Interpreting sensory data by combining p

Food Quality and Preference 20 (2009) 167–175

Contents lists available at ScienceDirect

Food Quality and Preference
journal homepage: www.elsevier.com/locate/foodqual

Interpreting sensory data by combining principal component analysis
and analysis of variance
Giorgio Luciano a, Tormod Ns a,b,*
a
b

NOFIMA FOOD, Matforsk, Oslovegen 1, 1430 Ås, Norway
Department of Mathematics, University of Oslo, Blindern, Oslo, Norway

a r t i c l e

i n f o

Article history:

Received 13 June 2008
Received in revised form 15 August 2008
Accepted 19 August 2008
Available online 11 September 2008
Keywords:
PCA
ANOVA
Sensory profiling
ASCA

a b s t r a c t
This paper compares two different methods for combining PCA and ANOVA for sensory profiling data.
One of the methods is based on first using PCA on raw data and then relating dominating principal components to the design variables. The other method is based on first estimating ANOVA effects and then
using PCA to analyse the different effect matrices. The properties of the methods are discussed and they
are compared on a data set based on sensory analysis of a candy product. Some new plots are also proposed for improved interpretation of results.
Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction
Sensory panel data can always be looked upon as three-way
data tables with assessors, objects/samples and attributes as the

three ‘‘ways”. In order to analyse differences and similarities between samples and assessors as well as the correlation structure
among attributes, the three-way structure of the data needs to
be taken into account. This can be done in various ways using different underlying ideas and philosophies.
A technique that can be useful in some cases is regular multivariate analysis of variance (MANOVA, Kent & Bibby, 1978) for
testing the effect of samples and/or assessors for all attributes
simultaneously. Usually one is, however, interested in more insight
than this method can give and therefore other techniques are to be
preferred. A much used method within the area of sensory analysis
is the generalised procrustes analysis (GPA), treating each assessor
slice as a matrix, followed by a principal component analysis of the
average or consensus matrix (Dijksterhuis, 1996). GPA is based on
the idea of making individual assessor data matrices as similar as
possible to each other by scaling and rotation. Another possible approach is regular principal components analysis (PCA) of all individual sensory profiles followed by a two-way ANOVA of the
most important components with assessor and products effects
as independent variables. (Ellekjr, Ilseng, & Ns, 2002). The rows
in the data table used for this analysis correspond to all samples * assessor combinations and the columns correspond to sen* Corresponding author. Tel.: +47 64 97 0333; fax: +47 64 97 0165.
E-mail address: tormod.naes@matforsk.no (T. Ns).
0950-3293/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.foodqual.2008.08.003

sory attributes. This method can be modified using the 50–50
MANOVA (Langsrud, 2002) method which handles significance
testing in a more elegant way. Using partial least squares regression (PLS-2) of all sensory profiles versus the two independent design variables assessors and products and their interaction is a
closely related approach (Martens & Martens, 2001). PCA based
on the alternative unfolding with objects and assessor * attributes
as columns and rows has been tested in for instance Dahl and
Ns (2006). In the same paper a generalised canonical correlation
analysis CCA (Carroll, 1968) analysis of individual sensory data was
tested and compared to PCA. Classical three-way factor analyses
such as Tucker-2 and PARAFAC have also found useful applications
within the framework of sensory analysis (Bro, Qannari, Kiers, Ns,
& Frøst, 2008; Brockhoff, Hirst, & Ns, 1996). Recently an alternative method for three-way analysis of variance (ANOVA) has been
proposed in the chemometric literature (ASCA, Jansen et al., 2005),
but the method is not yet tested for sensory data. ASCA is a method
which first uses regular two-way ANOVA for each attribute separately, estimates the effects (under regular ANOVA restrictions)
and then uses PCA on the main effects matrices and interaction
matrix separately for interpretation of results. The method has recently been combined with PARAFAC in the so-called PARAFASCA
(Jansen et al., 2008). Other important approaches and overviews
of alternative methods can be found in Qannari, Wakeling, Courcoux, and MacFie (2000, 2001) and in Hanafi and Kiers (2006).
The present paper is a comparison study of two of the ANOVA

based methods described above. In particular we will be interested
in comparing the newly developed ASCA method with traditional
PCA of the unfolded three-way data table followed by ANOVA (here

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175

168

called PC-ANOVA). As can be noted, the two methods are closely
related in the sense that they are both based on the same two basic
methodologies, two-way ANOVA and PCA, but with the difference
that the two methodologies are used in opposite order. These approaches have the advantage over other methods that they focus
both on the multivariate aspects of the sensory profiles and the explicit relation of the sensory data to the design of the study. The
methods will be compared conceptually and also with respect to
results obtained in an empirical illustration.
2. Theory
In the present paper we will consider a three-way data table
with I * M rows corresponding to M replicates of I samples, K columns corresponding to the attributes and with J slices corresponding to assessors. We refer to Fig. 1 for an illustration of the
structure of the data set.
Three way data of this type can always, for each attribute k, be

