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Growth curve modelling
Food Quli& and Prcfnmce Vol. 9, No. 3, pp. 91-93, 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain ELSEVlER
0950-3293/98 S19.00+0.00
PIIrS0950-3293197100036-O
GROWTHCURVEMODELLING Per M. Brockhoff Department of Mathematics and Physics, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK- 187 1 Frederiksberg C, Denmark (Accepted 15 April 1997)
doing each regression and then doing linear analysis on the regression parameters yields the proper maximum likelihood results under the random coefficient model. The interesting information in this case lies in the B-values. However, the individual consumer’s use of the scale will affect these #I-values. A consumer using a larger proportion of the scale will tend to have numerically larger /I-values than a consumer using only a small part of the scale. As this is not so interesting the result of each of the 4432 regressions were summarized in the correlation coefficients p$l.
ABSTRACT A two-step ‘growth curve’ or ‘random coeficie& approach was used. The preference for each single consumer in each country was related to each sensory attribute used by the panel of that country. These individual measures of relational strength were then subjected to three-way analysis of variance leading to significance statements about country, sex and age dzyerences in the consumers’ relation to the sensory projiles. In general these did not vary much with sex and age but depended heavily on the country. 0 1998 Elsevier Science Ltd. All rights reserved
GROWTH
ANALYSIS
For each sensory attribute I = 1, . . . . 16 there is now a correlation coefficient for each consumer in a country, where the panel used the attribute in question. The consumers are divided into groups by country, sex and age, so for each attribute a three-way analysis of variance was performed. Since correlation coefficients are known to be non-normal distributed, the Fisher’s <-transformation
CURVE
Letyp be the liking on a g-point hedonic scale of coffee j by consumer k in country i, k = 1, . . . . tli, j = 1, . . . . 8 and i = 1, . ..( 5, where ni ranges from 51 to 85 and let $1 be the average over assessors and replicates of the scoring of attribute I for coffee j in the panel from country i, 1=1 , . . . . pip where pi ranges from 5 to 16. For consumer k in country i the pi regressions, one for each attribute,
/prk =
Ai;)+ &p
+ El),
llog 1 + 2
pi’
1 - p?) ik
were used prior to the analysis. Note that the numbers that are analyzed measure relational strength between consumer preferences and sensory attributes as evaluated by a panel from the same country as the consumer. For bitter taste all 369 consumers enter the analysis. The analysis of variance table, Table 1, shows that only country differences are significant, no age or sex differences are seen in consumers relation of liking to sensory evaluated bitter taste. Table 2 shows the estimated bitter taste z-levels of each country in the model with only country differences. We see that for Germany, France and Denmark there are significantly negative relation between bitter taste and liking, for the UK no relation and for Poland significantly positive relation. We also see that there are no significant differences between the three former countries, whereas indeed the UK and Poland do significantly differ from the other four countries.
were performed. Note that all regressions are based on eight observations and that altogether (5 x 51) + (14 x 80) + (9 x 73) + (13 x 80) + (16 x 85) = 4432 linear regressions were performed. The regressions are the individual ‘curves’ and the underlying model is the ‘random coefficient regression’ Yjjk=
OF VARIANCE
j = 1, . . . . 8
(0 where A, (0 and B, are normal random variables themselves. It is well known, see Johansen ( 1983), that in the balanced case with equal number of observations for each individual, the two-step procedure of first 91
92
P. M. Brockhoj’-
TABLE 1. Analysis of Variance Table for Bitter Taste
df
Country Sex Age Country*sex Country*age Sex*age Error
4 1 5 4 19 5 330
F-value
SS 0.378 0.00566 0.0440 0.00999 0.0403 0.0302 1.723
18.11 1.08 1.68 0.48 0.41 1.16
TABLE 3. Results of the 16 Three-way Analyses of Variance Including all Main and Interaction Effects Using the SASo Type II Test See e.g. SAS/STATo User’s Guide (1989)
Pr>F < 0.0001 0.30 0.14 0.75 0.99 0.33
Analyses as above for bitter taste were performed for all 16 attributes. Overall, the analyses showed few age and sex differences, but extensive country differences (see Table 3). Looking a little into sex and age differences, based on least squares means not given here, one sees the following: for malty flavour there are mostly positive relations, whereas the old males show negative relations. For fruit flavour age groups around 2544 show positive relations, otherwise no relation is seen. For floral flavour the older males and younger females in the UK show positive relations, a pattern not seen in the other countries. Finally, the country differences are summarized in Table 4. Apart from the interesting fact that the consumer preferences seem to be quite oppositely related to single sensory attributes from country to country, it is seen that overall the UK consumer preferences are quite poorly related to single sensory attributes as compared to the other countries. The latter is even more interesting in the light of the other workshop contribution by the author, Brockhoff (1997), where the ICO panel was shown to be generally the most sensitive of the five panels.
