Wine Chemistry and Biochemistry

684 P.J. Mart ́ın-Alvarez ́

Table 13.5Results of S-N-K test for means comparison:

Homogenous Groups

(α= 0 .05):

Variety Mean 1 2 3
Malvar 2.49 ∗∗∗∗
Air ́en 2.58 ∗∗∗∗
Monastrell 5.87 ∗∗∗∗
Trepat 10.65 ∗∗∗∗
Means values with∗∗∗∗in the same homogenous group indicate that there are no significant
differences between them (P>0.05).

Table 13.6Means and standard error (Std. Err.) values:

Variety Mean Std. Err. 95% Confidence interval N

Malvar 2.49a 0.32 1.46 3.52 4
Air ́en 2.58a 0.63 0.58 4.58 4
Trepat 10.65c 1.02 7.40 13.90 4
Monastrell 5.87b 0.67 3.75 7.99 4
a−cFrom S-N-K test mean values with the same letter indicate that there are no significant differ-

ences between them (P>0.05).

They include: the ANOVA table (since the P-value associated with the F-value is

less than 0.05, the four means are significantly different,P<0.05), the results of

S-N-K test for means comparisons (there are three homogenous groups of means),

and the table with the mean and the standard error values of octanoic acid in the

wines of the four varieties. All means are different except for the means of the

Malvar and Air ́en varieties, and the higher concentration is that of the Trepat variety

(Pozo-Bay ́on et al. 2001).

13.1.4.3 One Factor Experimental Design or One-Way ANOVA

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 fromkfixed levels (treatments)ofa

factor, obtained from acompletely randomized design, and with normal distribution

(xi,j∼N(μj,σ) ,j = 1 ,...,k), we can consider for each observationxi,j,the

fixed effects model: xi,j=μ+αj+εi,j,j= 1 ,...,k,i= 1 ,...,nj,whereμis

the overall mean (μ=

∑

njμj/n),αjis a fixed quantity representing theeffect of

treatment jon the overall mean (αj=(μj−μ),

∑

j

njαj=0), andεi,jthe random

errors, independent and with normal distribution (εi,j∼N(0,σ)). For this model, we

are interested in testing the hypothesisH 0 ≡α 1 =...=αk=0 using the previous

statisticFcal=
SSfactor/(k−1)
SSerr or/(n−k) with aFk−^1 ,n−kdistribution, except that now the sources
of variation between and within groups are assigned to the factor and to the error

(H 0 ≡μ 1 =...=μkimpliesH 0 ≡α 1 =...=αk=0). After fixing the value of

α, if the null hypothesisH 0 ≡α 1 =...=αk=0 is rejected (Fcal>F 1 −α,k− 1 ,n−k

orP<α), we conclude that the factor influences the analyzed variable, and the

posterior tests for means comparisons canbe used to characterize the differences

(Afifi and Azen 1979; Jobson 1991).