modelled by the two-way ANOVA model

X kijm ¼ lk þ aki þ bkj þ abkij þ ekijm

ð1Þ

Here the l is the general mean, the a’s are main effects for products,
the b’s main effects for assessors, the ab’s the interactions and e is
the random error term corresponding to replicate variation. For
ANOVA purposes, the error terms are assumed to be uncorrelated
and normally distributed with the same variance. The usual way
of applying this model is to assume that the assessor and interaction effects are random, leading to a mixed model (see Ns &
Langsrud, 1998).
Sometimes experimental designs are used for the samples and
in such studies (Baardseth et al., 1992), the product effect can be
split in several components corresponding to the experimental factors in the design (Box, Hunder, & Hunter, 1978). How to handle
this extension within the framework of the methodologies presented here will be discussed below. How to handle structures in
the replicates will also be discussed in the same sections.
In the following we will use the symbol X to denote the unfolded three-way data table with I * M * J rows and K columns.
Using this symbol it is possible to rewrite the model in Eq. (1)

for all the attributes simultaneously as follows

X ¼ 1lt þ D1 B1 þ D2 B2 þ D12 B12 þ E

j. The same structure holds for the other two matrices. The matrix E is
the matrix of residuals. Correlations between the different elements
(columns) of this matrix are possible (Mardia, Kent, & Bibby, 1978),
but it is always assumed within multivariate ANOVA that the error
terms for different observations and replicates are independent.
The model (2) can be used directly, either for each attribute
(column) separately or for all simultaneously for testing hypotheses about product and assessor effects. An example of an important
hypothesis related to this model is H0:B1 = 0, which is the hypothesis of no product effects. For the univariate ANOVA, this hypothesis can be separated in K individual hypotheses, one for each
attribute. Similar hypotheses can be set up for assessor and interaction effects. If wanted, one can also construct a combined
hypothesis of for instance B1 and B12 as is done in the ASCA paper
(Jansen et al., 2005) and in Ns & Langsrud, 1998.
The main problems with regular ANOVA approaches is that they
only focus on hypothesis tests and provide little further insight
about the relations between the attributes. Therefore ANOVA will
usually be accompanied with some type of PCA for further interpretation of the relations between the variables. In this paper we
will discuss two alternative approaches proposed in the literature

for providing this type of additional insight by combining ANOVA
with PCA.
For the purpose of the methods to be discussed below, it is of
interest to estimate the effects matrices B in Eq. (2) above. This
is usually done by least squares (LS) fitting of the responses to
the design matrices, but in order to obtain unique results, one
needs to add a restriction on the parameter estimates (see e.g.
Lea, Ns, & Rødbotten, 1997). This can be done in various ways,
but the most common way is to use the restriction that all main effects of assessors and main effects of products sum to 0 and that
the same is true for the interactions summed either over assessors
or products. In this paper main attention will be given to balanced
designs, but how to extend the approach to more general data sets
will also be discussed. For the balanced case, the main effects and
interactions have a particularly simple expression based on simple
averages and subtraction, i.e.

a^ ki ¼ X ki  X k

ð3Þ

k  X
k
^k ¼ X
b
j
j

ð4Þ

ab^kij ¼ X kij  X ki  X kj þ X k

ð5Þ

ð2Þ

where l is the general mean vector for all K attributes simultaneously, 1 is a vector of 1’s, the D1, D2 and D12 are the dummy design matrices for the products, assessors and interactions between
assessors and products respectively and the B’s are the corresponding parameter matrices. The B1 corresponds to the a’s in Eq. (1), the
B2 to the b’s and B3 to the ab’s. The design matrix D1 will have one
column for each assessor and consists of 0’s and 1’s with a 1 in column j and row i if this line corresponds to an observation for assessor

 k is the average for product i and attribute k, X
 k is the averwhere X
i
j
 k is the total average for
age for assessor j and attribute k. and X
attribute k. Note that the interactions can be considered as obtained
by double centring of the original data matrix.
When PCA is used in this paper we will always use it on centred
data, i.e. data for which the average has been subtracted for each
column.

Sample

2.1. PCA-ANOVA

Assessor

Attribute
Fig. 1. The data structure for descriptive sensory data.

The simplest way of combining ANOVA with PCA is to use PCA
directly on the unfolded data matrix X in Eq. (2) where the number
of columns corresponds to the number of attributes and the number of rows corresponds to all assessor, product and replicate combinations. This implies that the PCA gives components that are
combinations of all the effects in the model (2). A possibility is to
average over replicates before computation of principal components, but generally this is not natural since the replicates are
needed for testing purposes in the subsequent ANOVA. Examples
of the use of this and similar methodologies can be found in
Ellekjr et al. (2002) and in Langsrud (2002).

