In the previous Block, we worked on univariate analysis, which involved examining variables one by one, separately. Univariate analysis is useful when conducting research aimed at describing or exploring a specific phenomenon.

We are now focusing on the other side of analysis, which is bivariate analysis. Bivariate analysis looks at variables in pairs, analyzing two variables simultaneously to examine the relationship between them and its strength.

The main areas of bivariate analysis involve identifying the relationship between two variables, quantifying the strength and direction of the relationship, modeling the relationship between two variables, and exploring the causal links between them. One topic of bivariate analysis concerns hypothesis testing, which we have placed in the last Block.

We have therefore divided this teaching into two sections: the first explores statistical distributions with two variables, and the second explains the statistical parameters of such a distribution.

1.1. The Elementary Table

An elementary table lists, for each individual \(i\) in the population, the values \(x_i\) and \(y_i\) of each of the studied variables in adjacent columns. One could say that it is a combination, a superposition, of two simple statistical tables (each with a single entry).
The following table represents an elementary table:

Example: The following table represents an elementary table featuring two variables: Gender and Education Level

We can see that for each individual in our example, the pair of modalities that represent them are placed side by side.

An elementary table is used when one wants to organize and present data in a way that allows for comparing the values of multiple variables for a set of individuals or observation units. An elementary table is not used for analysis; it is the bivariate equivalent of a single-entry statistical table.

1.2. The Contingency Table

The contingency table, also known as a cross table or correspondence table, defines a joint distribution. It relates a pair of variables \((x, y)\) by associating the frequencies corresponding to each pair of modalities.

A contingency table, unlike an elementary table, is used to present and analyze the relationship between two variables. It summarizes the frequencies of observations that lie at the intersection of the modalities of these two variables.

A contingency table is presented in the following format:

\(~~~~~~ ~~~~~~~~~~~~~~ Modalities ~~of ~~y \) \(Modalities ~~of ~~x \)	\(Y_1\)	\(Y_2\)	\(Y_3\)	\(...\)	\(Y_p\)
\(x_1\)	\(n_{11}\)	\(n_{12}\)	\(n_{13}\)	\(...\)	\(n_{1p}\)
\(x_2\)	\(n_{21}\)	\(n_{22}\)	\(n_{23}\)	\(...\)	\(n_{2p}\)
\(x_3\)	\(n_{31}\)	\(n_{32}\)	\(n_{33}\)	\(...\)	\(n_{3p}\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(x_k\)	\(n_{k1}\)	\(n_{k2}\)	\(n_{k3}\)	\(...\)	\(n_{kp}\)

Table II.2.3. Contingency Table.

Note:
The variable \(x\) has \(k\) categories listed in the left margin of the table [for an explanation of the nature and uses of statistical tables see Lahanier-Reuter, D. (2003)], each row of the table corresponds to a category of the variable \(X\) and contains the counts for that category. It is customary to denote the row index with the letter \(i\) (\(x_i\): \(i\) ranging from 1 to \(k\)). For the variable \(Y\), it is listed in the top margin of the contingency table, and each category of the variable \(Y\) corresponds to a column of the table (the column index is typically denoted by the letter \(j\) (\(y_j\): \(j\) ranging from 1 to \(p\)).

Example:

The following table shows the relationship between education level and employment status for a group of people:

Explanation and Note:
From the table, we can observe that:

• Each row corresponds to an education level: Primary, Secondary, and Tertiary. The Total row shows the total number of people in each employment status category;
• The columns Employed and Unemployed represent the employment status of individuals for each education level. The Total column to the right of each row represents the total number of individuals for each education level. We will revisit these concepts later in this session.

These observations will lead us, and this is one of the major objectives of the Module, to the Reading and Analysis of the contingency table. We can now provide an overview of this analytical work:

• Reading - Distribution of Employment (dependent variable): It is observed that the majority of individuals with a Secondary education level are employed (60 out of 80), while the number of unemployed is lower at each education level.
• Analysis of the Relationship: This table could be used to analyze whether there is a significant relationship between education level and employment status. For example, the chi-square \(\chi^2\) test could be used to test the independence between these two variables. (This topic will be covered in the third Block.)

