ten data sets and on the complete calibration set a stepwise, a backward and a forward multiple lo-
gistic regression procedure was performed. The most important variables were selected automati-
cally. Biologically important variables e.g. tem- perature were entered manually in the model.
The − 2 Log Likelihood statistic − 2 Log L and the Akaike Information Criterion AIC
Akaike, 1973 were used to compare the models. For both measures, lower values indicate a more
desirable model. Since − 2 Log L can only de- crease when including more explanatory variables,
it tends to select models overfitting the available data. The AIC criterion however includes a
penalty term for model complexity and therefore is more reliable to select good models McCullagh
and Nelder, 1989.
The odds ratio was used for the interpretation of the influence of the different variables on the
storage disorder. The odds ratio, associated with an explanatory variable, expresses by which factor
the odds of the event increases or decreases for a unit increase e.g. O
2
concentration increases from 2 to 3 of a particular explanatory variable,
keeping other variables constant. The odds of an event is defined as the ratio of the probabilities of
event to non-event and is calculated by exponenti- ating the parameter estimate for the explanatory
variable Agresti, 1996. An odds ratio can range between zero and infinity. A value of 1 indicates
that the variable has no influence on the incidence of the disorder. For instance, an odds ratio of 1.5
for the variable O
2
indicates that when the O
2
concentration is increased with one unit e.g. from 2 to 3 the odds corresponding with 2 O
2
needs to be multipled by 1.5 to obtain the odds for 3 O
2
.
3. Results and discussion
3
.
1
. Models for internal browning
3
.
1
.
1
. External explanatory 6ariables In general, the picking date, O
2
concentration, CO
2
concentration, storage time and temperature are strongly correlated with the incidence of inter-
nal browning. The model parameter estimates, the 95 lower and upper Wald confidence limits for
the parameter estimates, and the odds ratios were calculated Table 1. The picking date is an im-
portant variable in the model. Picking I is the first dummy variable. It can discriminate between the
early and the other picking dates. A change from optimal to early picking time corresponds to an
odds ratio of 0.092, indicating that internal browning occurs about ten times as often in
terms of odds for optimal picking time than for early picking time. Picking II is the dummy vari-
able used to discriminate between the late and the other picking dates. The odds for internal brown-
ing increase by a factor 3.7 for the over-mature picked pears compared to the optimally picked
ones. Since the picking time is confounded with cooling period preceding the controlled atmo-
sphere conditions, further research needs to be carried out to separate both variables. However,
the validation results prove that the error made by taking this parameter into the model is moder-
ate see further. An increase in storage tempera- ture by 1°C will increase the likelihood of
browning by 50. Since all the other variables left in the model occur in interaction terms, the inter-
pretation of their odds ratios and their coefficients is not straightforward. Not directly interpretable
odds ratios are indicated by an asterisk in Table 1. Oxygen, CO
2
and storage time, here trans- formed to logstorage time, interact. The relation
between the O
2
concentration and the occurence of internal browning is always negative in the
range of the model, indicating that the odds for browning increase with decreasing O
2
concentra- tion. However, the strength of this relation
depends on the CO
2
concentration and the stor- age time. The parameter estimate for the variable
O
2
is a function of the CO
2
concentration and the storage time: −7.95+1.61×CO
2
— 0.351× logstorage time×CO
2
+ 1.43×logstorage time.
Low CO
2
concentrations 0 – 1 and short stor- age times a few months, result in a large nega-
tive effect of the applied O
2
concentration on brown development, indicating a high O
2
sensitiv- ity of the pears for browning during the first 3
months of storage. The parameter estimates for CO
2
and storage time can be interpreted in an analogue way. An increase in CO
2
always corre-
sponds to more browning. However, this effect is more pronounced for low O
2
concentrations and long storage times.
Overdispersion was noticed during the analysis. This phenomenon causes underestimation of the
variance of parameter estimates. It can be as a result of variation between the response probabili-
ties variation for experimental units observed under the same conditions not explained by the
explanatory variables or correlation between the binary responses Collett, 1991. In our situation,
both causes might explain the observed overdis- persion. To correct for this effect, a heterogeneity
factor was estimated 1.409 and introduced into the analysis. This correction resulted in broader
confidence intervals.
