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RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 10
Research and Reviews: Journal of Dairy Science and Technology
ISSN: 2319-3409 (Online), ISSN: 2349-3704 (Print)
Volume 8, Issue 1
www.stmjournals.com
The Milklipidometric Model: A Mathematical Equation-
based on Electric Conductivity to Predict Butterfat
Concentrations in Bovine Milk
S. Muyambo1,*, J.A. Urombo2, M. Mudyiwa3, A. Musengi4
1,3Department of Food Processing Technology, Harare Institute of Technology, P.O. Box BE277
Belvedere, Harare, Zimbabwe
2Department of Mathematical Sciences, Harare Institute of Technology, P.O. Box BE277 Belvedere,
Harare, Zimbabwe
3,4Department of Biotechnology, Harare Institute of Technology, P.O. Box BE277 Belvedere, Harare,
Zimbabwe
Abstract
The milklipidometric model is a mathematical equation to predict butterfat in raw milk and it
is based on electric conductivity principle. The model differs from the existing equations in
that it uses variables (electric conductivity, and volume fraction of whole milk ) which
can be easily measured and it predicts fat at wide temperature, range of 5–60°C. Accurate
results can be obtained even using commercial electrical conductivity meters, hence making
the method easy and cheap, especially for small and medium dairy farmers in developing
countries. The model is effective for butterfat concentration between 0.60 and 5.50%. The
model has a correlation of determination R2 of 0.890 and root mean square error (RMSE) of
0.464. Validation of the model using Deming regression, Passing and Bablok regression and
Bland and Altman method indicate that the equation is capable of reproducing the butterfat
from the validation data (N=32) at 95% confidence interval. Further analysis of the model
accuracy and precision showed that the equation has a percentage bias of less than 2.9%, a
standard deviation of 0.63% and coefficient of variance of 19.19%.
Keywords: Electric conductivity, mathematical modelling, Butterfat determination.
*Author for Corresponding E-mail: shadiemyambo@hotmail.com
INTRODUCTION
Butterfat/milk fat concentration is an
important parameter for dairy farmers, dairy
companies/processors and regulatory boards. It
is commonly used for labelling purposes, food
safety and regulation purposes, farmer
payments, for providing the information
required for herd growers and for processing
of various products [1, 2]. The methods for
butterfat determination in bovine milk are
usually based on use of chemicals (Gerber
method), fluorescent light scattering (milk
scans) and infrared spectroscopy absorption
[3]. Although these methods are already used
in the dairy industry for analysis of butterfat,
however, many small and medium scale
farmers and dairy processors, especially in the
developing world, have found it difficult to
use these methods routinely. The instruments
or methods maybe highly expensive (buying
and maintenance cost) and they have high
demand of expertise and some are relatively
slow for routine use. It is then indispensable to
develop an analytical method, which is
inherently flexible for easiness of use and
considerably cheap, for accurate and precise
determination of butterfat in bovine milk. This
paper gives a general view, summarised by a
mathematical equation, for butterfat
determination in bovine milk using the electric
conductivity properties of milk.
The electric conductivity properties of milk
have been well studied and applied in the
evaluation of food properties, over the past
years. Jha et al. [4] and Venkatesh and
Raghavan [5] stated a non-destructive quality
measurement techniques based on electrical
Mathematical equation for butterfat prediction Muyambo et al.
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 11
conductivity properties (dielectric) for
evaluation of moisture content and
temperature of fruits and vegetables during
post-harvest. Mabrook and Petty [6] studied on
the effect of composition on the electrical
conductance of milk. Binnur and Serap [7]
showed the effects of temperature and milk fat
content on the electrical conductivity of Kefir
during incubation. Rendevski et al. [8]
indicated that the measurement of electrical
conductance of milk at high frequency can
give information of added water in the milk.
This is common in adulterated milk. The effect
of dilution and temperature on the electrical
conductivity of milk was studied by
Henningsson et al. [9] and Fernandez-Martin
and Sanz [10]. Henningsson et al. [11] specify
a dynamic system for determining milk
properties using a fusion of electric
conductivity, density and optical properties of
milk. Goodling et al. [12] and Ilie et al. [13]
indicated that milk electric conductivity
properties has been used as a main indicator of
mastitis.