1.3. The Joint Distribution

Joint distribution refers to the pairs \((x, y)\) formed by the left and top margins (the \(k\) rows and \(p\) columns). Simply put, it is the set of \((x_i, y_i, n_{ij})\) (\(i\) ranging from 1 to \(k\) and \(j\) ranging from 1 to \(p\)).

The joint frequency \(n_{ij}\) represents both the \(x_i\) modality of the \(x\) variable and the \(y_i\) modality of the \(y\) variable.

The set of joint frequencies constitutes the total frequency, represented by the following formula:

$$\sum_{j=1}^{p} \sum_{i=1}^{k} n_{ij} = \sum_{i=1}^{k}\sum_{j=1}^{p} n_{ij} = n $$

Sometimes it is more useful to replace frequencies with proportions; joint frequencies are then divided by \(n\) (the total frequency), which is called the joint proportion.
As in a univariate table, the sum of the joint proportions is equal to 1.

$$\sum_{j=1}^{p} \sum_{i=1}^{k} f_{ij} = \sum_{i=1}^{k}\sum_{j=1}^{p} f_{ij} = 1 $$

Note: For our previous example, linking education level to employment status, we can say that:

• Reading Joint Frequencies: Primary and Employed: \(40\) people have a primary education level and are employed, Tertiary and Unemployed: \(10\) people have a tertiary education level and are unemployed;
• Calculation of Total Frequency: The sum of the frequencies in all cells of the table is 170, which is the total frequency;
• Joint Proportion: To obtain the joint proportions, each joint frequency is divided by the total frequency. For example, the joint proportion for Primary and Employed is: \(\frac{40}{170} = 0.235\) \( (or ~~ 23.5 \%) \).

1.4. The Marginal Distribution

The marginal distribution concerns the distribution of a single variable \((x\) or \(y)\). In other words, a marginal distribution involves deducing the distribution of each variable considered in isolation, as seen in the previous lesson. It is the process of extracting single-entry tables.
The distribution related to the variable \(x\) alone is called the marginal distribution of \(x\); the distribution related to the variable \(y\) alone is called the marginal distribution of \(y\).

In the previous example, constructing the marginal distributions involves creating a single-entry table for the Education Level variable and another for the Employment Status variable. We will explain the theoretical principles of marginal distribution in the following lines.

Marginal Distribution of \(X\)

The marginal distribution of the variable \(x\) is defined using the pairs \((x_i, n_i)\), where \(i = 1, 2, 3, \ldots, K\) [\(x_i\) is the modality of the \(x\) variable and \(n_i\) is the corresponding frequency, referred to as the marginal frequency of the modality \(x_i\). This represents the number of individuals where the modality of the \(X\) variable is \(x_i\) and the modality of the \(Y\) variable is \(y_1, y_2, \ldots, y_p\); the frequency is equal to the total or sum of the frequencies in the \(i\)-th row, as shown in the following formula:

$$ n_{i\ } =\ n_{i1\ } +\ n_{i2\ } +\ n_{i3\ } +\ \ldots\ +\ n_{ip\ } =\ \sum_{j=1}^{p}n_{ij} $$

It should be noted that the sum of the marginal frequencies is equal to the total frequency \(n\) (or \(n..\)).

The following table shows the marginal distribution of the variable \(x\):

\(Modalities ~~of ~~the ~~variable ~~x\)	\(Marginal ~~Frequency\)
\(x_1\)	\(n_{1.}\)
\(x_2\)	\(n_{2.}\)
\(x_3\)	\(n_{3.}\)
\(...\)	\(...\)
\(x_k\)	\(n_{k.}\)
\(\sum\)	\(n\)

Table II.2.5. The marginal distribution of the variable \(x\).

