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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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C = Y
mean
− mX
mean
(6:7)
where Y
mean
depicts the mean of biological activity (Y), m depicts t he slope of the
curve, and X
mean
depicts the mean of the value of descriptor (log P)
Now, we have all the necessary formulas for the calculation of our simple regres-
sion model. Table 6.2 depicts the calculated parameters required for the generation of
the simple regression model.
The value of Pearson’s correlation coefficient (R) (equation (6.8)) can be calcu-
lated by plugging in the required values in the following equation:
R = Σ
0.7113
p
0.7056ðÞ0.7233ðÞ
(6:8)
From equation (6.8), the value of R is calculated to be 0.996. Likewise, the slope (m)
can be calculated from equation (6.3) and comes out to be 0.98. By using equations
(6.5) and (6.6) the values of SD
Xvar
are calculated to be 0.280 and SD
Yvar
is calculated to
be 0.283. C, representing the intercept, can be calculated from equation (6.7) and
comes out to be −2.05.
We now have all the values of m and C that are required for the generation of an
equation for the correlation between the biological activity and log P. From the above
values, an SLR equation can be written in the form of the following equation:
Biological activity = 0.98 log P − 2.05 (6:9)
From the above generated simple regression equation, a scatter plot ca n be derived
between the experimental values of biological activity and the calculated values from
the equation (Fig. 6.2).
SLR is nowadays seldom used as it is close to impossible to correlate the biological
activity of the molecules to a single parameter. With the emergence of numerous de-
scriptors (parameters), the use of other advanced statistical approaches like MLR and
PLS has become more prevalent in the QSAR model generation.
6.2.1.1 Advantages of simple linear regression (SLR)
– Simplicity: SLR involves only one independent variable (descriptor) and one de-
pendent variable (biological activity). This simplicity makes it easier to interpret
and understand the relationship between the descriptor and the activity.
– Interpretability: The relationship between the descriptor and the activity is rep-
resented by a straight line, making it straightforward to grasp the direction and
magnitude of the effect of the descriptor on the activity.
– Visualization: The relationship between the descriptor and the activit y can be
easily visualized through a scatter plot with the regression line, aiding in the intu-
itive understanding of the data.
6.2 Linear regression analysis 109
– Reduced multicollinearity: Since only one descriptor is used, issues related to
multicollinearity (high correlation between descriptors) are eliminated, leading
to more stable coefficient estimates.
– Feature selection: SLR can help in the initial screening and selection of descrip-
tors that show the most promising linear relationships with the activity. This can
guide more complex modeling efforts.
– Computational efficiency: Building and interpreting an SLR model is computa-
tionally less intensive than dealing with multiple descriptors in MLR, making it
quicker to analyze.
– Baseline model: SLR can serve as a baseline model for comparison with more
complex models. If the relationship between the descriptor and the activity is
close to linear, a simple linear model might provide sufficient accuracy.
– Initial insight: SLR can provide an initial insight into the potential influence of a
single descriptor on the activity, helping researchers decide whether to explore
more advanced modeling techniques.
– Useful for educational purposes: SLR is often used to introduce students and
newcomers to the concept of regression analysis and its application in QSAR.
However, it is important to note that SLR has its own limitations, particularly when
dealing with complex QSAR datasets that involve multiple descriptors with potentially
nonlinear relationships. In such cases, SLR might not capture the full complexity of
Fig. 6.2: A scatter plot obtained from the simple linear regression between the observed and predicted
biological activities.
110 Chapter 6 Statistical approaches used in the QSAR
the underlying interactions, and more advanced techniques might be necessary for
accurate modeling and prediction.
6.2.1.2 Limitations of simple linear regression (SLR)
– Simplistic model: SLR assumes a linear relationship between a single independent
variable (descriptor) and the dependent variable (biological activity). This simplic-
ity might not capture the complexity of interactions among multiple descriptors.
– Multivariate interactions: Biological activities are often influenced by multiple
molecular properties simultaneously. SLR overlooks the se multivariate interac-
tions, potentially leading to oversimplified or inaccurate predictions.
– Limited predictive power: SLR’s single-variable focus might not capture the full
variation in the dependent variable, resulting in limited predictive power. QSAR
models require more comprehensive approaches to make reliable predictions.
– Assumption of independence: SLR assumes that the independent variable is not
influenced by any other factors. In QSAR, molecular descriptors can be interre-
lated, violating this assumption and leading to biased coefficient estimates.
– Outliers: Outliers can disproportionately influence SLR models, leading to a
skewed fit. QSAR datasets are susceptible to outliers due to experimental errors
or unique compound behaviors, affecting the model’s accuracy.
– Data distribution assumptions: SLR assumes that the residuals (differences be-
tween predicted and actual values) are normally distributed. If the data does not
meet this assumption, the model’s reliability is compromised.