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175

The PCA model, with let use say A components, for this data set
(mean centred) can be written as

X ¼ TPT þ E

ð6Þ

where T now represents the A first scores for the M * I * J product * assessor combinations and P represents the loadings for the

K attributes for the same components. The E represents the rest,
i.e. the components with small variance, sometimes thought of as
noise. If we sort the X matrix according to assessors (in the vertical
direction), the first I * M rows of T will contain the scores for all the
products for the first assessor, the next I * M lines will contain the
scores for all observations for assessor 2 etc. A similar sorting can
be done for the samples.
As soon as the scores are computed, they can be interpreted by
the use of scatter plots and also related to the design matrix using
regular two-way ANOVA. The ANOVA model for the scores can be
written as in Eq. (2) with X replaced by T. All calculations of effects
and hypothesis tests are done as for regular ANOVA. Since the
T- scores are uncorrelated, running separate ANOVAs for each response can now be justified. For exact statements of joint significances etc. they can, however, should also be considered in a
joint approach.
The scores in Eq. (6) can always be written as (using Eq. (2))

T ¼ XP ¼ ðD1 B1 þ D2 B2 þ D3 B3 þ EÞP

ð7Þ

The last part of the equation clearly shows that the scores are
functions of all the effects in the data table including the errors
E. Note also that the noise part of T, i.e. EP, can be written as a linear function of the original noise matrix E. Since the error terms
contribution of the scores are linear functions of the original errors,
there are reasons to expect an error distribution closer to the normal for the principal components than for the original attributes.
The product, assessor and interaction effects can as above be
computed from averages of the T’s. For the purpose of improved
interpretation of the multivariate variability in the data and its
relation to the design variables we will in this paper propose to
plot these effects in the same PCA scores plot as the original observations (see also Langsrud, 2002). This is done by simply plotting
the factor effects for the different components against each other
in the PCA scores plot. For instance for the main effects for products, the estimated a values (see Eq. (3)) obtained by ANOVA of
score 1, are plotted against the corresponding estimated a-values
for score 2. Note that this is identical to doing the corresponding
averaging over X-values, projecting these averages onto P and then
plotting them in the same way as for the original scores T. This implies that it is meaningful to interpret them vs. the same loadings
as used for the interpretation of the original scores.
Another feature that will be proposed here for enhanced interpretation of the scores plots, is the superimposition of line segments in both plotting directions (for PCA) corresponding to the
level of noise in the same two directions. In the plots presented
here we use the square root of the residual error variance for the
ANOVA models for his purpose. For component number 1 this
means that we first compute the square root of the residual error
variance for the ANOVA model of the first component vs. the design variables and then present this value as a line segment along
the first component. The line segment is centred at 0. The same is
then done for component number 2. Alternatively, one can use the
least significant difference (LSD) values from multiple testing using
the same ANOVA model. In both cases, the line segments provide
the user with a visual tool for getting a quick and direct impression
of the importance of the effects seen in the plot.
Note that all the usual tools of ANOVA, such as different types of
sums of squares and corresponding tests (Type 1, Type II sums of
squares (SS) etc.) can be used also for the t’s. In this paper, how-

169

ever, with balanced data, all the SS-types will give the same results.
It should also be mentioned that as in regular ANOVA, multiple
comparison tests may be used for assessing which individual products and assessors that are different from each other along the different PCA directions. Combined hypotheses related to D1 and D12
as proposed in Ns and Langsrud (1998) and for ASCA (Jansen
et al., 2005 can also be used here. The method can also easily be extended to situations where the product effect is composed of different experimental factors, for instance according to a factorial
design. A simple example where the product effect is composed
of two experimental factors can be modelled as

X kijlm ¼ lk þ /ki þ dkl þ bkj þ /dkil þ /bkij þ dbklj þ ekijlm

ð8Þ

Where u and d are now the two experimental factors in the product
design and b is as before the assessor effect. A three-way interaction
is also possible to incorporate in the model. The PCA of X goes as before and the ANOVA of the scores T is performed by simply incorporating an extra factor in the model. All tests and computations of
main effects and interactions etc. go as usual.
Another extension which is easily handled by the PC-ANOVA is
the use of more complex error structure as discussed in e.g. Lea
et al. (1997). An example of such a structure is the quite common
replicate error structure

dkim þ ekijm

ð9Þ

where the d’s correspond to the systematic replicate effect
within each product and the e’s correspond to the regular random
error noise. This model is quite typical in situations where the
same physical sample, i.e. a replicate within product) (for instance
a fish) is served to all panellists. In such cases, each individual fish,
m, is a replicate within product (for instance a special treatment)
and will thus correspond to one of the d-terms in Eq. (9). The
superscripts and subscripts have the same meanings as before; k
denotes attribute, i denotes product, j denotes assessor and m denotes replicate. Multiplying this effect by P, one obtains a new error vector with the same structure

dim P þ eijm P

ð10Þ

The two error terms in the sum now represent the vectors of the
error contributions in Eq. (9) for all attributes considered simultaneously. In other words, one can easily see that the more complex
error structure in Eq. (9) is split in the same way for the scores as
for the original variables.
Imbalance with respect to the number of replicates is also simple to handle. The same effects can be used in the model (see Eq.
(1)) and the same ANOVA can be used with the appropriate correction for the degrees of freedom. The calculation of the effects is,
however, slightly more complex, but this is easily handled by modern ANOVA programmes. In situations with missing product and
assessor combinations the situation is more complex, but not more
complex than for regular ANOVA. The problem is then how to define and compute interaction and main effects. Different suggestion is proposed. One simple possibility is to eliminate the
interactions, but this is not advisable for sensory data at least
unless a pre-treatment has been done to reduce the scaling effect
(Romano, Brockhoff, Hersleth, Tomic, & Ns, in press).
Another possible modification of the PC-ANOVA was proposed
in Langsrud (2002), where a more elegant way of testing significance was proposed. In that case, however, less emphasis was
put on the effects of the factors for the different principal components. Thus, this technique is merely to be considered as a modification of classical MANOVA and lies somewhere between the
present approach and the classical MANOVA. Another interesting
possibility is the PLS-2 method advocated in Martens and Martens
(2001) with the sensory profile as the multivariate response and