The marginal frequency of a modality \(x_i\), denoted \(f_i\), is also defined as follows:

$$f_i\ =\ \frac{n_{i.}}{n}$$

The sum of the marginal frequencies \(f_{i.}\) is equal to 1 [\(\sum_{i=1}^{k}f_i\ =\ 1\)].

Marginal Distribution of \(y\)

The marginal distribution of the variable \(y\) consists of the pairs \((y_{j}, n_{.j})\) (\(j = 1, 2, 3, \ldots, p\)), where \(y_j\) is the modality of the variable \(y\) and \(n_{.j}\) is the corresponding frequency, referred to as the marginal frequency of the modality \(y_j\). This represents the number of individuals where the modality of the \(Y\) variable is \(y_j\) and the modality of the \(X\) variable is \(x_1\), \(x_2\), \ldots, \(x_k\); the frequency is equal to the total or sum of the frequencies in the \(j\)-th column, as shown in the following formula:

$$ n_{.j\ }\ =\ n_{1j\ }\ +\ n_{2j\ }\ +\ n_{3j\ }\ +\ \ldots\ +\ n_{kj\ }\ =\ \sum_{i=1}^{k}n_{ij} $$

It should be noted that the sum of the marginal frequencies is equal to the total frequency \(n\) (or \(n_{..}\)).

The following table shows the marginal distribution of the variable \(x\):

\(Modalities ~~of ~~the ~~variable ~~y\)	\(Marginal ~~Frequency\)
\(y_1\)	\(n_{.1}\)
\(y_2\)	\(n_{.2}\)
\(y_3\)	\(n_{.3}\)
\(...\)	\(...\)
\(y_p\)	\(n_{.k}\)
\(\sum\)	\(n\)

Table II.2.6. The marginal distribution of the variable \(y\).

The marginal frequency of a modality \(y_i\), denoted \(f_i\), is also defined as follows:

$$f_i\ =\ \frac{n_{.j}}{n}$$

The sum of the marginal frequencies \(f_{i.}\) is equal to 1 [\(\sum_{j=1}^{p}f_{i.}\ =\ 1\)].

Note: Contingency table and marginal distributions.
It should be noted that the marginal frequencies \(n_{i.}\) are displayed in an additional column, while the marginal frequencies \(n_{.j}\) are shown in an additional row of the joint distribution (x, y).

The following table illustrates this idea:

\(~~~~~~ ~~~~~~~~~~~~~~ Modalities ~~of ~~y \) \(Modalities ~~of ~~x \)	\(Y_1\)	\(Y_2\)	\(Y_3\)	\(...\)	\(Y_p\)	\(\sum\)
\(x_1\)	\(n_{11}\)	\(n_{12}\)	\(n_{13}\)	\(...\)	\(n_{1p}\)	\(n_{1.}\)
\(x_2\)	\(n_{21}\)	\(n_{22}\)	\(n_{23}\)	\(...\)	\(n_{2p}\)	\(n_{2.}\)
\(x_3\)	\(n_{31}\)	\(n_{32}\)	\(n_{33}\)	\(...\)	\(n_{3p}\)	\(n_{3.}\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(x_k\)	\(n_{k1}\)	\(n_{k2}\)	\(n_{k3}\)	\(...\)	\(n_{kp}\)	\(n_{k.}\)
\(\sum\)	\(n_{.1}\)	\(n_{.2}\)	\(n_{.3}\)	\(...\)	\(n_{.p}\)	\(n\)

Table II.2.7. Contingency table and marginal distributions.

We note that the marginal distribution of \(x\) is given by the left margin and the last column of the table, while that of the variable \(y\) is given by the top margin and the last row.

Note: Marginal statistics.
Each contingency table consists of two univariate distributions. By creating marginal distribution tables, we can calculate the indices seen in the previous chapters (central tendency, dispersion, and position …). In the following chapters, we will need to calculate these indices and make the necessary interpretations for the analysis.

