How To Evaluate Dependency Among Variables In R

Introduction

 
In this article I am going to demonstrate how to evaluate dependency between variables from datasets to verify which one of the following variables are dependent on other variables so that unnecessary variables can be removed for creating a best fit model in R. Using a combination of function aggregate with certain variables and datasets, we can select and evaluate dependencies between variables and evaluate dependencies between variables from a dataset to create a model in R. Using combination function, we can evaluate how a certain amount of change in one variable can affect another variable.
 

Evaluating the variables

 
In order to evaluate dependencies between multiple variables together to fit a model, we can use a combination function along with mathematical operators so as to evaluate dependencies between two or more variables together. For example, we can use colon operator inside aggregate function to evaluate how many variables are dependent on each other for the creation of best fit model. To create a model in R, we can use various mathematical operators to remove irrelevant variables.
 
We can define a formula along with arithmetic operators in lots of functions in R. One such functions is the aggregate() function using which we can remove different irrelevant variables to create a model.
 
Now I will demonstrate the use of cluster function to find dependencies between several variables together. We will be using gscars dataset to demonstrate the use of aggregate function.
  1. > gscars  
  2.                      mg cyl  disp  hp drat    wt  qsec vs bn gr carb  
  3. Mazda RX4           21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4  
  4. Mazda RX4 Wag       21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4  
  5. Datsun 710          22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1  
  6. Hornet 4 Drive      21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1  
  7. Hornet Sportabout   18.7   8 360.0 175 3.15 3.440 17.02  0  0    3    2  
  8. Valiant             18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1  
  9. Duster 360          14.3   8 360.0 245 3.21 3.570 15.84  0  0    3    4  
  10. Merc 240D           24.4   4 146.7  62 3.69 3.190 20.00  1  0    4    2  
  11. Merc 230            22.8   4 140.8  95 3.92 3.150 22.90  1  0    4    2  
  12. Merc 280            19.2   6 167.6 123 3.92 3.440 18.30  1  0    4    4  
  13. Merc 280C           17.8   6 167.6 123 3.92 3.440 18.90  1  0    4    4  
  14. Merc 450SE          16.4   8 275.8 180 3.07 4.070 17.40  0  0    3    3  
  15. Merc 450SL          17.3   8 275.8 180 3.07 3.730 17.60  0  0    3    3  
  16. Merc 450SLC         15.2   8 275.8 180 3.07 3.780 18.00  0  0    3    3  
  17. Cadillac Fleetwood  10.4   8 472.0 205 2.93 5.250 17.98  0  0    3    4  
  18. Lincoln Continental 10.4   8 460.0 215 3.00 5.424 17.82  0  0    3    4  
  19. Chrysler Imperial   14.7   8 440.0 230 3.23 5.345 17.42  0  0    3    4  
  20. Fiat 128            32.4   4  78.7  66 4.08 2.200 19.47  1  1    4    1  
  21. Honda Civic         30.4   4  75.7  52 4.93 1.615 18.52  1  1    4    2  
  22. Toyota Corolla      33.9   4  71.1  65 4.22 1.835 19.90  1  1    4    1  
  23. Toyota Corona       21.5   4 120.1  97 3.70 2.465 20.01  1  0    3    1  
  24. Dodge Challenger    15.5   8 318.0 150 2.76 3.520 16.87  0  0    3    2  
  25. AMC Javelin         15.2   8 304.0 150 3.15 3.435 17.30  0  0    3    2  
  26. Camaro Z28          13.3   8 350.0 245 3.73 3.840 15.41  0  0    3    4  
  27. Pontiac Firebird    19.2   8 400.0 175 3.08 3.845 17.05  0  0    3    2  
  28. Fiat X1-9           27.3   4  79.0  66 4.08 1.935 18.90  1  1    4    1  
  29. Porsche 914-2       26.0   4 120.3  91 4.43 2.140 16.70  0  1    5    2  
  30. Lotus Europa        30.4   4  95.1 113 3.77 1.513 16.90  1  1    5    2  
  31. Ford Pantera L      15.8   8 351.0 264 4.22 3.170 14.50  0  1    5    4  
  32. Ferrari Dino        19.7   6 145.0 175 3.62 2.770 15.50  0  1    5    6  
  33. Maserati Bora       15.0   8 301.0 335 3.54 3.570 14.60  0  1    5    8  
  34. Volvo 142E          21.4   4 121.0 109 4.11 2.780 18.60  1  1    4    2  
  35. >  
We will be using ~ operator inside aggregate function along with colon operator to find associativity among multiple variables in R. For example, if there are three variables named a, b and c, then a ~ b : c means that aggregate function will create a model a, after evaluating the dependency of variable b and c among each other.
  1. > aggregate(mg ~ gr : bn, data = data, mean)  
  2.   gr bn      mg  
  3. 1    4  0 14.10667  
  4. 2    5  0 27.05000  
  5. 3    5  1 24.27500  
  6. 4    6  1 22.38000  