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175

Xj ¼ TWj PT þ E

ð11Þ

where now Xj indicates assessor or slice number in the data matrix
in Fig. 1. The model (11) assumes that the different assessors share
the same dimensions and their relation to the external data, but it
allows for different weight to the two dimensions, represented by
the individual matrices Wj. The modification proposed in van der
Kloot and Kroonenberg (1985) is to relate the common product
scores T to external data, for instance a product design matrix as
was done for the PC-ANOVA. Representing T as a linear function
of the external design variables D for the products, the model in
(11) can be written as

Xj ¼ DBWj PT þ E

ð12Þ

where B is a matrix of regression coefficients and the D is the design
matrix for the products. Note that the way the assessors and products combine in this model is different from the PC-ANOVA. Here
the joint effect is a combination of an additive effect for the products and a multiplicative effect for the assessors.
An obvious advantage of the PC-ANOVA is its simplicity and
that it provides both significance tests based on ANOVA and visual
tools for direct interpretation of the tests (main effects, interaction
plots and multiple comparisons). The main disadvantage is that if
the factors span different multivariate spaces, the number of components to interpret may be high.
2.2. ASCA
Compared to the PC-ANOVA, the ASCA method is based on
reversing the order of the two operations ANOVA and PCA. The first
step is to use the regular two-way ANOVA model for each attribute
(model (1) above) and estimate the effects using the regular
ANOVA restrictions (sum equal to 0 over the levels). Considering
these values for all attributes at the same time gives us a matrix
of effects for the samples (dimension I * K), a matrix of effects for
the assessors (J * K) and a three-way matrix (I * J * K) for the interactions. The two former are regular two-dimensional matrices and
can then be analysed directly by the use of PCA. The latter is analysed by the use of PCA (also so-called Tucker-1. see Tucker (1966))
on an unfolded matrix as described above. Three different ways of
unfolding are possible, but here we will focus on the same unfolding as for the PC-ANOVA, i.e. the unfolded matrix has dimensions
I * M * J and K. Note that the matrices used in ASCA correspond to
the estimated versions of the matrices B1, B2 and B3 in the model
(2) above. As an example, the B1 matrix can be written as

1  X
1
X
1
B
:
B
B
^1 ¼ B
B
B
B
@
1
1  X
X
0

I

1
K  X
K
X
1
C
:
:
C
C
C
:
:
C
C
A
:
:
K  X
K
X

ð13Þ

I

As an alternative to using Tucker-1 for the interactions one can
use the PARAFAC as was suggested in Jansen et al. (2008). This corresponds to using the PARAFAC on the double centred matrix, i.e.
after subtraction of main effects. The method is called PARAFASCA.
Note that this approach is a direct extension to several dimensions
of the approach proposed in Mandel (1971).

The difference between ASCA and the PC-ANOVA approach is
that here the averaging is taken before PCA, while above it was taken after PCA. This is generally an advantage for ASCA since one
obtains PCA plots which are focused on the different effects and
not influenced by everything at the same time. It may thus possibly
give clearer conclusions for each of the separate effects. As will be
seen from the example below, however, the multivariate spaces
spanned by the different effects are rather similar for this data set.

Table 1
Percentage explained variance by the PCA model of the unfolded data organised as
object * assessors * replicates vs. attributes
Number of
components

Percentage variance
explained

Cumulative percentage variance
explained

1
2
3
4
5
6
7
8
9

74.69
9.38
6.47
2.90
1.98
1.89
1.28
1.08
0.33

74.69
84.08
90.55
93.44
95.42
97.30
98.58
99.67
100.00

a

20

1
2
3
4
5

15
10

PC 2 % exp var 9.385

the dummy design variables as the independent variables. This
method is also graphically oriented, but less developed when concerns significance testing.
There is also a close relation of PC-ANOVA to an extension of the
Tucker-2 model proposed by van der Kloot and Kroonenberg
(1985). The classical Tucker-2 model is one with both common
scores and common loadings and can be written as

5
0
-5
-10
-15
-20
-25

-20

-15

-10

-5

0

5

10

15

20

25

PC 1 % exp var 74.69

b

Sweet

0.6

Raspb.

0.4

PC 2 9.385 % exp var

170

0.2
Bites
Elastic
Hard
Sticky
Transp

0

Sugar
-0.2

-0.4

-0.6

Acid
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

PC 1 74.69 % exp var
Fig. 2. PC-ANOVA. Scores and Loadings plot for component 1 and 2 of the unfolded
sensory data.

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175
Table 2
Mixed model ANOVA performed on components 1,2,3,4, (effect of assessors and
interactions considered random and effect of sample considered fixed)
d.f.

