Cramer’s V

Cramer’s V measures the relation between two variables in categorical scale. It is implemented in the cramer() function. It returns the value in a range of 0 to 1 (1 - when the two categorical variables are linearly associated with each other, 0 - otherwise), Chi-Squared statistics from the chisq.test(), the respective p-value and the number of degrees of freedom. The tested hypothesis in this case is formulated as:

H₀ : V = 0 (the variables don’t have association);

H₁ : V ≠ 0 (there is an association between the variables).

Here’s what we get when trying to find the association between the engine and transmission in the mtcars data:

cramer(mtcarsData$vs, mtcarsData$am)
#> Cramer's V: 0
#> Chi^2 statistics = 0.3475, df: 1, p-value: 0.5555

Judging by this output, the association between these two variables is very low (close to zero) and is not statistically significant.

Cramer’s V can also be used for the data in numerical scales. In general, this might be not the most suitable solution, but this might be useful when you have a small number of values in the data. For example, the variable gear in mtcars is numerical, but it has only three options (3, 4 and 5). Here’s what Cramer’s V tells us in the case of gear and am:

cramer(mtcarsData$am, mtcarsData$gear)
#> Cramer's V: 0.7808
#> Chi^2 statistics = 20.9447, df: 2, p-value: 0

As we see, the value is high in this case (0.809), and the null hypothesis is rejected on 5%. So we can conclude that there is a relation between the two variables. This does not mean that one variable causes the other one, but they both might be driven by something else (do more expensive cars have less gears but the automatic transmission?).

Plotting categorical variables

While R allows plotting two categorical variables against each other, the plot is hard to read and is not very helpful (in my opinion):

plot(table(mtcarsData$am,mtcarsData$gear))

So I have created a function that produces a heat map for two categorical variables. It is called tableplot():

tableplot(mtcarsData$am,mtcarsData$gear)

It is based on table() function and uses the frequencies inside the table for the colours:

table(mtcarsData$am,mtcarsData$gear) / length(mtcarsData$am)
#>    
#>           3       4       5
#>   a 0.46875 0.12500 0.00000
#>   m 0.00000 0.25000 0.15625

The darker sectors mean that there is a higher concentration of values, while the white ones correspond to zeroes. So, in our example, we see that the majority of cars have automatic transmissions with three gears. Furthermore, the plot shows that there is some sort of relation between the two variables: the cars with automatic transmissions have the lower number of gears, while the ones with the manual have the higher number of gears (something we’ve already noticed in the previous subsection).

Measures of association

When you deal with datasets (i.e. data frames or matrices), then you can use cor() function in order to calculate the correlation coefficients between the variables in the data. But when you have a mixture of numerical and categorical variables, the situation becomes more difficult, as the correlation does not make sense for the latter. This motivated me to create a function that uses either cor(), or cramer(), or mcor() functions depending on the types of data (see discussions of cramer() and mcor() above). The function is called association() or assoc() and returns three matrices: the values of the measures of association, their p-values and the types of the functions used between the variables. Here’s an example:

