28z19 z29

z110 z210z111 z211z112 z212z113 z213z114 z214z115 z215z116 z216

Ξ1 =

-z11 -z11

-z12 -z12-z13 -z13-z14 -z14+z15 -z15+z16 -z16+z17 -z17+z18 -z18-z19 +z19

-z110 +z110-z111 +z111-z112 +z112+z113 +z113+z114 +z114+z115 +z115

+z116 +z116

Ξ2 =

-z21 -z21

-z22 -z22-z23 -z23-z24 -z24+z25 -z25+z26 -z26+z27 -z27+z28 -z28-z29 +z29

-z210 +z210-z211 +z211-z212 +z212+z213 +z213+z214 +z214+z215 +z215+z216 +z216

while the matrix X is the same as in the first experiment.

We have assumed that the variables (Z1, Z2) satisfy the “scale assumption” that is

Var(Z1)= Var(Z2)=v2 with v2 known constant. This assumption is not a restrictive one.

In fact suppose that the unwound roll of film is 5000 cm long and 50 cm high.

In this case in the original units we have -2500≤ξ1≤2500, -25≤ξ2≤25 and we can put

approximately v1= R1/3= 833.3, v2= R2/3= 8.333.

Then we can always transform the variables so that their standard deviations v1 and v2

are the same. For example if we consider the variables ξ1/v1 and ξ2/v2 they have both

variance equal to v2=1.

3. Effects estimators in the second experiment: obtaining their variances through

the first experiment

The effects of the factors, which are not noise factors, are estimated by the ordinary

least squares criterion according to expressions (2.7), h=1,2,…,m, with corresponding

variances (2.9). We now shall establish a formula which gives unbiased estimators of

these variances by using the observations of the first experiment. With regard to (2.1) let

ηi = E(Yi), i=1,2,…,n, denote the expected value of a generic observed variable Yi and

18

η̂i the corresponding estimator obtained by substituting the estimators β̂h, (2.3), γ̂j (2.4),

δ̂js (2.5) for the unknown parameters in (2.1). For later reference we note that in force

of (2.4), (2.5) we have

E(γ̂j) = E

∑i=1

nbji (ηi+Ei) = ∑

i=1

nbjiηi = γj, (3.1)

E(δ̂js) = E

∑i=1

ncjsi (ηi+Ei) = ∑

i=1

ncjsiηi = δjs, (3.2)

j=1,2…,q; s=1,2,…,p.

1) Since the errors Ε∼

i in the second experiment have the same variance σ2E as the errors

Ei in the first one, as an unbiased estimator of σ2E we shall assume the usual one, based

on residuals after fitting model (2.3) to the data of the first experiment, that is:

σ̂2E = ∑

i=1

n

(Yi-η̂i)2 / ν (3.3)

where ν=n-[1+p(p+1)/2+q(p+1)]. It ensues, according to (2.7’), that:

hσ̂2E* = σ̂2

E/hx2 . (3.4)

2) To estimate hσ2D, (2.7’’), we first consider the “natural” estimator defined by

replacing the unknown parameters by their estimators. Thus we obtain from (2.4), (2.5),

by taking (3.1), (3.2) into consideration:

hσ̂2D =

v2

hx4 ∑i=1

n x2

hi

∑j=1

qA( γ̂j+δ̂'

jxi)2 =v2

hx4 ∑i=1

n x2

hi

∑j=1

qA

γj+δ'jxi + ∑

k=1

n

bjk + ∑

s=1

p cjskxsi Ek

2

=

= v2

hx4 ∑j=1

q

∑i=1

nb

∑

k=1

n xhi

bjk + ∑

s=1

p cjskxsi (ηk+Ek)

2

= v2

hx4 ∑j=1

q Q2

j , (3.5)

say, h= 0,1,…,m; j=1,2,…,q. We want to find the expected value of Q2 j , ∀ j=1,2,…,q.

In order to do so it is convenient to write the expressions in square brackets Q2 j in

matrix form:

19

xh1 0 .. .. 0

0 .. .. .. 00 .. xhi .. 00 .. .. .. 00 .. .. .. xhn

⋅

bj1 bj2 .. .. bjn

bj1 bj2 .. .. bjn.. .. .. .. .... .. .. .. ..

bj1 bj2 .. .. bjn

+

x11 x21 .. .. xp1

.. .. .. .. ..x1i x2i .. .. xpi.. .. .. .. ..

x1n x2n .. .. xpn

⋅

cj11 cj12 .. cj1k .. cj1n

cj21 cj22 .. cj2k .. cj2n.. .. .. .. .. .... .. .. .. .. ..

cjp1 cjp2 .. cjpk .. cjpn

(η+ E)=

= A⋅(Bj + X*Cj)(η + E) = Gj (η + E) (3.6)

with obvious notations and where Gj are n×n square matrices, η, E are the n×1 vectors

summarizing the Yi expected values and the errors Ei respectively.

Then we can write:

Q2 j = (η + E)’ G′jGj (η + E), (3.7)

j=1,2,…,q. We can now prove some lemmas useful later on.

Lemma 3.1. i. If all elements xhi , i=1,2,…,n, are different from zero, the rank of Gj in

(3.6) is rj=p+1, otherwise rj≤p+1.

ii. The probabilistic structure of any Q2 j is the following:

Q2 j = σ2

E ∑i=1

rj

.jλ2i jT

2i (3.8)

j=1,2,…,q, where jλi are the rj non-null eigenvalues of G′jGj assumed to be the first rj

ones, without loss of generality; jT2i are independent noncentral chi-square random

variables each with one degree of freedom and noncentrality parameters jτ2i =

(η’oi/ σE)2, where oi is the corresponding column of an orthogonal matrix O, n×n,

such that O′G′jGj O = Diag (jλ21…,jλ

2r , 0, …, 0).

Proof. i) Note that as it is well-known the rank of a matrix X*Cj, n×n, defined by (3.6),

is equal to C′jX*′X*Cj = C′jDiag(1/1x, 1/2x,…, 1/px)Cj, which in turn is equal to C′j, which

is p if jx≠0, j=1,2,…,p. This follows from the orthogonality assumption which, in

particular, ensures that the p columns of C′j are linearly independent. On the other hand

B′j has only a distinct column, which, once more by the orthogonality assumption, is

orthogonal to the linear p-dimensional manifold spanned by the columns of C′jX*′; thus

the mentioned column of B′j cannot be linearly dependent on the latter ones and it

follows that the rank of B′j + C′jX*′, and likewise of Bj + X*Cj, is p+1. When A has all

elements different from zero it has rank n and consequently Gj has rank p+1; in the other

20

cases, by the orthogonality condition, the rank r of A must be 2, at least, and the product

leading to Gj has at most rank equal to the smallest rank shown by the two factors,

which cannot exceed p+1.

ii) Since G′jGj is a symmetric matrix, as it is known, it is possible to find an orthogonal

matrix O = (o1, o2, …, on) so that:

Q2 j = σ2

E [η+ E]/σE]′O[O′G j′GjO]O′[η+ E]/σE] = σ2E ∑

i=1

rj jλ

2i jT

2i (3.10)

where jλ2i are the non-null eigenvalues of G′jGj, and jTi = [(η+ E)′/σE] oi , i=1,2, … ,rj

are independent normal random variables with means (η′oi)/σE and unit variance, since

oi are columns of an orthogonal matrix and we assumed that the errors E1,E2,…,En are

zero mean normal random variables with variance σ2E and independent of each other.

Thus jT2i are noncentral chi-square random variables with one degree of freedom, with

noncentrality parameter (η′oi)2/σ2E , and relationship (3.8) is proved.

Lemma 3.2. Suppose, for simplicity, that the matrices Gj, j=1,2,…,q, have rank

r=(p+1); it is shown that all Q2 j , j=1,2…,q, are stochastically independent of each

other and also independent of σ̂2E , (3.3), and β̂h , (2.3), respectively, i.e. of the unbiased

estimators of σ2E and βh in the first experiment.

Proof. 1) By assumption the square matrix, Gj =(Gj1,Gj2, …, Gjn)′, where G′ji , i=1,2,…,n,

are the vectors defined by the rows of Gj, n×n, has rank r=p+1. Thus in ℜn the former

vectors define a linear vector space V(r) of dimension r. With reference to comma ii. it is

easily seen that any vector of V(r) is orthogonal to any vector of the linear vector space

V(n-r) spanned by the (n-r) columns oi, i=r+1,…,n, of the orthogonal matrix O. In fact we

have:

O′G′j GjO =

o′1

.. o′i .. o′n

⋅ [ ]Gj1,..,Gji,..,Gjn ⋅

G′j1

..G′ji..

G′jn

⋅[ ]o1 ..o2 .. on = Diag (jλ21…,jλ

2r , 0, …, 0),

(3.11)

which implies that the resulting diagonal terms must vanish, for i>r, that is:

21

∑k=1

n (G′jkor+1)2 = … = ∑

k=1

n (G′jkoi)2 = … = ∑

k=1

n (G′jkon)2 = 0, (3.12)

and (3.12) ensures that G′jkoi=0 , i=r+1,…,n; k=1,2,…,n, that is, the vectors oi,

i=r+1,…,n – and any linear combination of the same – are orthogonal to the vectors Gjk,

and to any linear combination of the latter, which belongs to V(r) by definition. Now, as

well-known, the other columns oi, i = 1,2,…,r, of matrix O span the orthocomplement of

V(n-r), say V*(r) , which is the r-dimensional vector space of ℜn that includes all vectors

orthogonal to V(n-r). It follows that V(r) ⊂ V*(r) , but, since also the dimension of V(r) is r,

this implies that any basis for V(r) is such also for V*(r), i.e. V(r)⊃V*

(r)and, thus, V(r)=V*(r).