Walstra et al. [14] depicted that the milk
compositional elements have impact on the
properties of milk for instance, viscosity,
electrical conductivity and optical appearance.
It was shown by Mabrook and Petty [6] that
the electrical conduction of milk is a result of
the presence of charge carrying
elements/conducting ions such as
(). Thus, the relative
mobility, of milk conducting ions determines
the conductivity capacity of the typical milk
sample. Ninfa et al. [15], Fernandez-Martin
and Sanz [10], Mabrook and Petty [6] and
McSweeney and Fox [16] indicated that the
relative mobility of conducting ions is
influenced mainly by the concentration and
distribution of butterfat and caseins proteins.
The electrical conductivity of milk is higher
than that of its fat- and casein-free phase due
to the obstruction of the charge-carrying ions
by the fat globules and casein micelles [16].
Thus, increase in concentration of butterfat
and caseins proteins lead to a decrease in the
mobility of the conducting ions and hence
milk electrical conductivity. Other milk
molecules, lactose, serum/whey proteins and
microelements, though equally important, they
have little influence on the electrical
conductivity of milk, unless their
concentration and physicochemical properties
in natural milk are significantly altered [6, 16].
As such the conductivity of milk is usually in
the range 0.40-0.55 Sm-1 at 25°C [16].
Common Mathematical Models for
Relating Electric Conductivity with Milk
Compositional Properties
According to Henningsson et al. [11] one of the
earliest mathematical model for determining the
relationship between fat content and
conductivity was established, theoretically, in
1935 by Bruggerman and confirmed
experimentally in 1962 by Prentice, hence, the
Bruggerman-Prentice model (Eq. 1).
(1)
where:-is the electrical conductivity of milk,
-is the conductivity of the fat-free milk and
-is the volume fraction of the conducting
medium i.e. the non-fat phase.
The Bruggerman-Prentice model becomes the
base of most of models, which illustrate the
relationship between conductivity and milk
components. Such models include Prentice
[17], Eq. 2, and Lawton and Petty [18], Eq. 3.
The mathematical model by Prentice [17]
relates butterfat content within the range of 2-
6.1% at room temperature, whilst that of
Lawton and Petty [18] relates butterfat content
up to 7% at 20°C.
, (2)
(3)
Where, is the volume fraction of
butterfat
Lawton and Petty [18] further developed a
more complex conductivity-butterfat
expression for butterfat in the range of 0.15–
51% (Eq. 4).
(4)
Where, and are constants and is the
volume fraction of the butterfat
A number of mathematical models for
correlating with the total milk solids and
solids non-fat have also been developed.
McSweeney and Fox [16] indicated a second-
degree polynomial function for the
relationship between electrical conductivity ,
and total solids (Eq. 5).
Research and Reviews: Journal of Dairy Science and Technology
Volume 8, Issue 1
ISSN: 2319-3409 (Online), ISSN: 2349-3704 (Print)
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 12
(5)
where, is the % total solids and are
parameter constants
The model for relating electrical conductivity
of milk with viscosity is shown below (Eq. 6).
The model developed experimentally by
Sudheendranath and Rao [19] using skim milk,
whole milk and half-half mixtures.
(6)
Where, is the milk viscosity; and are
proportionality constants
Fernandez-Martin and Sanz [10] stated a
modified Sudheendranath and Rao equation
(Eq. 6) which correlates total solids , to
electrical conductivity and viscosity ,
within temperature range of 15-60°C (Eq. 7).
(7)
Fernandez-Martin and Sanz [10] further
develops a hybrid model which relates
electrical conductivity , total solids , and fat
to solid non-fat ratio , at a definite
temperature range (Eq. 8).
(8)
Where, and are parametric coefficients
of the model obtained by regression and is
fat to solid non-fat ratio.
METHODOLOGY
Sample Collection and Standardization
Milk samples were collected from dairy
farmers and dairy companies, around Harare,
Zimbabwe. In most dairy companies not all
producers are able to deliver milk on a daily
basis; therefore, milk was randomly selected
due to its presence on the day of sampling.
