13 Statistical Techniques for the Interpretation of Analytical Data 683
TheFcalratio is also the result of a statistical method known as theone-way
Analysis of Variance(ANOVA), which assumes the modelxi,j=μj+εi,jfor each
observationxi,j,whereεi,jare the independent and normally distributed random
errors (εi,j ∼ N(0,σ)), and which has as objective to testH 0 ≡μ 1 =μ 2 =
...=μk. The results of the ANOVA procedure, which include the decomposition
of the total sum of squares of the deviation of all observations around the general
mean (SStotal=
∑k
j= 1∑nj
i= 1(xi,j−x ̄)^2 ) in two parts: the sum of the deviations between
the groups (SSbetween=
∑k
j= 1nj( ̄xj−x ̄)^2 ) and the sum of the deviations within the
groups (SSwithin=
∑k
j= 1∑nj
i= 1(xi,j−x ̄j)^2 ), with their df, (n−1)=(k−1)+(n−k), their
mean squares,MSSbetween=SSbetween/(k−1) andMSSwithin=SSwithin/(n−k),
and the value of the statisticFcal=
MSSbetween
MSSwithin, withk− 1 and n−kdf, along
with their associated probability values (P), are summarized in the corresponding
ANOVA table.
To check the assumptions of the model, Bartlett’s or Levene’s tests can be used
to assess the assumption of equality of variance, and the normal probability plot
of the residuals (ei,j =xi,j−x ̄j) to assess the assumption of normality. If either
equality or normality are inappropriate, we can transform the data, or we can
use the nonparametric Kruskal-Wallis test to compare thekgroups. In any case,
the ANOVA procedure is insensitive to moderate departures from the assumptions
(Massart et al. 1990).
13.1.4.2 Example of Application of One-Way ANOVA
One-way ANOVA has been used to determine if there are significant differences
between varietal wines (Pozo-Bay ́on et al. 2001), to compare the concentrations
from seven laboratories for a same reference sample (Massart et al. 1990), to
compare the biogenic amine contents inthe wines grouped according to the type
of aging (Marcobal et al. 2005), or from five wineries (Mart ́ın-Alvarez et al. 2006), ́
and more generally, to compare the mean values ofk groups of independent
observations. Tables 13.4 to 13.6 show the results of the application of one-way
ANOVA for octanoic acid content in four varietal wines, obtained with the
STATISTICA program (One-way ANOVA procedure in the ANOVAmodule).
Table 13.4Results of the application of one-way ANOVA for octanoic acid content in four varietal
wines. ANOVA table:
Source of Degrees of Sums of Mean square
variation: freedom (df) Squares (SS) (MSS) F-value P-Value
Variety 3 176.7010 58.9003 29.6947 0. 000008 ∗
Error 12 23.8024 1.9835
Total 15 200.5034
∗There is a statistically significant difference between the four means.