2) Now consider the vectors of ℜn corresponding to two generic rows, of two different

matrices Gj, Gj′, j≠j′, say G′ji , G′j′t , i, t=1,2,…,n. According to (3.5) their scalar product

is:

G′ji Gj′t = ∑k=1

nbxhixht

(bjk+∑

s=1

p cjskxsi ) (bj′k+∑

s=1

p cj′skxst ) = 0,

for a given pair i, t since in force of the orthogonality assumption:

∑k=1

n bjkbj′k = ∑

k=1

n bjkcj′sk = ∑

k=1

n cjskbj′k = ∑

k=1

n cjskcj′sk = 0

∀s, j≠j′. It follows that the vector spaces spanned by G′1i, G′2i, …, G′qi, i=1,2,…,n, say

jV(r), j=1,2,…,q, are all mutually orthogonal. Let us denote by jo′i , i=1,2,…,r, the set of

rows of the orthogonal matrix jO′ - that following (3.11) allows putting Q2j in diagonal

form – which corresponds to the non-null eigenvalues of G′jGj – and represents a basis

for jV(r) .

3) From the orthogonality assumption and with reference to (2.2) and (2.3) it also

ensues that:

xhi ∑k=1

nbahk (bjk+∑

s=1

p cjskxsi ) = 0

h=0,1,…,m; j=1,2,…,q; i=1,2,…,n, which implies that also the vectors of coefficients

ahk = xhk/hx2 , k=1,2,…,n, whose linear combinations define the random components of

the estimators β̂h , are orthogonal to the vector spaces jV(r), j=1,2,…,q, besides being

orthogonal to each other by the orthogonality assumption. Let us denote by o~′h,

22

h=0,1,…,m, the corresponding row vectors obtained through the transformation o~hk =

ahk/ha= xhi/hx, with ha=

∑k=1

n a2

hk 1/2

.

4) Further note that the (p+1) vectors, n×1, of ℜn, whose components are bjk, cjsk,

k=1,2,…,n; s=1,2,…,p, say bj, cjs , are orthogonal to each other by the orthogonality

assumption, and thus they are linearly independent; in particular each set (bj,cjs),

s=1,2,…,p, for a given j, spans of vector space of dimension (p+1) of ℜn . When the

rank of Gj is (p+1), which requires, in particular, that xhi≠0, i=1,2,..,n, it is easily seen

that they too form a basis for jV(r). In fact as it is shown by (3.5), (3.6), we have Gjk =

xhi (bj+∑s=1

p cjsxsi ), i=1,2,…,n, that is for each j the vectors Gji are linear combinations of

vectors bj, cjs and, by the hypothesis on the rank of Gj, there are (p+1) linear

independent vectors Gji, which, as we saw, span jV(r) = jV(p+1), in this case. It follows

that also the (p+1) vectors bj, cjs , s=1,2,…,p, form a basis for the vector space jV(p+1),

thus belonging to it, and this holds for j=1,2,…,q.

5) Now consider the following, n×n, orthogonal matrix:

O~

′ =

o~ 0

.. o~ m 1o′1 ..

1o′p+1 ..

qo′1 ..

qo′p+1

O′*

=

O

~′1

O′* (3.13)

where the first m+1+q(p+1) = 1+p(p+1)/2+q(p+1) rows are the orthogonal vectors

defined above and their number equals that of the parameters in model (2.1), O′* is a

further set of ν =n-[m+1+q(p+1)] rows which are orthogonal to each other and with

respect to the other ones. We note that in (3.10) we can assume O′=O~

′, (3.13), to put Q2 j

in diagonal form, whichever is j, because the orthogonal rows jo′i , i=p+2,…n, which are

23

the eigenvectors of matrix G′j Gj, corresponding to the null eigenvalues, can be arbitrarily

chosen, provided that they are orthogonal to jo′i , i=1,2,…,p+1, and the order change

only induces a corresponding displacement of the non-null diagonal elements of O′G′j

GjO (see e.g. Zurmühl (1961), p.167-168, 190). Thus we can conclude that: 1) O~

′(η+

E) still is a set of n independent normal random variables; 2) correspondingly the

quadratic forms Q2 j , j=1,2,…,q, are stochastically independent, since they are

functions of q independent sets of independent normal variables; 3) they are also

independent of any of the quantities hxβ̂h, h=0,1,…,m (2.3) – obtained from the first

(m+1) rows of matrix O~

′ – since these quantities are normal variables independent of

each other and also of any of the sets of normal variables used in the definitions of the

quantities Q2 j, j=1,2,…,q, implying the same property for the β̂h; 4) finally the rows of

matrix O~

′1 span the vector space to which the mean values ηi, i=1,2,…,n, are constrained

by the linear model (2.1) – recall the remark on the vectors bj, cjs – so that O′*η=0. This

implies that (3.3) is equivalent to:

σ̂2E = E′O*O*′ E/ν

which is distributed (as a (σ2E/ν)χ2

ν random variable) independently of Q2 j , j=1,2,…,q,

and of β̂h, h=0,1,2,…,m. Thus the statements of the lemma are proved.

Lemma 3.3. If |xhi|=1, i=1,2,…,n, ∑i=1

n b2

ji =1/jξ2 =1/n, ∑i=1

n c2

jsi=1/jsξ2=1/sx2, js=q+(j-1)p+s,

s=1,2,…,p, j∈{1,2,…,q}, see (2.4), (2.5), the matrix G′jGj is idempotent of rank (p+1)

and Q2 j /σ

2E has a noncentral chi-square distribution with (p+1) degrees of freedom and

noncentrality parameter:

τ2j = ∑

i=1

n (γj+δ'

jxi)2/σ2E. (3.9)

Proof. By using the notations defined through (3.6) we obtain:

G′j Gj = (B′j + C′j X*′)A′A(Bj+X*Cj) = B′j Bj + B′j X*Cj + C′j X*′Bj + C′j X*′ X*Cj =

= B′j Bj + C′j X*′ X*Cj , (3.14)

24

since the diagonal matrix A′A has unitary elements, x2hi =1, i=1,2,...,n, thus coinciding

with the identity matrix, n×n and all elements of matrix B′j X*, n×p, are zero, because, by

(3.6), any row of B′j has constant elements and is multiplied by a coloumn hx,

h=1,2,...,p, of X* which is orthogonal to the vector with unitary elements 0x; obviously

the same holds for the transpose X*′Bj . From (3.14) it ensues that:

G′j Gj ⋅ G′j Gj = (B′j Bj + C′j X*′ X*Cj ) ⋅ (B′j Bj + C′j X*′ X*Cj ) =

B′j (Bj B′j) Bj +C′j X*′ X*(Cj C′j)(X*′ X*)Cj , (3.15)

because all elements of matrix BjC′j, n×p, are zero as the rows of Bj are all “orthogonal”

to the coloumns of C′j , owing to the orthogonality condition, and obviously the same

holds for the transpose Cj B′j.

Now by the definition of Bj given in (3.6) and recalling (2.4) we have:

(Bj B′j) Bj =

bj1 .. bjn

.. .. .. bj1 .. bjn

bj1 .. bj1

.. .. .. bjn .. bjn

Bj = (1/jξ2)U*n Bj= (n/jξ2) Bj= Bj (3.16)

if U*n denotes an n×n matrix with all elements equal to unity and we take the hypothesis

jξ2 = n into consideration. Furthermore by the definitions of Cj and X* given in (3.6) and

the orthogonality condition, it follows that:

(Cj C′j) (X*′ X*) =

1/j1

ξ2 .. 0 .. .. .. 0 .. 1/jp

ξ2

1x2 .. 0

.. .. .. 0 .. px2

= Ip, (3.17)

if Ip denotes the identity matrix p×p and the hypothesis jsξ2 = sx2 is taken into account.

By inserting (3.16), (3.17) into (3.15) and in view of (3.14) the desired result:

(G′j Gj)2 = G′j Gj ⋅ G′j Gj = G′j Gj

is thus obtained , that is G′j Gj is an idempotent matrix and of rank (p+1), since xhi=1

(see remarks at the end of comma i.).

Finally, as Yi = ηi+Ei , i =1,2,…,n, are assumed to be independent normal random

variables with constant variance σ2E and mean ηi , a well-known result, see e.g. Graybill

(1961), p. 83, ensures that Q2 j/ σ2

E has a noncentral chi-square distribution with (p+1)

degrees of freedom and noncentrality parameter:

τ2 j = (η′G′j Gj η)/ σ2

E = ∑i=1

n x2

hi(γj+δ'jxi)2/σ2

E, (3.18)

25

if we recall (3.1), (3.2) and the second and third relationship (3.5). Since by hypothesis

x2hi =1, ∀ i, (3.9) follows from (3.18), thus the proof of the lemma is complete.

Now we can obtain a first main result.