1.5. Conditional Distributions

The conditional distribution of \(x\) given \(y\) is the distribution of \(x\) concerning only the individuals with the modality \(y_i\) of \(Y\). Similarly, the conditional distribution of \(y\) given \(x\) is the distribution of \(y\) concerning only the individuals with the modality \(x_i\) of \(X\).

Conditional Distributions of \(x\) given \(y\)

The variable \(y\) has \(p\) modalities, so the population could be divided into \(p\) sub-populations (individuals identified by modality \(y_1\), those identified by modality \(y_2\), … up to those identified by \(y = y_p\)). For each sub-population, we can have what is called a conditional distribution.

Following this reasoning, we obtain \(p\) conditional distributions of \(x\) given \(y\):

The conditional distribution of \(x\) given \(y = y_{1}\);
The conditional distribution of \(x\) given \(y = y_{2}\);
The conditional distribution of \(x\) given \(y = y_{3}\);
................................................... ;
The conditional distribution of \(x\) given \(y = y_{p}\);

Each distribution is defined by a pair (\(x_i\), \(n_{ij}\)) [\(i\) ranging from \(1\) to \(k\) and \(j\) being fixed]. The following table represents the idea of a conditional distribution:

\(Modalities ~~of ~~x \)	\(Conditional ~~frequencies ~~n_{ij}\)
\(x_1\)	\(n_{1j}\)
\(x_2\)	\(n_{2j}\)
\(x_3\)	\(n_{3j}\)
\(...\)	\(...\)
\(x_k\)	\(k_j\)
\(\sum\)	\(n_{.j}\)

Table II.2.8: Conditional distribution of \(x\) given \(y\).

The total frequency is given by the formula:

$$ n_{.j} = n_{1j} + n_{2j} + n_{3j} + ..... + n_{kj} = \sum_{i=1}^{k}n_{ij}$$

We could also calculate the conditional frequencies \(f_{xi/yj}\) using the formula:

$$ f_{xi/yj}\ =\ \frac{n_{ij}}{n_{.j}} $$

The sum of the conditional frequencies is equal to 1.

With the help of conditional distributions (or frequencies), we can create a column-profile table. A column-profile table contains the modalities of \(x\) in the left margin, the conditional frequencies of \(x\) given \(y = y_1\) in the first column, the conditional frequencies of \(x\) given \(y = y_2\) in the second column, …, the conditional frequencies of \(x\) given \(y = y_p\) in the last column. The following table illustrates the column-profile table.

\(~~~~~~ ~~~~~~~~~~~~~~ Modalities ~~of ~~y \) \(Modalities ~~of ~~x \)	\(Y_1\)	\(Y_2\)	\(Y_3\)	\(...\)	\(Y_p\)	\(\sum\)
\(x_1\)	\(f_{x1/y1}\)	\(f_{x1/y2}\)	\(f_{x1/y3}\)	\(...\)	\(f_{x1/yp}\)	\(f_{1.}\)
\(x_2\)	\(f_{x2/y1}\)	\(f_{x2/y2}\)	\(f_{x2/y3}\)	\(...\)	\(f_{x2/yp}\)	\(f_{2.}\)
\(x_3\)	\(f_{x3/y1}\)	\(f_{x3/y2}\)	\(f_{x3/y3}\)	\(...\)	\(f_{x3/yp}\)	\(f_{3.}\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)
\(x_k\)	\(f_{xk/y1}\)	\(f_{xk/y2}\)	\(f_{xk/y3}\)	\(...\)	\(f_{xk/yp}\)	\(f_{k.}\)
\(\sum\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)

Table II.2.9. Column-profile table.