Thus, each producer had an equal chance of
being selected or deselected, randomly, by
delivering milk (or otherwise) on the day of
sampling. Collection of the sample from churn
or bulk tank was done by following standard
techniques described by the International
Dairy Federation [20]. Antibiotic, freezing
point, pH and temperature of collected
samples were also tested and recorded.
Samples collected (N=9) from dairy
companies were coded (S1-S9), standardized
for butterfat before analysis. Standardization,
of collected milk samples, was done using
either one of these three milk samples (1)
skimmed milk, Sm; (2) low fat milk (L) and
(3) creamed milk (C), to obtain standardised
samples with butterfat concentration between
1.5% and 5.5%. Lactoscope (Delta
Instruments, Germany) was used to analyse
the compositional profile of the prepared milk
samples, for instant, % butterfat, % protein, %
lactose, % Total solids and freezing point.
Measuring of Temperature and pH of the
Milk Samples
The pH and temperature of the milk samples
were analysed and adjusted (if possible) before
each test. The ideal range of pH and
temperature of samples were 6.5-6.8 and
25.5±0.5°C respectively. The temperature of
the samples was set to 25.5±0.5°C using water
bath. The effects of temperature on electrical
conductivity , was also investigated. A
sample of constant concentration (butterfat)
was measured at temperatures between 5°C
and 60°C, in accordance to Fernandez-Martin
and Sanz [10]. Two water baths (cold and
warm) were prepared to control temperature of
samples before measurement.
Measuring of Electrical Conductivity
Calibration of the electrical conductivity meter
(Reliable Digital EC meter-bench) was done
with reference to the ‘conductivity of
standards electrolyte’ established by the
International Organization of Legal
Metrology, OIML R 56 [21]. The values
were measured at cell constant , of
1.0000±0.0002 cm-1. The sample was mixed
well before analysis by shacking gently and
the tests were done in triplets.
Mathematical Modelling and Validation
Data used for mathematical modelling
obtained from raw milk sample and
standardised samples (N=110) was checked
for presence of outliers (Grubbs test and Dixon
test) as shown in Figure 1. The data was split
randomly into training and validation data
(N=78), and the training further grouped
(randomly) into cluster 1 and 2 which were
used for model testing. Training data was used
for developing the mathematical model and
testing, whilst the validation data (N=32) was
used for selecting and determining the
efficiency of the best fit model. The validation
Mathematical equation for butterfat prediction Muyambo et al.
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 13
methods used include Deming regression,
Passing and Bablok regression and Bland and
Altman method. Accuracy and precision was
further determined using standard regression
method (SRM), Nash-Sutcliffe efficiency
(NSE), percentage bias (PBias), Error index
(RMSE and MAE), RMSE-observations
standard deviation ratio (RSR), coefficient of
variation ( ) and standard deviation of
predicted values . Statistical software
which was used for mathematical model
development and validation consist of Matlab
(Mathworks v2015a), SPSS (IBM, v26),
XLStat (Microsoft v2018-Trial) and Excel
(Microsoft, 2016).
RESULTS AND DISCUSSION
The summary of the results for the butterfat,
volume fraction of whole milk (v/v) and
electric conductivity of the milk samples are
shown in Table 1, at constant temperature of
25.5±5°C. The results indicated that the
butterfat of standardised milk samples (1st
dilution, W:Sm1) ranges from 1.49-1.63% and
the electric conductivity was within 0.347-
0.382 Sm-1 range. Second dilution
(W:Sm2/W:L1) has butterfat and electric
conductivity range of 2.34-2.60% and 0.356-
0.401 Sm-1 respectively; whilst for 3rd dilution
(W:C/WL2) the butterfat ranges from 4.10-
5.34% and electric conductivity from 0.309-
0.421 Sm-1. The distribution of the electric
conductivity of milk is shown in Figure 2. It
was observed that as fat concentration
increases from 2nd dilution through to 3rd
dilution the electric conductivity initially
increases and reaches a plateau then drops
down. The mathematical relationship of and
was observed to be a 2nd degree polynomial
equations (Eq. 9).