Proposition 3.1. i. With reference to relationships (3.5), (3.6), (2.7’), (2.7’’) and the

related hypotheses it is shown that the expected value of hσ̂2D has the following form:

E(hσ̂2D) = (v2/hx4) ∑

j=1

q E(Q2

j) = (v2/hx4) ∑j=1

q

∑

i=1

n x2

hi (γj+δ'jxi)2 +σ2

E tr (G′j Gj) =

= hσ2D + hσ

2E* (v2/hx2) [∑

j=1

q tr(G′j Gj)] , (3.19)

where tr(⋅) denotes the trace of a matrix (sum of the diagonal elements).

ii. It follows that an unbiased estimator, hσ̂2T , say, of Var(β̂

^h) = σ2

E* + hσ2D, (2.9), which is

the variance of the ordinary least squares estimator β̂^

h, (2.7), of the fixed effect

parameter βh, in the second experiment, is given by:

hσ̂2T = hσ̂

2E* [1-(v2/hx2)∑

j=1

q tr(G′j Gj)] + hσ̂

2D =

= σ̂2

E

hx2 ⋅[1-(v2/hx2)∑j=1

q tr(G′j Gj)] +

v2

hx4 ∑j=1

q

∑

i=1

n x2

hi (γ̂j+δ̂'jxi)2 (3.20)

where relationships (3.3), (3.4), (3.5) are taken into account and we can suppose that

[1-(v2/hx2)∑j=1

q tr(G′j Gj)]≥0, since v2 is substantially arbitrary, see Remark 3.1 below, so

that the estimator is certainly positive, i.e. admissible. Furthermore, assuming for

semplicity, rj=p+1 ∀j, that is that the rank of any matrix G′j Gj is maximum, Lemma 3.2

ensures that the estimator hσ̂2T is stochastically independent of β̂

^h = β̂h+β̂*

h , (2.8), also if

we assume that β̂h is calculated by means of the results of the first experiment through

(2.3).

Proof. i. From (3.5), (3.10) it follows that:

E(hσ̂2D) =

v2

hx4 ∑j=1

q E(Q2

j) = k2σ2E ∑

j=1

q ∑i=1

rj

jλ2i E (jT

2i ) =

k2σ2E ∑

j=1

q ∑i=1

rj

jλ2i [1+(=(η′joi/σE)2] = k2σ2

E {∑j=1

q [∑

i=1

rj

jλ2i (=(η′joi /σE)2] + ∑

j=1

q tr(G′j Gj)} =

26

= k2[σ2E (∑

j=1

q (η′G′j Gj η)/σ2

E + σ2E ∑

j=1

q tr(G′j Gj)], (3.21)

where k2= (v2

hx4 ); the notation joi is used, cfr. Lemma 3.2, 2), to indicate the eigenvector

of matrix G′jGj pertaining to jλ2i ; the fourth expression (3.21) is obtained by recalling that

the expected value of a noncentral chi-square variate, with ν degrees of freedom and

noncentrality parameter, τ2 j is ν+τ2

j and that in our case ν=1, τ2 j =(η′joi/σE)2; the fifth

expression (3.21) takes into account that the trace of a matrix equals the sum of its

eigenvalues; in the last expression the original coordinates system is restored, see

(3.10), which, for E =0, leads to the equivalent form used in (3.21).

On the other hand:

∑j=1

q (η′G′j Gj η) = ∑

j=1

q

∑i=1

n x2

hi (γj+δ'jxi)2 (3.22)

according to (3.1), (3.2), see also (3.5). Combining (3.21) with (3.22) we obtain (3.19)

as desired.

ii. Consider the expectation E(hσ̂2T). Remembering that E(hσ̂

2E*)=σ2

E/hx2 and that E(hσ̂2D)

has the expression given in (3.19), we see that the bias of hσ̂2D and the term

σ2E(v2/hx4)∑

j=1

qtr(G′j Gj) cancel out so that:

E(hσ̂2T) = hσ

2E* + hσ

2D = Var(β̂

^h)

as required.

With reference to (2.7) (2.8), we see that β̂h, even if calculated by means of the results

of the first experiment – Ei replace E∼

i, i=1,2,…,n – is stochastically independent of hσ̂2T,

since by Lemma 3.2, the latter quantity is a function of random variables independent of

β̂h. On the other hand β̂*h is a function of the noise variables Zi, which are assumed to be

independent of the errors E∼

i , Ei, i=1,2,…,n; thus β̂*h is independent of σ̂2

T (and of β̂h

whichever is the reference experiment). It follows obviously that the sum: β̂^

h = β̂h + β̂*h

remains stochastically independent of hσ̂2T. This completes the proof of the proposition.

27

The following corollary to Proposition 3.1 shows that, in particular, by an appropriate

choice of the scales for the noise factors, the estimator hσ̂2T can result proportional to a

noncentral chi-square variable. We shall use the notations χ2ν, χ2

ν(τ2) to indicate chi-

square random variables, respectively central and noncentral, with noncentrality

parameter τ2 and with ν degrees of freedom.

Corollary 3.1. Under the hypotheses of Lemma 3.3, which in particular ensure that

(Q2j /σ

2E) is a χ2

p+1(τ2j ) variate, suppose that the scale of the noise factors ξ1, ξ2, …, ξq are

chosen in such a way that the constant variance v2 of the noise variables Zj, ∀j, assured

by the scale assumption, has the value:

v*2 = n/(n-m-1) (3.23)

where n, m+1 are the number of observations and that of the parameters βh , pertaining

to the fixed effect factors, respectively. Then we have:

hσ̂2T = (σ2

E/n) [χ2ν* (τ2 )]/ν* (3.24)

that is the unbiased estimator hσ̂2T , (3.20), is proportional to a noncentral chi-square

variate with ν*=(n-m-1) degrees of freedom and noncentrality parameter τ2 defined by:

τ2 = ∑j=1

q ∑i=1

n (γj+δ'

jxi)2 /σ2E . (3.25)

Proof. By Lemma 3.2 and 3.3, since, in particular, hx2= ∑i=1

nx2

hi = n and, as well-known

the trace of an idempotent matrix of rank (p+1), as are the matrices G′j Gj, is p+1, we can

give to (3.20) the following form:

hσ̂2T = (σ2

E/nν)[1 - v2

n q(p+1)] χ2ν +

v2σ2E

n2 [χ2q(p+1) (τ2 )], (3.26)

where, because of (3.3), ν=n-[1+p(p+1)/2 + q(p+1)]=n-[m+1+q(p+1)] and the second

term on the right side of (3.26) follows from the fact that also independent noncentral

chi-square variates, as are Q2 j/σ

2 E by Lemma 3.2, have a distribution reproductive under

convolution. This implies that their sum leads to a similar variate with degrees of

freedom as well as a noncentrality parameters given by the corresponding sums so that

in our case the values are q(p+1) and τ2, (3.25), respectively.

By (3.23) we obtain:

28

[1- v*

2

n q(p+1)] = [1- n

n(n-m-1)q(p+1)] = n-m-1-q(p+1)

n-m-1 = ν

n-m-1 >0 ;

v*

2

n2 = n

[n2(n-m-1)] = 1

[n(n-m-1)] , (3.26’)

and (3.26) becomes

hσ̂2T = (σ2

E/n)[χ2ν +χ2

q(p+1) (τ2 ) ]/(n-m-1) =(σ2E/n) [χ2

ν* (τ2 )]/ν*

using once more the property that the chi-square distribution is reproductive under

convolution and thus ν* = n-[m+1+q(p+1)] + q(p+1) = n-m-1, while τ2 , (3.5), remains

unchanged, since the first variable in the sum is a central chi-square. Thus (3.24) holds

and the corollary is proved.

Remark 3.1. The further constraints on the scales of the noise factors introduced in

Lemma 3.3 and Corollary 3.1 deserve some comments. In general the noise factors

correspond to certain variables with values ξ*j , j=1,2,…,q expressed on “physical scales”

according to appropriate measurement units, like centimeter, degree, etc.. The scale

assumption of § 2, b) leads to introduce for each noise factor a conventional unit

(indicated as uconvj later on) defined through the relationship:

v × uconvj = Rj/3, (3.27)

j=1,2,…,q, where v is a positive constant, which represents the approximate

conventional standard deviation common to all noise factors when their values happen

at random as values of the variables Zj; Rj is the half range of the expected differences

(ξ*j - ξ*

j0) from a central value ξ*j0. It follows from (3.27) that

uconvj = Rj/3v, (3.28)

j=1,2,…,q, express the conventional units through the original physical ones. In the first

experiment the conventional levels of the noise factors are defined as:

ξji = (ξ*

ji - ξ*j0)

Rj/3v (3.29)

j=1,2,…,q; i=1,2,…,n, with ξ*j0 coinciding with the average of the ξ*

ji values, i=1,2,..,n,

because of the orthogonality condition. Note that the range of the possible conventional

values ξji is {-3v,3v}, which corresponds to {-Rj, Rj}, j=1,2,…,q, in the original units by

(3.27). Now consider the constraints jξ2=n, j=1,2,…,q, imposed by Lemma 3.3, which is

tantamount to requiring that:

29

∑i=1

n ξ2

ji/n1/2

= 1 (3.30)

j=1,2,…,q, that is, that the standard deviation of the noise factors conventional levels is

equal to 1. This only apparently puts a limit to the spread of the real factors levels if the

choice of v remains arbitrary. In fact a change in v, from v to v*, say, only represents a

different definition of the conventional units according to (3.28) (and correspondingly

the values of the parameters γj, δjs in model (2.6) will be multiplied by (v/v*) so that the

model is actually unchanged. Thus, if for example, we choose v=1/3, the conventional

ranges become {-1,1} and the constraints (3.30) allow one to choose the real levels of

the noise factors in such a way that they cover their physical ranges. The situation is

different if we have to fix the value v as required by Corollary 3.1, which imposes to

choose v= n/(n-m-1). This value is larger than 1, which ensures that the constraints

(3.30) are compatible with the allowed range {-3v,3v}. However in this case they

impose a true restriction of the spread of the noise factors real levels, since, in

consequence of conditions (3.30) and of (3.29), their standard deviations in the original

units must be (1)⋅ (n-m-1)/n(Rj/3), that is, smaller than those, equal to Rj/3 in the same

units, assigned through (3.27) to the corresponding random variables Zj. This is not a

serious drawback if the model is exactly linear in the variables ξ*j . In the example

presented in the paper n=16, m+1=4 and we get 0.289Rj.