– Limited descriptor exploration: SLR only explores the relationship between
one descriptor and the activity, potentially missing out on important molecular
properties that collectively influence the activity.
– Overfitting and underfitting: SLR can suffer from overfitting (capturing noise)
or underfitting (oversimplifying the relationship). Balancing model complexity
with data fit is challenging, especially in QSAR with limited data.
– Context dependence: Similar to multivariate models, SLR might not be univer-
sally applicable across diverse compound classes or biological systems, limiting
its generalization.
– Mechanistic insights: SLR provides a basic correlation analysis but lacks mecha-
nistic insights into the underlying interactions between molecular properties and
biological activity.
Given these limitations, researchers often move beyond SLR to embrace more sophis-
ticated techniques such as MLR, nonlinear regression, and machine learning methods.
These approaches are better equipped to handle the complexities inherent in QSAR
drug discovery, offering improved predictive accuracy and a deeper understanding of
molecular interactions.
6.2 Linear regression analysis 111
6.2.2 Multiple linear regression (MLR)
MLR is a pivotal statistical technique within QSAR analysis for drug discovery [15]. It
evaluates the relationship between multiple molecular descriptors (features) and the
biological activity of compounds [16]. MLR aims to model this connection by fitting a
linear equation, considering various descriptors simultaneously [17]. The coefficients
in the equation signify the impact of each descriptor on the activity. MLR aids in iden-
tifying essential descriptors influencing the activity and predicting activity values for
new compounds [18]. It serves as a valuable tool in optimizing drug design by eluci-
dating the quantitative link between molecular properties and biological outcomes,
fostering more informed decision-making in drug development [19].
MLR is a simple extension of the SLR in which biological activity (dependent, re-
sponse, or outcome variable) is correlated with more than one physicochemical pa-
rameter or descriptor (independent, predictor, or explanatory variable). MLR is the
choice of statistical approach in the traditional QSAR [20].
The general representation of the MLR is depicted in the following equation:
Y = m
1
X
1
+ m
2
X
2
+ m
3
X
3
................+m
n
X
n
+ C + ϵ (6:10)
where Y is the dependent variable, m
n
is the slope coefficient of each independent vari-
able, X
n
represents each independent variable, C is the intercept of the equation, and ϵ
represents the error in the calculation of the response variable, also known as residual.
Let us understand the underlying basics of the MLR with an extension of the pre-
vious example. In Tab. 6.3 we have given a hypothetical example of 10 compounds
with their biological activity data and values of two parameters, log P and polar sur-
face area. The MLR equation to be developed from this data will be in the form of the
following equation:
Biological activity= m
1
X
1
+ m
2
X
2
+ C (6:11)
Tab. 6.3: Hypothetical values of biological activity, log P and polar surface area of 10 hypothetical
compounds.
Compound Biological activity value
(negative log of IC
)(Y)
Log P value (X
) Polar surface area (X
)
. . .
. . .
. . .
. . .
. . .
. . .
. . .
112 Chapter 6 Statistical approaches used in the QSAR
From the above table, we know that we have response variable Y (biological activity)
and two independent variables (X
1
and X
2
), which are to be correlated using MLR. For
generating the MLR equation, we have to calculate the values m
1
, m
2
, and C that can
be determined from the following equations:
m
1
=
ΣX
2
2
ΣX
1
YðÞ− ΣX
1
X
2
ðÞΣX
2
YðÞ
ΣX
2
1
ΣX
2
2
− ðΣX
1
X
2
Þ
2
(6:12)
m
2
=
ΣX
2
1
ΣX
2
YðÞ− ΣX
1
X
2
ðÞΣX
1
YðÞ
ΣX
2
1
ΣX
2
2
− ðΣX
1
X
2
Þ
2
(6:13)
where Σ X
i
2
= Σ X
i
2
− ΣX
i
ðÞ
2
=N
hi
, Σx
i
Y = ΣX
i
Y − ΣX
i
ðÞΣYðÞ=N½, andΣX
1
X
2
= ΣX
1
X
2
− Σð½
X
1
Þ ΣX
2
ðÞ=N, and N is the total number of compounds in the study.
The other parameter C, representing the intercept of the equation, can be calcu-
lated from the slope value by employing the following equation:
C = Y
mean
− m
1
X
1 mean
− m
2
X
2 mean
(6:14)
Equations (6.12) to (6.14) at first seem to be very complicated but are much easier to
understand and calculate, as depicted in Tab. 6.4.
From the parameters obtained from Tab. 6.4, we can easily calculate the values of
ΣX
1
2
, ΣX
2
2
, ΣX
1
Y, ΣX
2
Y, and ΣX
1
X
2.
