Correlation

Pearson Correlation

So far we have compared two groups within the same numeric variable. But what about comparing two different numeric variables? We often accomplish this by assessing the correlation between the two numeric variables. First we will discuss Pearson Correlation which can be calculated by:

\[ r = \frac{\sum{(x - \mu_x)(y - \mu_y)}}{\sqrt{\sum{(x - \mu_x)^2} \sum{(y - \mu_y)^2}}} \]

Explanation of Terms

\(r\) : correlation coefficient
\(x\) : variable x
\(y\) : variable y
\(\mu_x\) : mean of variable of x
\(\mu_y\) : mean of variable of y

Pearson Correlation is called parametric correlation as it depends on the distribution of your data. This type of correlation is assessing the linear dependence of variables x and y. We can test the signifcance of this association using the t statistic for pearson correlation:

\[t = \frac{r}{\sqrt{1-r^2}}\sqrt{n - 2}\]

Explanation of Terms

\(r\) : correlation coefficient
\(n\) : number of observations

Calculating Pearson Correlation

Using our glioblastoma data, let's determine the correlation between height and weight in our patients. However, before we do this, let's plot height versus weight:

# let's plot height and weight
ggplot(meta,aes(x=HEIGHT,y=WEIGHT))+
  geom_point()+
  theme_bw()

Here we see there is a positive relationship between height and weight. Let's test the significance of this relationship:

# test the correlation between height and weight
cor.test(
  x = meta$HEIGHT,
  y = meta$WEIGHT,
  method = "pearson"
)

    Pearson's product-moment correlation

data:  meta$HEIGHT and meta$WEIGHT
t = 9.5779, df = 97, p-value = 1.094e-15
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5795233 0.7863597 
sample estimates:
     cor 
0.697175

Explanation of Results

our test statistic is 9.5779
our d.f. is 97
our p-value is 1.094e-15 giving us enough evidence to reject the null hypothesis that the correlation is equal to 0
the alternative hypothesis is that the true correlation is not equal to 0 appears supported
the 95% confidence interval around our correlation coefficient is 0.5795233 and 0.7863597
the correlation coefficient is 0.697175 indicating a positive association

Correlation Coefficient Interpretation

correlation coefficient = 0 : no association
correlation coefficient = 1 : positive assocation
correlation coefficient. = -1 : negative associtaion

Pearson Correlation Assumptions

Now that we have conducted our test we must test our assumptions:

the relationship is linear
is each variable normally distributed?

We saw from our plot that the relationship does appear linear, meaning that as one variable increases/decreases the other does too. To test the normality of each variable we will use the Shapiro-Wilk Test:

# test the normality of height and weight
shapiro.test(meta$HEIGHT)
shapiro.test(meta$WEIGHT)

    Shapiro-Wilk normality test

data:  meta$HEIGHT
W = 0.98735, p-value = 0.4689

    Shapiro-Wilk normality test

data:  meta$WEIGHT
W = 0.91388, p-value = 7.436e-06

Given our p-value for the Shapiro-Wilk Test on height is above 0.05 we do not have enough evidence to reject the null hypothesis of the Shapiro-Wilk Test; that the data are normally distributed. However, we note the opposite case for weight. The p-value is below 0.05 which indicates the data are not normally distributed.

Spearman Rank Correlation

Above we saw that the normality assumption was violated in our data. What are we to do! Another option we could pursue is non-parametric correlation which does not make the assumptions we discussed above. This correlation coefficient is referred to as Spearman Rank Correlation which can be calculated like so:

\[ r = \frac{\sum{(x\prime - \mu_{x\prime} )(y\prime - \mu_{y\prime} )}}{\sqrt{\sum{(x\prime - \mu_{x\prime} )^2} \sum{(y\prime - \mu_{y\prime} )^2}}} \]

Explanation of Terms

\(r\) : spearman rank correlation coefficient
\(x\) : ranked variable x
\(y\) : ranked variable y
\(\mu_x\) : mean of variable of x
\(\mu_y\) : mean of variable of y

Essentially, the only difference is that the values in the variables are ranked and then the correlation coefficient is calculated. Let's try this in R!

# test the spearman correlation between height and weight
cor.test(
  x = meta$HEIGHT,
  y = meta$WEIGHT,
  method = "spearman",
  exact = FALSE
)

    Spearman's rank correlation rho

data:  meta$HEIGHT and meta$WEIGHT
S = 46676, p-value < 2.2e-16
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.7113395

Explanation of Results

the p-value is less than 2.2e-16
the p-value is below 0.05 giving us enough evidence to reject the null hypothesis, that the true rho is equal to 0
the spearman correlation coefficient is 0.7113395 indicative of a positive association

Correlation

Pearson Correlation

Calculating Pearson Correlation

Pearson Correlation Assumptions

Spearman Rank Correlation

References