4. Testing whether the effects of the systematic factors may be detected when the

noise factors variability is taken into account.

We shall discuss two proposals.

a) The case when Corollary 3.1 holds. Consider the statistic:

β̂h

hσ̂T

2

≅

(n-m-1)χ21(

nβ2h

σ2E

)

χ2n-m-1 (τ2 )

= F1,ν*

nβ2

h

σ2E

, τ2 , (4.1)

30

ν*=n-m-1, with regard to (2.3), in particular to the implied assumption xhi=1,

i=1,2,…,n, and to (3.23), (3.24) with the corresponding notations, also remembering

that β̂h is a normal variate with mean βh and variance σ2E/n.

Since according to Lemma 3.2, β̂2h and hσ̂

2T are independent random variables, the

variable (4.1) follows a so-called doubly noncentral F-distribution with 1, ν* degrees of

freedom and noncentrality parameter β∼2

h = nβ2h /σ

2E, say, and τ2 (3.25). For a review of

the subject see Johnson and Kotz (1992), Chp. 30, § 7, p.499; more specifically in the

following, for the numerical applications, we shall have recourse to the software

DATAPLOT, and in particular to the commands DNFCDF, DNFPPF written on the

basis of the paper of Revee (1986), who refers to the series representation given in

Bulgren (1971).

We note that under the assumptions of Corollary 3.1 the variance of β̂^

h, (2.9), related to

the second experiment is:

Var(β̂^

h) = (σ2E/nν*)(ν*+τ2)= (σ2

E/n)(1+τ2/ ν*) =

(σ2E/n)(1+

n hσ2D

σ2E

) = (σ2E/n)(1+ζ2), (4.2)

say; this follows directly by recalling the definitions (2.7’), (2.7’’), (2.9) and expressing

hσ2T as the expected value of the unbiased estimate hσ̂

2T obtained from (3.26). We are

interested in assessing whether the effect βh can be distinguishable from random

fluctuations when their variance is the real one, which is present in the second

experiment and is given by (4.2). Thus, for instance, we are led to test a hypothesis like

the following:

H0 =

|βh|

σ2E/n+hσ

2D

= n |βh/σE|

1+ζ2 ≥ c0

|βh|/σE ≥ |βh0|/σE

0≤ζ2≤ζ20

(4.3)

where c0 is a chosen positive “threshold value of distinguishability”, for instance c0 =

1.5 or 2; ζ20 =

hσ2D

σ2E/n

is a chosen variance increment for the estimator of the effect βh when

31

the noise factors happen at random; βh0/σE = 1+ ζ2

0 c0

n is the standardized threshold

value of a distinguishable factor effect when ζ2 =ζ20 , see Fig. 2.

Fig. 2 Graphic illustration of the hypothesis H0 in the parameter space: dashed strip. The

region above the thin line corresponds to a subset of the parameter space where the

probability of accepting H0 is larger than 1-α=0.90; it follows that, if we assume the

Neyman-Pearson approach, the test is biased in the regions A, B. The bold line is the contour

of the region defined by considering in (4.3) the first condition only, for c0=1 and n=16.

For testing the hypothesis (4.3) we propose a test of significance which accepts H0 if:

β̂h

hσ̂T

2

≥ F1,ν*,1-α (c∼20 , ν*ζ2

0 ), (4.4)

otherwise rejects it, where F1,ν*,1-α ( ⋅ , ⋅ ) is the value which is exceeded with probability (1-α)

by the doubly non central F-variate with 1, ν∗= n-m-1 degrees of freedom and values of the

noncentrality parameters which are defined as β∼2

h=n⋅(βh0/σE)2=(1+ζ20)c

20= c∼2

0 and τ2=ν* hσ

2D

σ2E/n

= ν*ζ20 respectively, see (4.3). The test (4.4) is justified by the following proposition.

Proposition 4.1. Relationship (4.4) defines a test of significance of size α for testing H0

since

ξ0 = 4

c∼0=nβh0

σE

= 4.12

A

B

32

supθ∈Θ

P[

β̂h

hσ̂T

2

< F1,ν*,1-α (c∼20 , ν*ζ2

0 )] = α (4.5)

where P(⋅) denotes the probability, θ is the parameter vector [(βh/σE), ζ], Θ is the

domain in the parameter space defined by (βh/σE )≥(βh0/σE), 0≤ζ≤ζ0.

Proof. With reference to our notations, the theorem given by Scheffè (1959), p.136,

ensures that the probability:

P

ν*χ2

1(β∼2

h )χ

2ν* (τ2 )

> F1,ν*,1-α (c∼20 , ν*ζ2

0 ) = P(E0) (4.6)

for a fixed β∼2

h, is a strictly decreasing function of τ2, that is of ζ2, according to

(4.2). Thus, if we consider the low boundary of the domain Θ, defined by

(βh/σE)=(βh0/σE), 0≤ζ≤ζ0, see Fig. 2, we have that the probability of the

complementary event E_

0, defined by considering ≤ instead of > in (4.6), is an increasing

function of τ2. Correspondingly the supremum of P(E_

0) is the value of P(E_

0) at ζ=ζ0,

which is α because of the choice F1,ν*,1-α (c∼20, ν*ζ2

0).

It follows that for any 0≤ζ<ζ0: 1-P(E)=P(E-)<α. Now for such a ζ suppose that βh0/σE

is increased to obtain a larger value βh/σE, with a corresponding increment of the

noncentrality parameter, say from c∼20 to β

≈2h . We now extend Scheffe’s argument as

follows. The corresponding probability (4.6) can be equivalently written as:

P[(W+β≈

h)2> F1,ν*,1-α (c∼20 , ν*ζ2

0 ) χ2ν* (ν*ζ2 )/ ν*] (4.7)

where W is a normal random variable with mean zero and unit variance, independent of

χ2ν* (⋅). For a given value of this latter variable, (4.7) expresses a conditional probability,

which is the same as the unconditional probability because of the independence, we

have just remarked. For simplicity let us denote the expression on the right side of the

inequality in (4.7) by a2. Thus:

P(E∼

) = P[(W+β≈

h)2> a2] = P {(W+β≈

h)∉[-a,a]} (4.8)

33

is the probability that a normal variate of unit variance and “centered” at β≈

h assumes

values outside of the given interval [-a,a] and it is known that this is a function which

increases strictly when |β≈

h| increases. So if we multiply the probability (4.8) by the

probability density of χ2ν* (ν*ζ2 ), which does not depend on βh/σE, and integrate over

all possible values, to obtain the unconditional probability, we find a value P(E∼

)>P(E)

and still more 1-P(E∼

)=P(E∼_

)<α. Thus in any point of Θ, which is not on the boundary,

we have a probability of rejecting H0 smaller than α. For the aforesaid and continuity

considerations, which allow replacing < with ≤ in (4.5) we can conclude that this

probability has α as its supremum and (4.5) holds, as it was to be proved.

Remark 4.1. A thorough investigation of the properties of the test defined by (4.4),

when the Neyman-Pearson’s approach is considered, will be presented in another paper.

We only point out that the results obtained in the proof of Proposition 4.1 allows one to

forecast that regions like A and B of Fig. 2 will correspond to a subset of the parameter

space where the test is biased, that is, the probability of falsely accepting H0 could be

larger than (1-α). How extensive this subset is will be one of the subject of future

investigation. However Fig. 2 let one guess that the region where the test is unbiased

will not differ much from the set of true interest in the parameter space which is

composed of all distinguishable values and is defined by considering in (4.3) the first

condition only (see bold line in Fig. 2).

The following Table I gives an example of the elements which are required for the

practical application of the test (4.4). It refers to a 2-levels experimental design 24 of

n=16 runs in 4 factors, of which 2 are noise factors. The critical values F1,ν*,1-α (c∼20,ν*ζ2

0 )

are given for the values c0 = c∼0/ 1+ζ20 = 1, 1.5, 2, of the “distinguishability threshold”

and the values ζ20/n =

hσ2D

σ2E

= 0.25, 0.7, 1, 2, 4 of the error variance increments; the level

of significance α = 0.05, 0.10 are considered. We recall that, since we are applying

Corollary 3.1, the scale of the noise factors is supposed to be chosen in such a way that

Tab

le I.

Cri

tical

val

ues F

1,ν *

,1-α

(c∼2 0 , ν

* ζ2 0 ) n

eede

d fo

r app

lyin

g te

st (4

.4);

c∼ 0=c 0

1+ζ2 0,

ζ2 0=n(

hσ2 D/σ

2 E);

n =1

6, p

=2, q

=2; v

=n/

ν* ,

α=0

.05;

0.1

0.

α=0

.05

c 0

1 1.

5 2

hσ2 D/σ

2 E

0.25

0.

7 1

2 4

0.25

0.

7 1

2 4

0.25

0.

7 1

2 4

ν* V

12

1.16

0.

0751

5 0.

2763

0.

3561

0.

5023

0.

6261

0.

2368

0.

5564

0.

6677

0.

8641

1.

0259

0.

4506

0.

8654

1.

0035

1.

2432

1.

4380

13

1.11

0.

0751

3 0.

2766

0.

3565

0.

5028

0.

6267

0.

2370

0.

5572

0.

6687

0.

8653

1.

0271

0.

4514

0.

8672

1.

0055

1.

2453

1.

4399

14

1.07

0.

0751

2 0.

2768

0.