The calculated values are given in Tab. 6.5.
From the values obtained in Tab. 6.5, we can calculate the m
1
and m
2
parameters
required for the generation of our MLR equation by using the following equations The
calculated values of m
1
and m
2
were found to be 0.48 and −0.20, respectively:
m
1
=
3, 193.04ðÞ101.75ðÞ− 0.35ðÞ−639.83ðÞ
210.91ðÞ3, 193.04ðÞ− ð0.35Þ
2
m
2
=
210.91ðÞ−639.83ðÞ− 0.35ðÞ101.75ðÞ
210.91ðÞ3, 193.04ðÞ− ð0.35Þ
2
From the values of m
1
and m
2,
we can now calculate the intercept value (C)fromthefol-
lowing equation. The calculated value of C from this equation is 3.81:
Tab. 6.3 (continued)
Compound Biological activity value
(negative log of IC
)(Y)
Log P value (X
) Polar surface area (X
)
. . .
. . .
. . .
6.2 Linear regression analysis 113
Tab. 6.4: Values calculated from the data of the Tab. 6.3.
Compound Biological activity
(negative log of IC
)(Y)
Log P (X
) Polar surface
area (X
)
(X
)
(X
)
X
YX
YX
X
. . . . . .. .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
ΣY = . ΣX
= . ΣX
= . (ΣX
)=. (ΣX
)=. ΣX
Y = . ΣX
Y = . ΣX
X
= .
Y
mean
= . X
mean
= . X
mean
= .
114 Chapter 6 Statistical approaches used in the QSAR
C = 2.46 − 0.48ðÞ4.61ðÞ−−0.20ðÞ17.79ðÞ
From all these calculated values of m
1
, m
2
, and C, the correlation equation of MLR can
be written as in the following equation:
Biological activity = 0.48 log P − 0.20 PSA + 3.81 (6:15)
The scatter plot between the experimental biological activity and calculated biological
activity can be plotted from the MLR equation (Fig. 6.3).
Tab. 6.5: The values calculated from Tab. 6.4.
S. no. Parameter Values obtained
ΣX
. – [(./)] = .
ΣX
. – [(./)] = ,.
ΣX
Y . – [(. × .)/]=.
ΣX
Y . – [(. × .)/]=−.
ΣX
X
. – [(. × .)/]=.
Fig. 6.3: A scatter plot obtained from the multiple linear regression (MLR) between the observed and
predicted biological activities.
6.2 Linear regression analysis 115
The MLR approach for the development of the correlation between biological activity
and descriptors is widely used in the traditional QSAR. Almost a half-decade has
passed since the first QSAR model was developed by Hansch and Fujita and now
there has been an emergence of thousands of 2D descriptors. One major hurdle that is
observed for the development of a reliable and robust QSAR model is to choose the
best descriptors from the pool of descriptors to generate our MLR equation. For the
selection of appropriate descriptors, techniques such as best subset selection or ge-
netic algorithm (GA) approach are used. The number of variables to be used for the
development of the MLR equation depends upon the number of molecules available
for the generation of the model. Generally, it is believed that molecules in the MLR
study should be five times or more than the number of descriptors to be used. The
major drawback of the MLR is that while selecting the independent variables from
the pool of the descriptors, important descriptors could be neglected; hence, more so-
phisticated statistical approaches are now becoming popular in the selection of the
variables.
6.2.2.1 Advantages of multiple linear regression (MLR)
– Interpretability: MLR provides coefficients for each descriptor, allowing re-
searchers to interpret the impact of individual molecular properties on the bio-
logical activity. This aids in understanding the underlying relationships between
chemical structures and activity.
– Variable selection: MLR helps identify significant descriptors that contribute
most to the activity. This feature selection process guides researchers in focusing
on relevant molecular features and excluding irrelevant ones, improving model
accuracy.
– Prediction: MLR models can predict the activity of new compounds based on their
molecular descriptors. This predictive ability accelerates drug discovery by suggest-
ing potential candidates for synthesis and testing, saving time and resources.
– Simplicity: MLR is relatively straightforward to implement and comprehend,
making it accessible to researchers without advanced statistical expertise. It
serves as a foundational technique for those new to QSAR analysis.
– Insight generation: By analyzing the coefficients and statis tical significanc e of
descriptors, MLR provides insights into the structure–activity relationships, aid-
ing in the design of compounds with desired activity profiles.
– Model transparency: MLR models are transparent and offer a clear mathemati-
cal relationship between descriptors and activity. This transparency enhances the
trustworthiness of the model’s predictions and findings.
– Risk assessment: MLR can quantify the uncertainty and variability associate d
with the model’s predictions, enabling researchers to assess the reliability and po-
tential risks of using the model in decision-making.