(9)
It was also observed that in general the
conductivity of the samples increases with an
increase in the volume fraction of whole milk
, Table 1. is a dimensionless parameter
in the range 0.0 -1.0 and can be calculated as
follows (Eq. 10):
(10)
Modelling processes
Data Collection
N=110
Outliers Testing
• Grubbs test
• Dixon test
Training Data
N=78
Cluster 1
N=41 Cluster 2
N=37
MATHEMATICAL MODEL
Validation Data
N=32
Model testing
Model validation
Fig. 1: Outline for the Data Collection and Mathematical Modelling Process.
Research and Reviews: Journal of Dairy Science and Technology
Volume 8, Issue 1
ISSN: 2319-3409 (Online), ISSN: 2349-3704 (Print)
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 14
Table 1: Summary of Averaged Values and Values of Collected Milk Sample. All Analysis were
Carried out at 25.5±0.5°C.
% Butterfat ()
Electric Conductivity (S/m)
( v/v)
Sample
Average(±SD)
Range
Average(±SD)
Range
Value
Skimmed milk(Sm)
0.63(±0.0258)
0.61-0.66
0.365(±0.01320)
0.357-0.382
0
Low fat milk(L)
1.62(±0.0)
-
0.389(±0.0)
-
0
Cream (C)
38.0(±0.0)
-
-
-
0
Whole milk(W)
3.67(0.2639)
3.18-4.01
0.387(±0.0333)
0.314-0.416
1.0
W:Sm1
1.59(±0.0575)
1.49-1.63
0.369(±0.0167)
0.347-0.3817
0.26-0.33
W:Sm2/W:L1
2.47(±0.0909)
2.34-2.60
0.384(±0.0187)
0.356-0.401
0.39-0.73
W:C/W:L2
4.43(±0.3800)
4.10-5.34
0.384(±0.0361)
0.309-0.421
96-99.7
Fig. 2: Relationship of Electric Conductivity with increasing Butterfat at 25.5±5oC for Averaged
Sample Values. W:Sm1 is 1st dilution of Whole Milk (W) and Skimmed Milk (Sm); W:Sm2/W:L1 is the
2nd Dilution of Whole Milk with Skimmed Milk or Low Fat Milk (L); W is the Whole Milk Sample and
W:C/W:L2 is 3rd Dilution of Whole Milk and Creamed Milk (C) or Low Fat Milk.
Fig. 3: Changes of Electric Conductivity with Increase in Temperature for Whole Milk Samples S1,
S3 and S5. Data Fitting Statistics: S1- R2=0.996, RMSE=0.00945; S3-R2=0.998, RMSE=0.00727 and
S5-R2=0.999, RMSE=0.00571.
Mathematical equation for butterfat prediction Muyambo et al.
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 15
The effects of temperature on the electric
conductivity of milk can be shown in Figure 3.
The graph shows the results of the three
selected milk samples (S1, S3 and S5). A
linear relationship was observed for all
samples investigated. This means there was a
linear increase in the electric conductivity of
milk as temperature increase. Thus, in general
the mathematical equations for electric
conductivity , and temperature , was in the
form: (11)
Where, are constants.
Mathematical Modelling
The mathematical model for predicting %
butterfat , was obtained using independent
variables like temperature , electric
conductivity and volume fraction of whole
milk . Thus, the model was anticipated to be
of the form:
(12)
where: are proportionality constants
The mathematical models (Eq.14-17) for
predicting in milk are shown in Table 2. We
introduced the parameter (electric
conductivity-volume fraction factor) which
takes into consideration the effects of dilution
on the composite value of electric conductivity
of the milk samples. The parameter is
expressed as shown in Eq. 13, as below:
(13)
Since the volume fraction , is dimensionless
as explained using Eq. 10, the parameter
carries the units of (Sm-1).
In general, the models are a combination of
linear and power law equations. Eq. 16 shows
relatively high values of correlation
coefficients (R2 and R2adjusted), lower values of
root mean squared errors (RMSE) and sum of
squares errors (SSE) as compared to others.
However, we consider all the models during
the validation process.