3569

0.

5032

0.

6272

0.

2372

0.

5580

0.

6696

0.

8664

1.

0281

0.

4520

0.

8687

1.

0072

1.

2472

1.

4416

15

1.03

0.

0751

1 0.

2770

0.

3571

0.

5036

0.

6276

0.

2373

0.

5587

0.

6704

0.

8673

1.

0290

0.

4525

0.

8701

1.

0087

1.

2488

1.

4431

α=0

.10

c 0

1 1.

5 2

hσ2 D/σ

2 E

0.25

0.

7 1

2 4

0.25

0.

7 1

2 4

0.25

0.

7 1

2 4

ν* V

12

1.16

0.

1833

15

0.39

61

0.46

92

0.59

67

0.70

06

0.41

94

0.72

28

0.82

07

0.98

80

1.12

20

0.69

32

1.07

18

1.19

10

1.39

28

1.55

29

13

1.11

0.

1833

46

0.39

65

0.46

96

0.59

72

0.70

11

0.41

97

0.72

37

0.82

18

0.98

92

1.12

30

0.69

43

1.07

36

1.19

30

1.39

47

1.55

55

14

1.07

0.

1833

72

0.39

68

0.46

99

0.59

77

0.70

16

0.42

01

0.72

46

0.82

27

0.99

02

1.12

39

0.69

52

1.07

52

1.19

46

1.39

64

1.55

60

15

1.03

0.

1833

96

0.39

70

0.47

03

0.59

80

0.70

19

0.42

04

0.72

53

0.82

35

0.99

10

1.12

47

0.69

59

1.07

66

1.19

61

1.39

78

1.55

73

35

the corresponding random variables Zj have common variance v2=n/ν*. The different

degrees of freedom ν* considered in Table I with an obvious modification of the general

definition ν*=n-m-1, allows you also to take into account the cases when in the first

experiment not all estimates of the parameters βh are statistically significant, typically

on the basis of a T-test, so that by “pooling” we can increase the value ν*.

We note that also some estimates of parameters relating to the noise factors, like γj, δjs,

could be statistically not significant and be neglected in calculating the estimator hσ̂T

through formula (3.20). However the degrees of freedom ν* remain unchanged; what is

changing is the detailed definition of the noncentrality parameter τ2 , (3.25), which only

might lose some addends leaving unchanged the meaning of the ratio hσ

2D

σ2E

= τ2/n ν*.

After fixing the “level of distinguishability” c0, to apply the test (4.4) you also have to

choose a value ζ20 = n

hσ2D

σ2E

, that is, a value of the ratio hσ

2D

σ2E

.

A procedure to direct the use of test (4.4) for a given c0 can be the following.

For each value ζ20 considered, see for example Table I, in view of (4.3) and by taking

(3.26), (3.26’) into consideration, you can check whether ζ20 is consistent or not with the

data by having recourse to the test that accepts H0: ζ2 ≤ζ20 with ν*ζ2=ν*n hσ

2D/σ2

E, in

consequence of (3.25), (3.26), (3.26’) and (2.7’’), if:

F = (ν*n/ ν0)( hσ̂2D / σ̂2

E) = χ

2ν0

(ν*ζ2 )/ν0

χ2ν/ν

≤ Fν0,ν,α (ν*ζ20 ) (4.9)

otherwise rejects it and accepts H1: ζ2>ζ20 , where Fν0,ν,α(⋅) is the value which is

exceeded with probability α by the noncentral F-variate with ν0≤q(p+1), ν degrees of

freedom and noncentrality parameter of value ν*ζ20 ; ν*=n-m-1, as we told before, might

be augmented to take into account the possible elimination of some parameters βh from

the model, whose estimates did not result to be statistically significant; ν0≤q(p+1) is the

number of degrees of freedom of hσ̂2D, see (3.26), equal to the number of parameters

related to the noise factors, that is in the sets {γj, δjs}, s=1,…,p, j=1,…,q, retained in the

model since in each of them at least one of the corresponding estimates was found

36

statistically significant; ν is the value of the degrees of freedom of the error variance

estimate, (3.3), in the first experiment, which likewise can be larger than that referring

to the complete model in order to take into account possible non significant estimates of

some parameters βh as well as of some γj, δjs. The stochastic independence of numerator

and denominator in (4.9) is once more ensured by Lemma 3.2. The test (4.9) is

unbiased; this follows from the result given in Scheffè (1959), p. 136, according to

which the probability that F satisfies (4.9) is a continuously decreasing function of ζ2.

Table II shows some critical values which can be used to carry out test (4.9). Then with

regard to Table I for a given c0 you can use the minimum value ζ20 which still does pass

test (4.9) to carry out the test (4.4). Conceptually this procedure corresponds to having

recourse to an approximate lower bound of a confidence set for ζ2, defined through the

test (4.9), cfr. Lehmann (1986), p. 90-91.

b) A conditional indicator for more general settings: all assumptions of Corollary

3.1 might not be satisfied. Consider the estimator β̂^

h=β̂h +β̂*h , (2.8), which we could

obtain by carrying out the second experiment. An unbiased estimator hσ̂2T of Var(β̂

^h),

defined through the results of the first experiment, was given in Proposition 3.1, ii.,

formula (3.20). Now let us approximate the probability distribution of hσ̂2T by means of

a central chi-square distribution as hσ̂2T ≅ πχ

2

ν∼ / ν∼, where the constant π and the degrees

of freedom ν∼ are to be appropriately chosen. For this purpose we consider the two

moment approximation which entails to make coincide mean and variance of the two

variables in the former relationship. Precisely we put:

E(hσ̂2T) = hσ

2T = Var(β̂

^h) = π E(χ

2

ν∼ / ν∼) = π, (4.10’)

Var(hσ̂2T) = π2 Var(χ

2

ν∼ / ν∼) = 2π2 / ν∼

from which we obtain:

ν∼ = 2 [Var(β̂^

h) ]2 / Var(hσ̂2T). (4.10”)

T

able

II. C

ritic

al v

alue

s Fν0

,ν,α

(ν* ζ2 0

) nee

ded

for a

pply

ing

test

(4.9

); λ=

ν* ζ2 0= ν

* ⋅n⋅ hσ

2 D/σ

2 E; n

=16,

p=2

, q=2

; α=0

.05.

hσ2 D

/σ2 E

0.25

0.

70

1.00

2.

00

4.00

ν∗

12

13

14

15

12

13

14

15

12

13

14

15

12

13

14

15

12

13

14

15

λ 48

52

56

60

13

4.4

145.

6 15

6.8

168

192

208

224

240

384

416

448

480

768

832

896

960

ν 0

ν

1 13

22 22.. 66

11 24

.89

27.1

7 29

.46

72.3

9 78

.88

85.3

7 91

.87

105.

79

115.

08

124.

37

133.

66

217.

31

235.

91

254.

5 27

3.1

440.

5 47

7.7

514.

92

552.

1

12

22

.32

24.5

6 26

.8

29.0

5 71

.23

77.6

83

.98

90.3

6 10

4.03

11

3.15

12

2.27

13

1.39

21

3.52

23

1.78

25

0.04

26

8.3

432.

64

469.

17

505.

68

542.

2

11

22

24

.19

26.3

9 28

.6

69.9

5 76

.19

82.4

4 88

.69

102.

09

111.

02

119.

96

128.

89

209.

35

227.

24

245.

13

263.

01

424

459.

78

495.

57

531.

35

2 12

11.4

7 12

.59

13.7

1 14

.83

35.9

1 39

.1

42.2

8 45

.47

52.3

1 56

.87

61.4

3 65

.99

107.

05

116.

18

125.

31

134.

44

216.

61

234.

87

253.

13

271.

39

11

11.3

12

.4

13.5

14

.6

35.2

6 38

.39

41.5

1 44

.63

51.3

3 55

.8

60.2

6 64

.73

104.

96

113.

91

122.

85

131.

79

212.

28

230.

18

248.

06

265.

95

10

11.1

1 12

.18

13.2

6 14

.34

34.5

4 37

.59

40.6

4 43

.7

50.2

4 54

.61

58.9

7 63

.34

102.

65

111.

39

120.

12

128.

86

207.

5 22

4.99

24

2.46

25

9.94

3

11

7.

73

8.46

9.

2 9.

93

23.7

25

.78

27.8

6 29

.95

34.4

1 37

.39

40.3

7 43

.34

70.1

6 76

.12

82.0

9 88

.05

141.

71

153.

64

165.

56

177.

49

10

7.6

8.32

9.

03

9.75

23

.22

25.2

5 27

.28

29.3

2 33

.68

36.5

9 39

.5

42.4

1 68

.61

74.4

4 80

.26

86.0

9 13

8.52

15

0.17

16

1.82

17

3.48

9

7.45

8.

15

8.85

9.

55

22.6

7 24

.65

26.6

3 28

.61

32.8

6 35

.7

38.5

3 41

.37

66.8

9 72

.56

78.2

3 83

.9

134.

96

146.

31

157.

66

169

4 10

5.85

6.

38

6.92

7.

46

17.5

5 19

.08

20.6

22

.13

25.4

27

.58

29.7

6 31

.95

51.6

55

.97

60.3

4 64

.7

104.

03

112.

77

121.

51

130.

25

9

5.

73

6.25

6.

77

7.3

17.1

4 18

.62

20.1

1 21

.6

24.7

8 26

.91

29.0

3 31

.16

50.3

54

.55

58.8

1 63

.06

101.

36

109.

87

118.

38

126.

89

8

5.

6 6.

1 6.