Couutry : :
BITTERT BURNTF ASTRGNMF CHEMF FLORALF ACIDT CARAMF MALTYF SALTT SWEETT EARTHF FRUITF GRASSF RANCIDF SOURT * WOODF
t : : t
* * * * *
t
t t
x t
The tests for the main effects of sex and age and the interaction effects country*sex 5%
and country*age
level) for all of the 16 variables
omitted.
*Significant
tsignificant
on 5%
TABLE 2. Estimated z-values for Bitter Taste withp-values for the t Test of <=O (i.e. p=O) and the Results of Pairwise t Test Comparisons
Countries
z
level.
-0.368 -0.284 -0.272 -0.0313 0.206 sharing
and these columns
level, tsignificant
on 1%
(on were level,
on 0.1 %level.
panels from one country and consumers from another or the relations between all consumers and a single panel. Although the second step analysis of variance above mainly discovered country differences, it is clear that the approach would be relevant also within a single country when only a single panel and consumer study is performed. The present analysis was entirely univariate. One may, which is in line with the classical growth curve setting, see e.g. Keuls and Garretsen (1982)) do multivariate analysis of variance on the set of pi z-values
Country
Attribute
In the analysis above consumers were only related to sensory profiling from the same country. It may be interesting to do similar analysis on relations between
Germany France Denmark UK Poland
were non-significant
TABLE 4. The Ordering of the Countries (left=small values, right = large values) for the Significant Country Effects (on 5%)
DISCUSSION
Country
Country*sex*age
Sex*age
P
< 0.0001 < 0.0001 0.0002 0.58 0.0004
letters are not significantly
Pairwise comparison A A A B C different on 5%
G,-F,-D,-I,,P,+ G,-D,,,-Fb-It,P,+
BITTERT BURNTF ASTRGNMF CHEMF FLORALF ACIDT MALTYF GRASSF RANCIDF SOURT WOODF
Fa-IabGbPb F,-G,-IbP, + I-PFG P,-I&DbF, + I,G,Fb + Fa-PbIb F,-Ib-P: P,-Ib-G + Da-IbPc
The subscripts (on 5%).
show the results of pair-wise t test comparisons
Countries
in a listing are significantly
do not share subscripts. P = Poland, Superscripts country:
differences
(I = ICO,
F = France,
different if they G = Germany,
D = Denmark). show the results of the t test for z=O
+ = significantly positive, - = significantly
superscript = non-significant.
in each
negative,
no
Growth Curve Modelling within each country. profile information,
One may also compile by for example
analysis, before doing the individual the approach Mapping group
above is basically
with
differences
segments. relations
analysis rather
component
regressions.
a version
of variance than
the sensory
principal
searching
And as is the case for Preference need not be linear as supposed
consumer
for consumer Mapping above.
the
For the
present analysis no effort was done to search for non-linear relationships linear.
that might fit the data better
REFERENCES
As such,
of Preference
testing
93
than the
Brockhoff, P. M. (1997) Assessor modelling, Sensometrics Workshop, Question 2. Food Qua& and Preference9, 87-89. Keuls, M. and Garretsen, F. (1995) Statistical analysis of growth curves in plant breeding. Euphytica 31, 51-64. Johansen, S. (1983) Some topics in regression. Scandinavian Journal of Statistics 10, 161-194. SAS/STATo (1989) Version 6, 4th edn, Vol. 2, SAS Institute Inc., Gary, NC.