The model was tested for its predictive quality on five validation sets of different orchards and
different harvest years. Table 2 shows the inter- cepts and the slopes of the linear regression of the
predicted probabilities on the measured probabili- ties of browning for the different validation sets.
The coefficient of determination R
2
was calcu- lated to give a measure of correlation between the
measured and the predicted values. Theoretically, the slope parameter of the regression line should
be equal to one, whereas the intercept should be zero. All validation sets except for the Velm
orchard in 1997 have a high R
2
value, the inter- cepts of the linear regression curves are small
except for the Velm orchard in 1997 and the slopes of the regression curves are sufficiently
close to one except for Zellik, 1998. The differ- ences in R
2
and slope parameters between the validation sets are probably a result of factors not
included in the model such as orchard, year of harvest, soil type, temperature, rainfall, hours
sunshine and quality properties of the fruit. The difference in slope parameter between the Zellik
orchard in 1997 and the Zellik orchard in 1998 suggests a bias caused by the harvest year; the
pears of this orchard were more susceptible to core breakdown in 1998 than in 1997. The model
considerably underestimates the number of brown pears in 1998 slope = 0.55, but this happens for
each storage condition. The same tendency can be noticed for the Velm orchard over the 2 years: the
slope parameter decreases from 0.91 to 0.85. The effect of the orchard is illustrated by a compari-
Table 2 Validation results of the models constructed to predict browning and cavities in ‘Conference’ pears
a
Browning external+intrinsic factors Browning external factors
Intercept Slope
R
2
Intercept Slope
R
2
3.5 0.86
0.86 Rillaar 1997 calibration
1.19 0.92
0.75 0.88
0.75 0.65
Rillaar 1997 –
– –
– –
– Velm 1997
4.6 0.91
0.49 –
– –
Zellik 1997 2.5
0.79 0.80
0.81 1.04
1.17 0.81
Velm 1998 0.85
0.69 Zellik 1998
1.19 1.6
0.62 0.91
0.55 0.87
Cavities external factors Cavities external+intrinsic factors
Slope R
2
Intercept Intercept
Slope R
2
Rillaar 1997 calibration 0.57
2.6 0.72
0.87 0.82
0.90 1.7
0.76 0.90
Rillaar 1997 –
– –
Velm 1997 4
0.98 0.70
– –
– Zellik 1997
– –
– 0.92
2.1 1.00
– –
– Velm 1998
– –
– 3.5
1.04 Zellik 1998
0.75 4.2
0.94 0.69
a
The R
2
, the intercept and the slope of the regression curve of the measured probabilities on the predicted probabilities are given.
Table 3 Classification table for the calibration model for internal browning based on external parameters: sensitivity, specificity and the
corresponding number of correctly and incorrectly classified pears are given as a function of the cut-off value Correct
Incorrect Cut off value
Correct Sensitivity
Specificity No browning
Browning No browning
Browning 481
1535 23.9
100 0.0
934 601
10 26
455 68.9
94.6 60.8
1138 397
51 430
77.8 20
89.4 74.1
385 30
1264 271
96 81.8
80.0 82.3
1342 193
161 40
82.4 320
66.5 87.4
1411 124
198 283
84.0 50
58.8 91.9
225 60
1453 82
256 83.2
46.8 94.7
160 70
1496 39
321 82.1
33.3 97.5
1518 17
386 95
80.0 80
19.8 98.9
30 90
1533 2
451 77.5
6.2 99.9
1535 481
76.1 100
100
son of Rillaar 1997 and Zellik 1997. The slope parameters are almost the same, but the variabil-
ity on the predictions is higher for Zellik 1997 than for Rillaar 1997. Velm 1997 has a good
slope but the variability is too high to make good predictions.
