All about Correlation - Part - 2

Is the correlation significant ? 

Note:
This post is in continuation to the post All about correlation - Part - 1
We are using the mtdata set in R.

Now that you have computed the correlation between 2 variables, you need to decide if the value is significant. In other words, the null hypothesis states -

H_o : The correlation is zero
H₁: There is significant correlation

In R, we can test this using the function cor.test. This function does the test of association between paired samples, using one of Pearson's product moment correlation coefficient. Neither cor(), nor cov() produce tests of significance, but cor.test() function can be used to test a single correlation coefficient.

R Code

> cor.test(mtcars$hp,mtcars$cyl)
 
 Pearson's product-moment correlation
 
data:  mtcars$hp and mtcars$cyl
t = 8.2286, df = 30, p-value = 3.478e-09
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.6816016 0.9154223
sample estimates:
      cor 
0.8324475 

Scatterplots

R Code

plot(mtcars$wt~mtcars$mpg,main = "Scatterplot",xlab = "Mileage", ylab = "Weight", col = "Blue")

Interpreting Correlations

At times, non representative samples can cause wrong interpretation of the correlations. Thus, it is important to test out this effect by sub-setting the data.  

In the mtcars datset, we have transmission column. It is a categorical variable, which is set to 0 for no transmission and 1 for transmission. 

If we make subsets of our data and then calculate the correlations again, we will observe there will be a difference. Thus, our sample may not be a good representative of the population.

R Code

> #Needed for describeBy function 
> library("psych")
> 
> #To read the first few lines of the datset
> head(mtcars)
                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1
> 
> #Subseting the data
> subset_trans <- subset(mtcars, am == 1)
> subset_notrans <- subset(mtcars, am == 0)
> 
> #print the subsetted data 
> head(subset_trans)
                mpg cyl  disp  hp drat    wt  qsec vs am gear carb
Mazda RX4      21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag  21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4
Datsun 710     22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1
Fiat 128       32.4   4  78.7  66 4.08 2.200 19.47  1  1    4    1
Honda Civic    30.4   4  75.7  52 4.93 1.615 18.52  1  1    4    2
Toyota Corolla 33.9   4  71.1  65 4.22 1.835 19.90  1  1    4    1
> head(subset_notrans)
                   mpg cyl  disp  hp drat    wt  qsec vs am gear carb
Hornet 4 Drive    21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8 360.0 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1
Duster 360        14.3   8 360.0 245 3.21 3.570 15.84  0  0    3    4
Merc 240D         24.4   4 146.7  62 3.69 3.190 20.00  1  0    4    2
Merc 230          22.8   4 140.8  95 3.92 3.150 22.90  1  0    4    2
> 
> #Using the describeBy function, need to install the psych package for this 
> describeBy(mtcars,mtcars$am)
group: 0
     vars  n   mean     sd median trimmed    mad    min    max  range  skew kurtosis    se
mpg     1 19  17.15   3.83  17.30   17.12   3.11  10.40  24.40  14.00  0.01    -0.80  0.88
cyl     2 19   6.95   1.54   8.00    7.06   0.00   4.00   8.00   4.00 -0.95    -0.74  0.35
disp    3 19 290.38 110.17 275.80  289.71 124.83 120.10 472.00 351.90  0.05    -1.26 25.28
hp      4 19 160.26  53.91 175.00  161.06  77.10  62.00 245.00 183.00 -0.01    -1.21 12.37
drat    5 19   3.29   0.39   3.15    3.28   0.22   2.76   3.92   1.16  0.50    -1.30  0.09
wt      6 19   3.77   0.78   3.52    3.75   0.45   2.46   5.42   2.96  0.98     0.14  0.18
qsec    7 19  18.18   1.75  17.82   18.07   1.19  15.41  22.90   7.49  0.85     0.55  0.40
vs      8 19   0.37   0.50   0.00    0.35   0.00   0.00   1.00   1.00  0.50    -1.84  0.11
am      9 19   0.00   0.00   0.00    0.00   0.00   0.00   0.00   0.00   NaN      NaN  0.00
gear   10 19   3.21   0.42   3.00    3.18   0.00   3.00   4.00   1.00  1.31    -0.29  0.10
carb   11 19   2.74   1.15   3.00    2.76   1.48   1.00   4.00   3.00 -0.14    -1.57  0.26
-------------------------------------------------------------------------------- 
group: 1
     vars  n   mean    sd median trimmed   mad   min    max  range  skew kurtosis    se
mpg     1 13  24.39  6.17  22.80   24.38  6.67 15.00  33.90  18.90  0.05    -1.46  1.71
cyl     2 13   5.08  1.55   4.00    4.91  0.00  4.00   8.00   4.00  0.87    -0.90  0.43
disp    3 13 143.53 87.20 120.30  131.25 58.86 71.10 351.00 279.90  1.33     0.40 24.19
hp      4 13 126.85 84.06 109.00  114.73 63.75 52.00 335.00 283.00  1.36     0.56 23.31
drat    5 13   4.05  0.36   4.08    4.02  0.27  3.54   4.93   1.39  0.79     0.21  0.10
wt      6 13   2.41  0.62   2.32    2.39  0.68  1.51   3.57   2.06  0.21    -1.17  0.17
qsec    7 13  17.36  1.79  17.02   17.39  2.34 14.50  19.90   5.40 -0.23    -1.42  0.50
vs      8 13   0.54  0.52   1.00    0.55  0.00  0.00   1.00   1.00 -0.14    -2.13  0.14
am      9 13   1.00  0.00   1.00    1.00  0.00  1.00   1.00   0.00   NaN      NaN  0.00
gear   10 13   4.38  0.51   4.00    4.36  0.00  4.00   5.00   1.00  0.42    -1.96  0.14
carb   11 13   2.92  2.18   2.00    2.64  1.48  1.00   8.00   7.00  0.98    -0.21  0.60
> 
> #Finding correlation in both the groups
> corr <- cor(mtcars$mpg,mtcars$wt)
> corr
[1] -0.8676594
> corr_trans <- cor(subset_trans$mpg,subset_trans$wt)
> corr_trans
[1] -0.9089148
> corr_nontrans <- cor(subset_notrans$mpg,subset_notrans$wt)
> corr_nontrans
[1] -0.7676554
> #Combine all the results 
> Correlation <- cbind(corr,corr_trans,corr_nontrans)
> # Notice the difference in correlation for the same set of variables but differently grouped/ subsetted 
> Correlation
           corr corr_trans corr_nontrans
[1,] -0.8676594 -0.9089148    -0.7676554

Thus, the sample should always be representative of the population. Any kind of sub-setting can change the correlation that is derived as such.