Mean
Square

PC1
Assessor
Sample
Assessor * sample
Error
Total

224.9
31470.6
707.7
1109.3
33512.5

10
4
40
110
164

22.49
7867.65
17.69
10.08

PC2
Assessor
Sample
Assessor * sample
Error
Total

278.3
1573.83
1053.3
1305.18
4210.61

10
4
40
110
164

27.83
393.457
26.332
11.865

PC3
Assessor
Sample
Assessor * sample
Error
Total

1006.62
307.12
484.02
1104.61
2902.38

10
4
40
110
164

100.662
76.78
12.101
10.042

PC4
Assessor
Sample
Assessor * sample
Error
Total

108.27
29.62
357.43
803.89
1299.21

10
4
40
110
164

10.8275
7.4041
8.9357
7.3081

Square root
of mean
square

F

Prob > F

1
2
3
4
5

10
1.27
444.68
1.75

0.2789
0
0.0116

1.06
14.94
2.22

0.4166
0
0.0006

3.17

PC 2 % exp var 9.385

Sum Sq.

20
15

5
0
-5
-10

3.44

-15

8.32
6.35
1.21

-20
-25

0
0.0005
0.2229

3.17

b
1.21
0.83
1.22

All three replicates included in the analysis.

Since no information about random variation is available in the
plot, it is not obvious how to make a direct assessment of significance of the differences between products or assessors along the
different axes as could be done for the PC-ANOVA using multiple
comparisons. A possible extension of ASCA would be to add some
type of confidence ellipses based on for instance the bootstrap
(see e.g Pages & Husson, 2005). For the same reason it is natural
to use MANOVA and individual ANOVA’s before the ASCA to provide additional insight about significant effects that can be used
to interpret the ASCA plots.
Regarding imbalance, the same as stated for the PC-ANOVA can
be stated also here. Also with respect to more complex error structure, the ASCA method can be used. The only modification that has
to be done is that a restricted maximum likelihood (REML) estimate is needed for improved estimation of effects. REML is a method that takes the more complex error structure into account when
estimating the effects. The standard LS estimates can also be used
since they are unbiased, but the REML estimates are more precise.
Also incorporation of factorial designs in the product structure is
possible. As long as the effects can be estimated, the method can
be used. One simply ends up with more than three matrices to submit to PCA. For instance in the model (8) above, one will end up
with 6 different PCA analyses. As the number of factors increases,
the number of plots also increases. The ASCA method can also be
used to analyse the joint effect of for instance the main effects of
products and the interactions as was demonstrated in Jansen
et al. (2005).

-15

-10

-5

0

5

10

15

20

25

10

15

20

25

20
1
2
3
4
5

15

0.313
0.5149
0.2062

2.7

-20

PC 1 % exp var 74.69

10

PC 4 % exp var 2.896

Source

a

171

5
0
-5
-10
-15
-20
-25

-20

-15

-10

-5

0

5

PC 3 % exp var 6.469
Fig. 3. PC-ANOVA. Plot of scores averaged over assessors after performing PCA. (a)
PC1 vs. PC2. (b) PC3 vs. PC4. The horizontal and vertical bars close to the centre
represent the squares root of the MSE.

All variables were tested using a two-way ANOVA (model (1))
and all attributes were found to be significant for the separation
of the samples and therefore kept during the study.

4. Results and discussion
All calculations were performed using MinitabÓ, 15 UnscramblerÓ 9.2 and custom made Matlab/Octave routines which are
freely available for download from the first author’s website
http://www.chemometrics.it.

3. Data set

4.1. PC-ANOVA

The data set chosen for this paper is from sensory analysis of a
candy product. 5 different candies (I = 5). There are K = 9 sensory
attributes and J = 11 assessors in the panel and M = 3 replicates.
The attributes were transparency, acidity, sweet taste, raspberry
flavour, sugar coated texture tested with a spoon, biting strength
in the mouth, hardness, elasticity in the mouth, stick to teeth in
the mouth.

As can be seen from Table 1, the explained variances are quite
high for this data set (84% explained after 2 components). The first
factor is totally dominating with a percentage of explained variance equal to 75%. Scores and loadings plot of all the observations
are presented in Fig. 2. The scores are marked according to the 5
products. As can be seen from Fig. 2, there is some disagreement
among the assessors, but the overall agreement among the assessors

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175

172

and replicates seems to be quite good for each product as compared
to the difference between the products. The first component distinguishes between two groups of objects, objects 2, 3 and 4 on one
side and objects 1 and 5 on the other. The latter group has more
sugar flavour, higher sweetness, raspberry and acidity taste, while
the former group has more stickiness, hardness, transparency etc.
The second axis primarily distinguishes between samples 1 and 5
and 3 and 5. Objects 2 and 1 are the less acidic and the sweetest
and most raspberry flavoured among them. All the results were
in accordance with what could be expected from the design of
the samples.
As can be seen from the ANOVA tables in Table 2, the sample effect is the dominating effect for both the first two components,
while for the third component, the assessor effect is the strongest.
For the fourth component, none of the effects are significant.
ANOVA was also conducted for the rest of the components with
some significant effects here and there, but these results are not
considered further here due to their very low explained variance
(less than 5% in total). The interaction effect is significant for both
the first components.
All these results correspond well to what is seen in the plots
(Figs. 2–5). The advantage of the plots, however, is that one