0950-3293/98 S19.00+0.00
PIIrS0950-3293197100036-O
GROWTHCURVEMODELLING Per M. Brockhoff Department of Mathematics and Physics, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK- 187 1 Frederiksberg C, Denmark (Accepted 15 April 1997)
doing each regression and then doing linear analysis on the regression parameters yields the proper maximum likelihood results under the random coefficient model. The interesting information in this case lies in the B-values. However, the individual consumer’s use of the scale will affect these #I-values. A consumer using a larger proportion of the scale will tend to have numerically larger /I-values than a consumer using only a small part of the scale. As this is not so interesting the result of each of the 4432 regressions were summarized in the correlation coefficients p$l.
ABSTRACT A two-step ‘growth curve’ or ‘random coeficie& approach was used. The preference for each single consumer in each country was related to each sensory attribute used by the panel of that country. These individual measures of relational strength were then subjected to three-way analysis of variance leading to significance statements about country, sex and age dzyerences in the consumers’ relation to the sensory projiles. In general these did not vary much with sex and age but depended heavily on the country. 0 1998 Elsevier Science Ltd. All rights reserved
GROWTH
ANALYSIS
For each sensory attribute I = 1, . . . . 16 there is now a correlation coefficient for each consumer in a country, where the panel used the attribute in question. The consumers are divided into groups by country, sex and age, so for each attribute a three-way analysis of variance was performed. Since correlation coefficients are known to be non-normal distributed, the Fisher’s <-transformation
CURVE
Letyp be the liking on a g-point hedonic scale of coffee j by consumer k in country i, k = 1, . . . . tli, j = 1, . . . . 8 and i = 1, . ..( 5, where ni ranges from 51 to 85 and let $1 be the average over assessors and replicates of the scoring of attribute I for coffee j in the panel from country i, 1=1 , . . . . pip where pi ranges from 5 to 16. For consumer k in country i the pi regressions, one for each attribute,
/prk =
Ai;)+ &p
+ El),
llog 1 + 2
pi’
1 - p?) ik
were used prior to the analysis. Note that the numbers that are analyzed measure relational strength between consumer preferences and sensory attributes as evaluated by a panel from the same country as the consumer. For bitter taste all 369 consumers enter the analysis. The analysis of variance table, Table 1, shows that only country differences are significant, no age or sex differences are seen in consumers relation of liking to sensory evaluated bitter taste. Table 2 shows the estimated bitter taste z-levels of each country in the model with only country differences. We see that for Germany, France and Denmark there are significantly negative relation between bitter taste and liking, for the UK no relation and for Poland significantly positive relation. We also see that there are no significant differences between the three former countries, whereas indeed the UK and Poland do significantly differ from the other four countries.
were performed. Note that all regressions are based on eight observations and that altogether (5 x 51) + (14 x 80) + (9 x 73) + (13 x 80) + (16 x 85) = 4432 linear regressions were performed. The regressions are the individual ‘curves’ and the underlying model is the ‘random coefficient regression’ Yjjk=
OF VARIANCE
j = 1, . . . . 8
(0 where A, (0 and B, are normal random variables themselves. It is well known, see Johansen ( 1983), that in the balanced case with equal number of observations for each individual, the two-step procedure of first 91
92
P. M. Brockhoj’-
TABLE 1. Analysis of Variance Table for Bitter Taste
df
Country Sex Age Country*sex Country*age Sex*age Error
4 1 5 4 19 5 330
F-value
SS 0.378 0.00566 0.0440 0.00999 0.0403 0.0302 1.723
18.11 1.08 1.68 0.48 0.41 1.16
TABLE 3. Results of the 16 Three-way Analyses of Variance Including all Main and Interaction Effects Using the SASo Type II Test See e.g. SAS/STATo User’s Guide (1989)
Pr>F < 0.0001 0.30 0.14 0.75 0.99 0.33
Analyses as above for bitter taste were performed for all 16 attributes. Overall, the analyses showed few age and sex differences, but extensive country differences (see Table 3). Looking a little into sex and age differences, based on least squares means not given here, one sees the following: for malty flavour there are mostly positive relations, whereas the old males show negative relations. For fruit flavour age groups around 2544 show positive relations, otherwise no relation is seen. For floral flavour the older males and younger females in the UK show positive relations, a pattern not seen in the other countries. Finally, the country differences are summarized in Table 4. Apart from the interesting fact that the consumer preferences seem to be quite oppositely related to single sensory attributes from country to country, it is seen that overall the UK consumer preferences are quite poorly related to single sensory attributes as compared to the other countries. The latter is even more interesting in the light of the other workshop contribution by the author, Brockhoff (1997), where the ICO panel was shown to be generally the most sensitive of the five panels.