The construction of a model based on the or- chards of Velm 1997 and Zellik 1997, makes it
possible to study the orchard effect in 1997. An odds ratio of 1.7 was obtained for the orchard
factor, indicating that the odds for Zellik is 1.7 times higher than the odds for Velm. A model
based on Velm 1998 and Zellik 1998 shows that the orchard factor has an odds ratio of 1.6.
This implies a similar orchard effect over both harvest years: pears from Zellik are more suscepti-
ble to core breakdown than pears from Velm. The effect of harvest year was studied in an analogue
way. A model based on the data sets Velm 1997, 1998 and a model based on the data sets Zellik
1997, 1998 resulted in odds ratios for harvest year of 1.6 and 1.45, respectively. Within one
harvest year the orchard effect is more or less constant and the harvest year effect is more or
less constant over the different orchards.
Logistic regression models the probability of an event. By using a cut-off value, this probability
can be transformed again in presence and absence of the disorder. This is necessary to make classifi-
cation tables. However, the choice of the cut-off value is not easy. A receiver operating curve
ROC-curve can help in deciding the optimal value. This curve displays the sensitivity in rela-
tion to the specificity Collett, 1991. Sensitivity is a measure of accuracy for predicting events. It is
the proportion of event observations that the model predicts to be events for a given probability
cut-off value. Specificity is a measure of accuracy for predicting non-events. It is the proportion of
non-event observations that the model predicts to be non-events for a given probability cut-off
value. The cut-off value selection depends on the purpose of the model. Trying to eliminate almost
all pears with browning e.g. cut-off value 80 will result in wasting many undamaged pears.
Table 3 displays the specificity, the sensitivity and the number of correctly and incorrectly classified
observations as a function of the cut-off value. A total of 84 of the pears of the calibration set
were classified correctly for a cut-off value of 50, whereas 82 were classified correctly at a
cut-off value of 0.3. For the validation sets, the percentage of correctly classified pears at a cut-off
value of 30 ranged between 82 Rillaar, 1997 and 86 Zellik, 1997.
3
.
1
.
2
. External and intrinsic explanatory 6ariables In order to investigate the role of some intrinsic
fruit parameters in the development of internal browning, a model was built with both external
and intrinsic parameters. The heterogeneity esti-
mate 1.242 was used to correct for overdispersion. Table 1 summarises the model. This model con-
tains all the explanatory variables used in the previous model and is extended with three fruit
quality parameters: weight, Magness-Taylor firm- ness and soluble solids content. Pear colour and
size, because of its high correlation with weight, did not meet the 5 level of significance to enter
the model. Large and heavy pears have a higher probability for browning odds ratio = 1.009.
Harder pears are more sensitive to internal browning than softer ones: an increase of firmness
by 1 N increases the odds of browning with 2.1. Pears with a higher SSC tend to develop less core
breakdown than pears with a lower SSC. Since SSC is positively and firmness negatively corre-
lated with time, the coefficient of logstorage time changes somewhat with the introduction of the
intrinsic parameters. The parameter estimates of the other external parameters changed only
slightly with the addition of the intrinsic parame- ters, therefore the interpretation of the model is
similar to that of the previous model. In the previous model, the 16 pears per storage
condition made it possible to validate the model for the predicted percentages of browning. How-
ever, this is not possible in the present model because each pear has a different weight, firmness
and SSC. The measured probabilities were calcu- lated by taking all the pears of one storage condi-
tion and averaging out the values for weight, SSC and firmness. These probabilities were then used
for comparison with the predicted probabilities on internal browning. Validation was only possible
on the orchards of Velm in 1998 and Zellik in 1998 since only here intrinsic parameters were
measured. The model for core breakdown exter- nal parameters lowers the value of − 2 Log L
from 2215 to 1333. Inclusion of the intrinsic parameters decreases its value further to 1289,
although in the validation this improvement of goodness of fit was not pronounced. Table 2
shows slightly increased slope parameters for the models for browning orchards Zellik in 1998 and
Velm in 1998 with the intrinsic parameters. Fig. 1 shows the regression of the predicted percentages
Fig. 1. Predicted versus measured probabilities of internal browning for the validation sets Zellik 1998 external + intrinsic parameters and Velm 1998 external + intrinsic parameters.
on the measured percentages of internal brown discolouration. For Velm 1998 the slope is close
to one but the variation on the predicted values is larger. Zellik 1998 has a good correlation be-
tween predicted and measured values R
2
= 0.91,
but the calibration model underestimated the number of brown pears considerably slope =
0.62. This bias can only be solved by using a calibration set that contains different orchards and
different harvest years.
