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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5486_Библиотеки_им_академика_М_И_Перельмана
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variable is normalized to a common scale to avoid variability among the values. The
height of the plot represents the magnitude of the score possessed by each variable
and the shape of the plot represents the pattern of the variables. The variables em-
ployed for the cluster analysis are assumed to be independent and uncorrelated [48].
6.3.4.1 Measurement of proximity
Proximity measurement is used to define the parameters based on which clustering
or grouping will be done in the cluster analysis. Two types of such parameters are
used for this purpose: similarity indices and dissimilarity indices. The high value of
similarity indices between the molecules indicates they have common characteristics
and the same will be true if the value of dissimilarity indices is low between the mole-
cules. In other words, molecules with high similarity indices or low values of dissimi-
larity indices will be placed in the same group.
6.3.4.2 Algorithms of cluster analysis
There are many algorithms defined for carrying out cluster analysis. The three most
widely used approaches are described further.
6.3.4.2.1 Hierarchical algorithm
This is a popular means of performing the cluster analysis, which is further divided
into two types [49]:
Fig. 6.7: The plot between the score value of each variable and the variable itself.
6.3 Multivariate data analysis (MVDA) 129
– Agglomerative hierarchical clustering: This is a bottom-up clustering approach in
which initially all the data points are considered an individual cluster. Then, we start
merging the two nearest related data points and the process is continued till all the
data points are merged into one single cluster. In the end, the single cluster is divided
into small clusters as per the requirement (Fig. 6.8).
– Divisive hierarchical clustering: This is also known as the top-down approach. This
approach is totally opposite to the agglomerative hierarchical clustering, as in this
Fig. 6.8: Agglomerative hierarchical clustering of the cluster analysis.
130 Chapter 6 Statistical approaches used in the QSAR
method, all data points are considered as one single cluster and then dividing of the
same into smaller clusters is performed depending upon the requirement (Fig. 6.9).
6.3.4.2.2 Artificial neural network (ANN)-based clustering
This is another popular classification or clustering approach. The methodology is
analogous to the functioning of the human neurons present in the brain. A simple
model representing an ANN model (McCulloh-Pitts model) is given in Fig. 6.10. The
basic idea behind this type of clustering is that an external input (stimulus) is pro-
vided in the form of variables, and then with a combination of external weights (at
synapses), the desired output is obtained depending on the activation factor (linear,
hyperbolic, Gaussian, etc.) [50].
Fig. 6.9: Divisive hierarchical clustering (top-down approach) of the cluster analysis.
6.3 Multivariate data analysis (MVDA) 131
Fig. 6.10: Artificial neural network (ANN)-based clustering.
132 Chapter 6 Statistical approaches used in the QSAR
6.3.4.2.3 K-nearest neighbor (KNN) clustering
K-nearest neighbor-based clustering is a nonparametric approach for the classification
[51]. Here, K stands for the total number of neighbors required for clustering. In the ex-
ample given in Fig. 6.11, suppose there are two groups present in the study, for example,
groupArepresentedbycirclesandgroupBrepresented by squares. If a new data point x
is entered in the model, it can be grouped in either of the groups based on the value of K.
If K is 3, then the nearest three data points from x will be studied and it will be clustered
ingroupB.But,ifweconsiderK as 5, then it will be grouped in group A.
6.4 Statistical parameters for the validation of the QSAR models
Irrespective of the QSAR model, whether developed traditionally or through more so-
phisticated approaches such as 3D-QSAR, they are to be validated for assessing their
robustness and predictability. Certain common internal and external validation pa-
rameters are used for checking the quality of the QSAR model developed [52, 53].
6.4.1 Internal validation parameters for the QSAR model
In the internal validation of the QSAR model, an assessment of the QSAR model is
done from the molecules that are used for the development of that model [54]. It is
done through the following parameters:
Fig. 6.11: K-nearest neighbor (KNN) clustering.
6.4 Statistical parameters for the validation of the QSAR models 133
– Goodness of fit (R
2
): Goodness of fit, popularly known as the R
2
value, is one of the
most commonly employed parameters for the internal assessment of the QSAR model.
