Wine Chemistry and Biochemistry

682 P.J. Mart ́ın-Alvarez ́

Table 13.3Mean and the standard deviation values of the histamine concentrations in 30 commer-
cial wines analyzed by a direct ELISA and an HPLC method, and the results of t test for related
samples

Mean SD n Diff. (d ̄) SD. Diff. (sd) t-value df P

HPLC 4.51 3.90
ELISA 5.49 3.91 30 –0.978 1.722 –3.110 29 0.0042∗

t-value=value of the statistic tcal,df=degrees of freedom, P=associated probability
∗the two means are different (P<0.05).

Table 13.3 shows the mean and the standard deviation values of the histamine con-

centrations in 30 commercial wines analyzed by a direct ELISA and HPLC methods

(Marcobal et al. 2005) and the results of the t test obtained with the STATISTICA

program (procedureT-Test for dependent samples,intheBasic Statistics and Tables

module). The results revealed slightly higher results for ELISA (P<0.05).

13.1.4 Statistical Treatment to Compare More than Two

Independent Samples

Let

{

x 1 , 1 ,x 2 , 1 ,x 3 , 1 ,...,xn 1 , 1

}

,

{

x 1 , 2 ,x 2 , 2 ,x 3 , 2 ,...,xn 2 , 2

}

,...,

{

x 1 ,k,x 2 ,k,x 3 ,k,...,

xnk,k

}

bek(k>2) independent random samples ofnjobservations of a continuous

random variableX, fromkpopulationsWj,andletμjandσjbe the mean and

standard deviation ofXin populationWj,j= 1 ,...,k. For each sample the corre-

sponding estimates ofμjandσj( ̄xj andsj) can be calculated. Accepting normal

distributions, and for a fixedsignificance levelα, we can calculate the corresponding

confidence intervals for the parametersμj,σj, or to test hypotheses about them.

13.1.4.1 Hypothesis Test for k Means or One-Way ANOVA

Assuming normal distributions and equality variances (X∼N(μj,σ) in Wj),

the null hypothesisH 0 ≡μ 1 =μ 2 =...=μkmay be tested using the statistic

Fcal=

∑k

j= 1 nj( ̄xj−x ̄)

(^2) /(k−1)
∑k
j= 1
∑nj
i= 1 (xi,j−x ̄j)^2 /(n−k)
, which has an F-distribution withk−1andn−kdf,
underH 0 ,andwheren=
∑k
j= 1 nj,and ̄xis the sample general mean. For a fixed
value ofα,ifFcal>F 1 −α,k− 1 ,n−k,orifP<α,H 0 ≡μ 1 =μ 2 =...=μk
should be rejected and we conclude that there are some differences among thek
means (H 1 ≡not allμi(i= 1 ,...,k)are equal). If the null hypothesisH 0 is
rejected, the posterior tests for means comparisons, Least Significant Difference
(LSD), Scheff ́e, Tukey, Bonferroni or Student-Newman-Keuls (S-N-K) tests can
be used to characterize the differences. The error plots, with the 95% confidence
interval for the means in thekgroups, can be used for a graphical comparison; if
the mean value of a group remains inside the confidence interval for the mean of
another, it is accepted that the two groups have similar means.