Selection and Validation of the Best Fit
Model
The use of Deming regression, Passing and
Bablok regression and Bland and Altman
method, as techniques for mathematical model
validation has also been reported in Muyambo
and Urombo [22] and Bilić-Zulle [23]. Table 3
shows results from these 3 techniques of
model validation, for respective mathematical
equations.
Table 2: Mathematical Models for Predicting
% Butterfat () in Bovine Milk.
Equation
No.
Mathematical
model
RMSE
SSE
14
0.770
0.761
0.513
19.49
15
0.747
0.736
0.550
29.34
16
0.815
0.804
0.464
15.72
17
0.811
0.800
0.469
16.06
Where, , and are proportionality
constants
Table 3: Summary of Deming Regression,
Passing and Bablok Regression and Bland and
Altman Methods of Mathematical Model
Validation.
Deming
Passing and
Bablok
Bland and Altman
Eq.
p-value
p-
value
Bias
Standard
Error
14
0.869
0.415
0.183
0.107
0.444
0.62
15
0.877
0.415
0.753
0.025
0.443
0.66
16
0.890
0.415
0.172
0.099
0.399
0.52
17
0.888
0.211
0.328
0.071
0.401
0.52
* is Pearson correlation coefficient. All statistical
analysis where done at 95% confidence interval.
For Deming regression method we considered
the correlation coeffiency R2. The results in
Table 3 indicated relatively higher values of
R2 (>0.80), with Eq. 16 showing highest R2
value of 0.890. The assumption of the Deming
regression R2 value is that the higher the value,
the higher the correlation between the
observed and the predicted values. Thus, Eq.
16 predicts values which are close to the
observed values as compared to other
equations. We also make used of Passing and
Bablok regression and Bland and Altman
Research and Reviews: Journal of Dairy Science and Technology
Volume 8, Issue 1
ISSN: 2319-3409 (Online), ISSN: 2349-3704 (Print)
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 16
method t-test for comparing the significance
difference between the observed and predicted
values for each equation respectively. Thus,
there was no significant statistical difference
between the predicted and the observed values,
if the p-value was greater than the critical
value (, i.e. null hypothesis . The
results showed p-values were greater than
critical value (), for all mathematical
model (Eq. 14-17), which means that the mean
value of the observed were statistically the
same as those obtained from estimation. The
Bland and Altman results showed that Eq. 15
has minimum bias and Eq. 14 has the largest
and Eq. 16 had minimum standard error
values. The value was higher for Eq. 15 and
Eq.14 respectively as compared to Eq. 16 and
17 which carries the same value. In conclusion
the Deming regression, Passing and Bablok
regression and Bland and Altman validation
methods showed that Eq. 16 and 17 have
better predictive power though Eq. 16 was
slightly superior.
Accuracy Determination of the Models
The accuracy of the mathematical models was
determined using several methods which
includes standard regression method (SRM),
Nash-Sutcliffe efficiency (NSE) percentage
bias (PBias), Error index and RMSE-
observations standard deviation ratio (RSR).
The standard regression method SRM) is
based on the assumption that, for a perfect
model, the linear fit of observed and estimated
butterfat values will result in the gradient ()
value of 1.0 and the intercept () value of 0.0
[22, 24]. The results showed that Eq. 16 have a
better prediction power due to lower values of
and i.e. close to 1.0 and 0.0 respectively
as compared to other equations (Table 4).
Moreover, the value of Eq. 16 was the
highest, hence, there was a higher correlation
between the observed and predicted values.
The percentage bias (PBias) was used to
determine to what extend does the predicted
values differ from the actual or observed
values. The optimum value for PBias is 0.0,
positive values shows model underestimation
while negative values shows overestimation of
the model [24, 25]. PBias values showed that
the models were underestimating the value of
. Eq. 15 shows a minimum value PBias of
2.070% and Eq. 14 showed the highest value
of 5.902%. The Nash-Sutcliffe efficiency
(NSE) value was used to determine relative
residual variances according to Moriasi et al.
