Correlation measures the strength of the relationship between two variables.

Tip: Correlation indicates association, not causation.

Download the data used in this tutorial.

Load CSV data:

df<- read.csv("D:\\R4Researchers\\LAI_factors.csv")
#View(df)

Using the cor() function:

cor(df$LAI_India,df$LST_India, method="pearson") 
# -0.5035808 # inversely correlated

Using the cov() function:

cov(df$LAI_India,df$LST_India) #Covariance
# -0.0080975

Pearson correlation coefficient (r)

cor.test(df$LAI_India,df$LST_India, method="pearson")

#Pearson's product-moment correlation

#data:  df$LAI_India and df$LST_India
#t = -2.1809, df = 14, p-value = 0.04674
#alternative hypothesis: true correlation is not equal to 0
#95 percent confidence interval:
# -0.79966707 -0.01049536
#sample estimates:
#       cor 
#-0.5035808

In the above output:

• t is the t-test statistic value (t = -2.1809),

• df is the degrees of freedom (df = 14),

• p-value is the significance level of the t-test (p-value = 0.04674).

• conf.int is the confidence interval of the correlation coefficient at 95% (conf.int = [-0.79966707, -0.01049536]);

• sample estimates is the correlation coefficient (cor = -0.5035808).

Spearman's rank correlation coefficient (rho)

cor.test(df$LAI_India,df$LST_India, method="spearman")  

#data:  df$LAI_India and df$LST_India
#S = 1074.7, p-value = 0.01842
#alternative hypothesis: true rho is not equal to 0
#sample estimates:
#       rho 
#-0.5803983 

#Warning message:
#In cor.test.default(df$LAI_India, df$LST_India, method = "spearman") :
#  Cannot compute exact p-value with ties

Kendall rank correlation coefficient (tau)

Kendall's tau is the same as Pearson's r and Spearman's rho

cor.test(df$LAI_India,df$LST_India, method="kendall") 
#Kendall's rank correlation tau

#data:  df$LAI_India and df$LST_India
#z = -2.1475, p-value = 0.03175
#alternative hypothesis: true tau is not equal to 0
#sample estimates:
#       tau 
#-0.4109547 

#Warning message:
#In cor.test.default(df$LAI_India, df$LST_India, method = "kendall") :
#  Cannot compute exact p-value with ties

Extract correlation coefficient and p-value from correlation tests:

pe<- cor.test(df$LAI_India,df$LST_India, method="pearson")
# Extract the p.value
pe$p.value
# 0.04673796

# Extract (r) correlation coefficient
pe$estimate
#cor 
#-0.5035808

spe<- cor.test(df$LAI_India,df$LST_India, method="spearman")
# Extract the p.value
spe$p.value
# 0.01841523

# Extract (rho) correlation coefficient
spe$estimate
# rho 
# -0.5803983

ken<- cor.test(df$LAI_India,df$LST_India, method="kendal")
# Extract the p.value
ken$p.value
# 0.0317547

# Extract (tau) correlation coefficient
ken$estimate
# tau 
# -0.4109547

Interpret correlation coefficient

• -1 indicates a strong negative correlation.

• 0 means that there is no association between the two variables.

• 1 indicates a strong positive correlation.

Based on the previous tests, a significant negative relationship was found between land surface temperature and LAI vegetation index in India.

Create simple data and perform correlations between them:

cor.test(c(1,3,5,7), c(1,3,5,7), method="pearson")
cor.test(c(1,3,5,7), c(1,3,5,7), method='spearman')
cor.test(c(1,3,5,7), c(1,3,5,7), method='kendal')
cor.test(c(1,3,5,7), c(1,3,5,1), method="pearson")
cor.test(c(1,3,5,7), c(1,3,5,1), method='spearman')
cor.test(c(1,3,5,7), c(1,3,5,1), method='kendal')

Matrix of correlations with significance levels

Tip: Usually, statistically significant correlations should be considered.

library(Hmisc)
rcorr(as.matrix(df), type="pearson") # type can be pearson or spearman
rcorr(as.matrix(df), type='spearman') # data should be matrix

IQ score and weekly hours of watching TV

Create data:

iq<- c(86, 97, 99, 100, 101, 103, 106, 110, 112, 113)
tv<- c(2, 20, 28, 27, 50, 29, 7, 17, 6, 12)

Correlation test:

cor.test(iq, tv, method="pearson")
cor.test(iq, tv, method='spearman')
cor.test(iq, tv, method='kendal')

Based on all three tests, no significant correlation was found between IQ score and weekly hours of watching TV.

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