The R
2
value tells us about the reproducibility of the QSAR model but provides little
or no information regarding its robustness and predictability. An R
2
value greater
than 0.7 is considered acceptable in the QSAR modeling and R
2
calculation is done by
employing the following equation:
R
2
= 1 − RSS=TSSðÞ (6:27)
where RSS represents the residual sum of squares and TSS is the total sum of squares.
Both of the parameters are calculated from the following equations:
RSS = Σ ðY − Y
pred
Þ
2
(6:28)
TSS = Σ ðY − Y
mean
Þ
2
(6:29)
where Y is the biological activity of the molecule obtained experimentally, Y
pred
is the
biological activity value predicted through the developed QSAR model equation, and
Y
mean
is the mean of the experimental biological activity values of all the molecules.
The R
2
value can also mislead the user as a high value can also be obtained if the
number of descriptors used in the QSAR model generations is high. To overcome this
situation, a new parameter R
2
adj
is defined by the following equation:
R
2
adj
=
R
2
− pn− 1ðÞ
n − p + 1
(6:30)
where p represents the number of descriptors and n is the number of molecules em-
ployed for the QSAR model generation. For a reliable and robust QSAR model, the dif-
ference between R
2
and R
2
adj
should not be greater than 0.3.
– Cross-validation coefficient (Q
2
cv
)
The internal validation of the QSAR model is also performed by another parameter
known as the cross-validation coefficient (Q
2
cv
). The basic methodology behind this
type of cross-validation approach is that a specific amount of molecules are omitted
from the training set every time the QSAR model is built. Then the generated QSAR
model is used for the prediction of biological activity of the omitted molecules. This
step is repeated until all the molecules get omitted once. The parameter Q
2
cv
is calcu-
lated by using the following equation:
Q
2
cv
= 1 − ðPRESS=TSSÞ (6:31)
where PRESS is the predictive error sum of squares and TSS is the total sum of squares.
The PRESS can be calculated from
134 Chapter 6 Statistical approaches used in the QSAR
PRESS = Σ ðY − Y
p
Þ
2
(6:32)
where Y is the actual biological activity valu e of the omitted molecule and Y
p
is its
value of biological activity predicted by the generated QSAR model. An acceptable
QSAR model should have a Q
2
CV
value greater than 0.7.
6.4.2 External validation of the QSAR model
External validation is performed to assess the robustness of the QSAR model by using
an external test set of molecules that are not use d for the generation of the QSA R
model. Parameters used for external validation are:
6.4.2.1 Coefficients of determination (R
2
pred
)
Similar to the R
2
parameter employed for the internal validation of the QSAR model,
the parameter used for the external validation of the QSAR model is coefficients of
determination (R
2
pred
), which is calculated similarly to R
2
as follows:
R
2
pred
= 1 − ½Σ ðY − Y
pred
Þ
2
=Σ ðY − Y
mean
Þ
2
(6:33)
Here, Y is the experimentally observed biological activity of the molecule used in the
test dataset, Y
pred
is its value obtained by the QSAR model, and Y
mean
is the mean
value of the biological activity of the training set molecules. For external validation of
a QSAR model, a value greater than 0.6 is considered acceptable.
6.4.2.2 Mean absolute error (MAE)
This is another parameter given by Kunal Roy and his team for the external validation
of a QSAR model [55]. It is mathematically expressed as follows:
MAE = Σ Y
obs
− Y
pred
=n (6:34)
where Y
obs
and Y
pred
represent the experimen tally obtained and predicted value of
the test set molecule, and n represents the number of molecules in the test set. For the
external validation of the QSAR, MAE
95%
parameter is calculated by omitting the top
5% high residual molecules from the test set. A value of MAE
95%
less than 0.15 is de-
sired for rating a generated QSAR model as “not bad.”
Combining all the above approaches, Golbraikh and Tropsha proposed statistical
parameters for the internal and external validation of a QSAR model [56] (Tab. 6.9).
6.4 Statistical parameters for the validation of the QSAR models 135
All the parameters mentioned are already explained above except k or k′, which repre-
sent the regression slopes of the QSAR equation generated, which passes through the
origin.
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validation of the QSAR model.
S. no. Parameter Threshold value
Q
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Threshold value Q
> .
R
Threshold value R
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|R
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R
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138 Chapter 6 Statistical approaches used in the QSAR
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