13 Statistical Techniques for the Interpretation of Analytical Data 685
Accepting that previous values
{
x 1 , 1 ,x 2 , 1 ,...,xn 1 , 1}
,{
x 1 , 2 ,x 2 , 2 ,...,xn 2 , 2}
{ ,...,
x 1 ,k,x 2 ,k,...,xnk,k
}
correspond to the data fromkrandom subpopulations, withnormal distribution (xi,j∼N(mj,σ),j= 1 ,...,k), and assuming thatm 1 ,...,mj
represents a random sample from a population with distributionN(μ, σa), we can
consider for each observationxi,jtherandom effects model (components of variance
model) xi,j=μ+aj+εi,jwhereaj=mj−μare independentN(0,σa)andεi,j
are independentN(0,σ). For this model, we are interested in estimating the two
components of varianceσ^2 andσa^2 , and in testing the hypothesisH 0 ≡σa^2 = 0
using the same statisticFcal=
SSfactor/(k−1)
SSerr or/(n−k) with aFk−^1 ,n−kdistribution as in the
fixed effects case. The components of variance are estimated asσ^2 =MSSwithin
andσa^2 =(MSSbetween−MSSwithin)/q,whereq=k−^11 (
∑
nj−∑
∑n^2 j
nj)(Afifiand
Azen 1979; Massart et al. 1990; Gardiner 1997). Thisrandom effects modelis often
used in analytical chemistry to breakdown a total precision into its components such
as between-days and within-days, or between-laboratories and within-laboratories
in collaborative trials, to validate an analytical method using reference material.
13.1.4.4 Two Factor Experimental Design or Two-Way ANOVA
In the case of two factors A and B, withaandbfixed levels respectively, andm
replicate measurements of response variableXin each ofa·btreatment com-
binations, we accept for every observationxi,j,k,thek-th replicate of experimen-
tal units receiving factor A leveliand B levelj, the following model:xi,j,k =
μ+αi+βj+γi,j+εi,j,k, withi= 1 ,...,a, j= 1 ,...,b,yk = 1 ,...,m,
and whereμis a global mean,αirepresenting the effect on the response variable
Xof factor A at leveli,βjrepresenting the effect on the response variable of
factor B at levelj,γi,jrepresenting the combined effect on the response variable
Xof factor A at leveliwith factor B at levelj,andεi,j,kis the error with nor-
mal distributionN(0,σ). The two-way ANOVA breaks down the total variation
of the observations into four sources associated with both factors, their interaction
and with other factors not controlled, in agreement with the terms in the model
(SStotal=SSA+SSB+SSAx B+SSerr or). There are three null hypotheses asso-
ciated with two factors: factor A does not influence the response variableX,H 01 ≡
αi= 0 , ∀i, factor B does not influence the response,H 02 ≡βj= 0 , ∀j,and
no interaction effect occurs,H 03 ≡γi,j= 0 ,∀i,j, that can be confirmed with the
three following F-statistics:Fcal^1 =MSSA/MSSerr orwith (a−1) andab(m−1)df,
Fcal^2 =MSSB/MSSerr orwith (b−1) andab(m−1) df andFcal^3 =MSSAB/MSSerr or
with (a−1)(b−1) andab(m−1) df, respectively. The sums of squares associated
with the source of variation, their df and mean squares, and the F-ratios, along with
their associated probabilities, are summarized in the two-way ANOVA table (see
Table 13.7). After fixing the value forα,ifFcal^3 <F 1 −α,(a−1)(b−1),ab(m−1)orPAx B>α,
thenH 03 is accepted, the interaction does not exist, and the influence of one of the
factors will not depend on the levels of another factor. IfFcal^1 >F 1 −α,(a−1),ab(m−1)
orPA<α,thenH 02 is rejected, factor A influences the analyzed variable, and if