[24], and it is calculated as follows (Eq. 18):
(18)
The NSE has optimum value of 1.0, the
accepted values are those between 0.0-1.0 and
values ≤0.0 are unacceptable. The predicted
values from equation 14-17 showed NSE
values approx. equal to 0.7 with Eq. 16 and
Eq. 17 showing relatively higher values. The
results from Table 4 showed that the NSE
values for the models were within the accepted
range of 0.0-1.0.
Moriasi et al. [24] and Hyndman [26]
indicates that the root mean square error
(RMSE) and the mean absolute error (MAE)
are the common error indices used to evaluate
the prediction efficiency of mathematical
models. Thus, the closer the RMSE and MAE
to 0.0 the better the model. Table 4 showed
that the value of RMSE were relatively higher
as compared to MAE. However, in both cases
Eq. 17 has lower values of RMSE and MAE
which makes it a better fit model as compared
to others, making it the best model with
regards to error indices. Singh et al. [27]
indicated that the value of RMSE and MAE
can be regarded as low if it is below half of the
standard deviation value of the observed data
set (0.5*STDEVobs). For the validation data
used the standard deviation of the observed
data was 0.840 which makes the value of
RMSE and MAE (for Eq. 16 and 17) less than
0.5*STDEVobs (0.420), hence values were in
the acceptable level. Moriasi et al. [24] further
illustrate that the value of RMSE can be
determined if it’s lower or higher using the
RMSE-observations’ standard deviation ratio,
RSR. The RSR can be calculated as follows:
(19)
Mathematical equation for butterfat prediction Muyambo et al.
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 17
Table 4: Determination of the Accuracy and Precision of the Mathematical Models using Standard
Regression Method (SRM), Nash-Sutcliffe Efficiency (NSE), Percentage bias (PBias), Error Index
(RMSE and MAE), RMSE-Observations Standard Deviation Ratio (RSR), Coefficient of Variation
() and Standard Deviation of Predicted Values .
Standard Regression Method
Error Index
Eq.
RMSE
MAE
NSE
PBias
RSR
14
0.755
1.270
-0.791
0.450
0.370
0.714
5.902
0.535
17.32
0.575
15
0.769
1.325
-1.082
0.437
0.357
0.730
5.562
0.519
16.34
0.556
16
0.792
1.171
-0.469
0.405
0.318
0.768
2.876
0.482
19.19
0.639
17
0.789
1.174
-0.514
0.401
0.312
0.772
2.070
0.477
18.95
0.636
Thus, the optimum value for RSR is 0.0 which
denotes the value of 0.0 for RMSE or residual
and hence a perfect model [24, 27]. The RSR
for the mathematical models (Eq.14-17) were
approximately equal to 0.1 which denotes a
deviation from the optimum value of 0.0.
Nevertheless, Eq. 16 and 17 shows lower
values as compared to Eq. 14 and 15.
The precision of the mathematical model can
be determined by considering the variance of
the estimators obtained [28]. Thus, precision
determines the presence and absence of
random errors and the lesser the random errors
the more precise the mathematical model.
Walther and Moore [28] indicated that apart
from variance and standard deviation ,
coeffiency of variance , is also an important
parameter used to determine the precision of
the model. Coefficient of variance is usually
expressed as a percentage as follows:
(20)
Where, is the standard deviation
and is the sample mean of the predicted
butterfat values.
Table 4 showed that Eq. 14 and 15 have
relatively lower values of standard deviation
() and coeffiency of variation () as
compared to Eq. 16 and 17. Eq. 16 had the
highest values hence the least precise.
Nevertheless, we also tested if the values of
were lower enough for all Eq. 14-17. We
consider the classification method proposed by
Gomes [29]. Gomes [29] and Vaz et al. [30]
indicated that is classified as low (
); medium (); high
() and very high (
). Thus, according to this classification
the observed for the all the models falls
within the medium range, i.e. between 10 and
20%.
The analysis above suggests that both Eq. 16
and 17 have potential of being the best fit
model for predicting the butterfat
concentration of milk sample. In general, Eq.
16 correlates strongly with the observed or
measured values, whilst Eq. 17 predicts values
with better precision (low variance) and less
bias. The differences between some values
obtained from Eq. 16 and Eq. 17 were also
very small to be considered significant.