Conditional distributions of \(y\) given \(x\)

The variable \(x\) has \(k\) modalities, so we can divide the population into \(k\) sub-populations (individuals identified by modality \(x_1\), those identified by modality \(x_2\), … up to those identified by \(x = x_k\)). For each sub-population, we can have what is called the conditional distribution of individuals according to the modalities of the variable \(y\).
Following this reasoning, we will obtain \(p\) conditional distributions of \(y\) given \(x\):
The conditional distribution of \(y\) given \(x = x_1\);
The conditional distribution of \(y\) given \(x = x_2\);
The conditional distribution of \(y\) given \(x = x_3\);
.................................................. ;
The conditional distribution of \(y\) given \(x = x_k\);
Each distribution is defined by a pair (\(y_i\), \(n_{ij}\)) [\(j\) ranging from \(1\) to \(p\) and \(i\) being fixed].
The following table represents the idea of a conditional distribution:

\(Modalities ~~of ~~y \)	\(Conditional ~~frequencies ~~n_{ij}\)
\(y_1\)	\(n_{1j}\)
\(y_2\)	\(n_{2j}\)
\(y_3\)	\(n_{3j}\)
\(...\)	\(...\)
\(y_p\)	\(n_{ip}\)
\(\sum\)	\(n_{.j}\)

Table II.2.10: Conditional distribution of \(y\) given \(x\).

The total count is given by the formula:

$$ n_{i.} = n_{i1} + n_{i2} + n_{i3} + ... + n_{ip} = \sum_{i=1}^{k}n_{ij} $$

You can also calculate the conditional frequencies \(f_{yj/xi}\) using the formula: \(f_{yj/xi} = \frac{n_{ij}}{n_{i.}}\)

The sum of the conditional frequencies is equal to 1.

Using the conditional distributions (or frequencies), we can create a column-profile table. A column-profile table contains the modalities of \(x\) in the left margin, the conditional frequencies of \(y\) given \(x = x_1\) in the first column, the conditional frequencies of \(y\) given \(x = x_2\) in the second column, …, the conditional frequencies of \(y\) given \(x = x_k\) in the last column.
The following table illustrates the column-profile table:

\(~~~~~~ ~~~~~~~~~~~~~~ Modalities ~~of ~~y \) \(Modalities ~~of ~~x \)	\(Y_1\)	\(Y_2\)	\(Y_3\)	\(...\)	\(Y_p\)	\(\sum\)
\(x_1\)	\(f_{y1/x1}\)	\(f_{y2/x1}\)	\(f_{y3/x1}\)	\(...\)	\(f_{yp/x1}\)	\(1\)
\(x_2\)	\(f_{y1/x2}\)	\(f_{y2/x2}\)	\(f_{y3/x2}\)	\(...\)	\(f_{yp/x2}\)	\(1\)
\(x_3\)	\(f_{y1/x3}\)	\(f_{y2/x3}\)	\(f_{y3/x3}\)	\(...\)	\(f_{yp/x3}\)	\(1\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(1\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(1\)
\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(...\)	\(1\)
\(x_k\)	\(f_{y1/xk}\)	\(f_{y2/xk}\)	\(f_{y3/xk}\)	\(...\)	\(f_{yp/xk}\)	\(1\)
\(\sum\)	\(f_{.1}\)	\(f_{.2}\)	\(f_{.3}\)	\(...\)	\(f_{.p}\)	\(1\)

Table II.2.11. Column-profile table.

Conditional Averages

The conditional average of \(x\) is calculated for each of the \(p\) conditional distributions of \(x\). The conditional average of \(x\) given \(y = y_i\) is generally denoted by \({\bar{x}}_j\). It should be understood as a weighted average:

$${\bar{x}}_j\ =\ \frac{1}{n_{.j}}\ \sum_{i=1}^{k}{n_{ij}\ x_i}$$

By replacing the counts with frequencies, we get the following formula:

$${\bar{x}}_j\ =\ \sum_{i=1}^{k}{f_{xi/yj}\ x_i}$$

For the conditional average of \(y\) given \(x = x_i\), we have the following formula:

$${\bar{y}}_i\ =\ \frac{1}{n_{i.}}\ \sum_{j=1}^{p}{n_{ij}\ y_j}$$

By replacing the counts with frequencies, we get the following formula:

$${\bar{y}}_i\ =\ \sum_{j=1}^{p}{f_{yi/xj}\ y_j}$$

Note that for each variable, the marginal mean is equal to the mean of the conditional means. This relationship is expressed in the following formulas:

$$ \bar{x}\ =\ \frac{1}{n}\ \sum_{j=1}^{p}{n_{.j}\ {\bar{x}}_j} ~~~~ and ~~~~ \bar{y}\ =\ \frac{1}{n}\ \sum_{i=1}^{k}{n_{i.}\ {\bar{y}}_i}$$