assocValues <- assoc(mtcarsData)
print(assocValues,digits=2)
#> Associations: 
#> values:
#>        mpg  cyl  disp    hp  drat    wt  qsec   vs   am gear carb
#> mpg   1.00 0.86 -0.85 -0.78  0.68 -0.87  0.42 0.66 0.60 0.66 0.67
#> cyl   0.86 1.00  0.92  0.84  0.70  0.78  0.59 0.79 0.46 0.48 0.48
#> disp -0.85 0.92  1.00  0.79 -0.71  0.89 -0.43 0.71 0.59 0.77 0.56
#> hp   -0.78 0.84  0.79  1.00 -0.45  0.66 -0.71 0.72 0.24 0.66 0.79
#> drat  0.68 0.70 -0.71 -0.45  1.00 -0.71  0.09 0.44 0.71 0.83 0.33
#> wt   -0.87 0.78  0.89  0.66 -0.71  1.00 -0.17 0.55 0.69 0.66 0.61
#> qsec  0.42 0.59 -0.43 -0.71  0.09 -0.17  1.00 0.74 0.23 0.63 0.67
#> vs    0.66 0.79  0.71  0.72  0.44  0.55  0.74 1.00 0.00 0.57 0.57
#> am    0.60 0.46  0.59  0.24  0.71  0.69  0.23 0.00 1.00 0.78 0.19
#> gear  0.66 0.48  0.77  0.66  0.83  0.66  0.63 0.57 0.78 1.00 0.32
#> carb  0.67 0.48  0.56  0.79  0.33  0.61  0.67 0.57 0.19 0.32 1.00
#> 
#> p-values:
#>       mpg  cyl disp   hp drat   wt qsec   vs   am gear carb
#> mpg  0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.01
#> cyl  0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.01
#> disp 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.07
#> hp   0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.18 0.00 0.00
#> drat 0.00 0.00 0.00 0.01 0.00 0.00 0.62 0.01 0.00 0.00 0.66
#> wt   0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.02
#> qsec 0.02 0.00 0.01 0.00 0.62 0.34 0.00 0.00 0.21 0.00 0.01
#> vs   0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.56 0.00 0.01
#> am   0.00 0.01 0.00 0.18 0.00 0.00 0.21 0.56 0.00 0.00 0.28
#> gear 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.09
#> carb 0.01 0.01 0.07 0.00 0.66 0.02 0.01 0.01 0.28 0.09 0.00
#> 
#> types:
#>      mpg       cyl      disp      hp        drat      wt        qsec     
#> mpg  "none"    "mcor"   "pearson" "pearson" "pearson" "pearson" "pearson"
#> cyl  "mcor"    "none"   "mcor"    "mcor"    "mcor"    "mcor"    "mcor"   
#> disp "pearson" "mcor"   "none"    "pearson" "pearson" "pearson" "pearson"
#> hp   "pearson" "mcor"   "pearson" "none"    "pearson" "pearson" "pearson"
#> drat "pearson" "mcor"   "pearson" "pearson" "none"    "pearson" "pearson"
#> wt   "pearson" "mcor"   "pearson" "pearson" "pearson" "none"    "pearson"
#> qsec "pearson" "mcor"   "pearson" "pearson" "pearson" "pearson" "none"   
#> vs   "mcor"    "cramer" "mcor"    "mcor"    "mcor"    "mcor"    "mcor"   
#> am   "mcor"    "cramer" "mcor"    "mcor"    "mcor"    "mcor"    "mcor"   
#> gear "mcor"    "cramer" "mcor"    "mcor"    "mcor"    "mcor"    "mcor"   
#> carb "mcor"    "cramer" "mcor"    "mcor"    "mcor"    "mcor"    "mcor"   
#>      vs       am       gear     carb    
#> mpg  "mcor"   "mcor"   "mcor"   "mcor"  
#> cyl  "cramer" "cramer" "cramer" "cramer"
#> disp "mcor"   "mcor"   "mcor"   "mcor"  
#> hp   "mcor"   "mcor"   "mcor"   "mcor"  
#> drat "mcor"   "mcor"   "mcor"   "mcor"  
#> wt   "mcor"   "mcor"   "mcor"   "mcor"  
#> qsec "mcor"   "mcor"   "mcor"   "mcor"  
#> vs   "none"   "cramer" "cramer" "cramer"
#> am   "cramer" "none"   "cramer" "cramer"
#> gear "cramer" "cramer" "none"   "cramer"
#> carb "cramer" "cramer" "cramer" "none"

One thing to note is that the function considers numerical variables as categorical, when they only have up to 10 unique values. This is useful in case of number of gears in the dataset.

In addition, there is a parameter method in the function, which forces it to apply a specific measure of association to all the variables. This can be either Pearson / Spearman / Kendall correlation (using cor.test() function) or Cramer’s V (cramer() function from greybox). Note however, that it might not produce anything meaningful, when you apply the function with some methods to variables in categorical scales (e.g. correlation does not make sense between price and colour).