Couutry : :
BITTERT BURNTF ASTRGNMF CHEMF FLORALF ACIDT CARAMF MALTYF SALTT SWEETT EARTHF FRUITF GRASSF RANCIDF SOURT * WOODF
t : : t
* * * * *
t
t t
x t
The tests for the main effects of sex and age and the interaction effects country*sex 5%
and country*age
level) for all of the 16 variables
omitted.
*Significant
tsignificant
on 5%
TABLE 2. Estimated z-values for Bitter Taste withp-values for the t Test of <=O (i.e. p=O) and the Results of Pairwise t Test Comparisons
Countries
z
level.
-0.368 -0.284 -0.272 -0.0313 0.206 sharing
and these columns
level, tsignificant
on 1%
(on were level,
on 0.1 %level.
panels from one country and consumers from another or the relations between all consumers and a single panel. Although the second step analysis of variance above mainly discovered country differences, it is clear that the approach would be relevant also within a single country when only a single panel and consumer study is performed. The present analysis was entirely univariate. One may, which is in line with the classical growth curve setting, see e.g. Keuls and Garretsen (1982)) do multivariate analysis of variance on the set of pi z-values
Country
Attribute
In the analysis above consumers were only related to sensory profiling from the same country. It may be interesting to do similar analysis on relations between
Germany France Denmark UK Poland
were non-significant
TABLE 4. The Ordering of the Countries (left=small values, right = large values) for the Significant Country Effects (on 5%)
DISCUSSION
Country
Country*sex*age
Sex*age
P
< 0.0001 < 0.0001 0.0002 0.58 0.0004
letters are not significantly
Pairwise comparison A A A B C different on 5%
G,-F,-D,-I,,P,+ G,-D,,,-Fb-It,P,+
BITTERT BURNTF ASTRGNMF CHEMF FLORALF ACIDT MALTYF GRASSF RANCIDF SOURT WOODF
Fa-IabGbPb F,-G,-IbP, + I-PFG P,-I&DbF, + I,G,Fb + Fa-PbIb F,-Ib-P: P,-Ib-G + Da-IbPc
The subscripts (on 5%).
show the results of pair-wise t test comparisons
Countries
in a listing are significantly
do not share subscripts. P = Poland, Superscripts country:
differences
(I = ICO,
F = France,
different if they G = Germany,
D = Denmark). show the results of the t test for z=O
+ = significantly positive, - = significantly
superscript = non-significant.
in each
negative,
no
Growth Curve Modelling within each country. profile information,
One may also compile by for example
analysis, before doing the individual the approach Mapping group
above is basically
with
differences
segments. relations
analysis rather
component
regressions.
a version
of variance than
the sensory
principal
searching
And as is the case for Preference need not be linear as supposed
consumer
for consumer Mapping above.
the
For the
present analysis no effort was done to search for non-linear relationships linear.
that might fit the data better
REFERENCES
As such,
of Preference
testing
93
than the
Brockhoff, P. M. (1997) Assessor modelling, Sensometrics Workshop, Question 2. Food Qua& and Preference9, 87-89. Keuls, M. and Garretsen, F. (1995) Statistical analysis of growth curves in plant breeding. Euphytica 31, 51-64. Johansen, S. (1983) Some topics in regression. Scandinavian Journal of Statistics 10, 161-194. SAS/STATo (1989) Version 6, 4th edn, Vol. 2, SAS Institute Inc., Gary, NC.