This model classified up to 85 of the calibra- tion pears cut off-value = 50 in the right class.
Depending on the specificity and the sensitivity requirements, 80 – 84 of the validation pears were
classified correctly. Again a compromise has to be found between sensitivity and specificity depend-
ing on the purpose for which the model is used. The practical usefulness of including variables
such as SSC and firmness is doubtful, because both variables were measured destructively. Their
importance would increase if non-destructive tests could be used to predict firmness and SSC. Inclu-
sion of preharvest factors such as hours of sun- shine, rainfall, age of the trees, soil composition,
etc., and of intrinsic factors such as vitamin C Veltman et al., 1999 concentration before storage
should improve the prediction performances of the model.
3
.
2
. Models for ca6ities
3
.
2
.
1
. External explanatory 6ariables Parameters for the cavity model heterogeneity
factor = 1.271 were identified Table 1. Over-ma- ture picked pears are four times more susceptible
to cavity development, whereas early picked pears have ten times less probability of the disorder,
compared to optimally picked pears. Only one interaction term, between O
2
and CO
2
entered the model. Fig. 2 shows the validations of this model
on Zellik 1997, 1998. Both regression curves have a slope parameter close to one and a good R
2
value Table 2. However, the variability is higher for Zellik 1998 than for Zellik 1997. The vali-
dation results of the other orchards are listed in Table 2. The model classifies from 80 to 87 of
the pears correctly, depending on the choice of the cut-off value, the orchard and the year of harvest.
Fig. 2. Predicted versus measured probabilities of cavities for the validation sets Zellik 1997 external parameters and Zellik 1998 external parameters.
Fig. 3. Occurrence of the different stages of core breakdown disorder as a function of the storage time all storage conditions are summed to construct this plot.
3
.
2
.
2
. External and intrinsic explanatory 6ariables The intrinsic fruit parameters weight and firm-
ness are entered in the model Table 1. An increase in firmness of 1 N results in an increase in odds ratio
of 1.8. The odds ratio for weight equals 1.004. The other parameter estimates did not change
much by the addition of the intrinsic parameters: storage time and CO
2
concentration have a positive and the O
2
concentration a negative relation to cavities. Since the − 2 Log L value only decreases
with 9 units, the addition of the intrinsic parameters did not improve the model considerably. This can
also be seen from the validation results Table 2, which are comparable to those of the model pre-
sented in part 2.1. For this model a heterogeneity factor of 1.114 was applied. The prediction perfor-
mance of the model lies between 80 and 86 of correct classification, dependent on validation set
and cut-off value.
3
.
3
. Joint models for internal browning and ca6ities To study the hypothesis that internal browning
is followed in time by the development of cavities, and thus both are symptoms of core breakdown
disorder, a generalized logits model was fitted. The proportional odds assumption was rejected such
that the model could not be simplified Collett, 1991. Fig. 3 shows the predicted and measured
values for each of the four classes. During the first 4 months, the number of undamaged pears de-
creased, but reached a constant value of 70 afterwards. In the same period, the incidence of
internal browning plus cavities and cavities in- creased. After 4 months, the incidence of pears with
browning and cavities increased, whereas the inci- dence for internal browning decreased. The number
of pears with cavities stayed more or less constant. The model indicated strongly that the process of
pear core breakdown could be divided into four different stages. The undamaged pears first started
to develop internal brown spots, followed by the development of internal cavities. After a while the
brown spots disappeared and larger cavities re- mained. The incidence of undamaged pears did not
change after 4 months storage, indicating that disorders only switch between the different stages
after this storage time and no additional pear damage was initiated. Similar results were obtained
by Roelofs and de Jager 1997.
4. Conclusion
