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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