61

7.12

16

.67

18.1

1 19

.55

20.9

9 24

.08

26.1

4 28

.21

30.2

7 48

.83

52.9

6 57

.08

61.2

1 98

.35

106.

6 11

4.85

12

3.11

5

9

4.7

5.12

5.

54

5.95

13

.82

15

16.2

17

.39

19.9

4 21

.64

23.3

4 25

.04

40.3

5 43

.75

47.1

6 50

.51

81.2

88

94

.82

101.

62

8

4.

59

4.99

5.

4 5.

81

13.4

4 14

.59

15.7

5 16

.9

19.3

7 21

.02

22.6

7 24

.32

39.1

7 42

.47

45.7

7 49

.07

78.7

8 85

.38

91.9

9 98

.59

7

4.

46

4.85

5.

24

5.64

13

14

.12

15.2

3 16

.34

18.7

3 20

.32

21.9

1 23

.5

37.8

2 41

44

.19

47.3

8 76

.04

82.4

1 88

.76

95.1

4 6

8

3.91

4.

25

4.59

4.

93

11.2

9 12

.25

13.2

1 14

.17

16.2

3 17

.61

18.9

8 20

.35

32.7

3 35

.48

38.2

3 40

.98

65.7

4 71

.24

76.7

5 82

.25

7

3.

8 4.

13

4.46

4.

78

10.9

2 11

.85

12.7

8 13

.7

15.7

17

.01

18.3

4 19

.67

31.6

34

.26

36.9

1 39

.56

63.4

5 68

.75

74.0

6 79

.37

6

3.

67

3.98

4.

3 4.

61

10.4

9 11

.38

12.2

7 13

.16

15.0

6 16

.33

17.6

18

.87

30.3

32

.84

35.3

8 37

.92

60.7

9 65

.87

70.9

6 76

.04

Crit

ical

val

ues i

n gr

ey c

ells

cor

resp

ond

to th

ose

com

bina

tions

of ν

0 , ν

and

λ th

at c

an’t

be re

aliz

ed in

our

exa

mpl

e.

ν∗ : (n

umbe

r of o

bser

vatio

ns n

) – (n

umbe

r of p

aram

eter

s βh r

etai

ned

in th

e m

odel

as b

eing

sign

ifica

nt);

ν : n

umbe

r of o

bser

vatio

ns o

n w

hich

the

varia

nce

estim

ate

σ̂2 E is

bas

ed: (

num

ber o

f obs

erva

tions

n)-

(num

ber o

f all

para

met

ers β

h, γ j,

δjs re

tain

ed in

the

mod

el a

s bei

ng si

gnifi

cant

); ν 0

: nu

mbe

r of t

erm

s of t

ype

γ j+δ′ j

x i re

tain

ed in

the

mod

el a

s bei

ng si

gnifi

cant

.

38

It is easy to show that, if the matrices G′jGj, (3.7), are idempotent of rank (p+1) with

|xhi|=1, ∀i, see Lemma 3.3, and we refer to the complete model (2.6), it follows from

(3.25) , (3.26) that:

ν∼ = [ ]1+(n hσ

2D/σ2

E)2

v4

n2q(p+1)+1ν

1-

v2

n q(p+1)2+2

v2

n (n hσ2D/σ2

E) (4.11)

where, in particular, ν is defined with reference to (3.26). If in expression (4.11) we put

ζ2= n (hσ2D/σ2

E) and we consider the derivative with respect to ζ2 of the corresponding

function it is easily shown that ν∼, (4.11), is a monotonic increasing function of ζ2 for

ζ2=n (hσ2D/σ2

E) > 1.

With regard to (4.11) a guide to a choice of ζ2, which is consistent with the data, may be

obtained by applying the test which accepts H0: ζ2≥ζ20 see Lemma 3.3, and (3.26), if:

F = [n2/(v2ν0)] hσ̂2D/σ̂2

E = χ

2

ν0( nζ2

v2 )/ν0

χ2ν (ν)/ν

≥ Fν0

,ν; 1-α

nζ2

0

v2 , (4.12)

otherwise rejects it and accepts H1: ζ2<ζ20 , where Fν

0,ν; 1-α ( ⋅ ) is the value exceeded with

probability (1-α) by the noncentral F-variate with ν0≤q(p+1), ν degrees of freedom,

which are defined as in (4.9), and noncentrality parameter nζ20/v2, with regard to which

we recall that in general we put ζ2= n hσ̂2D/σ̂2

E. Also the test (4.12), which has a

probabilistic structure similar to that of test (4.9), is unbiased. With regard to (4.12), we

might consider a set of reasonable values for the ratio hσ2D/σ2

E , see Table I, and for

calculating ν∼, (4.11), use the largest value which still leads to accepts H0 according to

the test (4.12).

Alternatively we can estimate the ratio hσ2D/σ2

E by the method of moments replacing the

unknown parameters with the corresponding unbiased estimates of hσ2D, see (3.19),

(3.20) and (4.15) below, and of σ2E, (3.3). Now, by assuming the approximation defined

through (4.10’) , (4.10”), (4.11) with regard to (2.8) and (2.9) and remembering that the

39

random variable U is stochastically independent of hσ̂2T in force of Proposition 3.1. ii. we

have that:

T = (β̂^

h / hσ̂T ) = (β̂h +β̂*h ) / hσ̂ T = (βh + hσ

E*U1+ hσ

.DU2) / hσ̂ T =

= (βh + hσ2E*+ hσ

2D U ) / hσ̂T = [(βh / hσT) + U] / χ

2

ν∼ / ν∼ (4.13)

follows a T-distribution, in general noncentral, with ν∼ degrees of freedom and

noncentrality parameter |βh | / hσT regardless of the fact that either β̂^

h and thus β̂h ,β̂*h are

obtained together through the second experiment or β̂h is actually obtained from the first

experiment, while β̂*h would be implicitly obtained only by carrying out the second

experiment, see (2.7).

Let t

ν∼,α/2 be the value exceeded with probability α/2 by the central T random variable

with ν∼ degrees of freedom. Correspondingly under H0: βh = 0 consider the probability

that a possible value |β̂^

h|/hσ̂T, where the numerator would be actually available according

to (2.7) when carrying out also the second experiment, is not larger than t

ν∼,α/2,

conditional on the results of the first experiment, that is on β̂h, (2.3), and hσ̂2T , (3.20).

Note that the last estimates respectively give information on the “size” of the βh effect as

well as on the error variance increment, leading from hσ 2E* to hσ

2T, which arises when the

noise factors levels truly happen at random. If we take (2.8), (4.10’) and Proposition

3.1, ii. into considerations in explicit terms the above probability is given by:

P( - t

ν∼,α/2 hσ̂T ≤ β̂^

h = β̂h +β̂*h ≤ t

ν∼,α/2 hσ̂T | β̂h , hσ̂T) =

= P[ -( t

ν∼,α/2 hσ̂T + β̂h ) / hσD ≤ U2 ≤ ( t

ν∼,α/2 hσ̂T - β̂h ) / hσD | β̂h , hσ̂T)] =

= 12 π

⌡⌠

-( t

ν∼,α/2 hσ̂T + β̂h ) / hσD

( t

ν∼,α/2 hσ̂T - β̂h ) / hσD

exp(-u22/2)du2 = 1- α(β̂h, hσ̂T, hσD) (4.14)

where we took into consideration the normality assumptions on the errors E2i , from

which it followed that U2, with hσDU2=β̂*h , is a standardized normal variate, see (2.8),

40

stochastically independent of the variates β̂h , hσ̂T, which, in particular, ensues from the

stochastic independence of the random variables E∼

i , Zji among themselves and with

respect to the errors Ei of the first experiment, see § 2., b), and from Lemma 3.2. We

underline once more that the overall distributional characteristics of β̂h ,β̂*h, hσ̂T don’t

change if, besides hσ̂T, we consider β̂h as obtained from the first experiment too, while

β̂*h is meant to pertain to the second one.

As an indicator of distinguishability of an effect βh, on the basis of the first experiment

only, we propose to consider the value α̂(β̂h, hσ̂T, hσ̂D) obtained from (4.14) by replacing

the unknown parameter hσD with the square root of the unbiased estimator of hσ2D, which

can be obtained from (3.19) as:

hσ̂̂ 2D= hσ̂

2D - hσ̂

2E* (v2/ hx2 ) [∑

j=1

q tr(G′j Gj)] = hσ̂

2D - hσ̂

2E* (v2/ hx2 )[q(p+1)] (4.15)

where the last expression holds when all matrices G′j Gj , (3.7) are idempotent and the

complete model (2.6) is considered (for possible simplifications see the comments on ν0

following (4.9)).

The meaning of the proposed indicator is the following. Suppose that also the second

experiment is carried out. Then we could test the hypothesis H0: βh = 0 by applying the

usual two-sided T-test to the ratio β̂^

h /hσ̂T , where β̂^

h is the estimator (2.7) of βh

obtained from the second experiment, hσ̂2T the one of hσ

2T obtained from the first and,

thus, we would accept H0 if |β̂^

h| /hσ̂T ≤ t

ν∼,α/2. Instead of actually carrying out the second

experiment you can ask which is the probability, conditional on the results coming from

the first one leading to β̂h, hσ̂T, that the results coming from the second one would

produce non-significant values for the T-test. This probability is given by (4.14).