20
1
2
3
4
5
6
7
8
9
10
11

15

PC 2 % exp var 9.385

10
5
0
-5

a

0
-5

-15

-20

-15

-10

-5

0

5

10

15

20

25

PC 1 % exp var 74.69

-20
-25

-20

-15

1
2
3
4
5
6
7
8
9
10
11

-5

0

5

10

15

20

25

5
0
-5

b
1
2
3
4
5
6
7
8

6

4

PC 2 % exp var 9.385

10

-10
-15
-20
-25

-10

PC1 % exp var 74.69

20
15

PC 4 % exp var 2.896

5

-10

-15

b

1
2
3
4
5
6
7
8
9
10
11

10

-10

-20
-25

20
15

PC 2 % exp var 9.385

a

automatically sees which samples those are similar and also their
relation to the attributes. The product and assessor effects (see Eqs.
(3)–(5)) are plotted along the same axes and with the same units as
for the original scores. These results are presented in Figs. 3–5. The
Fig. 3 represents the products and shows a very clear tendency. As
compared to the square root of the corresponding MSE, i.e the standard deviation of the noise (presented as a straight line around 0 in
both directions), one can also get a visual interpretation of the differences as compared to the noise level. Multiple testing (Tukey’s
method) of the second axis (vertical) in Fig. 3a, shows that sample
1 is significantly different from all except sample 3 and that sample
2 is only slightly different from 5 and not significantly different
from the rest. The only sample which is significantly different from
all is sample 5. For this data set, the average score plot of the products does not provide new insight as compared to the overall plot,
but in a more complex situation with several more objects and
smaller differences between them this type of plot may simplify
interpretation considerably. In Fig. 4 is presented the differences
among the assessors and as can be seen, the differences are much
smaller than for the product effects. Along the third axis there is
some more variability among the assessors, however, which is also
reflected in the F-test for assessors (Table 2).

2

0

10
11

-2

-4

-20

-15

-10

-5

0

5

10

15

20

25

PC 3 % exp var 6.469

-6
-6

-4

-2

0

2

4

6

8

PC 1 % exp var 74.69
Fig. 4. PC-ANOVA. Plots of scores averaged over samples after performing PCA on
the raw data. The horizontal and vertical bars close to the centre represent the
squares root of the MSE.

Fig. 5. PC-ANOVA. PCA performed on data after double centring. Point is marked
according to assessors. Figure b is a magnification of the figure a.

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175

The interaction plot in Fig. 5 is presented both for the same
units as above and for units showing the results more clearly.
The main advantage of the interaction plot is probably that it can
shed some light onto which assessors that are the most similar
to the average and which are the most different. It is for instance
clear here that all points belonging to assessor 8 lie close to the
centre of the plot while assessor number 10 has some of the points
far away from the centre. This means that assessor 8 is much closer
to the panel average for all the samples than assessor number 10.
The latter is then more responsible for the interaction effect than
assessor number 8. More specifically, one can also identify for
which assessor and product combinations the interactions are largest. Comparing this with the loadings plot one can also obtain
information about which attributes that are involved in the
interactions.
A study of the normality of the residuals of the original data and
the PCA scores was done according to the ideas mentioned in Section 2.2. Generally, the residuals from the scores have a distribution closer to normality than the residuals from the original
variables. Two examples are presented in Fig. 6a and b. in Fig. 6a
is given a typical example related to one of the original variables

a

QQ Plot of Sample Data versus Standard Normal

173

(stickiness) while in Fig. 6b is presented the normality plot for
the residuals from the ANOVA of the first principal component.
As can be seen, the plot for the latter follows a much straighter line
than for the former except from a few outlying points on each side.
4.2. ASCA
The product plots, assessor plots and the interaction plots are
presented in Figs. 7–9. For the products in Fig. 7, more or less the
same interpretation as for the PC-ANOVA can be made. The plot
is almost identical except for a 180 degrees switch of the first component which has no effect on the interpretation. Since no random
variability is directly related to the plot, it is, however, hard to tell
what is significant, in particular along axis 2. For the assessors plot
in Fig. 8, one can see that there are strong similarities between the
loadings plot here and the loadings plot in Fig. 7, indicating much
of the same structure in the multivariate differences between the
assessors as between the products. There are, however, some small
differences related to for instance the attribute sugar coating. This
may indicate a possible advantage of the ASCA. It is, however, hard
to tell from the scores plot whether these differences are significant or not. For the interaction plot in Fig. 9, the same as stated
above is the case for the loadings. The interaction scores plot can
be used for the same purpose as above. Again we see that assessor

10

a

6

1
2
3
4
5

10

4

PC 2 % exp var 5.088

Quantiles of Input Sample

8

2
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-8
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-2

-1

0

1

2

3

-10

Standard Normal Quantiles
-15

b

-10

-5

0

5

10

PC 1 % exp var 94.24

QQ Plot of Sample Data versus Standard Normal
10

b

Sweet
0.4

Raspb.