Conditional Variances and Standard Deviations

The conditional variance of \(x\) given \(y = y_j\) is given by the formula:

$$ V_{j} (x) = \frac{1}{n_{.j}} \sum_{i=1}^{k} {n_{ij}} ~~(x_i - \bar{x}_{j}) ^ {2} = \frac{1}{n_{.j}} \sum_{i=1}^{k} {n}_{ij} ~~ {x}_{i}^{2} - {\bar{x}}_{j}^{2}$$

By replacing the counts with conditional frequencies, we get the following formula:

$$ V_{j} (x) = \frac{1}{n_{.j}} \sum_{i=1}^{k} ~~ {f_{xi/yj}} (x_i - \bar{x}_{j}) ^ {2} = \frac{1}{n_{.j}} \sum_{i=1}^{k} {f_{xi/j}} ~~ {x}_{i}^{2} - {\bar{x}}_{j}^{2}$$

The conditional standard deviation is calculated using the formula: \( \sigma_j\ (x)=\ \sqrt{v_j\ (x)}\)

For the variance and standard deviation of \(Y\) given \(X = x_i\), the formula is:

$$ V_{i} (y) = \frac{1}{n_{i.}} \sum_{j=1}^{p} {n_{ij}} ~~(y_j - \bar{y}_{i}) ^ {2} = \frac{1}{n_{i.}} \sum_{j=1}^{p} {n}_{ij} ~~ {y}_{j}^{2} - {\bar{y}}_{i}^{2}$$

By replacing the counts with conditional frequencies, we get the following formula:

$$ V_{i} (y) = \frac{1}{n_{i.}} \sum_{j=1}^{p} ~~ {f_{yj/xi}} (y_j - \bar{y}_{i}) ^ {2} = \frac{1}{n_{i.}} \sum_{j=1}^{p} {f_{yj/xi}} ~~ {y}_{j}^{2} - {\bar{y}}_{i}^{2}$$

The conditional standard deviation is calculated using the formula: \( \sigma_i\ (y)=\ \sqrt{v_i\ (y)}\)

It is also noted that the marginal variance equals the sum of the mean of the conditional variances plus the variance of the conditional means, as shown by the following two formulas:

$$v(x)\ =\ \frac{1}{n}\ \sum_{j=1}^{p}{n_{.j}\ } ~~ v_j\ (x)\ +\ \frac{1}{n}\ \sum_{j=1}^{p}{n_{.j}} ~~ ({\bar{x}_j\ -\ \bar{x}}^2) $$

$$ v(y)\ =\ \frac{1}{n}\ \sum_{i=1}^{k}{n_{i.}\ } ~~ v_i\ (y)\ +\ \frac{1}{n}\ \sum_{i=1}^{k}{n_{i.} ~~ {(\bar{y}}_i\ -\ \bar{y})}^2 $$

Independence

Two variables \(x\) and \(y\) are said to be independent when the conditional frequencies \(f_{yj/xi}\) are equal and equal to the marginal frequency \(f_{.j}\). Consequently, in the row profile table, all rows are exactly similar, and similarly for the column profile table, the conditional frequencies \(f_{xi/yj}\) are equal to the marginal frequency \(f_{i.}\).

It will be observed that in the contingency table, the rows are proportional to each other, and the same is true for the columns: \(n_{ij}\ =\ \frac{n_{i.}\ n_{.j}}{n} (f_{ij}\ =\ f_{i.}\ f_{.j})\).