Consequently if hσ 2D should be known, the indicator α(⋅) would express the probability

of rejecting H0 through the above test, that is, of distinguishing the effect βh from zero

also in the presence of the noise factors variability, given the values β̂h , hσ̂T observed in

the first experiment and for a generic random contribution β̂*h of the second one, if the

41

latter should be carried out. If α(⋅) is large, say, larger than the level of significance

α chosen for the T-test, this can be considered as an indication that, in the light of the

first experiment and without the need of carrying out the second one, the effect βh can

be considered as distinguishable from zero, even if the noise factors levels happen at

random. The proposed indicator α̂(⋅), which requires the estimation of hσ 2D, see (4.15),

obviously represents an approximation of the former one.

5. Completing the discussion of the case studied in the light of the theoretical

results - Conclusion.

Coming back to the example presented in § 2.c) if we want to obtain an unbiased

estimator of Var(β̂^

h) = σ2E* + hσ

2D = hσ

2T , (2.9), we need unbiased estimators of both σ2

E*

and hσ2D .

From (3.4) we have hσ̂2E* = σ̂2

E/hx2 , which, for the complete model and for the data of

the example leads to:

hσ̂2E* = σ̂2

E/hx2 = σ̂2E/16 = 0.051389583/16 = 0.003211848938, ∀ h=1,…,m=10, and 6

degree of freedom.

From (3.5), see also (3.19), we have that the “natural” biased estimator of hσ2D is hσ̂

2D =

v2

hx4 ∑j=1

q Q2

j , where Q2 j = (η+E)’G′jGj(η+E), (3.7), and Gj = A⋅(Bj+X*Cj), (3.6). To obtain

hσ̂2D, for example for h=1, by using the elements of Table 1, first we have to write down

G1 and G2. By taking (3.6) into consideration we obtain:

G1=A⋅(B1+X*C1) =

42

= 116

+3 -3

+3 -3+3 -3+3 -3-1 +1-1 +1-1 +1-1 +1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1

+3 -3+3 -3+3 -3+3 -3-1 +1-1 +1-1 +1-1 +1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1

+1 -1+1 -1+1 -1+1 -1-3 +3-3 +3-3 +3-3 +3-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1

+1 -1+1 -1+1 -1+1 -1-3 +3-3 +3-3 +3-3 +3-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1

+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+3 -3+3 -3+3 -3+3 -3-1 +1-1 +1-1 +1-1 +1

+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+1 -1+3 -3+3 -3+3 -3+3 -3-1 +1-1 +1-1 +1-1 +1

-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1+1 -1+1 -1+1 -1+1 -1-3 +3-3 +3-3 +3-3 +3

-1 +1

-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1-1 +1+1 -1+1 -1+1 -1+1 -1-3 +3-3 +3-3 +3-3 +3

,

since, according to the definitions of bji, c jsi which follow (2.4), (2.5),

• B1 = 116

-1 +1 -1 +1 -1 +1 -1 +1

.. .. .. .. .. .. .. ..-1 +1 -1 +1 -1 +1 -1 +1

-1 +1 -1 +1 -1 +1 -1 +1

.. .. .. .. .. .. .. ..-1 +1 -1 +1 -1 +1 -1 +1

with B1 a square matrix, 16×16, with 16 equal rows whose elements are those of 1ξ, that

is the elements of the first column of Ξ0;

• X*= [1x, 2x ]

with X* a 16×2 matrix where 1x, 2x equal respectively the second and the third column

of X;

• C1= 116

+1 -1 +1 -1 -1 +1 -1 +1 +1 -1 +1 -1 -1 +1 -1 +1

+1 -1 +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 -1 +1 ,

with C1 a 2×16 matrix where the elements of the two rows of the matrix equal those of

3ξ and 4ξ, respectively, that is the elements of the two columns of Ξ1;

for h=1, we also have:

• A =

x11 0 .. .. 0

0 .. .. .. 00 .. x1i .. 00 .. .. .. 00 .. .. .. x1n

where [x11, …, x1i, …, x1n]’ is the vector 1x’ which equals

the second column of X.

Likewise we obtain:

G2 = A⋅(B2+X*C2) =

43

= 116

+3 +3

+3 +3+3 +3+3 +3-1 -1-1 -1-1 -1-1 -1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1

-3 -3-3 -3-3 -3-3 -3+1 +1+1 +1+1 +1+1 +1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1

+1 +1+1 +1+1 +1+1 +1-3 -3-3 -3-3 -3-3 -3-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1

-1 -1-1 -1-1 -1-1 -1+3 +3+3 +3+3 +3+3 +3+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1

+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+3 -3+3 +3+3 +3+3 +3-1 -1-1 -1-1 -1-1 -1

-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-3 -3-3 -3-3 -3-3 -3+1 +1+1 +1+1 +1+1 +1

-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1-1 -1+1 +1+1 +1+1 +1+1 +1-3 -3-3 -3-3 -3-3 -3

+1 +1

+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1+1 +1-1 -1-1 -1-1 -1-1 -1+3 +3+3 +3+3 +3+3 +3

with

• B2 = 116

-1 -1 +1 +1 -1 -1 +1 +1

.. .. .. .. .. .. .. ..-1 -1 +1 +1 -1 -1 +1 +1

-1 -1 +1 +1 -1 -1 +1 +1

.. .. .. .. .. .. .. ..-1 -1 +1 +1 -1 -1 +1 +1

where the rows of the matrix are all equal to 2ξ,

• C2 = 116

+1 +1 -1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 +1 +1

+1 +1 -1 -1 +1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 +1 ,

where the two rows of the matrix are equal to 5ξ and 6ξ, respectively, and A and X* the

same as for G1.

With the data of the example we obtain, when v2=1:

Q2 1 = (η + E)’ G′1 G1 (η + E) = y’ G′1 G1 y = 0.00796875

Q2 2 = (η + E)’ G′2 G2 (η + E) = y’ G′2 G2 y = 0.82456875

and hσ̂2D =

v2

1x4 ∑j=1

q Q2

j = 0.0032520996, where 1x4=n2=162=256, a result which is shown

to hold not only for h=1 but also for h=2,3.

Following Lemma 3.1, it can easily be verified that the ranks r1 of G1 and r2 of G2 are

both equal to p+1=3, in fact ∀ h=1,..,3 all elements xhi , i=1,2,…,n, are different from

zero since they are the levels of the controlled factors recorded as –1,+1 in conventional

units (see comma i.). In this case, for Lemma 3.2, Q21 and Q2

2 are stochastically

independent and independent also of σ̂2E , β̂1, β̂2 and β̂12 .

44

The r1=3 eigenvalues of G′1G1 and of G′2G2 are all equal to 1. This holds since for the

experimental design of our example, ∀ h=1,..,3, the conditions of Lemma 3.3 are

satisfied. In fact: 1) |xhi| =1, i=1,2,…,16; 2) 1/1ξ2 =1/2ξ2 = 1/16; 3) 1/3ξ2 =1/5ξ2= 1/1x2 =

1/16; 1/4ξ2 =1/6ξ2= 1/2x2= 1/16. So the matrix G′1G1 and G′2G2 are idempotent of rank 3

and ∑j=1

2 Q2

j /σ2E is the sum of two independent noncentral chi-square random variables

each with 3 degrees of freedom and, thus, given the reproductivity of the corresponding

distributions has a noncentral chi-square distribution with 6 degrees of freedom and

noncentrality parameter τ21+τ2

2 , with τ2 j, j=1,2, defined by (3.18).

We note that as shown in Proposition 3.1 the natural estimator hσ̂2D of hσ

2D is biased.

However an unbiased estimator hσ̂2T of the overall variance hσ

2T , (2.9) of the estimate β̂

^h

is given in comma ii. of the same Proposition, formula (3.20). We remember that above

we obtained the values hσ̂2E* = 0.003211848938 and hσ̂

2D =

v2

1x4 ∑j=1

q Q2

j = 0.0032520996; to

apply formula (3.20) we note that [1-(v2/hx2)∑j=1

2 tr(G′j Gj)] = 0.625 for v2=1 and recalling

that the trace of an idempotent matrix is equal to its rank, which is 3 for both matrices

of interest. For the case under consideration with v2=1, we finally obtain hσ̂2T

=0.0052595052, h=1,2,3. From Lemma 3.2 it also follows that hσ̂2T is stochastically

independent of the effects estimates β̂^

h ∀h=1,…,3.

In view of Corollary 3.1, we remark that if we choose v*2=n/(n-m-1)=16/(16-3-1)=4/3,

we would obtain that hσ̂2T=(σ2

E/n)[χ2ν*(τ2 )]/ν*, (3.24), that is that hσ̂

2T follows a noncentral

chi-square distribution with ν*=16-3-1=12 degrees of freedom and noncentrality

parameter τ2 given by (3.25). In fact with any other value of v2, hσ̂2T is a linear

combination of noncentral chi-square random variables which can only be approximated

with a noncentral chi-square distribution.

What are the implications of fixing the value of v*2 = n/(n-m-1) = 4/3? As pointed out in

Remark 3.1 this condition added to (3.30) imposes a true restriction on the spread of the

noise factors real levels (see Remark 3.1). In fact their standard deviations in the

45

original units are not equal to the conventional standard deviation v1= R1/3= 833.3 and

v2= R2/3= 8.333 - where, as mentioned before, R1, R2 are respectively the lengths of the

longer and smaller side of the rectangle describing the unwound whole roll - but result

to be smaller and respectively equal to v1= 3/4 (R1/3) = 721.688 and v2= 3/4 (R2/3)=

7.21688 which correspond to 0.289Rj instead of conventional 0.333Rj. We note that this

means assigning conventionally a negligible probability to noise factor levels which

correspond to points outside a rectangle a little smaller than the real one related to a

whole roll.

For v2 = 43 we obtain the new estimates of hσ̂

2T =0.0059420573, as [1-(v2/hx2)∑

j=1

q tr(G′j Gj)]

= 0.5, and hσ̂2D = 0.0043361328.