However, for the sake of convenience Eq. 16
was selected as the best fit model.
(16)
The values of proportionality
constants/coefficient are shown in Table 5.
Table 5: Proportionality Constants or
Coefficient of Best Fit Model.
Coefficient
Value
6.842
-2757
23290
0.09298
-115.5
(5.89,
7.79)
(-3453,
-2061)
(16590,
29980)
(0.0725,
0.1135)
(-154.6,
-76.46)
*—confidence boundary of the proportionality
constants or coefficients
The surface plot showing the relationship of %
butterfat , with electric conductivity-volume
fraction factor (the parameter) and
temperature is illustrated in Figure 4.
The residuals from Eq. 16 were also analysed.
A one sample t-test statistics was carried out
following the null hypothesis that the mean
of residuals is equal to 0.0 at 95% confidence
interval. A p-value of 0.172 was obtained
using SPSS. Since p-value>0.05 (critical
value, ) it follows that we failed to reject the
null hypothesis. Hence, there was no
significant difference between the mean of the
residuals and 0.0. It is also recommended that
the distribution of the residuals should follow
normal distribution, which is an indication that
Research and Reviews: Journal of Dairy Science and Technology
Volume 8, Issue 1
ISSN: 2319-3409 (Online), ISSN: 2349-3704 (Print)
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 18
the model will be able to predict future values
of the dependent variable. Figure 5 shows a
histogram of the residuals (difference) with a
normal curve. The graph shows that the
residuals a somewhat normal distribution with
most values having a value of 0.0, i.e., the
highest frequency or density. The graph
(Figure 5) also shows that most values of the
residuals where within the (-1,+1) range.
The graph for comparing the predicted and the
observed values is illustrated in Figure 6.
The graph shows that first the model is
overestimation the values by a factor of
0.30% for butterfat values below 3.42%,
however, this value decreases to 0.0097% for
butterfat within the range of 3.42-3.9%. It is
observed that for butterfat values of 4.0% and
greater the mathematical model was
underestimating the fat by a factor of 0.51%.
Changes of temperature do not have a greater
influence on the prediction ability of Eq. 16,
thus they is a similar distribution between the
predicted and observed butterfat values.
Fig. 4: Surface Plot Showing the Distribution of Butterfat Values with Changes of Temperature and
the Parameter.
Fig. 5: Histogram of Residuals Obtained from the Best Fit Model.
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.5 0 0.5 1
Density
Difference
Histogram (Difference) Difference
Normal
Mathematical equation for butterfat prediction Muyambo et al.
RRJoDST (2019) 10-21 © STM Journals 2019. All Rights Reserved Page 19
Fig. 6: Comparison of Predicted and Observed Butterfat Values.
CONCLUSION
The milklipidometric model predicts the
butterfat of raw milk using electric
conductivity, volume fraction of whole milk to
skimmed milk/cream and temperature. The
model was valid at butterfat concentration
within 0.6-5.5% and over a temperature range
of 5-60oC. Common models which relate
butterfat to electric conductivity are restricted
to short range of temperature e.g. around room
temperature. Thus, the milklipidometric model
optimise butterfat determination by using
parameters which are easy to analyse at a wide
temperature range. The statistical methods for
determining the accuracy and precision of the
mathematical equations showed that the model
is capable of producing reliable results.
ACKNOWLEDGEMENTS
This project was supported by Harare Institute
of Technology (Food Processing Technology
Department, Mathematical Sciences
Department and Biotechnology Department),
Kefalos Cheese (Pvt) Ltd., YoMilk, Dairiboard
Zimbabwe and Aglabs Zimbabwe. There are
no areas of conflicts that have been recorded to
the best knowledge of the authors.
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Cite this Article
S. Muyambo, J.A. Urombo, M. Mudyiwa,
A. Musengi. The Milklipidometric Model:
A Mathematical Equation-based on
Electric Conductivity to Predict Butterfat
Concentrations in Bovine Milk. Research
& Reviews: Journal of Dairy Science and
Technology. 2019; 8(1): 10–21p.
