Choosing independent variables

Choosing independent variables
Three popular methods of choosing independent variables are:
Hellwig's method
Graphs analysis method
Correlation matrix method.

Hellwig’s method
Three steps:
Number of combinations: 2m-1
Individual capacity of every independent variable in the combination:
Integral capacity of information for every combination:

Hellwig’s method
1. Number of combinations
In Hellwig’s method the number of combinations is provided by the formula 2m –1 where m is the number of independent variables.

Hellwig’s method
2. Individual capacity of each independent variable in the combination is given by the formula:
where:
hkj – individual capacity of information for j-th variable in k-th combination

Hellwig’s method
2. Individual capacity of each independent variable in the combination is given by the formula:
where:
r0j – correlation coefficient between j-th variable (independent) and dependent variable

Hellwig’s method
2. Individual capacity of each independent variable in the combination is given by the formula:
where:
rij – correlation coefficient between i-th and j-th variable (both independent)

Hellwig’s method
2. Individual capacity of each independent variable in the combination is given by the formula:
where:
Ik – the set of numbers of variables in k-th combination

Hellwig’s method
3. Integral capacity of information for every combination
The next step is to calculate Hk – integral capacity of information for each combination as the sum of individual capacities of information within each combination:

Hellwig’s method
Q: HOW TO CHOOSE INDEPENDENT VARIABLES?
A: LOOK AT INTEGRAL CAPACITIES OF INFORMATION. THE GREATEST Hk MEANS THAT VARIABLES FROM THIS COMBINATION SHOULD BE INCLUDED IN THE MODEL.

Example
Let’s choose independent variables, using Hellwig's method.

Example
First we need to have vector and matrix of correlation coefficients.
❑ Correlation coefficients between every independent variable X1, X2 and X3 and dependent variable Y are provided in vector R0.

Example
First we need to have vector and matrix of correlation coefficients.
❑ Correlation matrix R includes correlation coefficients between independent variables.

Example
1. Number of combinations
We have 3 independent variables X1, X2 and X3. Thus we may have 2m-1 = 23-1= 8-1= 7 combinations of independent variables.
{X1} {X2} {X3} {X1, X2} {X1, X3} {X2, X3} {X1, X2, X3}

Example
2. Individual capacity of independent variable in the combination 1

Example
2. Individual capacity of independent variable in the combination 2

Example
2. Individual capacity of independent variable in the combination 3

Example
2. Individual capacity of every independent variable in the combination 4

Example
2. Individual capacity of independent variables in the combination 5

Example
2. Individual capacity of every independent variables in the combination 6

Example

Example
3. Integral capacity of information for each combination
The greatest integral capacity is for combination C4. Independent variables - X1, X2 - will be included in model.

Graph analysis method
Three steps
Calculating r*
Modification of correlation matrix
Drawing the graph

Graph analysis method
Q: HOW TO CHOOSE INDEPENDENT VARIABLES?
A: LOOK AT THE GRAPHS. THE NUMBER OF GROUPS MEANS THE NUMBER OF VARIABLES INCLUDED IN THE MODEL. IF THERE’S SEPARATED (ISOLATED) VARIABLE, YOU SHOULD INCLUDE IT IN THE MODEL. FROM EACH GROUP, THE VARIABLE WITH THE GREATEST NUMBER OF LINKS SHOULD BE INCLUDED IN MODEL. IF THERE’S TWO VARIABLES WITH THE GREATEST NUMBER OF LINKS, YOU SHOULD TAKE THE VARIABLE WHICH IS MORE STRONGLY CORRELATED WITH DEPENDENT VARIABLE.

Graph analysis method
Calculating r*
We start with calculating critical value of r* using the formula:
where tα is provided in the table of t-Student distribution at the significance level α and the degrees of freedom n-2 (sometimes r* can be given, so there’s no need to calculate it).

Graph analysis method
2. Modification of correlation matrix
The correlation coefficients for which are statistically irrelevant and we replace them with nulls in correlation matrix.
3. Drawing the graph
Using modified correlation matrix we draw the graphs with bulbs representing the variables and the links representing correlation coefficients of statistical significance.

Example
Let’s have an example (the same one as for Hellwig’s method, n=7)

Example
1. Calculating r* (n=7, tα, n-2=t0,05,5=2,571)

Example
Modification of correlation matrix

Example
3. Drawing the graph
Conclusion: Model will consist of X1 (as isolated variable) and x2 (cause is more strongly correlated with dependent variable – you may check it in R0 vector).

Correlation matrix method
Calculate r*
We start with calculating critical value of r* using the formula:
where tα is provided in the table of t-Student distribution at the significance level α and the degrees of freedom n-2 (sometimes r* can be given, so there’s no need to calculate it).

2. To eliminate Xi variables weakly correlated withY
3. To choose Xs where
[Xs is the best source of information]
4. To eliminate Xi variables strongly correlated with Xs

Example
Let’s have an example (the same one as for Hellwig’s method and graph analysis metod, n=7)

Example
1. Calculating r* (n=7, tα, n-2=t0,05,5=2,571)

2. To eliminate Xi variables weakly correlated withY
None of the variables will be eliminated

3. To choose Xs where

4. To eliminate Xi variables strongly correlated with Xs
None of the variables will be eliminated.
X1, X2, X3 will be included in model.