In the second experiment, when the noise factors levels happen at random the variance

of the effect estimate β̂^

h is Var(β̂^

h)=σ2E*+hσ

2D or, under the hypotheses of Lemma 3.3, and

of Corollary 3.1, it assumes the form: (σ2E/n)(1+

n hσ2D

σ2E

) see (4.2).

Then the variance of the parameter estimate becomes greater than that related to the first

experiment when we have Var(β̂h)=σ2E*=σ2

E/n. Now it is of interest to assess whether the

effect βh is actually distinguishable from random fluctuations even when the variance of

its estimate is the real one: so, for example, we may wonder whether conditioning

temperature actually shows a significant effect on the response η even if the test strip is

chosen at random in the roll as we assume to be the case in the real utilization of the

product. Then the variance to be considered is not hσ̂2E*=0.003211848938 but is hσ̂

2T

=0.0059420573.

The theory developed above in § 4 allows us to find out an answer to such a question in

two ways.

a) The case when Corollary 3.1 holds (v2=n/(n-m-1) and G′j Gj is idempotent ∀j=1,..,q).

The answer to the question we are interested in can be obtained by testing the following

hypothesis (4.3),

46

H0 =

|βh|

σ2E/n+hσ

2D

= n |βh/σE|

1+ζ2 ≥ c0

|βh|/σE ≥ |βh0|/σE

0≤ζ2≤ζ20

where we recall that c0 is a chosen threshold of distinguishability of the effect βh of

interest with respect to random fluctuations when the variance of the least squares

estimate β̂h of βh in the first experiment is augmented to take the noise factors also into

account; ζ20 = n hσ

2D/σ2

E is a chosen upper bound for said variance increment.

According to (4.4) we accept H0, (4.3), at a significance level α if:

β̂h

hσ̂T

2

≥ F1,ν*,1-α (c∼20 , ν*ζ2

0 ),

where hσ̂T is the total variance unbiased estimate obtained by (3.20), leading to (3.24)

under the assumed hypotheses, F1,ν*,1-α(⋅,⋅) is the (1-α), critical value of the doubly

noncentral F-distribution, with (1+ζ20)c

20, ν*ζ2

0 the values of the noncentrality parameters,

and degrees of freedom 1 and ν*=(n - number of the significant estimates parameter βh

retained in the model). In the sequel for simplicity we shall refer to the complete model;

thus for the case under consideration we have: ν*=n-m-1=16-4=12, since there are n=16

observations and 4 parameters βh.

To choose ζ20 we can follow the procedure suggested in §4, a). We have to consider the

test (4.9):

(ν*n/ ν0)( hσ̂2D / σ̂2

E) ≤ Fν0,ν,α (ν*ζ20)

which leads to accept the hypothesis H0: ζ2≤ζ20 when the former relationship is satisfied

otherwise to reject it, where Fν0,ν,α(⋅) is the critical value at a significance level α of the

noncentral F distribution which, in the case of a complete model, has degrees of

freedom ν0=q(p+1), equal to the number of parameters pertaining to the noise factors –

ν0=2⋅3=6 in the example – and ν, which express the number of degrees of freedom left

for the variance estimation σ2E after fitting the complete model to the data of the first

experiment – ν=16-10=6 in the example – while the value of the noncentrality

parameter is set equal to ν*ζ20.

47

With the help of a table like Table II and with reference to test (4.9) recalled above, for

example, one may choose as ζ20, or equivalently as corresponding ratio hσ

2D/σ2

E, the

largest value consider in the table which still allows to accept H0. In the example we are

considering we obtain

(ν*n/ ν0)( hσ̂2D / σ̂2

E) = (12/6)(hσ2D/σ2

E*) = 2⋅0.0043360.003211 = 2.7 .

In Table II for ν*=12 and ν=ν0=6 we read the critical value 3.67>2.7, corresponding to a

variance increment hσ2D/σ2

E=0.25.

We can now apply test (4.4), for example, to test the distinguishability from zero of the

conditioning temperature effect β2 also when we have a variance increment of the

random errors equal to hσ2D=0.25σ2

E. For this purpose we have to consider the value of

the statistic

β̂2

hσ̂T

2

=11.652 and compare it with the critical values given in Table I. We

see that for ν*=12, v=1.16 (note that this parameter has little influence on the upper

percent points), 2σ2D/σ2

E=0.25 (corresponding to ζ20=16⋅0.25=4), α=0.10 and for any

value c0=1,1.5,2 the value 11.652 is much larger than the corresponding upper percent

points and allows us to accept the hypothesis H0 that the effect β2 is distinguishable

even if the variability of its estimate is incremented to take the random effects of the

noise factors into consideration. According to the results given in Table II the same

conclusion is reached even if instead of the complete model we eliminate all not

significant effects.

b) A conditional indicator for more general settings (v2≠n/(n-m-1) and G′j Gj is

idempotent ∀j=1,..,q).

While the tests of significance discussed in a) are essentially based on the assumption

that the random errors Ei, (2.1), E∼

i, (2.6) i=1,2,…,n follow a same zero mean normal

distribution, besides being stochastically independent of each other and of the noise

variables Zji, j=1,2,…,q, i=1,2,…,n, only the stochastic independence of the latter plays

a role, but not the type of their distribution. On the contrary this second method

requires that the variables Zji also have a same zero mean normal distribution. Thus the

48

second method does not seem to be completely appropriate to deal with the example

considered in the paper, since two independent uniform distributions on the intervals

(-R1,R1), (-R2,R2) seem more suitable to describe the response η modification connected

with a random choice of a film strip from the whole roll. However also the second

procedure will be applied to the data of the example considered above for illustrative

purpose. So we can now consider the indicator of distinguishability 1-α(⋅) of an effect

βh defined by (4.14). We recall that α(⋅) expresses the probability of rejecting H0: βh=0,

that is, of distinguishing the effect βh from zero also in the presence of the noise factors

variability, that is when carrying out also the second experiment, given the values β̂h ,

hσ̂T observed in the first experiment. By applying (4.14) and having resort to the

variance components unbiased estimates, see (3.4), (3.20) and (4.15), in the case of the

complete model and for different values of v2, with regard to the example under study

we obtained the following results:

v2=4/3 (∗)

hσ̂2T hσ̂

^2D hσ̂

^2D /hσ̂2

E* ν∼ 1-α̂ (⋅)

β̂1 0.0059420573 0.0027302083 0.85004257 15.211331 0.99759854

β̂2 0.0059420573 0.0027302083 0.85004257 15.211331 0.029022987

β̂12 0.0059420573 0.0027302083 0.85004257 15.211331 0.99248746

v2=1 (∗)

hσ̂2T hσ̂

^2D hσ̂

^2D /hσ̂2

E* ν∼ 1-α̂ (⋅)

β̂1 0.0052595052 0.0020476562 0.63753193 15.939253 0.99886658

β̂2 0.0052595052 0.0020476562 0.63753193 15.939253 0.0078390572

β̂12 0.0052595052 0.0020476562 0.63753193 15.939253 0.99509812

v2=1/9 (∗)

hσ̂2T hσ̂

^2D hσ̂

^2D /hσ̂2

E* ν∼ 1-α̂ (⋅)

β̂1 0.0034393663 0.00022751736 0.070836881 7.4296307 1

β̂2 0.0034393663 0.00022751736 0.070836881 7.4296307 0

β̂12 0.0034393663 0.00022751736 0.070836881 7.4296307 1

(*) see Table 2 where:

49

• hσ̂2T is the unbiased estimator of Var(β̂

^h ) observed in the first experiment and

given in (3.20);

• hσ̂^ 2

D is the unbiased estimator of hσ2D given in (4.15), which represents the added

variability due to the second experiment;

• hσ̂^ 2

D /hσ̂2E* is the estimator of the error variance increment hσ

2D/σ2

E* obtained by the

method of moments, that is by replacing the unknown parameters with the

corresponding unbiased estimators;

• ν∼ is the approximate value of the degree of freedom obtained by substituting the

above estimator for hσ2D/hσ

2E* in (4.11).

In the last column of the former table we can read the value of 1-α̂ (⋅) which represents

an approximation to the proposed indicator (4.14) which expresses the probability of

accepting H0: βh = 0, that is, of not distinguishing from zero the effect βh when carrying

out the second experiment, that is in the presence of the noise factors variability,

conditional to the results obtained from the sole first experiment. Then, if 1-α̂(⋅) is small

or at least smaller than 1-α, where α is the level of significance chosen for the

underlying T-test, we can conclude that βh can be considered distinguishable from zero.

We can see that the results previously obtained for the case a) are confirmed by this new

indicator. In fact the factor x2 “conditioning temperature” turns out to affect the

response η even in the presence of noise factors, that is when the test strip is chosen at

random on the whole roll. We can also notice that the principal characteristic of this

indicator is its ability to discriminate the relevant factors from the others: in fact the

value of 1-α̂(⋅) for factor x2 is much smaller than those regarding x1 and the interaction

x1⋅x2, see Table 2.

Furthermore, the indicator is not sensitive to changes in v2 values (v2=4/3 is considered

to allow comparing these results with those of case a)).

Conclusion.

In conclusion both methods lead us to accept that x2, that is the conditioning

temperature, has an evident effect on the mean value η of the response, strength loss

50

under stress, Y, which is clearly distinguishable from zero even when it is assessed in

relation to the overall random variability which includes the noise factor effects. In the

opposite case a possibly costly effort to control this factor at its higher value during

production may appear as meaningless, in view of the subsequent real utilization and

behaviour of the finished product.

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