0.2

PC 2 5.088 % exp var

Quantiles of Input Sample

5

0

-5

-10

Bites
Transp
Hard
Sticky
Elastic

Sugar

0

-0.2

-0.4

-0.6
-15
-3

-2

-1

0

1

2

3

Standard Normal Quantiles

-0.8

Acid
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

PC 1 94.24 % exp var
Fig. 6. Normal probability plot for (a) stickiness and the (b) first principal
component.

Fig. 7. ASCA. PCA scores and loadings plot for the object effects matrix.

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175

174

a

2

1

0

1
2
3
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3

PC 2 % exp var 20.91

3

PC 2 % exp var 16.42

a

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Acid

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Transp

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Sticky

0.4

0.4
Elastic

0.2

PC 2 20.91 % exp var

PC 2 16.42 % exp var

2

PC 1 % exp var 32.92

PC 1 % exp var 55.21

Sticky
Sugar
0
Transp

Hard
Bites

-0.2

Raspb.
-0.4

Hard
Bites
Elastic

0.3
0.2
0.1

Raspb.

0
-0.1
-0.2

Sweet

Sugar

-0.3
-0.6
-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Acid

-0.4

Sweet
0.8

1

-0.6

-0.4

PC 1 55.21 % exp var
Fig. 8. ASCA. PCA scores and loadings plots for the assessor effects matrix.

number 8 is much closer to the average than is assessor number
10. The main conclusion is that for this data set, the two methods
gave practically the same results and interpretations.
The fact that all the three matrices give similar loadings is
strongly related to the fact that so few components are needed
for the PCA on the raw data (PC-ANOVA), In a situation with different multivariate structure for the different model effects, the number of components for PC-ANOVA would basically be a sum of the
dimensions for each of the effects and this is clearly not the case
here. An interesting research question is whether this finding is
generally true for sensory data.
For the two main effects plots, 99% and 70% of the variation is
explained by the first two axes while for the interaction plot, the
corresponding value is 54% (See Tables 3–5).
5. Conclusions
In this paper we have compared two approaches for analysis of
sensory data which combine PCA and ANOVA. One of them is based
on PCA of for the original data with subsequent ANOVA of the
scores to test the effects of the design factors on the scores (PCANOVA). The other one is based on using PCA on the matrices of
estimated main effects and interactions (ASCA). The difference

-0.2

0

0.2

0.4

PC 1 32.92 % exp var
Fig. 9. ASCA. PCA scores and loadings plot for the interactions data matrix.

Table 3
ASCA. Percentage variance explained by PCA performed on matrix of products effects
Number of
components

Percentage variance
explained

Cumulative percentage variance
explained

1
2
3
4

94.24
5.09
0.58
0.10

94.24
99.33
99.90
100.00

Table 4
ASCA. Percentage variance explained by PCA performed on matrix of assessor effects
Number of
components

Percentage variance
explained

Cumulative percentage variance
explained

1
2
3
4
5
6
7
8
9

55.21
16.42
14.34
6.30
5.00
1.44
0.87
0.25
0.17

55.21
71.63
85.96
92.26
97.26
98.71
99.57
99.83
100.00

G. Luciano, T. Ns / Food Quality and Preference 20 (2009) 167–175
Table 5
ASCA. Percentage variance explained by PCA performed on the interaction effects
Number of
components

Percentage variance
explained

Cumulative percentage variance
explained

1
2
3
4
5
6
7
8
9

32.92
20.91
15.72
10.21
8.05
4.60
4.20
2.92
0.48

32.92
53.83
69.55
79.76
87.80
92.40
96.60
99.52
100.00

between the methods lies in the fact that they use ANOVA and PCA
in opposite order. Both methods have a multivariate focus since
they consider all attributes simultaneously and they provide information about assessor effects and products effects as well as their
interactions on the multivariate structure. In this sense both methods are highly useful for analysing sensory data. Both methods are
easily computed using regular statistical software packages and
none of them need any iteration or other complex numerical procedures. Complex replicate structure and several factors in the
experiment can also easily be handled without modification of
the methods. Relations to other methods in the literature were
highlighted in the method section.
The main advantage of PC-ANOVA is that it is easier to use for
direct assessment of significant differences, also multiple comparison, directly related to the interpretation plots, i.e. PCA plots, obtained. Another advantage is that the PCA plots can easily be
equipped with additional information that can be used to give a direct assessment of the size of the effects as compared to the noise
level in the data, as measures by either the standard deviation of
the random noise or the LSD values from multiple comparisons.
The main disadvantage of PCA-ANOVA is that in cases where the
different effects have a different multivariate profile, one may
end up with many PCA components to analyse and interpret which
can be both a time-consuming and complex task. This is particularly true in situations with several factors in the design. This is exactly the point where ASCA has its main advantage. Since it
provides a separate PCA plot for each factor separately, each of
the PCA models will generally have a lower dimension than for
PC-ANOVA. Testing significance for the effects in the plots, is, however, less obvious for the ASCA, although some recent attempts
have been made to create such tests (Vis, Westerhuis, Smilde,
and van der Greef (2007)).
As was demonstrated in the example, the two methods gave the
same overall interpretation for this particular data set. It was detected that the assessor differences and product differences
spanned the same low-dimensional multivariate correlation structure leading to very similar loadings plots for the different effects
for the ASCA method. This phenomenon leads to a small number
of important components also for the PC-ANOVA. If this is not
the cases, the latter method will need several components and
the ASCA will need a different interpretation for each of the effect
matrices. An interesting problem to be investigated in the future is
how often this situation occurs in practice in sensory analysis.