Partial correlations

Sometimes you might want to get an idea of the pure correlation between two variables, x and y - the one that is free of the potential influence of other variables, for example, some variable z. This is called “partial correlation”. It is based on the two regression models: y = b0 + b1 z + e and x = a0 + a1 z + u - and then on the calculation of the Pearson’s correlations between the residuals of models, cor(e,u). This is what the function pcor() does, using all the provided variables for z. It returns a matrix of partial correlations and a matrix of the respective p-values from the cor.test():

pcorValues <- pcor(mtcars)
print(pcorValues,digits=2)
#> Partial correlations using pearson method: 
#> values:
#>        mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear  carb
#> mpg   1.00  0.10  0.16 -0.20  0.18 -0.40  0.48  0.02  0.29  0.16 -0.07
#> cyl   0.10  1.00  0.35  0.32  0.00 -0.17  0.40 -0.52 -0.17 -0.13  0.21
#> disp  0.16  0.35  1.00  0.54  0.10  0.76 -0.45 -0.10 -0.07  0.06 -0.66
#> hp   -0.20  0.32  0.54  1.00 -0.03 -0.30  0.02  0.27  0.12  0.16  0.50
#> drat  0.18  0.00  0.10 -0.03  1.00 -0.13  0.35 -0.01  0.22  0.26  0.18
#> wt   -0.40 -0.17  0.76 -0.30 -0.13  1.00  0.75 -0.06  0.01 -0.15  0.62
#> qsec  0.48  0.40 -0.45  0.02  0.35  0.75  1.00  0.40 -0.27  0.26 -0.46
#> vs    0.02 -0.52 -0.10  0.27 -0.01 -0.06  0.40  1.00 -0.23  0.00 -0.08
#> am    0.29 -0.17 -0.07  0.12  0.22  0.01 -0.27 -0.23  1.00  0.38 -0.07
#> gear  0.16 -0.13  0.06  0.16  0.26 -0.15  0.26  0.00  0.38  1.00  0.42
#> carb -0.07  0.21 -0.66  0.50  0.18  0.62 -0.46 -0.08 -0.07  0.42  1.00
#> 
#> p-values:
#>       mpg  cyl disp   hp drat   wt qsec   vs   am gear carb
#> mpg  0.00 0.60 0.38 0.27 0.33 0.02 0.01 0.92 0.10 0.37 0.71
#> cyl  0.60 0.00 0.05 0.07 0.98 0.34 0.02 0.00 0.36 0.47 0.25
#> disp 0.38 0.05 0.00 0.00 0.57 0.00 0.01 0.57 0.71 0.72 0.00
#> hp   0.27 0.07 0.00 0.00 0.88 0.09 0.92 0.14 0.52 0.39 0.00
#> drat 0.33 0.98 0.57 0.88 0.00 0.46 0.05 0.94 0.23 0.15 0.32
#> wt   0.02 0.34 0.00 0.09 0.46 0.00 0.00 0.73 0.94 0.41 0.00
#> qsec 0.01 0.02 0.01 0.92 0.05 0.00 0.00 0.02 0.13 0.14 0.01
#> vs   0.92 0.00 0.57 0.14 0.94 0.73 0.02 0.00 0.20 0.99 0.66
#> am   0.10 0.36 0.71 0.52 0.23 0.94 0.13 0.20 0.00 0.03 0.71
#> gear 0.37 0.47 0.72 0.39 0.15 0.41 0.14 0.99 0.03 0.00 0.02
#> carb 0.71 0.25 0.00 0.00 0.32 0.00 0.01 0.66 0.71 0.02 0.00

These values might be useful on the stages of the variables selection.

Plots of association between several variables

Similarly to the problem with cor(), scatterplot matrix (produced using plot()) is not meaningful in case of a mixture of variables:

plot(mtcarsData)

It makes sense to use scatterplot in case of numeric variables, tableplot() in case of categorical and boxplot() in case of a mixture. So, there is the function spread() in greybox that creates something more meaningful. It uses the same algorithm as assoc() function, but produces plots instead of calculating measures of association. So, gear will be considered as categorical and the function will produce either boxplot() or tableplot(), when plotting it against other variables.

Here’s an example:

spread(mtcarsData)

This plot demonstrates, for example, that the number of carburetors influences fuel consumption (something that we could not have spotted in the case of plot()). Notice also, that the number of gears influences the fuel consumption in a non-linear relation as well. So constructing the model with dummy variables for the number of gears might be a reasonable thing to do.

The function also has the parameter log, which will transform all the numerical variables using logarithms, which is handy, when you suspect the non-linear relation between the variables. Finally, there is a parameter histogram, which will plot either histograms, or barplots on the diagonal.

spread(mtcarsData, histograms=TRUE, log=TRUE)

The plot demonstrates that the disp has a strong non-linear relation with mpg, and, similarly, drat and hp also influence mpg in a non-linear fashion.