116 Chapter 6 Statistical approaches used in the QSAR
– Data exploration: MLR can handle multiple descriptors simultaneously, allowing
researchers to explore complex interactions and correlations between molecular
features and activity.
– Foundation for advanced techniques: MLR serves as a stepping stone for more
sophisticated modeling techniques. It helps researchers understand the basic
principles of modeling structure–activity relationships before delving into more
complex methods.
– Scientific insights: MLR aids in formulating hypotheses about the underlying
mechanisms responsible for the observed biological activity, contributing to sci-
entific understanding beyond just predictive modeling.
6.2.2.2 Limitations of multiple linear regression (MLR)
– Assumption of linearity: MLR assumes a linear relationship between the inde-
pendent variables (molecular descriptors) and the dependent variable (biological
activity). In QSAR, this assumption might not always hold true, as complex inter-
actions between descriptors can lead to nonlinear relationships.
– Multicollinearity: When descriptors are highly correlated, multicollinearity occurs.
This makes it challenging to discern the individual contribution of each descriptor to
the model’s outcome, potentially leading to unstable coefficient estimates.
– Overfitting: If the model includes too many descriptors relative to the available
data points, it might capture noise rather than true relationships, leading to over-
fitting. This results in poor generalization of new data and unreliable predictions.
– Limited descriptor space: QSARmodelsareconstrainedbythechoiceofde-
scriptors. Some relevant molecular properties might not be included, and irrele-
vant ones could be present, leading to biased or inaccurate predictions.
– Outliers: Outliers can significantly influence regression outcomes, pulling the
model’s fit towards extreme values. QSAR data sets often contain outliers due to
experimental errors or rare compound behaviors, affecting the reliability of the
model.
– Sensitive to data distribution: MLR assumes that the residuals (differences be-
tween predicted and actual values) are normally distributed. If the residuals devi-
ate from normality, the model assumptions are violated.
– Data quality and quantity: MLR requires a sufficient amount of high-quality
data to build robust models. Limited or noisy data can lead to unreliable parame-
ter estimates and poor predictive performance.
– Lack of mechanistic insight: MLR provides correlation information between de-
scriptors and activity, but it does not offer mechanistic insights into the underly-
ing biological interactions driving the observed relationships.
– Extrapolation risk: MLR models are sensitive to the range of data used for train-
ing. Predictions for compounds outside the training data’s descriptor space might
be unreliable due to extrapolation.
6.2 Linear regression analysis 117
– Context dependency: QSAR models might not be transferable across different bi-
ological systems or experimental conditions. A model developed for one class of
compounds might not perform well for structurally diverse compounds.
To address these limitations, researchers often turn to more advanced techniques
such as nonlinear regression, machine learning algorithms, or ensemble methods to
build more accurate and robust QSAR models.
6.3 Multivariate data analysis (MVDA)
Multivariate data analys is (MVDA) in QSAR for drug discovery is a pivotal approach
that enables the extraction of meaningful insights from complex chemical and biologi-
cal datasets [21, 22]. MVDA techniques play a crucial role in QSAR by handling datasets
with multiple variables, such as molecular descriptors and biological activities, simul-
taneously [23]. These methods include principal component analysis (PCA), partial
least squares (PLS), and cluster analysis, among others [24]. PCA reduces dimensional-
ity by t ransforming correlated variables into orthogonal components, highlighting
dominant patterns in the data [25]. PLS combines the information from both molecu-
lar descriptors and biological activities to develop predictive models, effectively iden-
tifying key descriptors influencing activity. Cluster analysis categorizes compounds
into meaningful groups based on similarity. MVDA not only enhances data explora-
tion and visualization but also guides feature selection, model building, and valida-
tion. It helps uncover hidden relationships, identify outliers, and optimize predictive
models [26, 27]. Through MVDA, QSAR empowers researchers to prioritize lead com-
pounds, design novel molecules with desired properties, and accelerate drug discov-
ery by narrowing down the chemical space for experimental testing. In essence,
MVDA in QSAR offers a systematic and efficient approach to transforming complex
multidimensional data into actionable insights, revolutionizing the field of drug devel-
opment [28].
6.3.1 Principal component analysis (PCA)
PCA is a fundamental technique within MVDA, which holds significant importance in
the realm of QSAR for drug discovery. With the burgeoning volume of chemical and
biological data, PCA serves as a potent tool to distill the essential information, en-
abling more efficient decision-making. PCA operates by transforming the original var-
iables (molecular descriptors) into a new set of orthogonal components, termed
principal components (PCs). These components capture the maximum variance pres-
ent in the data while being uncorrelated, simplify ing the interpretation of complex
relationships. By reducing dimensionality, PCA not only aids in data visualization but
118 Chapter 6 Statistical approaches used in the QSAR
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