175

In the example, only product and assessor effects were considered, but both methods discussed can be extended to situations
with several design factors and complex replicate structure.

Acknowledgments
We would like to thank Norwegian Research Council (NFR) for
financial support for this study. We would also like to thank Asgeir
Nilsen and Grete Hyldig for making the data available to us.
References
Baardseth, P., Naes, T., Mielnik, J., Skrede, G., Hølland, S., & Eide, O. (1992). Dairy
ingredients effects on sausage sensory properties studied by principal
component analysis. Journal of Food Science, 57(4), 822–828.
Box, G. E. P., Hunder, W., & Hunter, S. (1978). Statistics for experimenter. NY: Wiley.
Bro, R., Qannari, E. M., Kiers, H. A., Ns, T., & Frøst, M. B. (2008). Multi-way models
for sensory profiling data. Journal of Chemometrics, 22, 36–45.
Brockhoff, P. M., Hirst, D., & Ns, T. (1996). Analysing individual profiles by threeway factors analysis. In T. Ns & E. Risvik (Eds.), Multivariate analysis of data in
sensory science. Elsevier Science Publishers.
Carroll, J. D. (1968). Generalisation of canonical analysis to three or more sets of
variables. Proceedings of the 76th convention of the American Psychological
Association (vol. 3) (pp. 227–228).
Dahl, T., & Ns, T. (2006). A bridge between Tucker-1 and Carroll’s generalised
canonical analysis. Computational statistics and data analysis, 50(11),
3086–3098.
Dijksterhuis, G. (1996). Procrustes analysis in sensory research. In T. Ns & E. Risvik
(Eds.), Multivariate analysis of data in sensory science (pp. 185–217). Elsevier
Science.
Ellekjr, M. R., Ilseng, M. R., & Ns, T. (2002). A case study of the use of
experimental design and multivariate analysis in product improvement. Food
Quality and Preference, 7(1), 29–36.
Hanafi, M., & Kiers, H. (2006). Analysis of K sets of data, with differential emphasis
on agreement between and within sets. Computational statistics and data
analysis, 51, 1491–1508.
Jansen, J., Hoefsloot, J., van der Greef, M., Timmerman, E., Westerhuis, J., & Smilde, A.
K. (2005). ASCA: Analysis of multivariate data obtained from an experimental
design. Journal of Chemometrics, 19(9), 469–481.
Jansen, J., Bro, R., Huub, C., Hoefsloot, J., van den Berg, F. W. J., Westerhuis, J., &
Smilde, A. K. (2008). PARAFASCA: ASCA combined with PARAFAC for the
analysis of metabolic fingerprinting data. Journal of Chemometrics, 22, 114–121.
Langsrud, O. (2002). 50–50 multivariate analysis of variance for collinear responses.
Journal of the Royal Statistical Society: Series D, 51, 305–317 (The Statistician).
Lea, P., Ns, T., & Rødbotten, M. (1997). Analayis of variance of sensory data. J. Wiley
and sons.
Mandel, J. (1971). A new analysis of variance model for non-additive data.
Technometrics, 13(1), 1–18.
Mardia, K., Kent, J., & Bibby, J. (1978). Multivariate analysis. UK: Academic press.
Martens, H., & Martens, M. (2001). Multivariate analysis of quality: An introduction.
UK: Wiley Chicester.
Ns, T., & Langsrud, Ø. (1998). Fixed or random assessors in sensory profiling? Food
Quality and preference, 9(3), 145–152.
Pages, J., & Husson, F. (2005). Multiple factors analysis with confidence ellipses: a
methodology to study the relationships between sensory and instr8umental
data. Journal of Chemometrics, 19, 138–144.
Qannari, E. M., Wakeling, I., Courcoux, P., & MacFie, H. J. H. (2000). Defining the
underlying sensory dimensions. Food Quality and preference, 11, 151–154.
Romano, R., Brockhoff, P. B., Hersleth, M., Tomic, O., & Ns, T. (in press) Correcting
for different use of the scale and the need for further analysis of individual
differences in sensory analysis, Food Quality and Preference, doi:10.1016/
j.foodqual.2007.06.008.
Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis.
Psykometrica, 31, 279–311.
Van der Kloot, W. A., & Kroonenberg, P. M. (1985). External analysis with threemode principal component analysis. Psykometrika, 50(4), 479–494.
Vis, D. J., Westerhuis, J. A., Smilde, A. K., & van der Greef, J. (2007). Statistical
validation of megavariate effects in ASCA. BioMed Central (BMC) BioInformatics,
8, 322.