In the above aggregate function, there are three arguments. The first argument in the formula indicates that aggregate function represents mg as a function of the model calculating the mean and the second argument is a variable depicting the dataset. Using the above code, aggregate function creates a model in which model is evaluating the dependency between the gr and bn variables to verify the dependency among these two variables.
  1. > aggregate(mg ~ disp : hp, data = data, mean)  
  2.     disp  hp  mg  
  3. 1   75.7  52 30.4  
  4. 2  146.7  62 24.4  
  5. 3   71.1  65 33.9  
  6. 4   78.7  66 32.4  
  7. 5   79.0  66 27.3  
  8. 6  120.3  91 26.0  
  9. 7  108.0  93 22.8  
  10. 8  140.8  95 22.8  
  11. 9  120.1  97 21.5  
  12. 10 225.0 105 18.1  
  13. 11 121.0 109 21.4  
  14. 12 160.0 110 21.0  
  15. 13 258.0 110 21.4  
  16. 14  95.1 113 30.4  
  17. 15 167.6 123 18.5  
  18. 16 304.0 150 15.2  
  19. 17 318.0 150 15.5  
  20. 18 145.0 175 19.7  
  21. 19 360.0 175 18.7  
  22. 20 400.0 175 19.2  
  23. 21 275.8 180 16.3  
  24. 22 472.0 205 10.4  
  25. 23 460.0 215 10.4  
  26. 24 440.0 230 14.7  
  27. 25 350.0 245 13.3  
  28. 26 360.0 245 14.3  
  29. 27 351.0 264 15.8  
  30. 28 301.0 335 15.0  
In the above aggregate function, there are three arguments. First argument in the formula indicates that aggregate function represents mg as a function of disp and hp variables calculating the mean and the second argument is variable depicting the dataset. Using the above code, aggregate function creates a model in which model is evaluating the dependency between the disp and hp variables to verify whether any change in one variable affects another variable or not by mapping the dependency among these two variables.
  1. > aggregate(hp ~ mg : cyl, data = data, mean)  
  2.     mg cyl    hp  
  3. 1  21.4   4 109.0  
  4. 2  21.5   4  97.0  
  5. 3  22.8   4  94.0  
  6. 4  24.4   4  62.0  
  7. 5  26.0   4  91.0  
  8. 6  27.3   4  66.0  
  9. 7  30.4   4  82.5  
  10. 8  32.4   4  66.0  
  11. 9  33.9   4  65.0  
  12. 10 17.8   6 123.0  
  13. 11 18.1   6 105.0  
  14. 12 19.2   6 123.0  
  15. 13 19.7   6 175.0  
  16. 14 21.0   6 110.0  
  17. 15 21.4   6 110.0  
  18. 16 10.4   8 210.0  
  19. 17 13.3   8 245.0  
  20. 18 14.3   8 245.0  
  21. 19 14.7   8 230.0  
  22. 20 15.0   8 335.0  
  23. 21 15.2   8 165.0  
  24. 22 15.5   8 150.0  
  25. 23 15.8   8 264.0  
  26. 24 16.4   8 180.0  
  27. 25 17.3   8 180.0  
  28. 26 18.7   8 175.0  
  29. 27 19.2   8 175.0  
In the above aggregate function, there are three arguments. The first argument in the formula indicates that aggregate function represents hp as a function of mg and cyl variables calculating the mean and the second argument is variable depicting the dataset. Using the above code, aggregate function creates a model in which model is evaluating the dependency between the mg and cyl variables to verify whether any change in one variable affects another variable or not by mapping the dependency among these two variables.
  1. > aggregate(wt ~ gr : qsec, data = data, mean)  
  2.    gr  qsec     wt  
  3. 1     5 14.50 3.1700  
  4. 2     5 14.60 3.5700  
  5. 3     3 15.41 3.8400  
  6. 4     5 15.50 2.7700  
  7. 5     3 15.84 3.5700  
  8. 6     4 16.46 2.6200  
  9. 7     5 16.70 2.1400  
  10. 8     3 16.87 3.5200  
  11. 9     5 16.90 1.5130  
  12. 10    3 17.02 3.4400  
  13. 11    4 17.02 2.8750  
  14. 12    3 17.05 3.8450  
  15. 13    3 17.30 3.4350  
  16. 14    3 17.40 4.0700  
  17. 15    3 17.42 5.3450  
  18. 16    3 17.60 3.7300  
  19. 17    3 17.82 5.4240  
  20. 18    3 17.98 5.2500  
  21. 19    3 18.00 3.7800  
  22. 20    4 18.30 3.4400  
  23. 21    4 18.52 1.6150  
  24. 22    4 18.60 2.7800  
  25. 23    4 18.61 2.3200  
  26. 24    4 18.90 2.6875  
  27. 25    3 19.44 3.2150  
  28. 26    4 19.47 2.2000  
  29. 27    4 19.90 1.8350  
  30. 28    4 20.00 3.1900  
  31. 29    3 20.01 2.4650  
  32. 30    3 20.22 3.4600  
  33. 31    4 22.90 3.1500  
In the above aggregate function, there are three arguments. The first argument in the formula indicates that aggregate function represents wt as a function of gr and qsec variables calculating the mean and the second argument is variable depicting the dataset. Using the above code, aggregate function creates a model in which model is evaluating the dependency between the gr and qsec variables to verify whether any change in one variable affects another variable or not by mapping the dependency among these two variables.
 

Summary

 
In this article I demonstrated how to evaluate interaction between variables from datasets to verify which one of the following variables are dependent on other variables so that unnecessary variables can be removed for creating a best fit model in R.