Do not forget the example

The statistical independence of \(X\) and \(Y\) results in:

• The independence of \(Y\) with respect to \(X\), where the conditional frequencies of \(Y\) for \(X = x_i\) do not depend on \(i\);
• The independence of \(X\) with respect to \(Y\), where the conditional frequencies of \(X\) for \(Y = y_i\) do not depend on \(j\).

Functional Dependency – Reciprocal Functional Dependency

Variable \(y\) is functionally related to \(x\) when each modality (value or class) of \(x\) corresponds to a modality (value or class) of variable \(y\). It will be observed that in the joint distribution table, there is only one non-zero frequency per row [the same reasoning can be applied by analogy for a variable \(x\) functionally related to \(y\)].

Two variables \(x\) and \(y\) are said to be in reciprocal functional dependency (reciprocally dependent) when each modality (value or class) of variable \(x\) corresponds to a modality (value or class) of variable \(y\) and vice versa. In this case, there will be as many rows as columns in the joint distribution table, and exactly one non-zero frequency per row and column.

Relative Dependency

Two variables are said to be in relative dependency when they are neither independent nor functionally related (reserve a lengthy passage to introduce the rest of the chapters).

The topics covered in this section (i.e. covariance, correlation, and adjustment) are introduced briefly. We will revisit them in more detail in the final Block of our Course, where we will discuss the principles of statistical inference. They are introduced here for mastering the vocabulary of bivariate data analysis.

Covariance

Insert a brief definition

Covariance is given by the following formula:

\[ Cov(x,y) = \frac{1}{n} \sum_{\substack{1 \leq i \leq p \\ 1 \leq j \leq k }} n_{ij} ~ (x_{i} - \bar{x}) (y_{j} - \bar {y}) \]

The previous equation can be simplified as follows:

\[ Cov(x,y) = \frac{1}{n} \left( \sum_{\substack{1 \leq i \leq p \\ 1 \leq j \leq k }} n_{ij} ~~ x_{i} y_{j} \right ) - \bar{x} \bar {y} \]

Covariance provides us with the following properties:

• \(Cov~~(aX~+~b~,~cY~+~d)~~=~a~c~Cov~(X~,~Y)\)
• \(Cov~~(X~,~Y)~=~V~(X)\)
• \(\left|~Cov~(X~,~Y)~\right|~\leq~\sigma~(X)~\sigma~(Y)\)

Note: When the variables \(X\) and \(Y\) are independent, \(Cov~(X~,~Y)~ = ~0\); the reverse reasoning is not necessarily true.

From covariance, we can calculate the linear correlation coefficient of \(X\) and \(Y\), which is defined by the following formula:

\[ r = \frac {Cov(x,y)} {\sigma(X) ~ \sigma(Y)} \]

The value \(r\) is invariant to changes in origin and scale, it ranges between (-1) and (+1), and takes the value zero when the variables are independent.

The Least Squares Method

In a scatter plot \((x_{1}, y_{1})\), \((x_{2}, y_{2})\)... \((x_{p}, y_{p})\) with weighting coefficients \(n_1\), \(n_2\), ... \(n_p\) equal to 1, we identify a form of functional relationship between \(x\) and \(y\) depending on the appearance of the scatter plot (this relationship can take one of the forms: \(y = ax + b\); \(y = ax^{b}\), etc.).
The principle of adjustment is to determine the parameter values that minimize the distance between the points and the curve that represents the chosen model to account for the functional relationship.

Adjustment is said to be affine (\(y = ax + b\)), or linear (\(y = ax\)) when the points are, in a certain way, aligned; this is referred to as fitting by a straight line.
In the least squares method, the goal is to minimize the distance \(S\) defined as the sum of the squares of the vertical deviations: \(S = \displaystyle\sum_{i=1}^{p} n_{i} (y_{i} - ax_{i} - b)^{2}\)
The following figure illustrates the idea of affine adjustment using the least squares method: